Advances in Engineering Mechanics - At Chwang MH Teng DT Vale (2005 World Scientific)

ADVANCES IN Engineering Mechanics Relections and Outlooks

In Honor of Theodore Y-T Wu

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ADVANCES IN Engineering Mechanics Relections and Outlooks

In Honor of Theodore Y-T Wu

editors

Allen T Chwang The University of Hong Kong, China

Michelle H Teng University of Hawaii at Manoa, USA

Daniel T Valentine Clarkson University, USA

World Scientific 1: N E W J E R S E Y * L O N D O N * S I N G A P O R E * B E l J l N G - S H A N G H A I - H O N G K O N G * T A I P E I - C H E N N A I

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover illustration: The three frames illustrating the interaction of two right-going internal solitary waves in a density-stratifiedfluid (with the crest naturally pointing downward in this case) are the courtesy of Brian C. Barr. The density stratification is nearly two-layered; the details are described in Chapter2, Section 2 (pp. 195-212) of this book. Inspiredby Professor Wu’s workon nonlinear waves, Bruin C. Barr peformed the computational simulation of a stronger wave overtaking a weaker one in a time sequence as marked, from which thesefigures were created. The middlefigure exhibits the wave profile at time t = 20.3, just after the flashing instant when the two wave crests coalesce into a single peak, from which the stronger wave barely emerges in outrunning the weaker one.

ADVANCES IN ENGINEERING MECHANICS - REFLECTIONS AND OUTLOOKS In Honor of Theodore Y.-T. Wu

Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-144-7

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Editors:

Allen T. Chwang Michelle H. Teng Mechanical Engineering Civil Engineering The University of Hong Kong University of Hawaii at Manoa Hong Kong, CHINA Honolulu, HI, USA

Daniel T. Valentine Mechanical Engineering Clarkson University Potsdam, NY, USA

Editorial Board:

Alex Cheng Allen T. Chwang (Co-Chair) Norden Huang Tin-Kan Hung Chiang C. Mei (Co-Chair) Michelle Teng Daniel T. Valentine (Secretary) Keh-Han Wang

Scientific Committee:

S. K. Chakrabarti A. T. Chwang R. C. Ertekin N. E. Huang C. C. Mei M. H. Teng M. P. Tulin D. T. Valentine D. Weihs Theodore Y.-T. Wu (Guest & Honored Chair)

V


ACKNOWLEDGEMENTS

The Theodore Y.-T. Wu Symposium on Engineering Mechanics was held on 21-22 June 2004 in Vancouver. This symposium was held to honor Professor Theodore Yao-Tsu Wu for his 80th birthday. The papers presented led to the publication of this book. It was co-sponsored by the Ocean Engineering Committee of the Ocean, Offshore and Arctic Engineering Division of American Society of Mechanical Engineers (ASME) and the Fluids Technical Committee of the Engineering Mechanics Division of American Society of Civil Engineers (ASCE).

The support of the National Science Foundation (NSF) is acknowledged; Dr. Michael W. Plesniak, Director on Fluid Dynamics and Hydraulics in Division of Chemical and Transport Systems of NSF, is the Program Official at NSF who is responsible for providing partial support for the symposium and for the preparation of this book on the topics covered in the proceedings of the symposium under the Grant No. CTS-0405918. The help of Dr. Ken P. Chong, Director of Mechanics & Materials Program in the Engineering Directorate of NSF is also acknowledged. The Clarkson Space Grant Program provided partial support for this project under Grant No. 39555- 6524 via Cornell University from NASA. In addition, the help of Clarkson University and the University of British Columbia are acknowledged.

The help of the Organizing Committee for this special symposium, the organizers of the OMAE 2004, and the participants of this event have been greatly appreciated. Finally, the communications with Professor Theodore Y.T. Wu on the preparation, organization and management of this symposium were enjoyable and greatly appreciated. On behalf of the Organizing Committee and all of the participants, we would like to also thank Professor Wu and some of his family members for agreeing to attend this event.

vii


PREFACE

This book is a compilation of original papers based on the talks given during the special event held in Vancouver, British Columbia, Canada in honor of Theodore Yao-Tsu Wu for his 80th birthday. It was part of the International OMAE 2004 Conference. In the program it was listed as Track 8, The Theodore Y.-T. Wu Symposium on Engineering Mechanics. It was held on June 21-22,2004 at the Hyatt Regency Hotel in Vancouver. In addition to the publication of the manuscripts in the content of the book, a biographical sketch of Professor Wu and an essay describing the symposium is provided in the appendix.

This volume presents more than 45 original papers on recent advances in several topics in engineering mechanics. Several members of the National Academy of Engineering wrote papers describing the present state-of-the-art and directions for future work. The topics covered are cavitation, nonlinear water waves, swimming and flying in nature, biomechanics, data analysis methodology, and propulsion hydrodynamics. The areas covered are areas influenced significantly by Professor Wu in Engineering Science at Caltech. All of the authors offer their reflections on current work and outlooks on the future of the topics they discussed. Hence, this book will be useful for researchers and students interested in advancing the art of engineering mechanics.

The goals of the symposium are the same as the book: (1) To honor the significant accomplishments of Professor Wu in the fields of nonlinear waves, hydrodynamics, biomechanics, wave-structure interaction and other areas of fluid dynamics that have guided the community in their investigations of fluid mechanical phenomena. (2) To review the present state of engineering mechanics, and to chart the future of this area of investigations from the view point of civil engineering, biomechanics, geophysics, mechanical engineering, naval architecture, ocean and offshore engineering. Thus, the primary purpose of this book is to provide guidance and inspiration for those who are interested in continuing to advance engineering mechanics as we begin the 21st century. To quote Professor Wu,

“The value of a book publication lies in disseminating new knowledge attained with effort and dedication from all those who participate, and in having the useful results within ready reach to students and researchers actively working in thefield. ”

Allen T. Chwang Michelle H. Teng

Daniel T. Valentine

ix


Professor Theodore K-T. Wu

Waving in Acknowledgment at Resonance

Sincere invitations have been dispatched,

Simple feasting, and light drink, by dear fiiends to summon leamedfiiends long admired;

just so as to have mind open, and tongues loosened.

Let no expounding thoughts stop short, let no words for exploration be left unsaid;

All aimed at reaching a comprehension of the theme subject at hand

to a depth as profound as can be fathomed.

Thus enabling us, with gratijkation, to explain the results of great significance so attained

with value and merits everlasting, in the simplest way for all to comprehend.

Waving to my distinguishedfiiends and scholars, in acknowledgment at resonance with their creative mood,

appreciating having grasped the sound physical conception underbing the basic mechanisms to good truth,

and cherishing the results of essence in simplicity and elegance, for disseminating seeds to germinate vital new growth.

tyw mid-autumn day of 2004

xi


TABLE OF CONTENTS

Acknowledgments

Preface

Prologue: Cavitution On the theory and modeling of real cavity flows Marshall Tulin

Chapter 1 : Nonlinear Waves: Theoretical Considerations

vii

ix

1 3

27 Localization of dispersive waves in weakly random media 29 C. C. Mei, Jorgen H. Pihl, Mathew Hancock & Yile Li Water wave equations 48 Jin E. Zhang Wu’s mass postulate and approximate solutions of the 60 K d V equation S. S. P. Shen, Q. Zheng, S. Gao, Z. Xu & C. T. Ong Explicit analytic solutions of KdV equation given by the homotopy analysis method Chen Chen, Chun Wang & Shijun Liao Rigorous computation of Nekrasov’s integral equation for water waves Sunao Murashige & Shin ’ichi Oishi Numerical modeling of nonlinear surface waves and its validation W. Choi, C. P. Kent & C. J. Schillinger

The Davey-Stewartson system K. W. Chow, D. H. Zhang& C. K. Poon Rip currents due to wave-current interaction Jie Yu & A. Brad Murray Higher order Boussinesq equations for water waves on uneven bottom Hua Liu & Benlong Wang Waves on a liquid sheet S. P. Lin

70

84

94

111 Three dimensional wave for water waves on finite depth:

117

128

140

xiii

xiv

A different view on data from a nonlinear and nonstationary world 150 Norden E. Huang

Chapter 2: Nonlinear Waves: Experiments and Computations 171

Solitary-wave collisions 173 Joseph Hammack, Diane Henderson, Philippe Guyenne & Ming Yi Computer simulations of overtaking internal solitary waves Brian C. Ban- & Daniel T. Valentine Theoretical and experimental investigation of waves due to a moving dipole in a stratified fluid Shiqiang Dai, Gang Wei, Dong-Qiang Lu & Xiao-Sing Su Thin film dynamics in a liquid lined circular pipe Roberto Camassa & Long Lee Transverse waves in a channel with rectangular cross section L. M Deng &A. T. Chwang Long time evolution of nonlinear wave trains in deep water Hwung-Hweng Hwung & Wen-Son Chiang On the Zhang-Wu run-up model Hongqiang Zhou, Michelle H. Teng & Kelie Feng A numerical study of bore runup a slope Qinghai Zhang & Philip L.-F. Liu Studies of intense internal gravity waves: Field measurements and numerical modeling Hsien P. Pa0 & Andrey N. Serebryany Nonlinear internal waves in the South China Sea Antony Liu, Yunhe Zhao & Ming-Kuang Hsu A numerical predictive model of tides around Taiwan Hsien- Wen Li & Yung-Ching Wu New concepts in image analysis applied to the study of nonlinear wave interactions Steven R. Long

Chapter 3: Wave Structure Interaction Nonlinear wave loads acting on a body with a low-frequency oscillation Motoki Yoshida, Takeshi Kinoshita & Weiguang Bao Analytical features of unsteady ship waves Xiao-Bo Chen

195

213

222

239

247

257

265

286

297

3 14

327

353

355

37 1

xv

A note on the classical fiee surface hydrodynamic impact problem 390 Celso P. Pesce

Measurements of velocity field around hydrofoil of finite span with shallow submergence 5’. J. Lee

Chapter 4: Biomechanics: Medical Blood flow abnormalities in sickle cell anemia Anthony T. Cheung Does interfacial viscosity exist, its application to medical science S. C. Ling Unsteady flows with moving boundaries: Pulsating blood flows and earthquake hydrodynamics Tin-Kan Hung Interdisciplinary education and research experiences for Undergraduates in mathematics and biology George T. Yates In vitro study on the internal design of Provox 2TM voice prosthesis Horace H, Lam A potential role for muscle pump-generated intravascular solitons in maintenance of tissue-engineered bioreactors implanted in bone H. Winet, C. Caulkins-Pennell & J. Y. Bao

Chapter 5: Biomechanics: Zoological Creeping flow around a finite row of slender bodies Efi-ath Barta &Daniel Weihs Theory and numerical calculation of hovering flight of a dragonfly Hiroshi Isshiki A numerical study on fluid dynamics of backward and forward swimming in the eel Anguilla Anguilla Wenrong Hu, Binggang Tong & Ha0 Liu Impulse extremization in vortex formation for application in low speed maneuvering of underwater vehicles Kamra Mohseni

408

423 425

440

446

474

484

493

513

515

539

557

574

xvi

Chapter 6: Hydrodynamics: MHD, Viscous and Geophysical Flows MHD self-propulsion and motion of deformable shapes Touvia Miloh Nonlinear analysis on transition to form coherent structures Jun Yu & Yi Yang Large-Reynolds-number flow across a translating circular cylinder Bang-Fuh Chen, Yi-Hsiang Yu & Tin-Kan Hung A spectral method for the mass transport in a layer of power-law fluid under periodic forcing Lingyan Huang, Chiu-On Ng & Allen T. Chwang The cohesion and re-separation of particles in slow viscous flows Ren Sun & A . T. Chwang On coherent vortices in turbulent plane jets generated by surface Water Waves Chin-Tsau Hsu & Jun Kuang A dynamic model for strong vortices over topography on a ,8 Plane Hung-Cheng Chen, Chin-Chou Chu & Chien-Cheng Chang Sea ice floe tracking and motion analysis for S A R imagery in the marginal ice zone Jun Yu & Antony Liu

Reflections and Resolutions Reflections and resolutions to some recent studies on fluid Mechanics Theodore Y.-T. Wu

Appendix The origin of this book of scientific reflections: The other side of doing engineering science Daniel T. Valentine

601

603

613

625

633

65 1

660

669

68 1

691

693

715

Author Index 729

PROLOGUE

CAVITATION


ON THE THEORY AND MODELING OF REAL CAVITY FLOWS

MARSHALL P. TULIN Ocean Engineering Laboratory

University of California, Santa Barbam Santa Barbara, CA 93106

mpt60cox. net

In real cavity flows, vorticity must be produced in the near field and wake, consistent with the body forces. Therefore, vorticity considerations place constraints on their possible configuration. The theory necessary to understand and apply these concepts is developed, including considerations of vorticity creation and flux, and the behavior of discontinuity sheets, their growth and impingement. Partially cavitating flows and their oscillations are then considered. It is thought that the ‘partial cavity oscillation’ is forced, while the ‘transitional oscillation’ is an unsta, ble divergence. The “partially filled” cavity model is suggested and applied in the former case, and the “reentrant jet” model to the latter. It is concluded that in this latter supercritical case, the Iength of the cavity is determined by both the lift parameter (foil lift/cavitation number) and the growth rate of the cavity volume, and a diagram of the cavity history is shown, involving hysteresis. While the reentrant jet can be produced during cavity growth as a byproduct, it does not play an essential role during rapid growth. However, when the cavity growth slows down, then the jet can penetrate the cavity and cause its detachment. Finally, it is argued that viscous separation can play a very important role during partial cavitation on foils by inhibiting and interfering with the cavitation.

1. Introduction

The systematic investigation of the cavitating characteristics of foil sections was begun by F. Numachi during the period 1940-50, see Numachi, et a1 (1957). Much systematic work, both theoretical and experimental followed in the US, Europe, and Japan starting in the mid 50s. More recently there has been a renewed interest in partial cavitation on hydrofoils and foil cascades, and many new details have been revealed through experiments. These results give rise to a number of important questions, for example: What is the nature of the observed instabilities?; How can the observed frequencies be predicted?; What is the role of the observed re-entrant jets?; What is the real nature of cavity shedding and of cloud cavitation?; How

3

4

can the observed vortical flows be best modeled? Here we discuss these flows from a fundamental point of view, with an

emphasis on the role of vorticity and the nature of the vortical flows which are produced. Our intention is to create a sound basis for flow modeling. Some mechanisms are also suggested.

Early Models. The mathematical modeling of cavity flows began with the introduction into the “exact” free streamline theory, of the reflected image, termination body of Riabouchinsky (1920). This artificial but successful model of the finite cavity allowed the specification of an arbitrary constant speed, qc, on the free streamline, where qc > go, and qo is the flow speed at infinity. This rendered useful the original steady free streamline theory introduced by Kirchoff-Helmholtz on the assumption, qc = qo. In a modified version of the Riabouchinsky model, the reflected image is replaced by a small vertical plate, see Tulin (1964a).

In a subsequent model, Gilbarg and Rock (1946), Efros (1946), and Kreisel (1946), the free surface turns into a re-entrant jet at the rear of the cavity, introducing a flow into the cavity at a constant rate. This model cannot be consistent with a steady flow reality any more than the termination model of Rabouchinsky exists in practice.

Both these early models suffer from a further lack of reality: they do not include a trailing momentum wake to account for the drag on the forebody. This wake causes an outward displacement of the outer streamlines in the downstream field, proportional to the drag. The re-entrant jet, in fact, causes a displacement of the outer streamlines in the upstream field, quite in opposition to the real situation.

Linearized Theory. In the case of slender bodies, these inconsistencies were resolved, Tulin (1964a), who showed rigorously that these two models, Riabouchinsky and Re-Entrant Jet, reduced to exactly the same asymptotic theory, and that the wake vanished in the asymptotic limit, its thickness being of smaller order than that of the cavity.

This asymptotic, i.e., linearized, theory therefore allowed the option to ignore the real flow inconsistencies in the “exact” models. This has first been done in the case of supercavitation flows, Tulin (1953, 1954, 1955), and then in similar fashion for partial cavities, Acosta (1955) and Geurst and Timman (1956).

These applications of linearized theory were immediately successful and were followed by others, Fabula (1962); Geurst (1959); Hanaoka (1967); Hsu (1969); Tulin (1960); and Wade (1963, 1967), including applications to cascades. In the case of supercavitating flows, the theory was almost

5

immediately applied to the design of efficient supercavitating propellers, with great impact. In the case of partial cavities, Acostas theory for the sharp edged flat plate made several remarkable predictions, all without the benefit of prior experimental observations; this is rare in hydrodynamics,

These predictions were: that the cavity length depends on the variables (a,a) only through the combined parameter,(a/cr), and that there is a critical value, ( a , ( ~ ) * , beyond which no steady solutions exist; that two separate solutions corresponding to a short and a longer cavity exist for (a,a) < (a,a)*. He found

(a, a)* = 0.096; (L/c)* = 0.75, (1)

where L and c are the cavity and foil chord lengths, respectively. All of these flat plate predictions have been well confirmed by exper-

iments, except that only the shorter of the sub-critical cavities, (L/c) I (C/c)*, has been observed.

(L/c) = f(a/c). The dependence of cavity length on (a/o) for the thin plate seems at first mysterious, and its simple explanation does not seem to have been given in the literature. Consider a given partial cavity shape with cavity length L‘, on an inclined flat plate, with incidence, a‘, and cavitation number, u’. According to linearized theory, the pressures will everywhere scale by a constant factor when the vertical scale of the flow is changed. That means that if a = K(Y’ and 6 (local cavity thickness) = n6‘, then cp (local pressure coefficient) = 64. Since IT = -cp on the cavity, it follows that: (a/c) = (a’/o’) = constant for a fixed non-dimensional cavity shape; but, of course, this shape and therefore the constant will depend on the initial values of (a’, d) . Therefore,

(el4 = f(a/a). (2)

When the partial cavity occurs on a foil with initial thickness, and/or a rounded leading edge, then this rule will not rigorously apply, as these dimensions do not change with a change in incidence. Indeed, the dependence of cavity length on the nose radius, rn, has been explicitly demonstrated through calculations by Tulin-Hsu (1980) who found:

(tic) = f (a /c ; T n / C ) , (3)

(4)

and

(a/a)* = f ( rn / c ) > 0.096. The Double Branched Solution of Acosta. The reason for two

branches lies in the following. For a = 0 there are actually two separate

6

exact solutions: i) the undisturbed flow about the flat plate, which corresponds to the undisturb ed flow in the linearized theory, and ii) a flow in which the lower surface corresponds to the horizontal flat plate with a cavity at constant pressure covering its upper surface. This second, drag free, flow is a solution of the homogeneous problem (u = 0 ) , and exists for all values of qc > qo. The flow is front-back symmetric.

These two separate flow regimes can be seen in Acostas solution at u = 0. For an increase in a, a cavity forms at the leading edge in flow i), above, grows and leads to the lower branch solution. Similarly, with increasing a, the cavity detaches at the trailing edge in flow ii) and shortens, creating the upper branch.

Improved Asymptotic Theory: Thickness Effects. Early experimental results, Parkin (1958) for a thin wedge, and Meijer (1959) for a double circular arc, showed satisfactory agreement with the linearized theory predictions of Acosta (1955) and Geurst-Timman (1956). Experimental results for asymmetrical foils with thickness did not give good agreement, however. This eventually led Tulin-Hsu (1977) to devise a new asymptotic theory based on a perturbation on the fully wetted flow, and they applied it extensively. This theory allows the calculation of partial cavities based on a knowledge of the surface distribution of speeds on the non-cavitating body. In Tulin-Hsu (1980), hereafter called (T-H), they showed comparisons between their calculations and past experimental results on four planar foils tested by others. They have also shown how to modify their theory for application to wings of finite aspect ratio, taking into account the contribution separately of both lift induced and thickness induced surface speed, using the aspect ratio corrections of R.T. Jones (1941), which are applicable from large down to small aspect ratios. The effects of aspect ratio are very strong. They obtained quite good agreement with the observations of Kermeen (1960).

The calculation of (T-H) clearly show a variety of nonlinear effects occurring in the practical range of variables, due both to incidence (even for relatively small angles), camber, thickness, and leading edge roundness. For example, the effect of a leading edge radius to chord ratio of .01 was found to increase (L/c)* by fifty percent, but at the same time to decrease the volume of the partial cavities.

Therefore, theoretical predictions of steady short cavities on real foils, and certainly on wings, is best based on this revised asymptotic theory, rather than on the older linearized version, except perhaps for very thin foils without camber. Finally, the (T-H) theory can be adapted for various free

7

streamline models, including those with trailing wakes, where a moderate reduction in ( t /c)* was found.

Trailing Wake Models. In order to introduce the reality of a trailing momentum wake, Tulin (1964a) introduced several new models in which the cavity terminated in spiral vortices which give rise to streamlines bounding the wake. Subsequently these models were modified, "din (196413) to allow for the downstream displacement of the wake streamline, Figure 1. One of these models, the single spiral vortex, reduces to the closed linearized model in the limit of thin bodies and cavities, just as the Riabouchinsky and re- entrant jet models do. However, the other, the double spiral vortex, does not, and it models reality better to the extent that losses occur at the collapse region, as they do in reality. In their predictive calculations, (T-H) have embedded this model in their revised asymptotic theory, with very good results.

Figure 1. Cavity flow models.

Later in this paper we shall relate the double-spiral vortex cavity-wake model to a more detailed cavity termination model where the cavity is partially filled with fluid, starting with the collapse region.

2. FLOW FIELDS

We consider a cavitating flow field which has been established a long, but finite time in the past by a body in steady motion. Such a flow field consists, in general, of three distinguishable parts, Figure 2.

8

Figure 2. Mean flow fields, flat plate with partial cavitation.

Far Field. This field is free of vorticity and it is necessarily unsteady, as the wake is finite and lengthening at a mean constant rate; it may also possess a periodic or even spectral component.

A net vortex cannot be present in the far field because of Kelvins Law, preventing the creation of net vorticity. Therefore an asymptotic descrip tion of the far potential field can be written in terms of expansions in multipoles, plus a source representing temporal changes in the cavity volume.

It may be shown that the forces on the body in the near field result, of necessity, in momentum fluxes in the far field, and that these are due entirely to the presence of time-dependent dipoles in the far field. In general, and by definition:

2 = p 1 d d V = -rho ?;’ x (V x $ ) d V , I ( 5 )

where --+ F = d Z / d t ,

+ + and M , F , 7, and i? are the fluid momentum, body force, radial distance, and flow velocity, respectively, all vectors; and p , t , and V are the fluid density, time, and flow domain. It follows, Eqs. ( 5 ) and (6), that, in two dimensions,

where pi, the far field dipole strength is defined as the moment of vorticity:

9

We note again that

w d V = 0, J (9)

where 3 is the vorticity, V x 2, a scalar in planar flows. The far field dipole representation is:

where r here is the radial distance from an origin in the near field. It is clear that the magnitudes of the dipole moments, pi, are the

consequence of the detailed vortical flow processes in the near and wake fields which produce the forces on the body, Eq. (7). The vortical near field includes the imputed vorticity field associated with the surface velocities, qs, on the body. The vorticity moment for the body is therefore, p(body) = J q,h(z) dx, where h(z) is the body thickness.

Near Field. The near field contains the flow in the near vicinity of the body, including the cavity, whether partial or trailing (supercavitating). These flows are partially potential, but include embedded vortical flows which result from the action of viscosity and of free surface impact, breaking, and splashing (there is a direct corollary with the breaking of water waves). Viscous effects may be confined in the boundary layer, or flow separation may occur.

A particular complexity arises at the rear of the cavity where its collapse takes place. This may involve a re-entrant jet penetrating the cavity and breaking the cavity surface, and/or a recirculating eddy. In the case of steady cavity flows (or reasonable approximations to steadiness), considerable success has been realized with a completely potential flow description, as through the use of linearized and/or second order theories; these model the cavity collapse and embed it in potential theory. In the case of supercavitating flows, they have resulted in adequate and very useful estimations of lift and drag on thin foils, and of the shape of the cavities, Tulin, (l955,1964a,1964b), as discussed in Section 1. The theories have also produced a great success in the case of partial cavities on flat plates.

Nevertheless, a complete description of the flow processes requires a description of the details of vorticity generation in the near field, and this has become imperative as attention has become focussed on unsteady effects for partial cavities, and their associated instabilities. These flows have been investigated in a growing number of experimental observations, de Lange and de Bruin (1998); Sato, et al (2001); and Arndt, et a1 (2001), which

10

have led to descriptions of unsteady cavity behavior, and have given rise to questions of a fundamental nature regarding the underlying nature of the instabilities, and how the unstable flows are properly to be modeled. This is our central concern here.

Vortical Wake Field. This field is an expression of the entire history of force generation on the body. For example, lift generation on a foil near the initiation of motion is always accompanied by circulation around the foil, and therefore results in a shed counter-vortex which can be found at the furthest downstream extent of the wake,, Figure 2. Any fluctuation in lift, and therefore in circulation around the foil, will result in a corresponding spatial distribution of counter-vorticity left behind in the wake, in addition to the starting vortex.

Corresponding to the mean drag, both the mean momentum deficit and the mean vertical moment of vorticity, see Eq. ( 8 ) , will grow at a constant rate in the lengthening wake, and are reflected in a mean steady flux of momentum deficit through a transverse wake plane, Figure 2.

When the drag on the body is fluctuating, then unsteady shedding of vorticity at the body will result in spatial variations in the wake vertical moment of vorticity. For example, periodical flow separation on bluff bodies leads to periodical spatial wake patterns, as modeled by Kkm6ns double vortex street; there exists a substantial and coherent theory of these wakes in Goldstein (1938); Birkhoff and Zarantonello (1957). It is now known that periodical cavityshedding occurs in two separate regimes of cavitation on inclined foils, which have been called “partial cavity oscillations” for mean cavity lengths, ( l / c ) < ( l / c ) * , and “transitional cavity oscillations” for longer cavity lengths, Sato, et a1 (2001). It has also been observed that vortical patterns originate periodically in the near field and flow into the wake. These are particularly noticeable in the form of large regions of cloud cavitation, which can be very injurious. It is highly desirable to learn more about their origin and fundamental nature.

3. VORTICITY

Creation and Flux. The vorticity required to make the wake originated on the surface of the body, where:

w(wal1) = T(wall)/p’, (11)

where r is the wall shear stress and p’. the fluid viscosity. This wall vorticity diffuses into the thin viscous layer bounding the wall

and eventually flows into the wake. The negative vorticity on the top of the

11

body (the flow is from left to right) mixes in the wake with the negative vorticity from the bottom, while the wake widens, preserving the vertical moment of vorticity. In planar flows, conservation of vorticity takes the form:

a W

at - + v * [ 2 w - UVG] = 0

so that the vorticity flux,

has contributions from both convection and molecular diffusion.

solely from diffusion, and, At solid surfaces, where the no-slip condition applies, the flux arises

where n, s denote normal to and tangential to streamlines in the curvilinear co-ordinates in steady flow; u is the kinematic viscosity, p'/p. The Navier Stokes equations are:

pqaqlas + d p l d s = p 'aw/an , (15)

pq2K + a p / a n = p law/as , (16)

where q, p , and K are the flow speed, pressure, and streamline curvature. As a result of Eqs. (15) and (16), Eq. (14) becomes,

In the case where the boundary layer approximation applies:

d p / d n = 0,

where q* is the flow speed at the outer edge of the viscous layer in the potential flow.

12

This wall flux into the viscous layer,

1 a (4*12 2 as

F, (wall) = - - - must, in the steady case, be balanced by the integral of tangential flux over the thickness of the viscous layer. This leads to the result:

d F ds

F,, (wall) = - , 6 where F = so F,dn.

nected with the curvilinear co-ordinate system: Finally, taking into account Eq. (20), and the sign conventions con-

(21) 1

where (-) applies when the solid surface is to the right of the local flow velocity vector, and (+), when it is to the left.

It is remarkable, that in this approximation (thin viscous layer) both the vorticity flux normal to the wall, Eq. (20), and the integrated flux, Fs, along the wall are totally independent of viscosity. In the limit of the vanishingly thin viscous layer then, these fluxes may be considered as applying on the discontinuous interface itself, and the vorticity, w , as an abstract quantity which satisfies the conservation law, Eq. (12), in the limit, Y + 0, and which has definite physical consequences.

When flow detachment from the body occurs, then the flux of vorticity along the solid surface must be continued along the detached streamline. In the case of the free streamline in cavity flow,

I?, = Zk5 ( q * ) 2 ,

F,(cavity surface) = f (&2) . (22)

The cavity surface is therefore equivalent to a vortex sheet, not only in the well known mathematical sense, but in a physically consequential sense, as we shall see in the examples below:

Impingement. When almost parallel, but opposed discontinuity surfaces, speed q1, and q2, impinge, the resulting vorticity flux is the sum of the fluxes on each surface at the point of impingement. A spreading viscous shear flow originates there, with a total vorticity flux, integrated across the shear zone, always equal to the sum of the original fluxes:

p,(shear zone) = (q; - 4:) /2. (23)

The speed along the streamline dividing the flows originating in the two initially separate discontinuity surfaces is (q1 + 4 2 ) / 2 . After impingement,

13

the vorticity is a maximum on the dividing streamline, so there is no lateral diffusion across it. However, vorticity diffuses outwards on each side of the wake, widening it continually. However, the net vorticity flux is invariant and given by Eq. (23).

The particular case when q2 = 0 was treated theoretically by Tollmein in the turbulent case, see Figure 3. This example demonstrates how the fluxes of vorticity in the two discontinuity surfaces before impact are real, in the sense that upon impact, they are instantly converted to a physically actual shear flow involving vorticity and shear stresses.

Figure 3. Impingement of moving and stagnant fields, resultant shear zones.

14

Growth of Discontinuity Surfaces. The rate of change of the length of discontinuity surfaces in unsteady flow are constrained by the flux of vorticity along them. The convection of vorticity is governed by the Con- servation Law, Eq. (12). On the cavity surface, where w = qs, it follows (non-triviall y ) that,

Dqs - aq.3 a q s -- - + q s - - 0 , Dt at as and after integrating along the surface, that,

provided that the process is loss free. The terms on the right are the incoming and outgoing vorticity fluxes, respectively.

This law plays a crucial role in the behavior of unsteady cavities and re-entrant jets as discussed in Section 7. The growth of a reentrant jet, however, will be effected bywall stresses which produce vorticity flux of the opposite sign and retards the jet, as well as by losses at the cavity rear.

4. CAVITY CONFIGURATIONS AS CONSTRAINED BY VORTICITY

Vorticity Constraints. The cavity surfaces must originate in fluxes of vorticity from the wetted surfaces of the cavitating body, where these fluxes are independent of viscosity provided that the viscous layers are thin. The originating integrated wall fluxes must at each instant consist in equal parts of opposite signs, as the net flux from the body at any instant must be null according to Kelvins Law.

The growth in length of the cavity surfaces is limited by the respective vorticity fluxes along them, according to Eq. (25).

Topological Constraints. In principle, the cavity surfaces can terminate in one of several geometrically distinctive ways. For example, see Figures 4(a - c) and 5(d - e):

(a) by impingement on a surface of opposite flux resulting in cancelation; Figure 4a, and a wake-free flow.

(b) by reattachment and penetration into surrounding fluid, resulting in the creation of a fluid filled shear zone bisected by a dividing streamline, the extension of the cavity surface, Figure 4b. This results in a splitting of the vorticity flux, one part flowing downstream to create the wake, and

15

the other part feeding a recirculating eddy inside the fluid filled cavity rear, Figure 4b. This reattachment-penetration may also be regarded as the impingement of separate surfaces.

(c) by extension of its own length terminating in a reentrant jet, Figure 4c.

(d&e) by impingement on itself, forming enclosed regions with circulation, constituting the first stage of vortex-shedding. Then, by shedding of the enclosed regions to allow renewed growth of the cavity surfaces, as dictated by the vorticity flux, Figures 4d and 4e.

In the case of cavity self-impingement, the symmetrical, Figure 4d, or asymmetrical, 4e, impingement of the cavity surface must result in the separation of the enclosed regions, which are hollow vortices, and their convection downstream and subsequent collapse in the pressure rise there, while conserving moment of vorticity in the wake. In the asymmetrical case, a K & m h double sheet of sizeable bubbly clouds with alternating vortical sign and structure can result. The coherence of these structures is probably dependent on the Reynolds number of the flow.

Force Constraints. The temporal rate of change of the integrated vorticity moment, Eq. (8), must at every instant correspond to the body forces. In particular, because of the drag, the vorticity distribution in the mean and wake fields must contain vorticity separated vertically, with (in the mean) the negative vorticity above, and the positive vorticity below, as in a KAxmAn double vortex sheet, or as in a normal self-similar wake pattern. At the same time, the net vorticity over the entire field at every instant must be null.

The previous requirement restricts the flux cancelation configuration, Figure 4a, to the case of a drag-free forebody. Such special cases exist, Johnson and Starley (1962) and Tulin (1964b), but in the usual case where drag accompanies the cavity flow, this configuration is impossible.

Restraint of Vertical Motion. The natural tendency of flow separation on blunt bodies is for the separation process to oscillate vertically, creating asymmetric shedding and wake, as in Figure 4e. This oscillation can be prevented in the case of blunt bodies by the insertion of a thin horizontal plate on the centerline behind the body and into the wake.

In the experiment of Arie and Rouse (1956) the separated flow behind a vertical plate was created in this way. This resulted in a smaller wake under pressure and a correspondingly longer separation pocket than for the

16

Figure 4. Cavity termination models.

17

vertically unconstrained wake. In the case of partial cavitation on a foil, the foil acts to restrain the

vertical motion, acting as a splitter plate in the previous example. This means that the wake behind the partial cavity must more resemble the dipole structure, Figure 4d, than the Ktirmhn double vortex sheet.

5. VISCOUS SEPARATION VS. CAVITATION

Viscous Separation Pockets. Aerodynamic experiments on thin lifting airfoils as well as on bluff bodies with restraint of vertical motion, reveal the existence of mean flow patterns featuring closed pockets containing almost stagnant fluid at almost uniform pressure. In the particular case of a vertical flat plate, Arie and Rouse (1956), the pressure coefficient in the wake central region, c’p, was found to be, c’p = -0.57, and in the center of the separation region just behind the sharp leading edge on the upper surface of a thin inclined wedge, McCullough and Gault (1951), cb = -1.2, constant for incidence up to 6’.

Interference with Cavitation. It is evident that in water, cavitation in these separated flows would, if present, be inhibited or restricted to dis- persed bubble cavitation for u > Ic’pI. Under these conditions the cavity will be difficult to observe and measure. This was recognized in 1980 by (T-H), who saw clear evidence of it in the cases for which they compared predictions and measurements: “The theoretical over prediction at the higher incidences is likely due to viscous stall effects on the foil In fact, it must be recognized that if a separation bubble can exist at u(sep) < a(cavitation), then cavitation will be inhibited . . . . ”

This criterion implies that this effect will begin on a sharp nosed foil at a = 3.4’ for (a/.) = +0.05, and at a = 6.8’ for (a/.) = 0.10. Evidence of this interference can also be seen in more recent experiments, Sato, et al (2001), notably in observations of their partial cavity oscillation for a = 5”, which differed sharply and did not correlate for variations in u, in comparison with data at 1.5’, where no interference with cavitation could be expected.

It should be fruitful to pay more attention to this condition, a > & I , as well as other viscous separation effects, in future studies of partial cavitation. Note that the condition on u is more easily reached for smaller (ala), and thus more likely in the case of ‘partial cavity oscillation’ than in translational cavity oscillation.

Finally, we should mention here that the two cases of viscous separa-

18

tion pockets mentioned above, were successfully modeled by (T-H), who matched the free streamline solution in the outer field with the viscous flow in the near field. We return to this subject in Section 6 , below, where we present the partially filled cavity model, which can be used for all (T 5 IcIpI.

6. THE SUBCFUTICAL REGIME

Experiments, Sat0 et a1 (2001), reveal two distinct regimes of oscillation: partial cavity, ( l / c ) < ( l / c )* ; and transitional cavity ( t / c ) > ( l / c ) * . We propose a separate model for each regime, described in this Section and the next. Our view is consistent with Watanabe, et al (2001).

The Partially Filled Cavity (PFC). A model for ( l / c ) < ( l / c ) * , and 0 5 d. It was originated by Tulin-Hsu (1980), and then applied successfully by them to the quantitative prediction of pocket pressure and length in fully separated homogeneous flow, i.e., (T = 0'. The assumptions of the model are very consistent with wind tunnel measurements of fully separated, vertically constrained flows, see Section 5.

In this (in the mean) steady but turbulent flow, the cavity collapse region has fiIled with fluid from the rear, which penetrates and occupies the cavity volume, as necessary. The collapse and flow deceleration process therefore takes place in this filled region, the gas being restricted in the cavity between the back of the forebody and front of the filled region (pocket) as shown in Figure 4b.

The cavity surface meets the pocket at pt. a and continues on as a dividing streamline on which a turbulent stress acts. This streamline, a-b, separates the exterior flow, which goes on to form the wake, from the interior flow into the pocket. Near pt. b where the dividing streamline stagnates on the foil upper surface there is a rise of pressure which contributes to the form drag acting on the exterior surface of the pseudo-body consisting of the forebody and the space enclosed by the cavity surface and dividing streamline. It was pointed out by (T-H) that the drag on this pseudo-body must be identical to that on the entire forebody alone. Therefore by matching the drag of the two bodies for a given cavitation number (i.e., cavity pressure) the required length, lp , of the pocket may be determined. The drag of the pseudo-body includes both the frictional and form (pressure) drag acting on the dividing streamline. Using the analysis and constants in Tulin-Hsu (1980), the pocket length in the case of the partial cavity on

19

a sharp edged inclined flat foil is found to be,

as shown in Table I. Table I. (Pocketlength)/(Cavity length) vs. 0

(%/lC) a 0.13 0.2 0.35 0.4 0.46 0.5 0.74 0.8 1.03 1.2

The vorticity flux on the cavity surface, q,2/2, is split at impingement, pt. a, with a fraction, pq,2/2, flowing into the exterior, and the remainder into the pocket. Based on the Tollmein model, p M 0.5. In this case, half on the impinging vorticity flux flows into the wake and half into the pocket.

The free streamline counterpart of the PFC is the double spiral vortex. In this model the collapse and deceleration is shrunk into the point where the incoming (speed qc) and the outgoing (speed qo) spirals meet. The incoming vorticity flux is 4,212, and outgoing is $12, The difference in these fluxes can be interpreted as lost into the pocket in the PFC model. This implies that p = 1/(1+ a), which yields not unreasonable values.

The double spiral vortex may thus be used in potential flow calculations to represent the real model.

Forced Oscillation. If the eddy within the pocket is to be stationary, then the vorticity flux into the pocket must be canceled by a corresponding flux of opposite sign from the wetted foil surface which forms the bottom of the pocket. In principle, the eddy may spin up until this equilibrium exists. There is, however, nothing to hold the fluid in place at its front face, while it is subject to turbulent pressure fluctuations from the near wake acting on the dividing streamline. A broad-band forcing of the pocket can therefore be expected with a peak frequency (Hz), f , depending on the local parameters, qc and 1, (or, perhaps, l,), so f l c / q c = constant.

In fact, this behavior and scaling of the peak oscillation was found in Sato, et a.l (2001) for (Y = 1.5”, see their Figures 4(a) and 5(b). Tanimura, et a1 (1995) noted that oscillations were prevented by the presence of a bar inside the cavity on the foil upper surface. This suggests that the mean

20

steady cavity is only weakly forced in this regime, and that a steady model may be appropriate, as we are suggesting here.

Finally, the evidence for supporting the PFC as the appropriate sub- critical model arises from four sources:

0 the success of PFC in its use by (T-H) to predict 0’ for both the inclined flat foil and the vertical plate.

0 the success of its potential flow surrogate, the double spiral vortex, in its use by (T-H) for the prediction of mean lift and cavity lengths for a variety of foils, see Tulin and Hsu (1980).

0 the existence of a steady cavity solution in this regime, as first found by Acosta.

0 the calculated static stability of the short cavity solution by (T-H), providing return toward equilibrium under forced excitation.

7. THE SUPERCRITICAL REGIME

Unstable Oscillation. For the condition (a /a) > (a/c~)*, observations and pressure measurements reveal a periodic large amplitude oscillation at a sharply peaked fundamental frequency and its harmonics, Sat0 et a1 (2001). The cavity length grows from, for example, (C/c) = 0.5, to a value of ( l / c ) = 1.4, followed by a sharp collapse associated with tear-off, or detachment, of the main cavity volume, and its convection downstream as a cloud of cavitation.

Correlated tunnel pressure oscillations show a very rapid rise associated with the termination of the cavity collapse followed abruptly by renewed cavity growth. The same rapid rise and peaking of the radiated pressure had been observed earlier by Bark and van Berkelom (1979), in their study of cavity growth and collapse on an oscillating hydrofoil. This is consistent with the fact that the radiated pressure due to an unsteady source representing the growth of the cavity volume, d V / d t , is proportional to d2V/d t2 . The peak pressure pulse is therefore to be expected at the point of tear- off where the cavity collapse suddenly ends and its renewed growth very quickly begins. This is shown very clearly in the measurements of both Bark and van Berkelom, and of Sato, et al.

It has been pointed out by (T-H) that the parameter (a/c~) is more correctly represented by a parameter based on the actual lift coefficient, CL. The reason for this is that the cavity is actually generated by the real flow speeds on the foil, as shown clearly in their revised asymptotic theory. They therefore recommended to replace (a/.) with (c~/27ra). This means

21

that changes in lift, whether due to changes in incidence or not, will result in changes in the cavity length and shape.

In their theory of unsteady cavity motions they extended Acostas theory, based on linearization, and found,

t /c = F [(cd2na); (V/7%4] (27)

where V, is d V / d t , the rate of temporal growth of the cavity volume. The double branched curve of Acosta is therefore augmented by a set of curves, each associated with the volume growth. They found that the space inside Acostas curve is reserved for collapsing cavities and the space outside for growing cavities. They therefore concluded that the cavity history under foil oscillations would involve hysteresis, Figure 5, and that for sufficiently large oscillations that the cavity would tearoff. This had, in fact, been observed by Bark and van Berkelom (1979),

Figure 5. from Tulin-Hsu (1980).

Cavity history due to foil pitching oscillation showing hysteresis (schenmatic)

It was at that time not appreciated that a closely related behavior, the unstable transitional oscillation, could occur for supercritical conditions, (c./a) > (cr/a)*, even without foil oscillation. This oscillation is driven on the growth side by the requirement that all cavity lengths in the supercritical regime corresponds to growing cavity volumes. The effect is exaggerated by the increase in foil lift with cavity length.

An increase in lift of 30-40% is predicted (and measured) for the inclined plate as the cavity grows to its critical length. However, as it grows further into the beginning of the supercavitating regime, the lift must rapidly decline since the supercavitating lift for long cavities is only 25% of that on the non-cavitating plate. A sketch of the estimated trajectory of the

22

changing cavity is shown in Figure 6, which is only schematic, in the case where the critical value of (alg) is 0.13.

i v.=o

Figure 6. Transitional cavity oscillation (schematic).

The shape of this curve is dictated by the quasi-steady solution, Eq. (27), and by the variations in lift which accompany changes in (Clc). Start- ing with a supercritical value of (ct/27ra) the cavity is required to grow from the beginning, and this is accompanied by an increase in lift. After reaching the critical length, the cavity growth continues while the lift eventually declines. The growth slows up markedly upon entering the supercavitating regime. This eventually results in the penetration of the re-entrant jet into the cavity and its subsequent tear-off. Then the cycle is renewed.

The period of the transitional cavity oscillations is determined largely by the growth phase, as the collapse is much quicker.

The Role of the Unsteady Re-Entrant Jet. In a steady cavity, the re-entrant jet will grow from the collapse region and impinge on the forebody in the time, t = C/qc, resulting in its destruction; losses will increase this time.

In a growing cavity, where the jet is of finite length inside it, the vorticity flux on the cavity surface provides new surface both to feed the cavity growth itself, and then to feed the jet. It follows from Eq.(25),

where qj is the speed of the jet. For dotC/qc < 1/2 therefore, the jet moves upstream toward the forebody and cavity detachment can occur.

23

These results do not take into account the retardation of the jet by friction on the foil surface along which it is flowing or of vorticity flux losses in the collapse region. Nevertheless, they show that cavity growth will delay, or even prevent the penetration of the cavity by the jet.

Examination of limited data in Sato, et a1 (2001) suggests that the growth of the cavity takes place in two stages, the much faster stage terminating when the cavity reaches the trailing edge. The initial growth rate seems sufficient to prevent the penetration of the cavity. On entering the supercavitating regime, the growth rate slows significantly, allowing the penetration and eventual detachment of the cavity by the jet, as in Figure 4c.

Finally we can suggest the following understanding of the supercritical, unstable oscillation regime:

0 The growth of the partial cavity follows the law of cavity length including the effect of lift and of cavity volume growth. For this law we now have the calculations of (T-H) based on linearized theory, but these should be improved and extended into the supercavitating regime in a continuous way. See also Watanabe, et al (2001).

0 The vorticity flux on the cavity surface is sufficient to feed a re-entrant jet, but the penetration of the jet is prevented by the growth of the cavity in its initial phase to the trailing edge

0 The growth then significantly slows, i < qc/2, and the jet quickly penetrates the cavity and impinges on it, leading to its detachment and convection downstream as a collapsing cavitation cloud.

0 The shortened cavity now begins to grow quickly, renewing the cycle.

0 Therefore, the re-entrant jet is a byproduct of the cavity growth, but it plays an essential role in the destruction of the cavity.

0 The greatest part of the period of the transitional cavity oscillation is spent in the supercavitation regime, ( t / c ) > 1.0, in a low growth and reversal phase, and it is limited by the time it takes the penetrating jet to reach the cavity front. This process, including the retarding effect of the wall, should be studied by simulation.

For students and researchers interested in initiating a detailed investigation of cavity flows, the review paper by Wu (1972) and the books by Robertson (1965) and Brennen (1995) are good starting points. Finally, the author’s dedication to his colleague, Ted Wu, is included in the Appendix

ti > qc12).

24

of this book.

Acknowledgement. The author wishes to thank Prof. Yoshinobu Tsu- jimoto of Osaka University for drawing his attention to this problem and for providing stimulating discussion and references. He also wishes to thank Ms. Eileen Horton for her help in the preparation of the text for printing.

References 1. Acosta, A.J. (1955), “A Note on Partial Cavitation of Flat Plate Hydro-

foils,” Calif. Inst. of Tech. Hydrodynamics Lab. Report No. E-19.9. 2. Arie, M. and H. Rouse (1956), “Experiments on TweDimensional Flow

Over a Normal Wall,” J. Fluid Mechanics, Vol. 1, Part 2. 3. Arndt, R., C. Song, M. Kjeldsen, J. He and A. Keller (ZOOO), “Instability

of Partial Cavitation: A Numerical/Experimental Approach,” Proc. 23rd ONR Symp. on Naval Hydrodynamics, Val de Reule.

4. Bark, G. and W.B. van Berkelom (1979), “Experimental Investigations of Cavitation Noise,” Proc. 12th Symposium on Naval Hydrodynamics, pp. 470493.

5. Birkhoff, G. and E.H. Zarantonello (1957), Jets, Wakes, and Cavities, Aca- demic Press.

6. Brennen, C. E. (1995) Cavitation and Bubble Dynamics, Oxford University Press.

7. de Lange, D.F. and G.J. de Bruin (1998), “Sheet Cav- itation and Cloud Cavitation, Re-Entrant Jet and Three Dimensionality.” In: In Fascination of Fluid Dynamics, Kluewer.

8. Efros, D. (1946), “Hydrodynamical Theory of Two-Dimensional Flow With Cavitation,” Dokl. Akad. Nauk USSR, Vol. 51, pp. 267-270.

9. Fabula, A. (1962), “Thin Airfoil Theory Applied to Hydrofoils With Single Cavity and Arbitrary Free Streamline Detachment,” Fluid Mechanics, Vol. 12, Part 2.

10. Geurst, J.A. (1959), “Linearized Theory for Partially Cavitated Hydrofoils,” Inter. Shipbuilding Progress, Vol. 6, No. 60.

11. Geurst, J.A. and R. Timman (1956), “Linearized Theory of Two- Dimensional Cavitation Flow Around a Wing Section,” IX International Congress of Applied Mechanics.

12. Gilbarg, D. and D. Rock (1946), “On Two Theories of Plane Potential Flows With Finite Cavities,” Naval Ord. Lab. Memo, 8718.

13. Goldstein, S. (1938), Modern Development in Fluid Dynamics, Dover Pub- lications Inc. (1956), Clarendon (1938).

14. Hanaoka, T. (1967), “Linearized Theory of Cav- ity Flow Past a Hydrofoil of Arbitrary Shape,” Ship Research Institute, Japan.

15. Hsu, C.C. (1969), “Flow Past a Cascade of Partially Cavitating Cambered Blades,” Hydronautics, Incorporated Technical Report 703-6.

16. Johnson, V. and S. Starley (1962), “The Design of Base-Vented Struts for

25

High Speed Hydrofoil Systems,” Hydronautics Tech. Report 001-16. 17. Jones, R.T. (1941), “Correction of the Lifting Line Theory for the Effect of

Chord,” NACA T N 817. 18. Kermeen, R.W. (1960), “Experimental Investigations on ThreeDimensional

Effects on Cavitating Hydrofoils,” Calif. Inst. of Tech. Engineering Report

19. Kreisel, G. (1946), “Cavitation With Finite Cavitation Numbers,” Great Britain Admiralty Research Lab. Report No. R 1/H/36.

20. McCullough, G.B. and D.E. Gault (1951), “Examples of Three Representa- tive Types of Airfoil-Section Stall at Low Speed,” NACA T N 2502.

21. Meijer, M.C. (1959), “Some Experiments on Partly Cavitating Hydrofoils,” International Shipbuilding Progress, Vol. 6, No. 60.

22. Numachi, F., K. Tsunoda and I. Chida (1957), “Cavitation Test on Hydro- foil of Simple Form (Report l),” Report Inst. High Speed Mechanics, Japan,

23. Parkin, B.R. (1958), “Experiments on Circular Arc and Flat Plate Hydro- foils,” J. Ship Research, Vol. 1, No. 4.

24. Riabouchinsky, D. (1920), ‘‘On Steady Fluid Motion With Free Surface,” Proc. London Math Society, Vol. 19, Ser. 2, pp. 206215.

25. Robertson, J. M. (1965) Hydrodynamics in Theory and Practice, Prentice- Hall.

26. Sato, K., M. Tanada, S. Monden and Y. Tsujimoto (2001),“Observations of Oscillating Cavitation on a Flat Plate Hydrofoil,’’ Proceedings, CAV 2001, Pasadena.

27. Tanimura, M., Y. Tagaya, H. Kato, H. Yamaguchi, M. Maeda, and Y. Kawanami (1998), “Mechanism of Cloud Cavitation and Its Control”, Jour- nal of the Society of Naval Architects of Japan. Vo1.178, pp.41-50 (in Japanese)

28. Tulin, M.P. (1953), “Steady Two-Dimensional Cavity Flows About Slender Bodies,” David Taylor Model Basin %port No. 843.

29. Tulin, M.P. (1954), “Hydrodynamic Characteristics of Supercavitating Hy- drofoil Sections,” Proc. of Joint Admiralty-U.S. Navy Meeting on Hydrobal- listics, NAVEXOS P-l452(c).

30. Tulin, M.P. (1955), “Supercavitating Flow Past Foils and Struts,” Proc. NPL Symp. on Cavitation in Hydrodynamics. Also Philosophical Library, New York, 1956.

31. Tulin, M.P. (1964a), “Supercavitating Flows - Small Perturbation The- ory,” J. Ship Research, Vol. 3, No. 3. Also Proc. IUTAM Symposium on the Applications of Analytic Functions in Continuum Mechanics, Tiblisi, USSR, 1963.

32. Tulin, M.P. (1964b), “The Shape of Cavities in Supercavitating Flows,” Proc. 11th International Congress of Applied Mechanics, Munich, (IUTAM), Springer-Verlag, pp. 1145-1155.

33. Tulin, M.P. and C.C. Hsu (1977), “The Theory of Leading Edge Cavitation on Lifting Surfaces With Thickness,” Proc. of Symp. Hydrodynamics of Ships and Offshore Propulsion Systems, Det Norske Veritas, Oslo.

NO. 47-14.

Vol. 8, pp. 67-88.

26

34. Tulin, M.P. and C.C. Hsu, (1980), “New Applications of CavityFlow The- ory,” Proc. 13th ONR Symp. on Naval Hydrodynamics, Tokyo, pp. 107-131.

35. Wade, R.B. (1963), “Flow Past a Partially Cavitating Cascade of Flat Plate Hydrofoils,” Calif. Inst. of Tech. Engineering Report No. E79-4.

36. Wade, R.B. (1967), “Linearized Theory of a Partially Cavitating Plano- Convex Hydrofoil Including the Effects of Camber and Thickness,” J. Ship

37. Watanabe, S., Y. Tsujimoto and A. Furukawa (2001), “Theoretical Analysis of Transitional and Partial Cavity Instabilities,” Trans. ASME, Vol. 123, pp.

38. Wu, T. Y.-T. (1972) “Cavity and wake flows.’’ Annual Review of Fluid

Res., Vol. 11, NO. 1, pp. 20-27.

692-697.

Mechanics, vol. 4, pp. 243-284.

CHAPTER 1

NONLINEAR WAVES: THEORETICAL CONSIDERATIONS


LOCALIZATION OF DISPERSIVE WAVES IN WEAKLY

RANDOM MEDIA

CHIANG C. ME1

Department of Civil and Environmental Engineering Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

JORGEN H. PIHL, MATHEW HANCOCK & YILE LI

Department of Civil and Environmental Engineering Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

Dedicated to Professor Theodore Y-T. Wu on his 80th birthdag

We apply the method of multiple scales for studying nonlinear dispersive waves through a weakly random medium. Specifically for a nearly periodic wave the evolution equation is a Schrodinger equation with a complex damping term. For transient long waves in shallow water, the result is a KdV equation with additional terms representing diffusion and modification of the phase velocity and dispersion. Sample numerical results for related nonlinear theories will be cited from published sources.

1. Introduction

In sea wave modeling considerable efforts have been devoted to the effects of nonlinear wave-wave interactions, wind forcing and dissipation due to turbulence and breaking. In coastal waters, friction at the bottom contributes to additional dissipation. Until recently possible energy loss due to wave scattering by irregular bathymetry has received scant attention. In view

*Note by CCM: It is my heart-felt pleasure to salute Professor Wu to whom my life- long indebtedness is due. His tireless and stimulating tutelage at Caltech helped me to start my own career. In addition, he has been a constant source of inspiration and an exemplary scholar to emulate. A greater teacher than him is hard to find.

29

30

of the improving technology in remote sensing for bathymetric data, more theoretical work is needed to predict the effect of such radiation losses.

Deterministic theories for wave diffraction and refraction have seen remarkable advances in recent decades. For a gentle bed slope covering a large area, ray approximation is well developed for both linear and weakly nonlinear waves. For abrupt changes of depth involving a few large scatters, effective theoretical and numerical tools of prediction also exist. Multiple scattering by many scatters may be classified in two groups. Periodic scatters leads to Bragg resonance (which is frequency-selective) and can be treated by asymptotic theories14. Less is known on random irregularities distributed over a large area of the seabed.

Linear wave propagation in a random medium is an old topic in physics. Example light through sky with dust particles, sound through water with bubbles, elastic waves through a solid with cracks, fibers, cavities, hard or soft grains (see, e.g., Chernov5, and Ishimarug). It is known in one- dimensional propagation that random scattering leads to changes in the wavenumber (or phase velocity) as well as an amplitude attenuation, if the inhomogeneities extend over a large distance12. These changes appear as a complex shift of the wavenumber vector with the real part corresponding to the phase change and the imaginary part to spatial attenuation. In particular, the spatial attenuation due to randomness is effective for a broad range of incident wave frequencies, in sharp contrast to the frequency-selective Bragg scattering by periodic inhomogeneities. Anderson' was the first to show in solid-state physics that a metal conductor with disorder can turn to an insulator if the disorder is sufficiently high. This transition, called Anderson localization, is now known to be important in classical systems also (see Sheng23i24).

Earliest theories on sea waves over disordered seabed are due to Hasselman" and Long13 and Elder & Molyneaux7 who employed the technique of Feynman diagrams. Kawahara" used multiple scales to deduce the evolution equations for water waves, as did Benilov & Pelinofskii3 for mathematically related systems. Stimulated by the experiments of Belzons et a14 and the linear theory of Devillard et a16, Nachbin & Papanicola~u'~ and others, the present authors have studied the physics of weakly nonlinear sea waves for narrow-banded sinusoidal waves over intermediate depth, and for transient waves in shallow water. Analytical and numerical investigations have been carried out by solving a damped nonlinear Schrodinger equation for the former casel6I2l, and a KdV-Burgers equation for the latter17. For long periodic waves in shallow water, the mutual influence of localization

31

and harmonic generation has been studied by Grataloup & Meis. It is found that the amplitudes of the fundamental and higher harmonics are governed by coupled nonlinear equations similar to those in optics2, but with additional terms whose complex coefficients are deterministic and related to certain correlation functions of disorder.

In this paper we review some of the essential ideas and results with a view to suggesting further extensions to other problems of technological interest. The basic analysis will be explained for two problems. Some numerical results will be cited from recent publications.

2. Nearly sinusoidal waves

We illustrate the derivation for the lateral displacement of a taut string buried in an elastic medium,

a2v a2v

Pdt2 ax2 - T - + K(1+ E M ( x ) ) V + f26V3 = 0,

where V denotes the lateral displacement, p the mass per unit length, T tension in the string, K the mean spring constant of the surrounding medium. We assume that the spring force contains a weak random component e K M ( x ) where M has zero mean and the typical length scale of O ( l l k ) *

2.1. The envelope equation

We consider narrow-banded sinusoidal waves propagating toward a large region of randomness from x N 00. Anticipating that wave oscillations, dispersive modulation and the modulation by randomness, are vastly different, three different time scales, w - ' ( l , e - l , c2) and space scales k- ' ( l , c - l , E - ~ )

must exist where e << 1. Let us introduce fast and slow variables x , x1 = ex, 2 2 = e2x, t , tl = et and t 2 = e2t, and further assume the perturbation series

V = PV,, with V, = V n ( x , x 1 , x 2 , t , t 1 , t 2 ) . ( 1 ) n=0,1,2,. . .

Allowing M to depend on both x and 5 2 , we obtain the following perturbation equations at orders O(eo), O(E), and O(e2),

32

p--T-+KV2+p ah a2v2 ++2-) d2 v1 a t 2 a x 2 dtatl

a%, awl ax: dxdx2 dXdX1 +KMVl -T (3 +2- + 2-) + 6v; = 0. (4)

Let the leading-order solution be a progressive wave

= A ( ~ 1 x 2 ; t l t 2 ) ei(kx--wt) ( 5 ) which is a coherent wave unaffected directly by disorder. Eq. (2) implies the dispersion relation

1 / 2

(6) pw2-K

so that the phase velocity is

We shall assume the frequency to be always above cutoff, w > that k is real and positive, and increases monotonically with w.

so

At the order O ( E ) Eqn. (3) can be written

(8)

To avoid unbounded resonance we set the coefficients of ei(kx-wt) to zero, d A T k d A dA dA - +-- = -+c - = o (9) at, pw ax1 at1 9 d X l

where dw T k c - - -_ ' - d k - pw

is the group velocity. The remaining part of (8) is

a w l a 2 v l

d X 2 pw - T- + KV1 = - K M ( X ) A ~ ~ " - ~ " ~ .

where the forcing term on the right is a random function of x through M ( x ) . The solution can be obtained by Fourier transform or Green's function method,

33

which is random (incoherent) with zero mean, but radiates energy to rtoo. Since the ensemble average of M(x) vanishes, we have, (V1) 5 0. The

ensemble average of (4) becomes

Assuming that the disorder is statistically homogeneous so that the correlation function (M(x)M(E)) is a function of x - E and 5 2 , the integral above is constant in x.

Using Eqs. ( 6 ) and ( 9 )

where

(14) d2w T K dk2 p2w3

from (6) and (10). Equating the sum of all secular terms in (13) to zero, we get the evolution equation for A,

-=-

(15) d2w dA 36 (g ax2 dk2 ax? pw

2i -+cg") - -- - -IAl2A + 2pA = 0

where

is a complex coefficient depending only on x2. Combining this result with Eq. ( 9 ) we obtain

36 -lA12A + 28A} = 0. (17)

which can be reduced to the Schrodinger equation

dA d2wd2A 36 d-r dk2 d.Cz pw

22- + -- - -lAl2A + 2pA = 0.

by the standard transformation to the moving coordinate system:

= x1 - Cgtl, r = t 2 = Etl (19)

34

Alternately we can take

to get

+ 2ipiB = 0. aB d2wd2B 2i- + -- ar dk2 at2

For a wave packet with IAI,IBI --+ 0 as [' -+ fool the total energy in the coherent wave can be defined by

Multiplying ( 2 1 ) with -iB*, adding the complex conjugate equation, and integrating the sum with respect to 5 from -co to co, we find

Thus the total energy in the coherent waves decays due to multiple scattering by randomness. If the extent of constant randomness S is infinite and pi is constant, then

E(T) = E(0) e-2PaT (24)

The time scale for total energy attenuation e 2 T ~ / 2 0; pi can be reinter- preted as the localization distance of the averaged amplitude by

L = C g T ~ 0; l / ~ ' p i ( 2 5 )

If instead the envelope is periodically modulated with the period S, (23) and (24 ) still hold if the range of integration extends only over the period S of modulation.

As a specific example we take the correlation to be Gaussian

( M ( x ) M ( ~ ) ) = a2(x2)e-a12--EI, (26) where EU corresponds to the root-mean-square of the fluctuation amplitude and a-l the correlation length. It can be shown 25 that

a(a2 + 4k2) + a2 + 4k2 2 (a2 + 2k2)

(27)

00 eiklz-<l a e 2 --(YIz--Ele-ik(z-€)g = a2

35

We omit the discussion of physical implications and turn to water waves which can be treated in very similar manner.

2.2. Namw-banded water waves over intermediate depth

Similar analysis has been carried out for nearly sinusoidal waves in water of intermediate depth with slow modulations in one l6 or two horizontal directions21.

For unidirectional waves (two dimensional flow), the envelope of the short waves is governed by

IAI2A w” a2A wk2 (cosh 4q + 8 - 2 tanh2 q)

16 sinh4 q

where h is the mean depth, q = kh, and w and k are related by w2 = gk tanh kh, so that

C, stands for the group velocity of the short waves. The short wave envelope is coupled with a long wave whose potential 410 is governed by

at: 8x1 2ksinh2q 8x1 4sinh2q dtl d2&0 - g h T d2+10 = w3 cosh2 q dlAI2 w2 aIA12. (31)

2.2.1. Localization of steady Stokes waves

As a first application of the evolution equations, we examine the limiting case of a steady wave train. There is no dependence on (xl , t l , tz), so that A = A(x2). Eq. (15) reduces to

aA ax2

C - + ialAI2A - $A = 0,

where wk2(cosh4kh+8-2tanh2 kh)

a = > 0. 16 sinh4 kh

The solution to (32) is a modified Stokes wave exponentially attenuated (localized) in the direction of propagation,

36

where a0 is the real amplitude at 2 2 = 0 and pT, pi are the real and imaginary parts of p, respectively.

From (33), the amplitude of A decays exponentially as IAI = aoe-Pix2/'g. Note that the spatial attenuation is exponential and is independent of nonlinearity. If the extent of disorder is L in the 2 2 scale, the amplitude at the transmission end is clearly reduced from the incident amplitude by a factor exponentially diminishing in L. Thus the physical consequence of random scattering here is the same as in the simplest cases of localization, i.e., exponential attenuation in space. Our localization distance can be defined by

For Gaussian correlation

r(E) = e-c2/e2, SO that ?(2k) = C.\/j;exp (35)

where ! is the Gaussian correlation distance. Substituting (35) into (34) yields

Lloc - (2kh + sinh2kh)2 -- h 2J;F~2kh(~/e)2(ke)3 (1 + e- (ke )2 ) *

Large u (strong disorder) or large a/! (steep roughness) lead to short localization distances and fast attenuation.

Belzons et a14 performed experiments on the localization of infinitesimal water waves over a random bathymetry in a small wave flume of length 4 m and mean water depth h in the range 1 to 4 cm. Bathymetric irregularities were represented by 58 discontinuous steps of random length and amplitude. The step height and step length were uniformly distributed, respectively, between -Ah and Ah (zero mean), and between !B - A t and !B + A!. The main results for localization were reported for h = 1.75 cm, Ah = 1.25 cm, eB = 4.1 cm and A! = 2.0 cm. Thus, the height of the steps was not small compared to the mean depth. Also, the recorded data on the localization length exhibit very large scatter, due in part to averaging over several realizations of the random bed and in part to vortex shedding at the step corners. Therefore their data only give qualitative indication of localization. Strictly speaking, our theory for small and gently sloped roughness cannot be compared with these data. A tentative comparison is shown in Figure 1 showing mutually consistent trends. Decisive checks must await new experiments for small-amplitude randomness, common in many oceanographic situations.

37

T

25

Figure 1.

zons et a1 for tall steps. Comparison of present theory for gentle roughness and experiments by Bel-

2.2.2. Nonlinear instability of side-band disturbances

Following the standard procedure (see e.g. 15, p. 614616), (31) and (29) can be combined and transformed to a nonlinear Schrodinger equation, which reads, in physical variables

W C X ~ d2A’ -2 ( $ + cg $) A’ + - - + w k 2 a 2 lA’I2 A‘ - i& ( kno)2 A‘ = 0, (37) k2 a x 2

where (TO is the dimensional root-mean-square bottom roughness height and & = Pi/ (kn)2 . Transforming to moving coordinates and introducing the dimensionless variables

B = A’/Ao, X = k2Ao (X - Cgt) Jm, 7 = 1~x21 ( k - 4 0 ) ~ wt.

(38) Equation (37) is reduced to the canonical form

where

38

signifies the relative importance of random and nonlinear effects and can be of order unity.

The special solution of (39) uniform in X is equivalent to (33),

In the moving frame of reference, the amplitude decays in time. Let us first examine how Bs reacts initially to side-band disturbances,

and substitute B = Bs (1 +a') into (39). Retaining first order terms in B', we obtain

Substituting 13' = R+il into (42) and separating real we obtain

dR 6'1 _--- 87- ax2 - O ,

and imaginary parts,

(43)

(44)

For a spatially sinusoidal disturbance with modulational wave number K,

R = Re (&r)eiKX) , I = Re (f(r)eiKX) . (45)

(43) and (44) can be combined to give

Instability is possible initially only if a2 > 0, corresponding to deep water with kh > 1.37, as in the case without disorder. However, since the carrier wave Bs decays in time, the side-band is unstable only if

Kd2e-28T - K2 > 9. (47)

Thus, over a random seabed, both the range of instability and the growth rate diminish in the course of propagation. Clearly, if 0 is large, attenuation takes over quickly and an initially unstable side-band is unlikely to grow significantly. However, if the randomness is weak relative to nonlinearity, nonlinear effects can still be important for some time.

As an example, we have solved an initial-value problem for the NLS equation (39) with a2 > 0 subject to periodic boundary conditions, by a

39

finite difference scheme26. At r = 0, the wave envelope contains a carrier wave and a pair of small, symmetric side-bands,

(48) l - i B(0, X) = 1 + 6 - cos x, fi

where 6 << 1 is a constant. Numerical results are shown in Figure 2, for a case of strong nonlinearity relative to randomness, 8 = 0.075. Here, the wavenumber of the side-bands is taken to be 1, which maximizes the left-hand-side of (47). It can be seen that unstable sidebands grow and then oscillate as they exchange energy with the carrier wave. However, over longer times, both the side-bands and the carrier wave decay due to random scattering. For larger values of 0, monotonic decay due to radiation damping dominates the evolution after a short time. Indeed, for K = 1, Eq. (47) implies that instability occurs only when

i

Thus, monotonic decay begins at r = 0 if 0 2 1, and at r = 0.94 if 0 = 0.5.

More numerical examples of nonlinear evolution with localization can be found in Mei & Hancock''. When long-crested Stokes waves pass over an elongated area of random seabed, diffraction occurs. it is shown in Pihl et a121 that in the wake of a slender triangular region of randomness, dark solitions are formed along straight rays emanating from the tip of the triangle.

3. Transient long waves in a shallow sea

3.1. The Kd V- Burgers integvo-differential equation

We begin with the well-known Boussinesq equations for long waves in shallow water. In physical dimensions, the horizontal and vertical coordinate are denoted by x*, z*, time by t+, the depth-averaged velocity by u*, the free surface displacement by q*, the constant mean depth by H and the depth variations by b*. Let K-l be the characteristic horizontal length and a the characteristic wave amplitude, we assume that E = a / H << 1, p = K H << 1, and E = O(p2). In addition we assume the depth fluctuations be small b*/H = O ( K H ) << 1, and introduce normalized variables (without primes) as follows:

40

3

Figure 2. Sideband instability affected by random roughness. At T = 0 the periodically

modulated Stokes wave enters the zone of disorder. In the top figure the carrier wave (zeroth harmonic) first loses energy to the first and higher harmonics of the side bands, then recovers to some extent. Ultimately all harmonics are localized. Dashed curve shows the monotonic decay of a uniform Stokes waves. Lower figure shows the envelope evolution.

By further choosing 1/K be the length scale of the incident soliton, i.e., K = m, then e = p2. The Boussinesq approximation for mass and momentum conservation are, to the first order in p2,

a 3 + - [(l - pb)u] = 0 at ax

41

and

au 2 au a7 p2 a3u

at ax ax 3 w a t - + p u - + - = - -

We assume that the depth fluctuation b(x ) is a stationary and random function of x, with zero mean.

Within the stated accuracy, the preceding two equations can be combined to give the stochastic differential equation,

3.1.1. The evolution equation

We shall now derive the statistical average of (53). In anticipation that the small disorder affects the leading order after a long distance inversely proportional to the mean square of the disorder, we introduce two space variables x and X = p2x and expand u and 11 as power series of p:

~ ( x , X ; t )=70f~171+~~~72+0(~1~) , ~ ( 2 , X ; t)=uo+pu1+p2~2+0(p3) (54)

The perturbation equations are easily found to be

At the leading order, 0 ( 1 ) , the governing wave equation (55) is homogeneous. We limit ourselves to a right-going wave with vanishing amplitude at x - t N -m,

V O ( X , x; t ) = uo(x, x; t ) = C(X; a), (58)

where (T = x - t .

forcing, and can be solved by means of Green’s function At the first order O(p), the inhomogeneous equation (56) has random

(59) 1 2

G(x, t; x’, t’) = -H [(t - t’) - I X - x ’ I ]

42

where H ( z ) is the Heaviside step function. The formal solution is

d ( dC(2’- t ’ ,X’) 00

71 = -J” d t ‘ / dx’G(x, t ;x’ , t ’ ) - b(x’) --oo -03 ax’ dX

where ~1 = q ~ ( x , X ; t ) . The ensemble average of the O ( p 2 ) equation is

By virtue of (58), the last three terms are all functions of a = IC - t , hence are homogeneous solutions of the averaged wave equation. Together they can be transformed to

a 2 c 3a2c2 1 a 4 c 2- + -- + -- d a d X 2 a 0 2 3da4

We now examine the first forcing term by using (60),

. , By assuming the disorder to be homogeneous in the short scale, the auto- correlation of depth fluctuations is

(b(x)b(x’)) = I?(<) with < = x’ - X . (64)

It can be shown that

( - b ( x ) a ) = 1 lrn p ( k , X ) t ( X ; k)eik(”-t) d k 7r --oo

where ((k) is the Fourier transform of C(<) and p ( k , X ) is the complex coefficient defined by

In view of (65), we get

(67) -- d (b(x)%) = Id e ikop(k , X ) t ( k , X ) dk dX 7rda --oo

Thus all forcing terms on the right of (61) are function of a = x - t . To avoid unbounded resonance for ( ~ 2 ) ~ their sum must vanish. After integrating this solvability condition with respect to (T, we obtain the asymptotic equation

43

for the leading order displacement, as seen by an observer traveling at the linear phase speedb during a very long course of propagation,

-+-'-+ " a' --=-- a3' Srn P ( k , X ) j ( k , X ) e i k " d k (68) ax 2 aff 6da3 2 n --M

This is just a linearized KdV equation modified by the additional term on the right representing the effect of random scattering.

In Mei & Li17 it is shown that (68) can be brought to a more explicit form by partial integration, Denoting by the exponential Fourier transform of f, and

Note that

Rep = k2 { ?f(0) 1 + ? ? ( 2 k ) } 1 > 0.

Substituting (70) into (68), we get, after invoking the convolution theorem,

a' 3 a' 3 a' 1 a3' r(o) a' +f (o ) a2' -+-'-+-'-+ --=-- -- dX 2 dn 2 da 6da3 2 do 8 da2

The effects of disorder are represented by terms on the right-hand side. It can be shown that all coefficients of the derivatives of C are positive. The first term implies a reduction of phase velocity, in the coordinate system moving at the linear wave speed. The second and third terms stand for diffusion, making the evolution equation a combination of KdV and Burgers. The fourth term signifies a reduction of dispersion.

For better insight we choose the correlation function to be Gaussian, with the correlation length taken to be 1* . The dimensional correlation function is,

bThe linear phase speed is unity in dimensionless form and in physical scale.

44

where B* is the root-mean-square amplitude of the random roughness. The normalized correlation function is

-O(l) . (74) D*’ 1 H2 p2 ( il:) where D2 = -- - r([) = D2exp --

It can be shown that (72) becomes

a5 3 a5 3 a( 1 a35 ~ + -c- + -c- + -- dX 2 da 2 da 6da3

(75)

3.1.2. Sample numerical results

Mei & Li l7 have discussed analytical approximations for small D and large X > 0 (which is the distance traveled by the soliton, or the fetch). We cite only the numerical results for a soliton entering a semi-infinite region of disorder. If the seabed were smooth everywhere, the solition would preserve its profile given by

< = sech2-((a 8 - X ) 2 In the coordinate system moving at the linear long wave speed, the soliton would travel at a constant speed forward with the same amplitude. The effects of increasing roughness height D2 can be seen in Figures 3.a to 3.d. For any finite D, the soliton is slowed down and its height diminishes with X. For large enough D , the soliton loses so much speed that it is slower than the linear long wave, hence the crest falls behind a = 0. Thus slowing down and flattening of the coherent wave are the main consequences of random scattering.

If the width of the random region is finite, the transmitted wave may disintegrate into several smaller solitons. These and other aspects are discussed in Mei & Li17.

4. Concluding remarks

It should be worthwhile for oceanographic interest to apply and extend the theory here to surface waves with a broad frequency band. Modification of

45

-60 -40 -20 0 20 1

0.8 0.6

Q 0.4 0.2 0

. . . . . . . . . . : . . . . . . . . . . ? . . . . . . . . . . . . . . . I . . . .

-60 -40 -20 0 20

-60 -40 -20 0 20

-60 -40 -20 0 20

Figure 3. Soliton evolution over a semi-infinite random seabed. The total computed

range of propagation is 0 < X < 100. Wave profiles are shown for every AX = 10.

The correlation length is I = 1. In figures (a) to (d), the square roughness heights are D2 = 0.1,0.25,0.5,1.0.

Zakharov’s theory is needed. Interfacial solitons in a two-layered Auid over a randomly rough bottom can be treated in ways similar to the surface soliton here. Finally in optical fibers for the transmission of electro-magnetic waves often span long distances of hundreds of kilometers. There can be significant losses due to imperfect connections or slight departure of the fiber cross section from a perfect circle. Theoretical analysis of losses due to wave

46

scattering by random inhomogeneities should be valuable. These problems and many others are interesting challenges for the future.

Acknowledgements

The financial support from the US Office of Naval Research (Grant N00014- 895-3128, Dr. Thomas Swean) and the US National Science Foundation (Grant CTS-0075713, Drs. John Foss, C. F. Chen and Michael W. Plesniak) is gratefully acknowledged.

References

1. Anderson, P. A. 1958, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492-1505.

2. Armstrong, J. A. Blombergen, N. Dushing, J. & Perhshan, P. S. 1962, Inter- actions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918- 1939.

3. Benilov, E. S. & Pelinovskii, E. 1988, Dispersionless wave propagation in nonlinear fluctuating media, Sou. Phys. JETP, 67, 98-103.

4. Belzons, M., Gauazelli, E. & Parodi, 0. 1988. Gravity waves on a rough bottom: experimental evidence of one dimensional localization J. Fluid Mech.

186, 539-558. 5. Chernov, L. A. 1960, Wave Propagation in a Random Medium Dover. 168

PP. 6. Devillard, P., Dunlop, F. & Souillard, J. 1988. J. Fluid Mech. Localization

of gravity waves on a channel with a random bottom, J. Fluid Mech. 186: 521-538.

7. Elter, J . F. & Molyneux, J. E., 1972. The long-distance propagation of shallow water waves over an ocean of random depth. J. Fluid Mech. 53: 1-15.

8. Grataloup, Geraldine & Mei, C. C., 2003. Localization of harmonics generated in nonlinear shallow water waves. Phys. Rev.E, 68, 026314-1-026314-9.

9. Ishimaru, A. 1997 Wave Propagation and Scattering in Random Media, 10. Hasselman, K., 1966, Feynman diagrams an interaction rules of wave-wave

scattering processes. . Rev. Geophys. 4, 1-32. 11. Kawahara, T. 1976, Effects o f random inhomogeneities on the propagation

of water waves. J. Phys. SOC. Japan, 41, 1402-1409. 12. Keller, J. B., 1964, Stochastic equation and wave propagation in random

47

media, Proc. 16th Symp. Appl. Math., 145-170. Amer. Math. SOC. Rhode Island.

13. Long, R. R. 1973, Scattering of surface waves by an irregular bottom, J. Geophys. Res. 78, 7861-7870.

14. Mei, C. C., 1985, Resonant reflection of surface waves by periodic bars, J. Fluid Mech., 152, 315-335.

15. Mei. C. C., 1989, Applied Dynamics of Ocean Surface Waves, World Scien- tific, Singapore. 700 pp.

16. Mei, C. C., & Hancock, M. 2003, Weakly nonlinear surface waves over a random seabed. J. Fluid Mech. 475, 247-268.

17. Mei, C. C. & Li, Y-L. 2004, Evolution of solitons over a randomly rough seabed. In press in Phys. Rev. E.

18. Mei C. C. & Pihl, J. H. 2001, Localization of nonlinear dispersive waves in weakly random media, Proc. Roy. SOC. Lond. 458, 119-134.

19. Nachbin, A., & Papanicolaou, G.C., 1992, Water waves in shallow channels of rapidly varying depth. J. Fluid Mech. 241: 311-332.

20. Nachbin, A., 1997. The localization length of randomly scattered water waves. J . Fluid Mech. 296: 353-372.

21. Pihl, J. H., Mei, C. C. & Hancock, M. 2002. Surface waves over a two- dimensional random seabed. Phys. Rev. E, 66, 0166011-1 -0166011-11.

22. Rosales, Rodolfo R., & Papanicolaou, G. C. 1983, Gravity waves in a channel with a rough bottom. Stud. Appl. Math. 68: 89-102.

23. Sheng, Ping, 1995. Introduction of Wave Scattering, Localization, and Meso- scopic Phenomena, Academic, 339 pp.

24. Sheng, Ping,(ed), 1990. Scattering and Localization of Classical Waves in Random Media, World Scientific.

25. Soong, T. T., 1973, Random Differential Equations in Science and Engineer- ing, 327pp. Academic.

26. Yue, D. K. P. & Mei, C. C. 1980. Forward Diffraction of Stokes Waves by a Thin Wedge. J. Fluid Mech. 99: 33-52.

WATER WAVE EQUATIONS

JIN E. ZHANG The University of Hong Kong, Pokfulam Road, Hong Kong

E-mail: [email protected]

In this chapter, we review some recent developments on water wave models, especially Wu’s4 unified theory. We discuss a few variations of Boussinesq models and their properties. We analyze the features of some integrable water equations, including the bidirectional solitons in water developed by Zhang and Li12. Finally we propose two problems for future research.

1. Introduction

During the period of my Ph.D. study in Caltech in 1991-1996, I was lucky to have a chance to learn Theodore Ym-tsu Wu’s work on water wave^^^^^^^^. In my Ph.D. t h e ~ i s ~ ~ ~ ~ ~ , I studied run-up of ocean waves on beaches by using Wu’s generalized Boussinesq modell. Ever since, I have been following Wu’s unified theory4 for modeling water waves and working with Yishen Li and others on the integrable water wave equations8~10~11~12~13~14~15

In Caltech, Dr. Wu often told me that being a scientist, one should always remember to serve engineering. This is where the name of the Department, Engineering Science, comes from. When solving a physical problem, one should use mathematics the simpler the better. After all, mathematics is for solving problems. Dr. Wu loves the mixing of basic and applied research. He was often heard to say that the best pure research grows from efforts to solve applied problems, and that the best applied research to solve problems grows from intellectual curiosity. This philosophy of Engineering Science has guided me well in the development of my research. I dedicate this chapter to Theodore Yao-tsu Wu for his philosophy and ways of learning that have set up a model for my life long emulation.

2. Wu’s Unified Theory for Modeling Water Waves

We study three-dimensional long waves on a layer of water of variable depth h(r,t), which may vary with the horizontal position vector r = (z,y,O)

48

49

and possibly also with the time t for representing submerged moving bodies, submarine landslides, drifting sandbars, or seismic activities along the seafloor. The water moves with velocity (u, w) = (u, w, w) in the flow field bounded below by the seabed at z = -h(r , t ) and above by the free water surface at z = C(r, t ) measured from its rest position at z = 0. Under the assumption that water is incompressible and inviscid and the flow is irrotational, the evolution of nonlinear water waves is governed by the Euler equations of continuity, horizontal and vertical momentum as follows:

v . u+ 20, = 0, du 1 - = Ut +u. vu+ wu, = --vp, dt P

dw 1 - = Wt + u-vw f ww, = --p* - g, dt P

where V = (az, a,, O), (az = a/ax, etc.) is the horizontal gradient operator, p is the pressure, p the constant water density, and g the gravitational acceleration. Here, the subsripts t and z denote partial differentiation. The boundary conditions are

6 = Dc P = P,(T,t) + mv * n

6 = -Bh

(D = at + G - V , on z = ~ ( r , t ) ) , (on 2 = S ( T , t ) ) ,

(B = at + G . V, on z = -h(r, t ) ) ,

(4) ( 5 )

(6)

where p , ( r , t ) is an external pressure disturbance gauged over the ambient pressure (which is set to zero), py is the uniform surface tension, n is the outward unit vector normal to the water surface, and G, ii, 6, 6 use the definition that f ( r , t ) and j ( r , t ) denote the value of an arbitrary flow variable f(r, z, t ) at the free surface and at the seabed, namely,

f(r,C(r,t),t) = f(r,t), f ( T , - q r l t ) , t ) = h t ) . (7)

The existence and the uniqueness of solutions of the Euler equations have been well-established by Wu18. One finds it difficult to describe the solutions analytically. Even for the simplest case of a single solitary wave on a layer of water with uniform depth, one finds it very challenge to present the solution in a closed formlg. It is necessary to develop simplified models.

The literature of water wave models traces back to 130 years ago. The works of BoussinesqZ0 and Korteweg-de VriesZ1 are now classical. An interesting description of their contributions is available in Ref.22. Later developments include Peregrine23, Wu1z3p4, and N w o g ~ ~ ~ among others.

50

When developing water wave models, two propositions given below are very useful.

Proposition 1: The average velocity ti satisfies following depth-mean continuity equation:

<t + v - ((h + C)E] = 0. (8) Prooj Taking the average of (1) over the water column -h < z < < under

the kinematic boundary condition (4) and (6), we get (8).

Proposition 2 (Wu 1998): The surface velocity u satisfies following projected momentum equation:

Du + [g + D'CIVC = --vp, - ~ V V . n. (9) 1 P

Prooj See Wu3.

With these two propositions, one can develop a water wave model by simply building up a relation between the depth-mean velocity, ti) and the surface velocity, 0. Wu4 presents two ways to close the system: perturbation series closure and boundary integral closure. Wu3 demonstrates that all the existing models can be generated with his procedure systematically. Furthermore, Wu's4 unified theory can be used to produce a new water wave model of arbitrary depth at any level of nonlinearity and dispersion. In other words, Wu's theory can be used to study the propagation of Tsunamis waves from very deep oceans to coastal area.

3. Boussinesq Models

Boussinesq Models play important roles in water wave models because they are useful in studying wave dynamics near the coast. Some of them are linked with an integrable soliton equation, which is an active research topic in mathematical physics. In Boussinesq class of water wave models, there are two important parameters associated with long waves. One is the ratio of amplitude to depth, represented by a, and the other one is the ratio of depth to wavelength, represented by E . Under the assumption that

a << 1, E << 1, and a = O(E'), (10)

(11)

one may derive the original Boussinesq equations

<t + v . [(l +<)El = 0,

(12) 1 3

- Ut + E * VE + v< = -vp, + -v2,i?;,

51

from the Euler equations by using a perturbation scheme. The Boussinesq equations can be used to model the propagation of nonlinear long waves on a layer of water with constant depth, normalized to be 1.

Peregrine23 derived a Boussinesq model

Ct + V * [(h + <)El = 0, (13) - h d h2 d

2 % 6 dt ut+n*Vzl+Vc=-Vpa+--V[V*(h?i ) ] - - -V(V-Z) (14)

for long waves in water of depth varying in space, i.e., h = h(x,y), and used the model to study wave evolution near a beach.

Wul derived a generalized Boussinesq (gB) model

ct + V * [ (h + C)n] = -ht, (15) h d h2 d 2 at 6 at Et + T i . VE + Vc = -Vpa + --V[ht + V * (ha)] - (16)

which included depth variations in space and time, i.e., h = h(z, y, t ) . Wu’s gB model can be used to study not only long wave propagation over variable depth (as Peregrine’s standard Boussinesq model) but also the forced generation of long waves by external moving disturbances2. WU’ also rewrote the Boussinesq model in terms of wave elevation and velocity potential in order to reduce the number of unknowns in the equations. This new form has proved to be more efficient for numerical simulations and has been widely applied. For example, Wu’s gB model is used to study the three- dimensional interaction between a solitary wave and a vertical cylinder25, and a breakwater26. It is also used to study the turning of a solitary wave in a curved channel2? and run-up of ocean waves on beaches6t7y9.

N w o g ~ ~ ~ derived an extended Boussinesq model by introducing a velocity, u, at an arbitrary depth, z,(z,y), (z, = 0 is on the surface, z, = -h(z,y) is at the bottom.)

ct + v . [(h + C)ul

+ V { ($ - :) hV(V u) + z, + - hV[V - (hu)] } = 0, (17) ( 9 U t + u VU + VC + Z, { ?V(V * ~ t ) + V[V * (hut)]} = -Vpa. (18)

He found that if the velocity at depth z, = -0.531 ( h = 1) is used, the dispersion relation of the Boussinesq model can be improved to be closer to the dispersion relation based on the potential theory. This helps to make the extended Boussinesq model applicable to broader range of water depth, i.e., not only for shallow water but also for deeper water. This has

52

been justified numerically by Wei and Kirbyz8, and Wei et a129 with a fully nonlinear Boussinesq model.

The Boussinesq model (11, 12) has been extended to higher-order by Wu and Zhang5 for constant water depth h = 1. They are presented here in three different velocity systems: (A) the { C , ii} system, - the depth-mean velocity basis

ct + v . [(l + C)ii ] = 0, (19) - 1 1 Ut + i i . vii + VC = -vp, + - (1+ C)zvzEt + zv4Tit + (VC)V. i i t

3 1

(B) the {C, ii} system, - the seabed variable basis

+-V[ii 3 * v2ii - (V * i i )2] ; (20)

Ct + v . [ (1+ C)G] = v . <)3VZii - -V4G 5! , (21) l l

1 1 2 4! Gt + ii . vii + VC = -vp, + - (1+ C)2V2iit - -v4?it + (0C)V.

+!V[ii 2 * v2ii - (V * i i ) Z ] ; (22)

(C) the { C , ii} system, - the surface variable basis

Ct + v . [ (1+ C)&] = -v *

iit + ii . vii + VC = -vp, + (VC)V . i i t ,

[)3VZii + -v40 ] 7 (23)

(24)

15

A leading order reduced version of system (C),

(25)

(26)

1 3 Ct + v * [(l + C > i i ] = --v . vzii,

fi t + 0. vii + VC = 0,

is regarded as Wu-Zhang equation and studied with a PainEve analysis approach by Chen, Tang and Lou3O. Its Lie symmetry analysis and some new exact solutions are provided recently by Ji et all4.

If the water wave is uniform along y direction, all the dependent variables are functions of z and t only, the leading order Boussinesq models are reduced to (a) the { C , ii } system ~

53

(b) the { C , G } system -

1- Gt + GGx + Cx = p x x t .

(c) the { C , G } system -

( 3 1 )

( 3 2 )

1, ct + [(I + C)G]x = - p x x x ,

Gt + GGx + CX = 0.

System (c) is well-known to be integrable. Its multi-soliton solution will be discussed in the next section.

We may write the three sets of Boussinesq model in N W O ~ U ’ S ~ ~ unified form as follows (d) A unified 1+1 leading order Boussinesq model,

where

( 3 5 ) 1 1 1

2 a = - - 2 ( e 2 - ?), b = -(I -e2),

and the velocity u is taken at depth 8 = z, + 1 (8 = 0 is at the bottom, 8 = 1 is on the surface). Depth-mean system is equivalent to a system at a water depth 8 = l/&. Zhang, Chen and Li15 study the mathematical properties of system (d), such as integrability, symmetries and solitary wave solutions. In particular, Ref.15 presents an analytical solitary wave solution as follows:

C = - 3(2b 4b + a)sech2 [ q k ( z - A t ) ] ,

3a(2b + a) 2bX(2b - a)

U =

where

2 b + a 2 a2 k2 = - A = 4ba ’ b(2b - a) *

( 3 7 )

54

The solution is not physical in general, but it can be used to test a numerical code. R13f.l‘ also identifies a integrable modified Boussinesq model

1 ct + [(I + C)U]z = - p z + k z t t + b(UUz t ) z , (39)

(40) U t + UUx + c x = bUzzt,

which can be obtained by applying a simple transformation C + C - bust to system (c).

4. Integrable Water Wave Equations

Integrable water wave equations are particularly interesting because of their analytical tractability.

4.1. The KdV equation

Among various integrable water wave models, the most popular one is the Korteweg-de Vries (KdV) equation

The KdV equation is used to model weakly nonlinear and weakly dispersive waves traveling from the left to the right on a uniform layer of water with a depth scaled to be 1. Its single-solitary wave solution can be written analytically as

2 6 U <(x, t; u) = u sech -(x - A t ) , X = 1 + -, 2 2

where u measures the wave amplitude relative to the quiescent water depth, and X measures the non-dimensional wave speed. This analytical solution serves as a good leading order approximation of the solution of the Euler equations. Due to the integrability of the KdV equation, the overtaking interaction of such solitary waves is elastic. Hence the term “soliton” is coined. But the KdV equation is a unidirectional model and, hence, it cannot be used to model the bidirectional water wave interaction.

4.2. The Boussinesq single-equation

The Boussinesq single-equation

55

is also used to model weakly nonlinear and weakly dispersive water waves traveling in a direction either from the left to the right or from the right to the left. Its single-solitary wave solution -

&a a c(z, t; a ) = a sech2-(z - At ) , A = < 1 + 5 (44)

2 has the same shape as that of the KdV equation with the same amplitude, but travels at a slightly slower speed.

The Boussinesq single-equation is integrable, has a bi-Hamiltonian structure and allows an exact bidirectional N-soliton solution including overtaking and head-on collisions, but few people realize that its solutions of head-on collision are not physically meaningful for water waves. In fact, the maximum run-up of a soliton with amplitude a << 1 on a vertical wall predicted by equation (43) is 2a - 3a2/2 + 0 ( a 3 ) , which is much smaller than the physically correct result 2a + a2/2 + O(a3) . Its prediction of the maximum run-up is not physical because the contribution from the potential energy of the soliton ( -3a2/2) is negative, see Ref.12 for a complete discussion on the details of the problem. The problem can be very serious if the Boussinesq equation ( 4 3 ) is used to model the interaction of a solitary wave with a vertical wall.

4.3. The classical Boussinesq equation

The classical Boussinesq equation 1

ct + [ ( I + C ) 4 Z = - p x x x ,

U t + uuz + e x = 0, (45)

seems to be the only serious candidate that can be used to model bidirectional solitons in water. The single-solitary wave solution of the classical Boussinesq equation is written analytically as

2(A2 - 1 ) ( 1 + A cosh d m ( z - A t ) )

(A + cosh d m ( z - At ) ) cB( (5 -A t ;A ) = 2 7 (46)

The wave speed-amplitude relation is the same as that of the KdV equation, i.e., X = l + a / 2 . Expanding wave elevation formula (46) for small amplitude a gives a leading order approximation that is the same as the single-soliton solution (42) of the KdV equation.

56

The classical Boussinesq equation is known to be integrable and equivalent to Broer-Kaup (BK) system that has a tri-Hamiltonian structure, but its exact bidirectional N-soliton solution has not been found until recent papers8~10~11~12. As shown in Ref.12, its solution of head-on collision is physically meaningful for water waves. In fact, the maximum run-up of a soliton with amplitude a < 1 on a vertical wave predicted by equation ( 4 5 ) is equal to the physically correct result 2a + a 2 / 2 . Therefore the equation can be used to model the run-up of ocean waves on dykes and dams.

We now briefly review the major result of Ref.8~10~11~12. With scaling transformation

- d3 d3 2 x + x , -t+t, 2

equation ( 4 5 ) becomes

Introducing the following transformation

or

where lnq is velocity potential, we have an equivalent system for q and r ,

qt + vqxz - q2r - q = 0, { l rt - Zr,, + qr2 + T = 0,

which is a member of the AKNS system. The multi-soliton solution of ( 5 2 ) , obtained with a Darboux transformation in Ref.12, is given as follows

1

( 5 2 )

q = 1 + 2a2, T = -1 - 2b1, ( 5 3 )

where a2 and bl are determined by two linear systems of equations

57

where X i # X j are assumed for i # j , i , j = 1,2 , ..., 2n. Xis are separated into two groups according to their signs. In each group, they are ranked according to their values. Q and 41,j, 4 2 j are given by

41,j = coshtj, 4 2 j = c j sinhtj + X j coshtj, j is odd, (56) 41j = sinhtj, 42j = cj CoShtj + X j sinhtj, j is even, (57)

where tj = cj(z + Xjt) and cj = JG. Han, Zhang and Li17.

Proposition 3 (Zhang and Li 2003): For the N right-going overtaking soliton solution given by (51, 53), the asymptotic behavior of the solution i s

Following proposition, proposed by Zhang and Li13, has been proved by

N N

t+-m lim c(z, t ) = xcB(z-Xjt+Aj) , )i~i~c(x, t ) = xcB(z-Xjt-Aj), j=1 j=l

(58) where <B(X - At; A) i s the solution of a single soliton, the total phase shift of the j t h soliton is given by

2 X j X i - 1 N 2Aj = x sign(Xj - X i )

i = 1, i # j

The phase shift f o r N head-on colliding soliton solution has a similar result.

4.4. The Camassa-Holm equation

Modifying the dispersive term of the KdV equation (41) without changing its order of magnitude yields the Benjamin-Bona-Mahoney3' (BBM) equation

3 1 6

which is not integrable. By including some nonlinear dispersive terms, the BBM equation can be made integrable

(60) ct + cz + ,Ct - - G z t = 0,

58

Equation (61) is known as the Camas~a-Holrn~~ (CH) equation with a p o p ular property of peaked solitons for degenerated case, i.e., without C; term. Its multi-soliton solution for the non-degenerated case of water wave was not available until a recent paper by Li and Zhang16.

5. Two Problems

We conclude our review by proposing two problems:

(1) Are there any integrable equations that can be used to model obliquely interacting solitons in water? Wu-Zhang equations (25, 26) are physical extensions of the classical Boussinesq equations (45). They seem to be good models for obliquely interacting solitons. But we still don’t know whether Wu-Zhang equations are integrable or not.

(2) What kind of role do nonlinear dispersive terms play in an integrable water wave equation? Can we make Boussinesq models (33, 34) integrable by adding some nonlinear dispersive terms just like what Camassa and Holm32 did for the BBM equation?

These two questions are open for us to study.

References 1. T. Y. Wu, J. Engrg. Mech. 107, 501 (1981). 2. T. Y. Wu, J. Fluid Mech. 184, 75 (1987). 3. T. Y. Wu, Physica D, 123, 48 (1998). 4. T. Y. Wu, Adv. Appl. Mech. 37, 1 (2001). 5. T. Y. Wu and J. E. Zhang, in Mathematics is f o r solving problems, edited by

L. P. Cook, V. Roytburd, and M. Tulin (SIAM, Philadelphia, 1996), p. 233. 6. J. E. Zhang, I . Run-up of ocean waves on beaches, II. Nonlinear waves in

a fluid-filled elastic tube, Ph.D. thesis, California Institute of Technology, Pasadena, 1996.

7. J. E. Zhang and T. Y. Wu, ASCE J. Eng. Mech. 125(7), 812 (1999). 8. Y.-S. Li, W.-X. Ma and J. E. Zhang, Phys. Lett. A 275(1-2), 60 (2000). 9. J. E. Zhang, T. Y. Wu and T. Y . Hou, Adv. App. Mech. 37, 89 (2001). 10. Y.-S. Li and J. E. Zhang, Phys. Lett. A 284(6), 253 (2001). 11. Y.-S. Li and J. E. Zhang, Chaos, Solitons & h c t a l s 16(2), 271 (2003). 12. J. E. Zhang and Y.-S. Li, Phys. Rev. E 67, 016306 (2003). 13. J. E. Zhang and Y.-S. Li, in Proceedings of Nonlinear Evolution Equations

and Dynamical System, ICM 2002 Satellite conference, edited by Y . Cheng, S. Hu, Y.-S. Li, and C.-K. Pen (World Scientific, Singapore, 2003), p. 135.

14. X.-D. Ji, C.-L. Chen, J. E. Zhang and Y.-S. Li, J. Math. Phys. 45(1), 448 (2004).

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15. J. E. Zhang, C.-L. Chen and Y.-S. Li, Phys. Fluids 16(5), 1287 (2004). 16. Y.-S. Li and J. E. Zhang, Proc. R . SOC. Lond. A , 460, 2617 (2004). 17. W.-T. Han, J. E. Zhang and Y.-S. Li, preprint, (2003). 18. S.-J. Wu, J. Amer. Math. SOC. 12(2), 445 (1999). 19. S. A. Pennell and C. H. Su, J. Fluid Mech. 149, 431 (1984). 20. J. Boussinesq, J. Math. Pures Appl. 17(2) 55 (1872). 21. D. J. Korteweg and G. de Vries, Phil. Mag. 39 422 (1895). 22. J. W. Miles, J. Fluid Mech. 106, 131 (1981). 23. D. H. Peregrine, J. Fluid Mech. 27, 815 (1967). 24. 0. Nwogu, J . Wtnuy., Port, Coast., and Oc. Engrg 119(6), 618 (1993). 25. K.-H. Wang, T. Y. Wu and G. T. Yates, J. Wtnuy., Port, Coast., and Oc.

26. K.-H. Wang, J. Wtnuy., Port, Coast., and Oc. Engrg 119(1), 49 (1993). 27. A.-M. Shi, M. H. Teng and T. Y. Wu, J . Fluid Mech. 362, 157 (1998). 28. G. Wei and J. T. Kirby, J. Wtrwy., Port, Coast., and Oc. Engrg 120, 251

29. G. Wei, J. T. Kirby, S. T. Grilli and R. Subramanya, J. Fluid Mech. 294, 71

30. C.-L. Chen, X.-Y. Tang and S.-Y. Lou, Phys. Rev. E 66, 036605 (2002). 31. T. B. Benjamin, J. L. Bona and J. J. Mahoney, Phil. h n s . R . SOC. Lond.

32. R. Camassa and D. Holm, Phys. Rev. Lett. 71, 1661 (1993).

Engrg. 118(5), 551 (1992).

(1995).

(1995).

A 227, 47 (1972).

WU’S MASS POSTULATE AND APPROXIMATE SOLUTIONS OF THE FKDV EQUATION

S. S. P. SHEN Department of Mathematical and Statistical Sciences, University of Alberta

Edmonton, Alberta T6G ZG1, CANADA E-mail: shen@ualberta. ca

Q. ZHENG

Department of Meteorology, University of Maryland College Park, Maryland, USA

S. GAO Institute of Atmospheric Physics, Chinese Academy of Sciences

Beajang, CHINA

z. xu Institute of Physical Oceanography, Ocean University of Qingdao

Qingdao, CHINA

C. T. ONG

Department of Mathematics, University of Technology of Malaysia Johor, MALAYSIA

Wu’s remarkable finding of upstream-advancing solitons in water flows over a topography revived nonlinear wave research in the 1980s. Wu and his colleagues numerically and experimentally found that a transicritical water flow over bump generates a train of upstream-advancing solitons, a depression zone at the downstream of the topography, and a wave zone further downstream. Wu attributed this intriguing phenomenon to the solutions of several mathematical models, including the forced Kortweg-de Vries (fKdV) equation. Wu (1987) postulated that the excess mass of the upstream-advancing solitons comes almost entirely from the region of surface depression (pp.81-82). With this postulate, the depth of the downstream depression zone can be found from the solvability condition of a boundary value problem of an ordinary differential equation.

60

61

Further, when the topography base is relatively short compared to its height, the depression’s depth can be explicitly written as a function of the upstream flow speed and the topography’s cross-section area but not the shape. Then an approximate solution of the fKdV equation can be found. A satellite observation of clouds in the form of upstream-advancing solitons and a downstream-propagating wake over Hainan Island, China, shows the existence of the fKdV transicritical waves in the real atmosphere. This paper argues that the fKdV model is inappropriately named and should be changed to “Russell-Wu model.”

1. Introduction

This paper studies the appromiate solution of the forced Korteweg-de Vries (fKdV) equation

(1) 3 1 2 6 rlt + Xrlz - -rlrl1z - -rl1zzz = f’(z)

with

q(&oo, t ) = r],(foo, t ) = 0, and q(x, t = 0) = 0,

when the time t is large. The above expressions are all in terms of dimensionless variables and parameters. The unknown variable ~ ( x , t ) in the equation models the free-surface wave profile of a water flow over a topography in a two-dimensional channel. The x-axis is along the channel’s longitudinal direction. The topography is modeled by the function f (x), and the upstream flow speed c* is related to the parameter X via the speed of shallow water waves m; i.e., c* = m ( l + ~ X ) . Here, g = 9.8 [m][~ec]-~] is the gravitational constant, H [m] is the upstream uniform depth of the flow, and E is a parameter defined by the square root of the ratio of the topography’s height to the upstream fluid depth. This parameter’s range is about (0.4,0.7) for our problem (Shen, 1992). The wave profile in terms of the dimensional variable is q* = E H ~ [m]. Figure 1 shows a schematic solution of the above initial boundary value problem in dimensionless coordinates when t is sufficiently large. In this figure, the topography forcing is represented by a Dirac delta function, and this situation is valid when the topography’s length is very short compared with the wave length of the free-surface. This free-surface consists of a train of uniform solitons upstream, a flat depression zone immediately behind the topography, and a wake zone further down the stream. This profile is a nonlinear wave discovered by Professor T. Y . Wu’s group. It first discovered this type of wave numerically (Wu and Wu, 1982) and then verified it experimentally in a lab (Lee et al., 1989). This group also developed mathematical models to describe the wave. One of the models is the fKdV equation (Wu, 1987).

62

Y

-1, X

Figure 1. chernatic solution of the fKdV equation (1) with zero initial and boundary conditions. The curve in the figure represents the transcritical free-surface wave solution of equation (l), and the topography is denoted by a Dirac delta function Pb(r) .

The experimental set-up can be described as follows. Set a bump as the bottom topography on the bottom of a flume. This bump can slide freely along the bottom. Instead of having a uniform upstream flow, one can move the bump upstream at a uniform speed. In a two-dimensional channel, if one moves the bump at a speed near @, then one can observe that solitons are periodically generated in front of the bump and surge ahead at a speed faster than the speed of the bump. Immediately behind the bump, there is a uniform depression zone, and behind this zone, there is a zone of wake propagating downstream. The number of the upstream radiated solitons, the length of the depression zone, and the length of the wake zone are all increasing at a constant rate. Wu postulated that the excess mass of upstream solitons comes solely from the downstream depression zone, while the mass of the wake zone stays balanced. With this postulate, the approximate and analytic profiles of the upstream solitons and the downstream depression can be found.

Researchers have commonly credited Korteweg and de Vries for being the first to theoretically confirm Scott Russell’s discovery of a solitary wave in the Union Canal near Edinburgh, Scotland, in 1834 (Dodd et al., 1982). Wu’s discovery implied that, strictly speaking, the credit to Korteweg and de Vries was inaccurate because the KdV model does not allow an external forcing, but Russell’s solitary wave was produced by a moving boat that accumulates water mass near the prow of the boat in the canal. Russell observed only one solitary wave. Because of the mechanism of his soli-

63

tary wave generation, if he had observed a second solitary wave, then it would have been the same size as the first one. However, the KdV model cannot generate solitary waves of the same size, for the KdV soliton sizes decrease when new solitons are generated. The Wu model, or the fKdV model, generates upstream radiating solitons of the same size. Therefore, it is appropriate to credit the Wu model as the first theoretical confirma- tion of Russell’s 1834 observation of a solitary wave in the Union Canal. Considering that solutions of the fKdV model are sufficiently different from those of KdV and that Russell’s observation was first repeated successfully by Wu’s group in a lab, it is appropriate to name the simple model for upstream-radiating solitons as the “Russell-Wu model.”

2. Amplitude of the upstream solitons

We discuss the approximate solutions of the Russell-Wu equation (1) when the forcing f(x) can be expressed in terms of the Dirac delta function. This expression can occur in the case of local forcing; i.e., when the support of the forcing is very short in comparison with the typical wave length.

The horizontal length scale is L. The dimensional horizontal coordinate is z* = xL. The free-surface is assumed to be q* = ~ H q ( x , t ) + O ( E ~ ) [ ~ ] . When c* = (gH)ll2(1 + eX)[m][sec]-’, the function q(x,t) satisfies the Russell-Wu equation:

3 1 P rllt + Xrllz - -qq1, 2 - -qlzzS 6 = +4, --oo < z < 00. (2)

Here, P = e-3/2S/H2 is the dimensionless area of the bump, S[mI2 is the actual area of the bump, and S(z) is the Dirac delta function. The solution to an initial value problem for the Russell-Wu equation gives an approximation to the free surface profile by rl* M eHv(x, t ) . The approximate velocity field is

(u*, v*) x (--~77,~~/~771,y)(gH)~/~. (3)

A schematic solution of equation (2) is shown in Fig. 1. Many examples have shown that the above fKdV is a very good model for the flows under investigation when the base length of the bump is on a comparable scale with the bump height (i.e., in the case of local forcing), even when E takes a relatively large value, say 0.5 (Shen, 1995).

We are concerned exclusively with the initial condition q* (z*, t* = 0) = 0. The mass conservation property of the wave motion gives the following

64

identity, 00

p r l * ( ~ * , t*) dx* = 0, (4)

for any t* 2 0. Here, the dimension for the density p is [ m a s ~ ] [ a r e a ] - ~ . Now, it is appropriate for us to estimate the soliton amplitude a,.

The following two first integrals are obtained by doing s!&(2)dx and

s:, rl(x, t ) ( 2 ) dx

(6) 1 1 2 + --rl(O, 3 t)rl1zz(0-,t) + p z ( O - , t ) ,

where M c ) is the horizontal momentum of an upstream soliton. The average height of the upstream soliton zone is denoted by h,. We adopt the following approximations:

Then, the long time average of the above two first integrals becomes

The k-th upstream soliton solution of the Russell-Wu equation (2) may

(14)

be expressed in the following form

~(~)(z,t) = 2(X + s)sech2{[(3/2)(X + s)]’’~(x + s t - xk)},

65

where s is the upstream advancing speed of the solitons, a, = 2(X + s) is the amplitude of the soliton, and Zk is the phase shift. For each soliton ~ ( ~ 1 , one has

J-00 00

Mif’ = [, ( v ( ~ ) ) ~ dx = 8 (horizontal momentum of a soliton).

The operation (13)/(12) results in

3. Mass postulate and the depth of the downstream depression

The depression depth hd may be determined by the mass postulate that the soliton mass comes solely from the depression (Wu, 1987, p. 87, before equation (38)) . The average height of the upstream is h,; that is, the average of q(z, t ) with respect to z over a period d,, the distance between the two peaks of any two adjacent solitons. When time is large, we regard h, as an upstream uniform state which falls to the downstream uniform state hd. Both of these uniform states extend to infinity as time t t 00

and form an “imaginary” stationary state w(z), governed by the following boundary value problem:

(16)

v(-00) = h, and v(00) = -hd. (17)

3 1 P 2 6

X v x - - w v x - - v x x x = , S x ( z ) , - 00 < z < 00,

Let ~ ( z ) = C(z) + h,. Then the above two equations become the hydraulic fall type of boundary value problem for C(z) (Shen, 1993 and 1996):

C(-0O) = 0,

C(m) = -(hs -k hd).

This boundary value problem is solvable only when C(z) is a smooth fall from the upstream zero solution to a downstream solitary wave tail. Hence, X - i h s < 0, and <(z) = 0 for all z in (-00,o). The first integral of the

66

above boundary value problem in ( 0 , ~ ) results in

3 1 (A - qh,) C - ;C2 - ;CZ5 = 0 when x > 0,

Another first integral of this boundary value problem results in

This problem is solvable only when the third order polynomial on the right hand side of the differential equation has a double zero. This double zero condition is

(18) 3 2 4 3

h, = (-P2)l13 + -A.

The amplitude of the fall of the downstream solitary wave tail is

-! 3 (A - qh,) ,

which has to be equal to h, + hd. This leads to the important formula for hd

In terms of the downstream depression depth hd, the soliton amplitude can be written as

To find the time period of soliton generation T,, perform the operation !_":(2)dx where XD is any point in the uniform depression zone. This integral yields

4. Geometry of the waves on the x-t plane

It is well known that a soliton propagates at a uniform speed. Wu's mass postulate implies that the length of the downstream depression zone increases at a constant speed as well. Thus, on the x-t plane, a triangle of

67

the soliton zone and a triangle of the depression zone appear. These two triangles can be determined as follows.

The upstream solitons are of the same size and hence move at the same speed toward upstream. The speed is

s = - - A. 2

Thus, the total length of the upstream soliton train is x8 = st and can be expressed in terms of P and A.

The length of the downstream depression zone 21 may be determined by the same mass postulate

N8mLk) = Xdhd, (23)

where N8 is the number of upstream solitons and is equal to t/T8. Thus,

which can also be expressed in terms of P and A:

The ratio of the length of the upstream soliton zone and the length of the downstream depression zone is

When A = 0, the ratio is 4 : 3; i.e., the upstream soliton zone is 30% longer than the downstream depression zone (Xu and Shen, 1997). Numerical solutions of the Russell-Wu equation have verified this conclusion.

5. Solitons in the atmosphere over the Hainan island

Zheng et al. (2004) observed the upstream-radiating solitons and a downstream propagating wake in the real atmosphere near Hainan island in southern China. In an image of the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) onboard the Orbview-2 satellite, two packets of orderly wave clouds on two sides of Hainan Island in the South China Sea were observed (Fig. 2). A packet of 23 wave clouds stretches southward from the island.

68

Figure 2. A SeaWiFS satellite image taken on March 19, 1999. Aerosol contamination is evident to the west of Hainan Island. Two groups of wave clouds in white arrayed on the two sides of the island are interpreted as signatures of upstream (Pl) and downstream (P2) solitary wavetrains generated in the real atmosphere by topographic disturbances. The solid line represents the wind direction at 850 mb.

A second packet of more than 20 wave clouds stretches northeastward off the northeast coast of the island. The concave orientation of the wave cloud lines implies that both packets are propagating away from the island. A chart of the geopotentid height and velocity at 850 mb shows a south- westerly air flow over the island; hence, the two wave cloud packets are propagating upstream and downstream, simultaneously.

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council of Canada and the US NOAA Office of Global Programs. Shen also thanks the US National Research Council for the Associateship award, MITACS (Mathematics of Information Technology and Complex Systems) for a research grant, and the Chinese Academy of Sciences for an Overseas

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Assessor’s research grant and for the Well-Known Overseas Chinese Scholar award.

References

1. R. K. Dodd, J. C. Eilbeek, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1982).

2. S. J. Lee, G. T. Yates and T. Y. Wu , Experiments and analyses of upstream- advancing solitary waves generated by moving disturbances, J. Fluid Mech.

3. S . S . P. Shen , Forced solitary waves and hydraulic falls in two-layer flows, J. Fluid Mech. 234, 583-612 (1992).

4. S. S. P. Shen, A Course on Nonlinear Waves, Kluwer Academic Publishers, Boston (1993).

5. S. S. P. Shen, On the accuracy of the stationary forced Korteweg-de Vries equation as a model equation for flows over a bump, Quarterly Appl. Math. 53, 701-719 (1995).

6. S. S. P. Shen, Energy distribution for waves in transcritical flows over a bump, Wave Motion 23, 39-48 (1996).

7. D. M. Wu and T. Y. Wu, Three-dimensional nonlinear long waves due to moving surface pressure. In Proc. 14th Symp. on Naval Hydrodynamics, pp. 103-125, National Academic Press, Washington, DC (1982).

8. T. Y. Wu , Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech. 184, 75-100 (1987).

9. 2. Xu and S. S. P. Shen, Physical universals in problem of precursor soliton generation, Science in China D40, 306-314 (1997).

10. Q. Zheng, S. S. P. Shen, Y. Yuan, N. E. Huang, V. Klemas, X-.H. Yan, F. Shi, X. Zhang, Z. Zhao, X. Li and P. ClementeColon, Evidence of upstream solitons and downstream solitary wavetrains coexistence in the real atmosphere, Int. J. Remote Sensing 25, 4433-4440 (2004).

199, 569-593 (1989).

EXPLICIT ANALYTIC SOLUTIONS OF KDV EQUATION GIVEN BY THE HOMOTOPY ANALYSIS METHOD*

CHEN CHEN, CHUN WANG AND SHIJUN LIAO School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University, Shanghai 200030, China

E-mail: sjliaoOsjtu. edu.cn

An analytic technique, namely the homotopy analysis method, is applied to solve shallow water waves governed by the nonlinear KdV equation. Explicit analytic solutions of periodic and solitary waves are given and compared with exact ones. This verifies the validity of the homotopy analysis method for nonlinear wave problem.

1. Introduction The homotopy analysis method L2 is an analytic technique for nonlinear problems. Compared with perturbation techniques and non-perturbation techniques like Lyapunov’s artificial small parameter method 4, the 6- expansion method and Adomian’s decomposition method 6 , the homotopy analysis method has several advantages. First, the homotopy analysis method is independent of small or large parameters. Second, the homotopy analysis method provides us with a convenient way to control and adjust the convergence region and rate of solution series. Third, the homotopy analysis method provides us with great freedom to select the base functions and initial approximations. Besides, the homotopy analysis method logically contains Lyapunov’s artificial small parameter method 4 , the 6- expansion method 5 , and Adomian’s decomposition method 6, as shown by Liao l. So, it is more general, and valid for more problems. All of these make the approximation of a nonlinear problem convenient and efficient.

The homotopy analysis method has been successfully applied to many nonlinear problems in science and engineering, such as the magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet 7 , nonlinear progressive waves in deep water 8 , free oscillations of positively damped systems with algebraically decaying amplitude 9, free oscillations of self- excited systems lo, similarity boundary layer equations ’’, boundary layer

*THIS WORK IS SUPPORTED BY NATIONAL SCIENCE FUND FOR DISTINGUISHED YOUNG SCHOL ARS OF CHINA (APPROVAL NO. 50125923)

70

71

free convection 12J3914, Cheng-Chang equation 15, a third grade fluid past a porous plate 16, the flow of an Oldroyd 6-constant fluid 17, and so on. All of these successful applications verified the validity, effectiveness and flexibility of the homotopy analysis method.

In this paper, we apply the homotopy analysis method to the Korteweg and de Vries (KdV) equation

where CO = m, g is the gravity acceleration, H and q denote the water depth and the wave elevation, respectively. Eq. (1) describes the motion of long waves in shallow water under gravity l 9 l2O. In this paper, explicit analytic solutions of periodic and solitary waves of the KdV equation are given by means of the homotopy analysis method. The validity of these solutions is then verified by the exact solutions.

2. Solution of periodic wave Under the transformation

~ ( z , t ) = Af(8) , 8 = kx - w t , (2) where A is the wave amplitude, k = 27r/Lw denotes the wave number, L, the wave length, and w the angular frequency, Eq. (1) becomes

(1 - A) f’ + yf”’ + pff’ = 0,

f(0) - f(.) = 2,

( 3 )

(4)

subject to the restriction

where the prime denotes derivatives with respect to 8, and

k2 H2 3A C CO 2H , p = - , A = - , y = - ( 5 )

in which C = w / k is the wave velocity.

base function The periodic solution of Eqs. ( 3 ) and ( 4 ) can be expressed by a set of

{cos(me)Im = 1,2,3, ...} (6)

(7)

in the form +oo

f(8) = c Qm cos(mQ), m= 1

where am is a coefficient. This provides us with the so-called Rule of Solution Expression of periodic waves, as defined by Liao l y 2 . Under the Rule of Solution Expression, it is natural for us to choose

fo(e) = case (8)

72

as the initial approximation of f(8), and

as the auxiliary linear operator, which has the property L [Cl + C2 cos8 + C3 sin81 = 0, (10)

(1 - q)L[F(6; q) - f O ( U = Q "(8; A(q)l, (11)

F(0; Q ) - F ( r ; Q ) = 2, (12)

N [ F ( e ; q), q q ) ] = [I - ~ ( q ) ] ~ ' ( e ; q) + - F f ' ( 8 ; q) + PW; q)F'(8; q),(13)

where (71, C2, and C3 are constants. Then, we construct the zero-order deformation equation

subject to the restriction

under the definition

where the prime denotes the partial differentiation with respects to 8, q is an embedding parameter, h is a non-zero auxiliary parameter. Obviously, when q = 0, it is clear from (8), (ll), and (12) that

When q = 1, Eqs. (11) and (12) are equivalent t o the original Eqs. (3) and (4), provided

Using (14), we have the Taylor series with respect to the embedding parameter q as follows

F(O; 01 = fo (e) . (14)

~ ( e ; 1) = f(e), ~ ( i ) = A. (15)

+m

m=l +m

m=l

where A0 is an initial guess of A, and

Assume that h is properly chosen so that these series are convergent at q = 1. Using (15), we have at q = 1 that

+m

m=l

A = A o + Exm. m= 1

73

For conciseness, define the vectors

f7, = Cfo(e),fl(e),fi(e),... , f k ( 0 > } , (21)

x'k = {AO,Al,A2, . ' . ,Ak}. (22) Differentiating the zero-order deformation equation (11) and boundary conditions (12) m times with respect to q, then setting q = 0, and finally dividing them by m!, we gain the mth-order deformation equation

where

m-1

n=O m- 1

n=O

and 1,m > 1,

x m = { 0,m = 1.

L . i c r the Rule o can be expressed by

Solution Expression denote(

where Sm,n is a coefficient. To obey the Rule of Solution Expression denoted by (7), the solution of Eq. (23) should not contain the so-called secular term 8sin0 . To ensure this, the right-hand side term R, of (23) should not contain the term sine, i.e., the coefficient of sin0 must be zero. This leads to an algebraic equation

t m , l ( i m - l ) = 0, (28) which determines Am-l . Thereafter, the general solution of Eq. (23) is

+C1 + C2 cos 0 + C3 sin 8, (29)

74

where C1, C2, and C3 are the inte ral constants. According to the Rule of Solution Expression denoted by (77, we have

c1 = c3 = 0. The coefficient C2 is determined by (24). It is found that f m ( 8 ) obtained in this way can be expressed by

m+l

(30)

f m ( 8 ) = C Pm,ncos(ne), m 2 1 (31) n=l

where /3m,n is a coefficient. Substituting (31) into the mth-order deformation equations (23) and

(24) , and due to (29) and (30), we obtain the following recursive formulas

Cm,n n(n2 - 1) Pm,n = f i + XmPm-l,nPm-l,n

for 2 5 n 5 m + 1, and

[tl Pm,l = Xm~m- l ,1~m- - l , l - Xm Pm-l,2k+l~m-l,2k+l

k=O

75

where

1 m-1 rnin{s+l,m-s+n}

Qm,n = C C z( i - n)Ps,iPs,iPm-s-l,i-nPm-s-l,i-n, s=n-1 i=rnax{l+n,l}

m-n min(s+l,m-s-n) .

(44) for 1 5 n I m + 1. Using (28) and (35), we obtain the recursive formula

m-2

i = O r m , l + y R m , 1 + p n m , l + C XiPm--i-1,1Prn-i-1,1

P0,l (45)

P0,l = 1. (46)

xm-l = -

for m 2 1. Erom the initial guess (8 ) , it is easy to get the first coefficient

Thus, Pm,,, and Am can be calculated recursively by using only the first one. The corresponding Mth order approximations of (19) and (20) are

m=O M

A = E x m .

(47)

(48) m=O

When M - +m, we have explicit periodic wave solutions of Eqs. (3) and (4)-

3. Solution of solitary wave The solitary wave solution of the KdV equation can be written as

V(Z, t ) = Ag(19), 29 = (Z - C t ) / H , (49) where A is the elevation of the solitary wave and C denotes the wave velocity. Substituting Eq. (49) into Eq. (1) yields

(50) 1 6

(1 - X)g' + -g'" + pgg' = 0,

76

where the prime denotes derivatives with respect to 6, the non-dimensional wave velocity X and non-dimensional wave amplitude p are defined by (5).

Write

g(6) M uexp(-d), 6 -+ 00. (51)

K = JW. (52)

Substituting the above expression into (50) and using asymptotic analysis, it is found

So, under the transformation

Eq. (50) becomes

(1 - X)(g/ - g,,,) + pgg’ = 0. (54) Due to the symmetry of the solitary wave, we only need consider the solution at r 2 0. Assume that the non-dimensional wave elevation g(T) arrives its maximum at the origin. Note that g(T), along with its derivatives, tends to zero as r -+ 00. Thus, the boundary conditions are

g(0) = 1, g’(0) = 0, g(+0O) = 0. (55) From the boundary conditions (55), the solution of g(7) can be expressed

by

m=l

where a, is a coefficient. This provides us with the Rule of Solution Ex- pression for solitary waves.

Under the Rule of Solution Expression denoted by (56), we choose

as the auxiliary linear operator, which have the following property

Z[C, exp(-.r) + CZ exp(7) + C ~ I = 0,

go(7) = 2exp(-7) - exp(-2~)

(58)

(59)

where C1, CZ and Cs are constants. Similarly, we choose

as the initial approximation of g(7). Let XO be the initial approximation of A. In the similar way, we can construct the zeroth-order deformation equation,

(1 - q ) @ m Q ) - 90(7)1 = f i 4 fi [G(T dl A(d1 I (60)

G(0; q) = 1, G’(0; 4 ) = 0, G(+w; q) = 0, (61)

subject to the boundary conditions

77

where

fi P(7; q) , A(!?))] = [1 - A(dI[G(T 4) - G1% dl +PG(T dG'(7; 4, (62)

G(T; 0) = go(7) (63)

G(7; 1) = 9(T), A(1) = A, (64)

and A(q) are defined by (17). When q = 0 and q = 1, we have

and

respectively. Assume that G(T; q ) and A(q) are analytic in q E [0,1] so that they can be expressed in the Taylor series with respect to q as follows

m=O m=O

where

and Am is defined by (18). Besides, assume that ti is properly chosen so that the series converge at q = 1. Using (64), we have at q = 1 that

+W

m=l m=l

Similarly, we obtain the high-order deformation equation - L[gm(T) - xrngm-1(~>1= 6 Em(gm-1, im-11, m L 1, (68)

subject to the boundary conditions

gm(0) = gL(0) = gm(+m) = 0,

- 1 d r n - l f i W ; 41, A(d1 (m - l)! aqm-1

Rm(Grn-1, Am-1) =

(69) +

where Xm and Am-l are defined by (26) and (22), respectively,

n=O

and

78

The solution of Eqs. (68) and (69) can be expressed by

gm(T) = g*(~) + C1 exp(-.r) + Czexp(.r) + C3. (71)

where C1, C2, and Cs are integral constants, g*(~) is a special solution of Eq. (68), which contains an unknown Xm-l. Due to the boundary condition (69) at infinity, the integral constants C2 and C3 should be zero. The unknown Xm-l and the constant C1 are determined by the two boundary conditions (69) at T = 0. it is found that gm(7) can be expressed by

- where Pm,k are coefficients. Similarly, we can obtain the recursive formulae to calculate the coefficients Pm,k by substituting (72) into the mth-order deformation equation (68). Using the first two coefficients given by the initial guess (59), i.e.

- F0,l = 2, P0,2 = -1, (73)

M all the coefficients can be recursively calculated. We take g m ( 7 ) as the Mth-order approximation of g(~) and Cz=OXm as the Mth-order approximation of A. When M - +m, we have an explicit analytic solitary wave solution of (50) and (55).

4. Validation of the solutions We verify the explicit analytic solutions of periodic and solitary waves of the KdV equation with the exact ones given by Korteweg and de Vries '*. Based on different lateral boundary conditions, the exact solutions for Eq. (1) leads to two different wave trains, i.e. the cnoidal wave and solitary wave.

4.1. Cnoidal waves The cnoidal wave solution of Eq. (1) and the non-dimensional wave velocity X are given by

(74)

(75)

79

where cn (8 , m) is the Jacobi elliptic function with modulus m, and

with c3 and <2 corresponding to the elevation at the crest and trough of the cnoidal wave, respectively, and <3 - <2 = 2. Here, K m) and E(m) are

the modulus m is determined by the Ursell number the complete elliptic integral of the first and the second k ind, respectively,

AIH v, = - ( k H ) 2 '

As m + 1 , under any small but finite value of A I H , the expressions (74) and (75) lead to solitary wave solution as kH tends to zero.

Notice that the solutions given by the homotopy analysis method contain the auxiliary parameter h. It is ti that provides us with a simple way to ensure that the solution series given by the homotopy analysis method converge, as shown by Liao '. In general, by means of plotting the so-called &curves of X N ti by means of the high-order approximations of X via the auxiliary parameter h, one can choose a proper value of f i in the region corresponding to a line segment of the curve. For example, the &curves of X N ti in case of kH = 1 , A / H = 0.05 and kH = 0 .25 ,A /H = 0.2, which correspond to modulus m = 0.508895 and m = 0.999992, respectively, are as shown in Fig. 1 . In this way, it is easy for us to find a proper ti to ensure that the corresponding solution series is convergent. Our analytic approximations agree well with the exact ones, as shown in Fig. 2.

4.2. Solitary waves The exact solutions of solitary wave of the KdV equation are

g(8) = sech2 (E8)

and 1 A X = l + - - . 2 H

Similarly, by means of plotting the corresponding &curves of X N 6, we can find a proper value of ti to ensure that the solution series converges. Our analytic approximations agree well with the exact ones, as shown in

80

h

Figure 1. The h-curves of C/Co - h. Solid line: 20th-order approximation of C/Co when k H = 1 and A / H = 0.05; Dotted line: 50th-order approximation of C/Co when k H = 0.25 and A / H = 0.2; Dashed line: exact solution.

Fig. 3 in case of A I H = 0.6, ti = -6. The non-dimensional wave velocities given by our approach match well with the exact ones, as shown in Fig. 4, which clearly indicates that C/Co is a linear function about A / H . This verifies the validity of the homotopy analysis method for solitary wave in shallow water.

5. Conclusions In this paper, we applied the homotopy analysis method to give the explicit analytic solutions of periodic and solitary waves of the KdV equation (l), and verified the validity and effectiveness of the solutions by comparing with the exact ones. This indicates that the homotopy analysis method is valid for nonlinear wave problems with both periodic and solitary solutions.

References 1. Lim, S.J., Beyond Perturbation:Introduction to the Homotopy Analysis

Method, Chapman & Hall/CRC, Boca Raton (2003).

81

2.5

/

I I

I-

c I \

ig 0.5 [ ‘L, \

-0.5 :j / I

-1.5 o/ 6

e

Figure 2. Solution of periodic wave compared with the exact ones. Solid line: 20th-order approximation when kH = 1, A/H = 0.05 and the modulus m = 0.508895 by means of R = -10; Dashed line: 50th-order approximation when kH = 0.25,A/H = 0.2 and the modulus rn = 0.999992 by means of R = -5; Circle: exact solution.

2.

3.

4.

5.

6.

7.

8.

9.

Liao, S.J., On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147, 499-513 (2004). Nayfeh, A.H., Perturbation Methods, John Wiley €4 Sons, Inc., New York

Lyapunov, A.M., General Problem on Stability of Motion (English translation), Taylor €4 Francis, London (1992). Karmishin, A.V., Zhukov, A.T., and Kolosov, V.G., Methods of Dynamics Calculation and Testing for Thin-walled Structures (in Russian), Mashinos- troyenie, Moscow (1990). Adomian, G., Nonlinear Stochastic Differential Equations, J. Math. Anal. and

Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non- Newtonian fluids over a stretching sheet, J. Fluid Mechanics, 488, 189-212 (2003). Liao, S.J. and Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water. Journal of Engineering Mathematics, 45, 105-116 (2003). Liao, S.J., An analytic approximate techniqe for free oscillations of positively

(2000).

Applic., 55, 441-452 (1976).

82

7

Figure 3. Solution of solitary wave at the 20th-order of approximation compared with the exact one in case of A / H = 0.6 and R = -6. Solid line: homotopy analysis solution; Symbol: exact solution.

damped systems with algebraically decaying amplitude. Int. J. of Non-Linear Mechanics, 38, 1173-1183 (2003).

10. Liao, S.J., An analytic approximate approach for free oscillations of self- excited systems, Int. J . Non-Linear Mechanics, 39, 271-280 (2004).

11. Liao, S.J. and Pop I., Explicit analytic solution for similarity boundary layer equations, Int. J . Heat and Mass Transfer, 47, 75-85 (2004).

12. Xu, H., An explicit analytic solution for free convection about a vertical flat plate embedded in a porous medium by means of homotopy analysis method, AMC, 158, 433-443 (2004).

13. Wang, C., Liao, S.J., and Zhu, J.M., An explicit analytic solution for the combined heat and mass transfer by natural convection from a vertical wall in a non-Darcy porous medium, Int. J. Heat Mass Transfer, 46, 4813-4822 (2003).

14. Wang, C., Liao, S.J., and Zhu, J.M., An explicit analytic solution for non- Darcy natural convection over horizontal plate with surface mass flux and thermal dispersion effects, Acta Mechanical 165, 139-150 (2003).

15. Wang, C., Zhu, J.M., Liao, S.J., and Pop, I., On the explicit analytic solution

83

Figure 4. solution; Solid line: homotopy analysis solution.

Homotopy analysis solution of C/Co compared with exact one. Symbol: exact

of Cheng-Chang equation, Int. J. Heat Mass 'Pransfer, 46, 1855-1860 (2003). 16. Ayub, M., %heed, A., and Hayat, T., Exact flow of a third grade fluid

past a porous plate using homotopy analysis method, Int. J. Engng. Sci., 41,

17. Hayat, T., Khan, M., and Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J . Engng. Sci., 42, 123-135 (2004).

18. Korteweg, D.J. and Vries, G.de, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39, 422-443 (1895).

2091-2103 (2003).

19. Whitham, G.B., Linear and nonlinear waves, Wiley, New York (1974). 20. Mei, C.C., The Applied Dynamics of Ocean Surface Waves, Wiley-

Interscience (1983), World Scientific (1989).

RIGOROUS COMPUTATION OF NEKRASOV’S INTEGRAL EQUATION FOR WATER WAVES

SUNAO MURASHIGE Department of Complexity Science and Engineering,

Graduate School of Fbontier Sciences, The University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8656 Japan

E-mail: s-murashigeQnds. k. u-tokyo. ac.jp

SHIN’ICHI OISHI Department of Computer and Information Science,

School of Science and Engineering, Waseda Univerity 3-4-1 Okubo, Shinjyuku, Tokyo, 169-8555 Japan

E-mail: oishiQwaseda.ac.jp

Nekrasov’s integral equation is a mathematical model for two-dimensional, periodic, symmetric and progressive waves on the surface of water. Although some iterative numerical methods have been developed for this type of integral equation, high dimensional approximation is required for the case of large wave height. This paper proposes the method of error estimate for this problem using a numerical verification technique based on a fixed point theorem and interval analysis. This method enables us to show existence of exact solutions in a neighborhood of an approximate solution.

1. Introduction

Water wave problems are often formulated using differential or/and integral equations with nonlinear free surface conditions. It is well known that this nonlinearity produces some interesting phenomena such as deformation, soliton, breaking, bifurcation and so on. Most of these nonlinear problems require numerical calculation, in particular, for fully nonlinear cases. The objective of this work is to rigorously compute these nonlinear problems using the numerical verification method which has been developed with a fixed point theorem and interval analysis[l] [2]. Here ‘rigorous computation’ is to show existence of exact solutions near a neighborhood of an approximate solution.

This paper considers Nekrasov’s integral equation as a simple mathe-

84

85

matical model for two-dimensional steady, periodic, symmetric and progressive water waves with the fully nonlinear free surface condition [3][4]. In the case of infinite depth, this equation can be written in the form a

where l-t denotes the Hilbert transform defined by

00

, (2)

G = SA,

and the singular integral operator S is given by

where cp is continuous and 2n-periodic. This integral equation can be obtained in the mapped plane as shown in fig.1.a where the unit circle corresponds to the water surface and the argument x denotes the position of water surface. The slope of the water surface 6' has the period 2n, and is assumed to be symmetric about the crest, e (x ) = -O(-x). Figure 1.b shows some examples of approximate solutions of wave shapes in the physical plane for different values of p. The parameter p of physical interest ranges from 3 to 00. From symmetry of the wave shape, the approximate solution in is expressed as the sinusoidal series form of TI terms, and obtained by the iterative method it") = Git) (v = 0,1,2,. . a ) . We can see that the crest and the trough of wave becomes sharper and flatter, respectively, with increase of p, namely increase of the wave height. It has

aEquation (1) is slightly different from the original form of Nekrasov's integral equation which includes an indefinite integral for which numerical verification requires computai tional costs ([4] Chapter 3).

86

been reported that, for large value of p, numerical solutions of Nekrasov’s integral equation show some interesting nonlinear phenomena such as bifurcation, and part of them has been elucidated analytically. Okamoto and ShZiji [4] gave a review on them in detail.

X

(a) Mapped plane (b) Computed wave shape

Figure 1. Two dimensional water waves. The flow field in the physical plane is mapped into the unit circle (a) where C = exp(-i2nf/(Xc)), and X denotes the wavelength, c the wave speed, and f the complex velocity potential, respectively. Some approximate solutions of wave shape are shown in (b).

Rigorous calculation with numerical verification has recently been developed with fixed point theorems and interval analysis. This is one of important topics in the field of numerical analysis. Fixed point theorems are often used to show existence of solutions of equations, and are generalization of the intermediate value theorem, namely

I f f : R 4 R is a continuous map on I = [a, b] and f ( a ) . f ( b ) 5 0, then there exists at least one solution o f f (x) = 0 in I .

This work applies the generalized one, Schauder’s fixed point theorem as follows:

If T : M + M is a compact operator and M is a bounded, closed, convex, nonempty subset of a Banach space X , then T has at least one fixed point in M.

Keady and Norbury [5] and Toland [6] investigated existence of solutions of Nekrasov’s integral equation theoretically. Chandler [7] examined the

87

numerical computation of Nekrasov’s integral equation in detail. Dobner [8] showed the numerical verification method for linear and singular integral equations. In this work, we propose a method of numerical verification for Nekrasov’s integral equation using some properties of the singular integral operator S in (1).

2. The singular integral of trigonometric polynomials

The singular integral operator S defined by (4) has the following properties ([9] Lemma 8.21):

(5)

From this relation and (2), trigonometric polynomials may be suitable for expression of solutions 8. Also, since e(z) = -e( -z) from symmetry of the wave shape, 8 can be written in the sinusoidal series form

00

m= 1

where (e), denotes the m-th Fourier coefficient of e(a:). From (Ad)(%) = - (Ae)( -z ) and ( 5 ) , we get the following alternative form of (1)

A fixed point theorem requires a suitable function space for 6 expressed as (6). In this work, we use the periodic Sobolev space H1[-.rr,n] for odd functions e ( x ) defined by

This is the Hilbert space with the norm

88

For the operators S , A and G, the following theorem holds.

Theorem 2.1. Since the singular integral operator S : H1[-n,n] -+

H2[-n, n] an (4) is continuous and the embedding H2[-n, n] H'[-n, n] is compact (91, S : H1[-n, n] ---f H1[-n, n] is compact. Also A : H1 [-n, n] 4

H' [-n, n] is continuous. Therefore G(= SA) : H' 1-n, n] --t H 1 [-n, n] in (I) is compact.

3. The method of verification

The aim of this work is to show existence of exact solutions 8, ( E H' [-n, n]) of (1) in the neighborhood of an approximate solution #,(z). For that, we transform (l), 8 = GO, into

F B = B - G B = O ,

and apply Schauder's fixed point theorem to the fixed point form

B = TO with TB = 8 - J- lF8 , (11)

which is equivalent to (1). Here T is the simplified Newton operator and is defined by

- - J = FIB = I - P,(G'8,)Pn = I - P,(SA'B,)P, , (12)

where G = S A is defined by (l), I : X = H1[-n,n] -+ X the identity operator, and P, : X -+ X, the projection operator from X into the finite dimensional space of trigonometric polynomial of n-th order, X,, respectively. The Fr6chet derivative A'B : X 4 X can be expressed as

Lemma 3.1. The operator J" transforms an odd function p(x)

C ( ( P ) ~ sin mx as follows:

= 00

m= 1

15 m I n and 1 I k 5 n , 1 m (16)

6mk - - ( f lken)m T

{hi 7 otherwise . Pmk =

Here f l k is given by

( f l k e ) ( x ) = (@O)(x). (coskx - 1) + ( W ) ( x ) -sinkx , (17)

Similarly, the inverse operator J-' transforms an odd function +(x) =

C (+)m sinmx into ( J + ) ( x ) = 00 00

( J $ J ) ~ sinmx with m=l m=l

for m > n

where ( p i ; ) is the inverse of the n x n mat* (Pmk) (1 I m, k I n).

90

The neighborhood N of an approximate solution 8, is defined by the product of the finite dimensional and the truncated space N = Nn x Nr with

where an approximate solution 8n = Ck=l(8n)msinmzl x = H1[-~ ,n ] , and w m and W p denote a non-negative real number, respectively.

The operator T : H'[-.rr, n] -+ H1 [-n, n] is compact and a fixed point of (ll), O = TO, corresponds to a solution of (1). If we can show

then the inclusion relation

T N G N ,

holds and, from Schauder's fixed point theorem, there exists at least one exact solution in N .

3.1. The truncated pard of (20)

The inclusion relation ( I --P,)(TO) E Nr for VO E N of (20) is equivalent to the inequality

where wr is given by (19). We can show the following theorem for the norm II (1 - Pn) (TO) II H I *

91

Theorem 3.1. For V8 E N , the norm II(I - Pn)(T8)1IH1 can be bounded as follows:

3.2. The finite dimensional part of (20)

The inclusion relation Pn(Tf?) E N, for VB E N is equivalent to

(TO - 8n)m E [ - ~ m , ~ m ] for 1 I Vm 5 n, ve E N . (25)

We can show the following theorem for the Fourier coefficient (TO - 8n)m.

Theorem 3.2. The Fourier coefficients (TO - 6n)m can be expressed as

1 where A& = 0, - en, 0; = 8, + t ( 0 - gn), fo = so f ( t ) d t , respectively. For V8 E N , each term in the right hand side of (26) can be estimated as follows:

92

with

n

where a, Q, R,, p i : are defined by (l4), (l7), and (18), respectively.

4. Results

We can try to show the inclusion relations in (20) using Theorems 3 and 4, and with control of rounding for double precision floating point numbers of IEEE 754 standard and automatic differentiation for higher order derivatives such as Here the result for the case of p = 10 is shown. In this case, the inclusion relations in (20) is satisfied with n = 128 where n is the dimension of the finite dimensional part. Also the error sup lee -8,l can be estimated as follows:

- - X _ < X < A

where Be denotes an exact solution.

93

For larger values of p, high dimensional approximation, namely large number of TI, is required. Thus computational complexity is the problem of this method. This remains as a future work.

5. Conclusions

This paper has shown the method of rigorous calculation with numerical verification for Nekrasov's integral equation (1) which is a mathematical model of two-dimensional water waves with the fully nonlinear free surface condition. The verification method utilizes the property of the singular integral for trigonometric polynomials and applies Schauder 's fixed point theorem in the periodic Sobolev space H1[-.rr, r]. The present method of numerical verification works well for small values of the parameter p.

Acknowledgments

The authors thank Dr Nobito Yamamoto of the University of Electro- Communications for enlightening discussions.

References 1. M.T. N a b , Numer. f inc t . Anal. and Optimiz. 22, 321 (2001). 2. U. Kulish, R. Lohner and A. Facius (eds.), Perspectives on Enclosure Meth-

ods, (Springer Wien New York, 2001). 3. L. M. Milne-Thomson, Theoretical Hydrodynamics, (Fifth ed., Dover, New

York, 1996, Sects. 14.65 and 14.70). 4. H. Okamoto and M. Shijji, The Mathematical Theory of Permanent Pro-

gressive Water- Waves, (World Scientific, 2001). 5. G. Keady and J. Norbuy, Math. Proc. Camb. Phil. SOC. 83, 137 (1978). 6. J.F. Toland, Proc. R. SOC. Lond. A 363, 469 (1978). 7. G.A. Chandler and I.G. Graham, SIAM J. Numer. Anal. 30, 1041 (1993). 8. H.-J. Dobner, Numer. f i n c . Anal. and Optimiz. 20, 27 (1999). 9. R. Kress, Linear Integral Equations, (Second ed., Springer-Verlag New York,

1999).

NUMERICAL MODELING OF NONLINEAR SURFACE WAVES AND ITS VALIDATION

W. CHOI, C. P. KENT AND C. J. SCHILLINGER Department of Naval Architecture €4 Marine Engineering

University of Michigan, Ann Arbor, MI 481 09, USA E-mail: wychoiOengin.umich. edu

We study numerically the evolution of nonlinear surface gravity waves in infinitely deep water using both the exact evolution equations and an asymptotic model correct to the third order in wave steepness. For onedimensional Stokes waves subject to perturbations at sideband frequencies, the numerical solutions of the third-order nonlinear model found using a pseudo-spectral method are carefully validated with those of the exact equations, and it is found that the thud-order model describes accurately the development of spectral components in time. For two-dimensional waves, we study resonant interactions of two mutually-orthogonal gravity wave trains and compare our numerical solutions with available theory and experimental data. We also simulate the evolution of a realistic surface wave field, characterized initially by the JONSWAP spectrum, and examine the occurrence of a larger wave compared with the background wave field.

1. Introduction

Accurate prediction of nonlinear surface wave fields is important to many engineering applications in both coastal and deep oceans. Recently, much attention h.a been paid to rogue waves, which often give rise to wave heights much greater than the significant wave height of a given spectrum. The occurrence of these intermittent waves is actively being investigated, but it is not yet well understood.

In order to describe such waves, fully nonlinear computations of surface waves are desirable, but even with an idealized potential flow assumption, solving the three-dimensional free surface hydrodynamic equations is still computationally problematic. Therefore, simpler mathematical models have to be adopted for practical applications, but such models have been developed mainly for long waves in the shallow water regime (see Wu 1998, 2001 for a more recent development for dispersive nonlinear long waves).

Despite significant advances in the understanding of nonlinear wave in-

94

95

teraction and wave instability in the 1960’s, progress in numerical modeling of deep water waves, on the other hand, is relatively slow. Although the wave action approach for slowly varying wave fields is somewhat successful, the description of nonlinear interaction of different wave components mostly relies on the integral formulation proposed by Hasselmann (1962) and Zakharov (1968). The complexity of its formulation in the spectral space nevertheless keeps the formulation from being used in operational wave prediction models.

A much more effective formulation to study the evolution of nonlinear surface waves in infinitely deep water wits first proposed by West et al. (1987). By expressing the solution of the Laplace equation via an integral operator and expanding the free surface boundary conditions about the mean free surface, they derived a closed system of nonlinear evolution equations in infinite series. Similar approaches have been also used in Mat- sun0 (1992), Craig & Sulem (1993), Choi (1995), Smith (1998), and Choi & Kent (2004) to derive more general evolution equations that include the effects of finite water depth, bottom topography, and wave-body interaction. Although the system written in infinite series is valid for arbitrary wave steepness, the series has to be truncated for numerical computations, depending on the desired accuracy and computational efficiency.

In this paper, we consider a truncated model correct to the third order in wave steepness and examine its validity by comparing numerical solutions of the third-order model with fully nonlinear solutions of the Euler equations and with available experimental data. To find fully nonlinear solutions for the two-dimensional Euler equations, we use the system of exact evolution equations first derived by Ovsjannikov (1974) using an unsteady conformal mapping method (see also Dyachenko, Zakharov & Kuznetsov 1996; Choi & Camassa 1999). To solve both the exact and the third-order systems numerically, we adopt a pseudo-spectral method based on Fourier series, as described in Kent & Choi (2004).

Three different numerical experiments are described. First , one- dimensional progressive waves subject to perturbations at sideband frequencies are chosen to test the third-order model and the time evolution of spectral components is carefully examined. Secondly, we study nonlinear resonant interaction among two primary wave trains propagating in mutually perpendicular directions. It has been known that resonant interactions can occur at third order so that a mode not initially present may be excited by the nonlinear interaction between two existing modes (Phillips 1960; Longuet-Higgins 1962). We numerically reproduce the experiments

96

of McGoldrick e t al. (1966) and compare our numerical solutions with their observations. Lastly, we present the evolution of a more realistic random, directional, oceanic sea state characterized initially by the JONSWAP spectrum with directional spreading.

2. Mathematical Formulation

On the free surface of an ideal fluid, the boundary conditions can be written, in terms of the surface elevation <(z, y, t ) and the free surface velocity potential a(%, y, t ) = p(z, y, 6 , t ) (Zakharov 1968), as

at + [email protected] = (1 + lV(I2) w , (1)

(2) aa at - + ;pal2 + g(' = ; (1 + 1v612) w2,

where ' p ( z , y , z , t ) is the three-dimensional velocity potential, g is the gravitational acceleration, V is the horizontal gradient defined by V = (a/&, a/ay), and W is the vertical velocity evaluated at the free surface,

2.1. Asymptot ic expansion

By expanding the free surface velocity potential about the mean free surface and using a property of harmonic functions, it has been shown (West e t al. 1987; Choi 1995; Kent & Choi 2004) that the vertical velocity surface, W , can be expanded as

00 n

at the free

(3)

where Wn = O ( P ) with E being the wave slope defined by €=wave ampli- tudelwave length, and ' p j = O ( E ~ ) can be found recursively, as a function of C and a, from

j - 1

P I ( ~ , Y , ~ ) = @ ( z , y , t ) , ' p j ( z , y , t ) =-Cdj-l[~p~] for j L 2. (4) k 1

In (3)-(4), two operators R, and Cn are defined, for even n = 2m (rn =

0,172, * * * 1, by

97

a n d , f o r o d d n = 2 m + l ( m = 0 , 1 , 2 , . . . ) , by

(6) where the linear integral operator L is defined by

2.2. Nonlinear evolution equations

Since pj, and therefore W , are functions of c and @, by substituting (3) for W into (1)-(2), we have a closed system of nonlinear evolution equations for c and @ in the form of:

In (8), Qn and R, representing the terms of O ( P ) are given by

Q1 = WI, Q2 = [email protected]< + "2, Q, = W, + JVcI2Wn-1 for n 2 3,

n-2

R, = ; c Wn-j-1 Wj+l j = O

n-4

(9)

j = O

Although the system valid to arbitrary order can be found from (8)- (lo), we consider the third-order system correct to O(e3):

5 + L[@] + V.(CV@) + L [cL[@]] at +v2 (ac2L[@]) + L [cL[cL[@]] + $C2V2@] = 0, (11)

a@ at - + gc + [email protected]@ - 4(L[@])2

-L[@] (cv% + L[cL[@]]) = 0, (12)

and the numerical solutions of this system will be compared with the Euler solutions.

98

2.3. Pseudo-spectral method

To solve the system (11)-(12) numerically, the surface elevation < and the free surface velocity potential are represented by the double Fourier series:

M N 2 2

- -

n=-E m,--M

M N 2 2

- -

1 (14) einKl s+imKzy a ( ~ , t ) = C C bnm(t) n=-& m=--M

where N and M are the numbers of Fourier modes in the x- and y-directions, respectively, and K1 = 21r/L1 and K2 = 21r/L2 with L1 and L2 being the computational domain lengths in the x- and y-directions, respectively. When acting on a Fourier component, the integral operator C in (7) is defined by

7 (15) [eik*r] = -keik'"

where k = (kl, kz) and k = (kl, for which An and Cn in (5)-(6) can be simplified (West et al. 1987) to

(16) R , = - k n C" , cn = - k n + l * C"

n! n! After evaluating the right-hand sides of (11)-(12) using a pseudo-spectral method (Fornberg & Whitham 1978), we integrate the evolution equations using the fourth-order Runge-Kutta method. We check the accuracy of our numerical solutions by monitoring conservation of energy:

3. Validation of Models for One-dimensional Waves

3.1. Exact evolution equations

For a two-dimensional flow, using the conformal mapping technique, the evolution equations for the surface elevation v(E, t ) G C(x(E, t ) , t) and the velocity potential at the free surface q5(E, t ) = a(x(5, t ) , t) parameterized in terms of 5, can be found (Ovsjannikov 1974; Dyachenko et at. 1996) as

99

where the subscript denotes differentiation with respect to E , the Hilbert transformation 'H is defined by

and

In (18), x(E, t ) and $(c , t ) are the complex conjugates of q(E, t ) and qb(6, t ) , respectively, and can be found as

where c is the velocity at infinity.

3.2. Numerical method for exact evolution equations

The system of exact evolution equations given by (18) are solved using a pseudo-spectral method similar to that for the third-order system. To reduce aliasing error, we add diffusive terms of uD[qc,~] and uD[qbt~] to the right-hand sides of (18), where constant u is chosen to be 0.05AE and D is a high-pass filter defined in the Fourier space which is 0 on the lower 1/2 Fourier modes and 1 for the higher 1/4 modes; there is a linear transition between two regions. Therefore, there is no energy dissipation on the lower 1/2 modes. The detailed description of the method can be found in Yi, Hyman & Choi (2004). In order to make sure that no significant physical energy is dissipated by these ad-hoc terms, we carefully monitor conservation of total energy defined by

where L is the length of the total computational domain.

3.3. Progressive waves

For initial conditions, we choose the progressive wave solutions ( C p , aP) of the exact system (18) for amplitudes a/X = 0.02 and a/A = 0.04, found via the Newton-Raphson method (Choi & Camassa 1999), for both the third- order model (11)-(12) and the exact system (18). For these computations, to fix time and length scales, we choose g = 1 and X = 1. Using N = 128 and At=0.001, a typical error in energy conservation is less than 0.001%.

100

0.04

0 0.2 0.4 0.6 0.8 1 X X

(a) ( b ) Figure 1. Progressive waves: Numerical solutions of the third-order model for < (-) compared with fully nonlinear solutions of the Euler equations (. . .) at t = 50 (approximately, 25 wave periods). (a) a/X = 0.02, ( b ) a/X = 0.04.

Two solutions at t = 50 (approximately, 25 wave periods) are compared in figure 1. Notice that there is a small error in phase velocity, but the wave profiles are well preserved for both wave amplitudes. For example, for the wave of a = 0.04, whose wave speed is c N 0.4117, the error in amplitude is less than O . O l % , and the error in phase velocity is less than 0.2%. Therefore, the third-order model can be regarded as a good approximation to the Euler equations for these wave amplitudes. It is interesting to notice that, compared with the exact solutions, the wave of the third-order model travels slightly slower for the smaller wave amplitude of a = 0.02, but faster for the larger wave amplitude of a = 0.04.

3.4. Benjamin- Feir instability

In order to validate the third-order model for time-dependent problems, we first consider the evolution of a progressive wave train subject to sideband perturbations which are known to be unstable (Benjamin & Feir 1967).

For initial conditions, the amplitude of the wave train is slightly modulated so that the free surface elevation < is given by

C ( G 0) = [I + A cos(Kz)] C p ( k 4 , (23)

where k = 27r/X and K = 2 r / L are the wave numbers of the carrier wave and its envelope, respectively, and C p represents the exact progressive wave solution of the Euler equations of wave amplitude a. Initially, no perturbation is added to the free surface velocity potential aP. In our computations, X = 1, g = 1, a = 0.02, A = 0.1, and L = 8. Notice that the wavelength of the wave envelope is the same as the length of the total computational

101

0.02

0.01

5 0 t=O

-o'oll,v -0.02 v , v v , v v , v 1.1 0 2 4 6 8

X

0.02

0.01

5 0

-0.01

-0.02

0.03

0.02

0.01

5 0

-0.01

-0.02

0.04

0.02

5 0

-0.02

t=100

) 2 4 6 8 X

t=200

7 1

t I 4 6 8 X

t=240

2 4 6 8 X

Figure 2. Benjamin-Feir instability: Numerical solutions of the third-order model for < (-) compared with fully nonlinear solutions of the Euler equations (. . .). The initial wave profile is given by (23).

102

n=8 r=O ,

1

n=9

0.01

0.008

0.006

0.004

0.002

0.008

0.006

0.004

0.002

t=240

n=VK

Figure 3. Wavenumber spectra at t = 0 and t = 240 for the wave profiles shown in 2. Numerical solutions of the third-order model (-) are compared with exact solutions of the Euler equations ( a . .).

domain. We use the total number of Fourier modes to be N = 21° = 1024 (27 = 128 modes per wavelength, the same resolution as in the previous comparison for progressive waves) and At = 0.001. Again, the total energy is conserved to 0(10-3%) at t = 240 for both the third-order and the exact systems. Two solutions for wave profiles at different times are compared in figure 2, and they show excellent agreement in both wave amplitude and phase. Notice that the highest wave amplitude is more than twice the initial wave amplitude of the carrier wave. In figure 3, we also compare the wavenumber spectra defined by the amplitude of the complex Fourier coefficients. At t = 0, we can observe the first harmonic component of the carrier wave at k / K = 8 and small perturbations at two sideband wavenumbers of k / K = 7 and 9, as well as the higher harmonics of the carrier wave at k / K = 16 and k / K = 24. At the end of computation (t = 240), the spectrum becomes wider with the wave amplitude at the lower sideband mode being higher than that of the primary mode, as noted in the experiments of Lake et al. (1977). Except for a small discrepancy in amplitude of the primary mode, the third-order system well captures the development of all spectral components.

As time increases, both systems continuously spread energy to higher harmonics and, therefore, the total energy in our computation starts to decrease at t = 250 due to our filter on higher harmonics to eliminate the aliasing error. It is well known that the nonlinear Schrodinger (NLS) equation serves as a reliable mathematical model for this situation, under the assumption that the wave spectrum remains narrow-banded. Based on our observation that the wave spectra become wide, a long-term prediction using the NLS model might be inaccurate for the evolution of a modulated

103

wave envelope.

4. Numerical Solutions for Two-dimensional Waves

4.1. Nonlinear resonant wave-wave interactions

As originally demonstrated by Phillips (1960), three trains of gravity waves in deep water, with wave numbers kl , kp, and k3, are capable of interacting such that energy is transferred to a fourth wave number, k4. In order for this continuous energy exchange to occur, it is necessary that the following conditions be simultaneously satisfied:

kl f k2 f k3 f k4 = 0 , W1 f w2 f W3 f W4 = 0, (24)

where wi2 = gki (i = 1,2,3,4) with ki = Ikil. For the special case in which two of the primary gravity trains coincide

(kl = kg), and are orthogonal to the third wave train (-k2), the resonant condition (24) between the primary wave trains and the resonant, or tertiary, wave (-k4) can be written as

2 k l - k2 - k4 = 0, 2 ~ 1 -wp - w 4 = O , (25)

and, from the dispersion relation between w4 and kq, it must be true that

Upon simplifying the equation with the substitution T = W ~ / W Z , the only non-trivial solution is given (Longuet-Higgins 1962) by

T , = w1/wp = 1.7357, kl /kz = 3.0123. (27)

For resonant conditions, Longuet-Higgins (1962) showed that tertiary wave amplitude will grow with time according to

(28) G

k4a4 = ,(k,a1)2(k,a2) W'I t ,

where G is a non-dimensional function of the angle between the two primary waves and the frequency ratio T and, for the case of mutually perpendicular primary waves, G = 0.442. From this equation, it is evident that the amplitude of the tertiary wave is expected to grow linearly with interaction time between the primary wave trains.

In order to verify this resonant mechanism, two experiments were carried out independently by Longuet-Higgins & Smith (1966) and McGoldrick et al. (1966) for two mutually-perpendicular wave trains. Both experiments confirmed the linear growth of resonant waves, but the measured growth

104

0.05.

0.04.

0.03

0.02.

0.01.

rate of Longuet-Higgins & Smith was lower than the predicted value of Longuet-Higgins (1962), while the opposite trend was observed in the experiments of McGoldrick et al.

To numerically model the interaction of two intersecting wave trains, we initially superimpose two third-order progressive gravity waves, whose surface profile and velocity potential are given by

1 ci = ai [ (1 + i(liiai)2) c o s ~ i + T(kiai) 1 cos(20i) + -(lciai)2cos(3ei) 3 , 8 (29)

@i = a i a e k i c i sinei, (30) where Bi = ki . z with kl = (k1,O) and k2 = (0, k2). In our numerical experiment, we solve the third-order equations given by (11)-( 12) with g = 1, A1 = 1, A2 = 3.0123, k l a l N 0.094, and k2a2 fli 0.085. The number of primary waves in each direction is equal to 4 and the number of Fourier modes used is N = M = 128.

,/ [

*,**ti .*

,.d

0 ,,..Jf'

:

A,P-"

o,*' ,,

off" .

Figure 4. Numerical solutions (0) for the growth of resonant wave ( a d ) for r(= U ~ / U Z ) = 1.7357 compared with the theoretical prediction (- - -) by Longuet-Higgins (1962). For references, experimental data of McGoldrick et al. (1966) are shown for r=1.73 (A) and r = 1.74 (0).

As shown clearly in figure 4, the amplitude of the resonant wave in our numerical computation does grow linearly with time. As might be expected, the resonant wave amplitudes are closer to the theoretical values at smaller times. As time increases, the resonant wave continues to grow and eventually begins to interact with the two primary waves, rendering the theoretical prediction invalid. At the end of computation (t N 335),

105

the resonant wave is nearly 113 the amplitude of a l , suggesting that a comparison of theoretical vs. numerical results beyond this time would have little significance. Compared with our numerical results, the growth rates measured by McGoldrick et al. (1966) are still higher and, currently, the origin of this discrepancy is not yet known. Notice that the frequency ratios for the earlier experiment and our computation are slightly different , but we have verified that this difference is not a source of discrepancy.

Table 1. Numerical results for the evolution of primary and resonant wave amplitudes

0 0.09425 0.08489 0 111.8 0.09362 0.08493 0.01987 223.6 0.09173 0.08497 0.03732 335.4 0.08954 0.08505 0.05044

As shown in table 1, another notable trend is that a2 grows slowly as a1 decays. This is an indication of an energy transfer from k l to k 2

accompanying the resonant interaction, as was noted by McGoldrick et al. (1966).

4.2. Nonlinear evolution of random wave fields

To examine the nonlinear evolution of a random wave field, an initial condition generated using the JONSWAP spectrum is considered. This simulation is similar to simulations carried out by Onorato, Osborne & Serio (2002) who used an equation similar to the Zakharov equation.

The JONSWAP spectrum is given in terms of the wave frequency w by

Y (31) a s (w) -

e- 1.25 ( u p / w ) ~ X P [ - ( w - ~ P 1 / (2 c2uW;: 11, 2 1c4

where k = w2/g , wp is the frequency of the dominant (principal) wave, a and y are constants that shape the spectrum, and u is a simple function of w given by

for w 5 wp f f = { u2 for w > wp '

To create a multi-directional wave spectrum, the JONSWAP spectrum S(w)

106

given by (31) is multiplied by the following function for angular dependence:

H(8) = ( g cos2 ($8) for -p 5 e 5 p otherwise

7 (33)

where 8 represents the angle between the x-axis and the direction of wave propagation, and /3 is an angle specifying the maximum directional spread in the spectrum.

To generate an initial condition from the JONSWAP spectrum (31) with the directional spreading function (33), we first compute the wave frequency w, at a given wave number k, using the linear deep-water dispersion relationship between w and k, and the wave propagation direction 8 for l l n l N , l < m < M :

knm = dnK12 + mKz2, Wnm = JK, On, = tan-'(mK2,nK1),

where N and M are the numbers of Fourier modes in the x- and y-directions, respectively, and K1 = 2.lr/L1 and K2 = 21r/L2 with L1 and L2 being the computational domain lengths in the x- and y-directions, respectively, as defined in (13)-(14). From (31) and (33)-(34), one can find the Fourier coefficients for initial conditions for C and @ in (13)-(14) as

(34)

anm(0) = An, ei6, , bnm(O) = iAn, Fei'r , (35) knm

where 6, is a random phase, and An, is defined by

The physical initial conditions for C and @ are then found by taking the inverse fast Fourier transform of the Fourier coefficients described by (35).

Using the dimensional third-order equations given by (11)-(12), a simulation is carried out on a square domain with sides of length 2260 m, and the initial wave field obtained from (35) is allowed to evolve for one hour. Other physical parameters involved are p = 1025kg/m3, g = 9.81m/s2, A, = 220m (w, = 0.529rad/s), a = 0.015 (significant wave height=9.5 m), y = 5.0, q = 0.07, ~ 7 2 = 0.09, and p = 8". In this simulation, we use 256 Fourier modes for both the x- and y-directions, and 64 time steps per wave period of the peak wave.

Figure 5 shows the time evolution of the maximum wave elevation (elevation of the highest crest) observed inside the computational domain. One can see a peak in the maximum wave elevation (almost 1.7 times the significant wave height) occurring at approximately 378 seconds. Notice that

107

I

I .roo rmo 1.1m Z m , LVM 3mo 3240

The ($1

Figure 5. spectrum simulation.

Maximum wave elevation versus time for the multi-directional JONSWAP

relatively large amplitude waves can be observed over short initial periods for which nonlinear wave interactions are assumed to be active. As time increases, the maximum wave elevation decreases. A series of instantaneous free surface elevations are shown in figure 6 , in which the localized nature of the large wave can be observed.

5. Discussion

We have presented a nonlinear formulation for surface waves in an infinitely deep fluid and have shown that the evolution of the free surface is governed by a pair of closed nonlinear evolution equations for two surface variables: the free surface elevation and the velocity potential at the free surface. The truncated form of the evolution equations correct up to third-order nonlinearity (or wave steepness) has been chosen for numerical experiments and solved using a pseudo-spectral method.

For validation of the truncated system, the one-dimensional solutions of the third-order equations have been compared with fully nonlinear solutions of a pair of exact evolution equations, derived from the two-dimensional Euler equations, for progressive waves as well as for a wave train subject to sideband perturbations. Based on the fact that the third-order equations approximate surprisingly well the fully nonlinear Euler equations for one- dimensional waves, one can anticipate that the third-order equations serve as a reliable theoretical model even for two-dimensional waves, for which

108

2260 "

Figure 6. Instantaneous surface elevation for the multidirectional JONSWAP spectrum simulation at (u) t=O s, (b ) t=378 s. The dominant direction of wave propagation is the positive r-direction.

fully nonlinear solutions are difficult to obtain. As a test of our numerical model for two-dimensional waves, we have

examined nonlinear resonant wave interactions between two waves traveling in perpendicular directions. The numerical model describes the excitation of the initially absent resonant wave, as predicted by the theory of Phillips (1960) and Longuet-Higgins (1962). Initially, the resonant wave amplitude grows linearly with time, but the computed growth of the resonant wave

109

eventually deviates from the theoretical prediction. In this paper, we have considered the simplest resonant wave interaction, but the numerical model should be equally applicable to more general nonlinear wave interactions. Numerical experiments for a wide range of wave slopes, frequency ratios, and propagation directions are currently underway.

Finally, we have simulated the generation of large amplitude waves in a random wave field, initialized by the JONSWAP spectrum multiplied by a directional spreading function, as a possible scenario of the occurrence of rogue waves. Our numerical solutions indeed have shown the creation of relatively short-lived local waves that are far larger than the background wave field. More systematic numerical experiments would help us better understand the physical mechanisms driving the creation of these intermittent large amplitude waves.

References 1. Benjamin, T. B. and Feir, J. E. (1967). The integration of wave trains on deep

water. J. Fluid Mech. 27, pp. 417-430. 2. Choi, W. (1995). Nonlinear evolution equations for two-dimensional waves in

a fluid of finite depth. J . Fluid Mech. 295, pp. 381-394. 3. Choi, W. and Camassa, R. (1999). Exact evolution equations for surface waves.

J. Eng. Mech. 125, pp. 756-760. 4. Choi, W. and Kent C. P. (2004). A pseudo-spectral method fir non-linear wave

hydrodynamics, Proceeding of the 2Sth Symposium on Naval Hydrodynamics St. John's, Newfoundland.

5. Craig, W. and Sulem, C. (1993). Numerical simulation of gravity waves. J. Cornput. Phys. 108, pp. 73-83.

6. Dyachenko, A. L., Zakharov,V. E., and Kuznetsov, E. A. (1996). Nonlinear dynamics of the free surface of an ideal fluid, Plasma Phys. Rep. 22, pp. 916- 928.

7. Fenton ,J. D. and Rienecker, M. M. (1982). A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. J. Fluid Mech. 118, pp. 411-443.

8. Fornberg, B. and Whitham, G. B. (1978). Numerical and Theoretical-Study of Certain Non-linear Wave Phenomena. Phil. Trans. Royal SOC. A 289,

9. Hasselmann, K. (1962). On the nonlinear energy transfer in a gravity wave spectrum, Part 1. General theory. J . Fluid Mech. 12, pp. 481-500.

10. Kent, C. P. and Choi, W. (2004). A hybrid asymptotic-numerical method for nonlinear surface waves. Part 1. Waves generated by a pressure forcing. Submitted for publication.

11. Lake, B. M., Yuen, H. C., Rungaldier, H. and Ferguson, W. E. (1977). Non- linear deep water waves: theory and experiment: Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, pp. 49-74.

pp. 373-404.

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12. Li, Y . A., Hyman, J. M. and Choi, W. (2004). A numerical study of the exact evolution equations for surface waves in water of finite depth. Studies in Applied. Math. 113, pp, 303-324.

13. Longuet-Higgins, M. S. and Smith, N. D. (1966). An experiment on third- order resonant wave interactions. J. Fluid Mech. 25, pp. 417-435.

14. Longuet-Higgins, M. (1962). Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, pp. 321-332.

15. Matsuno, Y . (1992). Nonlinear evolutions of surface gravity waves on fluid of finite depth. Phys. Rev. Lett. 69, pp. 609-611.

16. McGoldrick, L. F., Phillips, 0. M., Huang, N. E. and Hodgson, T. H. (1966). An experiment on third-order resonant wave interactions. J. Fluid Mech. 25,

17. Onorato, M., Osborne, A. R., & Serio, M. (2002). Extreme wave events in directional, random oceanic sea states. Physics of Fluids 14, pp. 25-28.

18. Ovsjannikov, L. V. (1974). To the shallow water theory foundation, Arch. Mech. 26, pp. 407-422.

19. Phillips, 0. M. (1960). On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, pp. 193-217.

20. Smith, R. A. (1998). An operator expansion formalism for nonlinear surface waves over variable depth. J. Fluid Mech. 363, pp. 333-347.

21. West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. and Milton, R. L. (1987). A New Numerical method for Surface Hydrodynamics. J. Geophys. Res. 92, pp. 11803-11824.

22. Wu, T. Y . (1998). Nonlinear waves and solitons in water. Physica D 123, pp. 48-63.

23. Wu, T. Y . (2001). A unified theory for modeling water waves. Adv. Appl. Mech. 37, pp. 1-88.

24. Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, pp. 190-194.

pp. 417-435.

THREE DIMENSIONAL WAVE PATTERNS FOR WATER WAVES ON A FINITE DEPTH:

THE DAVEY - STEWARTSON SYSTEM

K. W. CHOW' Department of Mechanical Engineering

University of Hong Kong Pokfulam Road, Hong Kong

D. H. ZHANG Department of Mechanical Engineering

University of Hong Kong Pokfblam Road, Hong Kong

C. K. POON Department of Mechanical Engineering

University of Hong Kong Pokfulam Road, Hong Kong

The Davey - Stewartson system (DS) governs the evolution of weakly nonlinear surface wavepackets on a fluid of finite depth, with long wavelength modulations in two mutaully perpendicular directions. Doubly periodic solutions are derived in terms of the Jacobi elliptic functions for a special range of parameters of DS.

1. Introduction

The generation and propagation of weakly nonlinear waves in shallow water and fluid of a finite depth are of great importance in fluid dynamical, marine and military applications. For surface waves in the shallow water regime, the dynamics is governed by a free or forced Korteweg - de Vries (KdV) model, where Professor T. Y. Wu and colleagues have made eminent contributions [l , 2, 31. Upstream advancing solitary waves have been studied intensively theoretically, numerically and experimentally. In particular, flows in channels of various geometry and configurations have been investigated.

For a fluid of finite depth, dispersive effects are much stronger. The focus is usually on the propagation of wave packets to minimize the effects of dispersion, and one studies the evolution of the packet on a slow time scale.

I Corresponding Author, Email: [email protected]

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With the presence of modulations in two mutually perpendicular horizontal directions, the dynamics and evolution of a wave packet are governed by the Davey - Stewartson system (DS) [4, 51. The assumptions employed in arriving at the DS system are, in some aspects similar to those of KdV, weakly nonlinear wave packets, finite water depth, long wave modulations in the streamwise as well as the spanwise directions, and a proper balance of these effects in a certain slow time scale [6] . More precisely, the DS system is

A is a slowly varying envelope and Q is the induced mean flow. The competing physical effects in (1.1) are dispersion and cubic nonlinearity.

Theoretically, DS is an important example of a family of higher dimensional evolution equations which can be solved exactly by techniques from the modem theory of nonlinear waves, just like the KdV case [7]. As expected DS admits solitary waves, or solitons, which are permanent, propagating structures resulting from a balance of nonlinear and dispersive effects. With the extra degree of freedom provided by the added horizontal spatial dimension, DS also admits a huge variety of waves not found in the KdV case, e.g., algebraic solitons, dark solitons, exponentially localized waves, and doubly periodic (periodic in two mutually perpendicular directions) waves and many others [8-151.

The goal of the present work is to present several new families of doubly periodic, standing wave patterns of DS. They are obtained by the Hirota bilinear method, a well established technique in the modem theory of nonlinear waves. Theoretically, the classical Jacobi elliptic functions will be employed. Computer algebra software, e.g., MATHEMATICA, will be heavily involved. A long wave limit will yield localized wave patterns.

The stability of such patterns, an important open question here, must be studied by direct numerical simulations and awaits more efforts in the future.

In realistic oceanic situations, the coefficients of DS system might not assume values necessary for exact integrability [6, 71. A rather comprehensive map of the parameter space for the modulational instability of plane waves has been given earlier [6 ] . Here we believe studies of the integrable case provide important insight for the general situation.

2. Formulation

The Hirota bilinear forms of (1.1, 1.2) are

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where D is the Hirota bilinear operator

D,"D:g. f = ( ~ - ~ ~ ( - - - ~ g ( x , t ) f ( x r , t i ) l x = x ~ , l = , , . a a ax ax# at ati

C is a constant to be determined as part of the solution process. o2 =+1 (02 =-1) corresponds to the DSI (DSII) system, where the governing structure in the mean flow equation is hyperbolic (elliptic).

Doubly periodic wave patterns for DS will now be presented. The intermediate calculations are conducted using theta functions (the Appendix), but the final results will be presented in terms of elliptic functions. As an additional check, all solutions are verified by direct differentiation and substitution in (1.1, 1.2) with the computer algebra software MATHEMATICA. In one of our earlier works we have presented solutions in terms of rational functions of quadratic expressions of elliptic functions [16]. Thls is related to the existence of this class of waves for nonlinear evolution equations in general [17, 181.

Two important advances are made in the present work

(a) We show that simpler rational functions of elliptic functions are also possible solutions. This reduces the analytical complexity and minimizes the ensuing computational resources necessary to handle such waves.

(b) We show that periodic waves also exist for the regime v < 0, whereas our earlier work [ 161 is explicitly restricted to the case v > 0.

With technical details described in the Appendix, the new doubly periodic solution of DSI (02 = 1) is

exp(-iQt) , (2.4) 2 112 rkl(1-k cn(sy,k,) iskcn(rx,k)dn(sy,k,) +

(l-k;)1'2 dn(rx,k) Wrx , k)

+ r2(2 4 2 ) + 3 2 , 2r2E

Q=-- K

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k, kl are different, independent moduli of the elliptic functions involved in the x, y directions respectively. K, E are the complete elliptic integrals of the first and second kind respectively. (2.4) is illustrated in Figure 1.

Figure 1. Intensity lA12 for the pattern of standing waves of DSI, (2.4), vo = 1, r = 1, s =

1.25,k=0.8,kl=0.6.

The wave numbers r and s are not independent, but must be related by

A long wave limit (k + 1 ) yields a periodic wave in the spanwise coordinate

dn(sy,k1)]exp(-iC2t)

R = s 2 ( 2 - k ; ) , Q = O .

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In conclusion, we have derived expressions for a pattern of doubly periodic standing waves for a special case of the Davey - Stewartson system (DSI, or o2 =+1 case,and v=-vo < O of(l.1)).

Appendix

The theta finctions Qfl(x) [19,20], n = 1, 2, 3 ,4 in terms of the parameter q (the nome) are defined by

Relationships between the theta and elliptic functions are

Arguments of the theta and elliptic functions are related by a scale factor. The modulus of the elliptic functions, k, is related to the theta constants by (A.1). Normally the dependence on k is omitted. However, solutions presented in this paper involve two different independent parameters. Hence the argument k is included explicitly in the text, e.g.,

sn(u) = sn(u, k), cn(u) = cn(u, k), dn(u) = dn(u, k ) .

The Hirota derivatives of theta functions are treated in a manner similar to our earlier work [ 161.

Acknowledgement

Partial financial support has been provided by the Hong Kong Research Grants Council through contracts HKU7 184/04E and HKU7006/02E.

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References

1.

2. 3.

4.

5.

6.

7.

Sh, M. H. Teng and T. Y. Wu, Journal of Fluid Mechanics 362, 157 (1998). M. H. Teng and T. Y. Wu, Physics ofFluids 9,3368 (1997). M. H. Teng and T. Y. Wu, Journal of Fluid Mechanics 266, 303 (1994). D. J. Benney and G. J. Roskes, Studies in Applied Mathematics 48,377 (1969). A. Davey and K. Stewartson, Proceedings of the Royal Society of London, Series A 338, 101 (1974). M. J. Ablowitz and H . Segur, Journal of Fluid Mechanics 92, 691 (1979). M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (London Mathematical Society, 1991). J. Satsuma and M. J. Ablowitz, Journal of Mathematical Physics 20, 1496 (1979). M. Tajiri and T. Arai, Physical Review E 60,2297 (1999).

8.

9. 10. M. Tajiri, K. Takeuchi and T. Arai, Physical Review E 64, 056622

11. M. Tajiri, H. Miura and T. Arai, Physical Review E 66,067601 (2002). 12. J. Hietarinta and R. Hirota, Physics Letters A 145, 237 (1990). 13. C. R. Gilson, Physics Letters A 161,423 (1992). 14. K. W. Chow, Journal of the Physical Society of Japan 65, 1971 (1996). 15. R. Takashima and M. Tajiri, Journal of the Physical Society of Japan

16. K.W. Chow, Journal of the Physical Society of Japan 69,1313 (2000). 17. E. V. Krishnan, Journal of the Physical Society of Japan 51, 2391

18. E. Fan, Journal of the Physical Society of Japan 71,2663 (2002). 19. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions

20. D.F. Lawden, Elliptic Functions and Applications (Springer Verlag

(2001).

68,2246 (1999).

(1982).

(Dover, 1965).

1989).

RIP CURRENTS DUE TO WAVE-CURRENT INTERACTION

JIE W School of Mechanical, Aerospace and Civil Engineering,

The University of Manchester, M60 IQD, UK

A. BRADMURRAY Division of Earth and Ocean Sciences

Nicholas School of the Environment and Earth Sciences Duke University, Durham, N C 27708, USA

In this linear instability study, we examine the incipience of depth averaged circulations, which are related to rip currents in the surf zone, due to interactions of waves and currents on alongshore uniform beaches. We consider normally incident waves, subject to infinitesimal perturbations which are alongshore periodic. Circulations are described by the depth-time averaged shallow water equations which include wave forcing and bottom friction. The slow modulation of the wave field is described by the ray theory. We assume linear shallow water waves and a saturated surf zone. The properties of the instability are then examined using two parameters, representing the effects of the incident wave height and bottom friction dissipation.

1. Introduction

When a wave climbs up a beach and breaks, the wave momentum is trans- fered into the water column, inducing a kind of nearshore current whose flow structures are predominantly horizontal. Rip currents, the narrow seaward flows, are of this origin. They occur when waves are nearly at normal incidence, and often have a regular alongshore spacing which can range approximately from 50 meters to 1000 meters (Short, 1985). They are associated with depth averaged circulation cells, which are characterized by narrow offshore flows (rip currents), broad onshore flows in between and intense alongshore flows 'feeding' the rips close to the shoreline.

The importance of alongshore variations in the wave field for the generation of rip currents has long been established (Shepard et al., 1941, Shepard and Inman, 1951). The causes of such alongshore variations are likely to

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be diverse, in view of the large range of observed rip spacing. It has been argued that on a planar beach alongshore variations may occur as a result of an instability due to interactions of waves and currents. Fkom the wave hydrodynamic point of view, when a wavetrain of uniform crests encounters a current, refraction by the current can cause the wave crests to bend, resulting in focusing and defocusing of the wave energy in the alongshore direction. In a study of rip currents driven by topography, Yu and Slinn (2003) showed that the alongshore variability of the wave field can be altered significantly even when circulations are weak. Therefore, the question put forward is: If an alongshore uniform wave field is perturbed, can the subsequent development of circulations enhance the alongshore variability in the wave field? a

Early studies to explore this mechanism include LeBlond and Tang (1974), Iwata( 1976), Mizuguchi (1976). Realizing the weaknesses and faults of previous analyses, Dalrymple and Lozano (1978, hereafter called DL) attempted this approach again, with a focus on the neutrally stable circulation cells of small amplitude on a planar beach. They found that equilibrium cells do not exist if the effects of currents on the wavenumber field are not taken into account. This conclusion was subsequently confirmed by FalquCs et al. (1999) with a nice argument using a Lyapounov functional. Upon including the effects of currents on both the wavenumber and wave energy fields, DL did find linearized steady circulations. However, there are uncertainties about their results. First, though we have used the same basic set of equations, we cannot recover the solutions far offshore asserted in DL. Second, they have neglected the perturbation in the breaking line location, assuming its effect is insignificant. This however is not the case; it plays a crucial role in determining the normal modes. Since DL considered only neutral modes, FalquCs et al. (1999) made an attempt to investigate if there is indeed an actual instability. They consider a constant still water depth in front of a vertical wall, as an attempt to avoid dealing with wave breaking hence any matching at the breaking line. However, they did not consider any reflection from the vertical shore face, nor any boundary condition at the wall. While they expressed some reservations about their model, they concluded that instability could occur, and estimated a typical growth time about 0.06 seconds for a water depth of lm. This is unreasonably fast.

aMurray and Reydellet (2001) studied rip currents on planar beaches based on a hypothesized wave-current interaction process which causes waves to be dissipated more in the presence of rip currents. That hypothesized wave-current process is outside the scope of this study.

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The purpose of this report is to present some results of a systematic analysis of the linear instabilities due to the interaktions of waves and currents. We will, in particular, address the issues of the boundary conditions at infinity, the appropriate unboundedness at the shoreline, and the matching conditions at the breaking line where the basic state is not smooth. Some of the main results will be discussed, followed by concluding remarks.

2. Formulation

We consider monochromatic waves, with no variations along their crests, normally incident on an alongshore uniform beach. In the horizontal plane, 2 points offihore and y alongshore. In the absence of waves, the shoreline is at x = 0; the topography is described by the water depth h(z). The usual model for nearshore horizontal currents, derived by averaging vertically and over a wave period (Mei, 1989), is employed. For the current velocity u and mean surface elevation (set-down/up) q, we have the shallow water equations,

qt + V * (du) = 0 (2.1)

ut + u ' vu = - g v q + 7~ - r b (2.2)

where d = h(z) + q is the total water depth, g the gravity. rw is the wave forcing, representing the transfer of wave momentum to the currents. It is modeled as the convergence of the wave radiation stress tensor after Longuet-Higgins (1970), i. e.,

rW = - (pd)-' V . S where Sij = E ( - lCi5 + 16,) (2.3) k - k 2 -

and E = i p g H 2 is the wave energy, H the wave height. The bottom friction Tb is parameterized as a linear function of the velocity, r b = $ C f H d - 2 f i d u .

Let us assume that both topography and the current vary slowly such that the modulations of the averaged wave field can be described by the ray theory, using wavenumber k and angular frequency w which satisfy (conservation of wave crests)

kt+Vw=O (2.4)

and the linear shallow water dispersion relationship, (w - u . k)2 = gd (k . k). In addition, V x k = 0. Before the wave breaks, the wave

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energy equation is

1 2

E* + v . [(C, + u) E] + -sijui,j = 0

where C, = k (k k)-1'2 &d is the group velocity vector. The last term represents the work done by radiation stresses on the strain of the currents. After breaking, we assume a saturated surf zone and write

H = yd

where the empirical breaking index y can vary from 0.3 to 0.78. Taking the curl of (2.2), one can see that the vorticity dynamics is

controlled by V x T, - V x T b , the balance between the curls of the wave forcing and the bottom friction. If the wave forcing has no curl, there is no vorticity generation, hence no circulation. For normally incident waves and in the absence of alongshore variations, the wave forcing is across-shore, i e . , S,,,,, due to breaking and refraction by the topography, but it has no curl and does not contribute to the generation of vorticity. This in fact is the scenario for the basic state; see section 3. An inhomogeneity alongshore, for instance due to waves responding to a disturbance, will lead to V x T, # 0 and the initiation of circulation.

We consider a topography which consists of two sections: h(z) = ,bx for 3: 5 z, and h = h, for 2 > z,. Two dimensionless parameters can be identified. & = Hin/hm, the ratio of the wave height at incidence Hi, to the maximum water depth h,. It measures how energetic the incoming waves are. R1 = &/3c,'&, the ratio of the rate of vorticity generation by curl 7, to the rate of its dissipation by bottom friction. The bottom slope /3 shows up because the scale for the depth is h, and the scale for the horizontal lengths is x, = h,/P.

3. Basic state: the 0th order solutions

For normally incident waves and in the absence of any alongshore variations, the basic state is motionless, independent of 9 and t . The across-shore wave momentum flux, resulting from wave shoaling and breaking over the topography, is balanced by a variation of the surface elevation. Denoting the variables of the basic state by the subscript 0, we have uo = wo = lo = 0, and wo uin where win is the wave angular frequency at incidence. Ho(z) , ko(x) and qo(z) can be obtained easily by integrating the equations. Details of obtaining them can be found in Mei (1989) and other research papers.

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The solutions presented below are normalized using the scales: 2, for 2, Hi, for H , h, for d and r ) , and ~ i , ( g h , ) - ~ / ~ for k . For x > 1, one obtains 4 = 1, Ho = 1, ko = -1, and 770 = 0. Inside x = 1 it is convenient to express ko and HO in terms of the total water depth Q = x + w. It is found that ko = -dg1I2. The breaking line is located at x = xm = (1 +y2/16) (&/7)4/5 - @/16. For 1 2 x 2 xbo, HO = di1j4 and Q is implicitly given by

1 16 4 = x + -G (1 - dg3l2) .

For x < xbo, Ho = yl$-l&, and

Q = p ( x - x S o ) where p 3 (1 + 3y2/8)-l (3.2)

and xso = -& [@ - 5y2 (~&,/y)~'~] is the location of the shoreline. Note that ko is unbounded at the shoreline because the wavelength approaches zero there. The solutions for the basic state are not smooth in their first derivatives at the breaking line xbo.

It is interesting to note that the basic state has no current, but it has a surface set-down/up 70 (or a pressure gradient) due to the presence of waves. It is the instability of r)o that is of interest here, and the instability depends on the interaction of waves and currents. This is analogous to Ftayleigh-Bdnard instability in which the instability of the basic temperature gradient generates motions.

4. Linear instability: the 1st order solution

We consider the linear instability of the basic state subject to infinitesimal alongshore disturbances, i .e., the perturbed variables have the form fl = f^(x)eias+iat, where the alongshore wavenumber cr is real and u the complex frequency. In this report, we shall focus on u = 0 and real iu. The equations for the perturbations can be easily obtained by linearizing (2.1) - (2.6) around the basic state. Apart from scaling, they are the same as in DL, and will not presented here.

Once the circulations develop, the actual shoreline becomes a moving boundary, at which the kinematic boundary condition must be applied. Upon linearization, this condition is approximated by -

u1 = -@-lr)l,t at x = (4-1) Note that the velocity is scaled by ( ~ h , ) l / ~ and the time by h,@-' (gh,)-lI2. The appropriate conditions as x + 00 for a hyper-

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bolic system are the ’radiation condition’, which states that disturbances must propagate outward at infinity (Courant and Hilbert, 1962).

4.1. Solution procedure

Since the equations inside and outside the surf zone are different, they must be integrated separately in their own regions with the appropriate boundary conditions. The solutions are then matched at the breaking line.

The asymptotic behavior as x + xSo and as x + 00 are first determined to provide the appropriate data needed by the mentioned integrations. The system is singular at x = 280 because 4 approaches zero there. Of the three families of solutions admitted by the equations, one family has unbounded flow field, thus is not physical. The second is unbounded only in the alongshore wavenumber I1 , and the third is bounded. Both of these are physically admissible and will be used to construct the solution inside the surf zone. For x > 1, the analysis of the hyperbolic system shows that among five characteristic curves three are shoreward directed. According to radiation conditions, the quantities on these three curves must vanish at infinity. On the other hand, the solutions admissible by the equations for real io behave exponentially in x for x > 1. For o = 0 and for sufficiently small io, even though two families of spatially decaying solutions are found, only the faster decaying one is physically allowed. This is the only way to construct a solution which uniformly satisfies the radiation conditions. For sufficiently large in, the equations admit only one family of decaying solutions.

For the same equations, DL reported the existence of three families of decaying solutions at large x. All of them were used to determine the solution outside the surf zone. Close to the shoreline, they reported only one family of solutions.

Matching is then done at the breaking line xb which is defined by H+(Xb) = y&-ld+(zb), where the superscript ”+” indicates the variables obtained for outside the surf zone. Correspondingly, the superscript ”-” is used below for the inside. Setting X b = xw + dxbleicry+iut, we then have

It is necessary to require the continuity of physical variables at the breaking line. Let f represent a variable to be matched, the continuity of f at the xb can be approximated by

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where Afo,% = fcz(zbo) - f&(xao). Since the flow is inviscid, w1 is not necessarily continuous at the breaking line. Second, kl is linearly dependent on G I , and & because of V x k = 0. Third, with the use of 6Xbl in (4.2), when ;il is matched, so is g1. Therefore, (4.3) represents only three h

independent conditions at the breaking line x = XbO, regarding 61, ql and 21.

We then have a homogeneous system of linear equations for the coefficients of the three families of the solutions which are physically admissible. To have non-trivial coefficients, the matrix of the homogeneous system must have zero determinant. This defines the dispersion relation between o and a, or the growth rate curve.

5. Results and discussion

Typical growth rate curves are shown in Figure 1 for & = 5 , 5 and $, which correspond to the incident wave height Hi, = l.Om, 1.2m and 1.5m if the maximum water depth is taken to be hm = 3.0m. RIG-' = 13.744, which may be obtained, for instance, by choosing /3 = tan4" = 0.07 and cf = 0.001. For given parameters, the instability properties can be described by the maximum growth rate ( i ~ ) , ~ ~ corresponding to the most unstable mode a,, and the marginal instability, acr. A, = 27r/a, is referred to as the preferred alongshore spacing of the circulation cells.

5.1. Eflects of incident wave height

By varying &, the effects of incident wave height on the instability can be studied. In Figure 2(a) and (b), ( i ~ ) , ~ ~ , a, and a,, are shown as functions of &. By fixing Rlh- l , the beach condition is unchanged in terms of /3/cr. Note that & must be smaller than y, otherwise the incoming wave is broken before it climbs the beach. From Figure 2(a), 2 0 when & 2 0.29. So the basic state becomes unstable only when the incoming wave is sufficiently energetic, e.g., Hi, 2 0.87m if hm = 3.0m. ( i ~ ) , ~ ~ increases with &. The dependence is slightly nonlinear. As & increases, the range of the unstable wavenumbers widens, and a, decreases slightly, see Figure 2(b). So the preferred alongshore spacing of the circulation cells tends to be longer when the incident waves are more energetic. This trend is weak, but consistent with observations (Short, 1985, Huntley and Short, 1992)

For /3 = 0.07 and h, = 3.0m, the horizontal length scale is x, = 42.857m. The preferred alongshore spacing of the circulations is then from 374.0m to 427.4m for Hi, = 0.87m to 1.78m. The time scale is

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2 a

Figure 1. Growth rate curves for different values of RQ. R I R ~ ' = 13.744 and 7 = 0.6

RO

1 .o

0.8

0.6

0.4

0.2

6

Figure 2. and most unstable mode a, as functions of &. RIR;;' = 13.744 and 7 = 0.6.

(a) Maximum growth rate as a function of &. (b) Marginal instability acp

h&-l (gh,)-li2 = 7.8976 seconds. The estimated e-folding times based on the most unstable modes are 22.51min for Hi, =1.5m (& = 0.5), and 15.93min for Hi, =1.7m (& = 17/30).

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5.2. Effects of bottom friction

Here the effects of bottom friction are examined for a given incident wave height, by varying R1 while fixing &. In Figure 3(a) and (b), ( i ~ ) , , , ~ ~ , a,,, and acr are shown as functions of 1/R1 for & = 17/30. Recall that l/R1 - c f . The case without any bottom friction, i e . , cf = 0 or R1 + 00,

is also included. The following observations can be made. (a) The basic state is unstable when 1/R1 < 0.476, which corresponds to c f < 0.00371 for /3 = 0.07. (b) Below this threshold value, ( i ~ ) , , , ~ ~ increases linearly as cf (or 1/R1) decreases. (c) At the limit c f = 0, ( i ~ ) , , , ~ ~ = 0.01139, which gives an e-folding time of 11.56min for /3 = 0.07 and h,,, = 3m. (d) a,,, = 0.632 and is independent of cf. This is due to the fact that the bottom friction appears as a body force, rather than a viscous dissipation. (e) The range of the unstable wavenumbers increases with decreasing c f . At the limit c f = 0, the wavelength of one of the two neutral modes becomes infinitely long.

0.004 -

1/R, 5

0.8 -

1 /R,

Figure 3. and most unstable mode a, as functions of R1. Rij = 17/30 and 7 = 0.6.

(a) Maximum growth rate as a function of R1. (b) Marginal instability aCc

Values of cf found in the literature are quite diverse, ranging from O(O.001) to O(O.01) (Church and Thornton, 1993), and largely based on studies of well developed alongshore currents. In the present study, the instability can occur at values of c f comparable to those in the literature, but nonetheless it tends to require sufficiently small bottom friction, par-

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ticularly on beaches of mild slope. In view of the uncertainty of c f , the extent of applicability of the mechanism considered here to the formation of rip currents on planar beaches in nature remains an open question.


With physically plausible parameter values, this linear analysis shows that depth averaged circulations can be initiated on alongshore uniform beaches due to an instability process in which the effects of currents on wave and waves on currents are essential. Several conclusions emerged. (i) The instability occurs when the incident wave is sufficiently energetic and bottom friction is sufficiently small. (ii) The growth rate increases with the incident wave height, and as the bottom friction decreases, being approximately a linear function of cf. At the theoretical limit cf = 0, the growth rate remains finite. (iii) The most unstable wavenumber decreases weakly with the incident wave height, but it is not affected by the bottom friction.

The present analysis predicts an alongshore spacing of a few hundreds of meters for the circulation cells on a beach of typical slope and with a typical water depth. This prediction is within the range of the observed rip current spacing cited in the literature. The estimated growth time is a few tens of minutes. While definitive observations on this seem to be lacking in the literature, this estimate is not totally unrealistic. However, one of the authors, who sat on beaches a lot observing rip currents, suggests that the predicted growth time tends to be longer than his visual observations. The other one, who sits more in front of the computer than on the beach, suggests that the discrepancy is within the uncertainties of other model parameters, such as bottom friction coefficient c f . We agree in saying to the reader “caveat emptor!” .

Acknowledgments

Support from the Andrew W. Mellon Foundation is gratefully acknowledged. JY wishes to say ‘Happy birthday, Grandpa Wu, and many happy returns.’

References 1. J. C. Church and E. B. Thornton, Coastal Eng., 20 1 (1993). 2. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I1 (1962). 3. R. A. Dalrymple and C. T. Lozano, J . Geophys. Res. 83, 6063 (1978).

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4. A. FalquBs, A. Montoto and D. Vila, J. Geophys. Res. 104(C9), 20605

5. D. A. Huntley and A. D. Short, Coastal Eng. 17, 211 (1992). 6. N. Iwata, J. Oceanogr. SOC. Jap. 32, 1 (1976). 7. P. H. LeBlond and C. L. Tang, J. Geophys. Res. 79, 811 (1974). 8. M. S. Longuet-Higgins, J. Gwphys. Res. 75, 6790 (1970). 9. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (1989). 10. M. Mizuguchi, !i?ans. Jap. Soc. Cavil Eng. 248, 83 (1976). 11. A. Brad Murray and G. Reydellet, J. Coastal Res. 17, 517 (2001). 12. F. P. Shepard, K. 0. Emery and E. C. LaFond, J. G w l . 49(4), 337 (1941). 13. F. P. Shepard and D. L. Inman, Proc. 1st Conf. Coastal Eng. 50 (1951). 14. A. D. Short, Mar. Gel. 65, 47 (1985). 15. J. Yu and D. N. Slinn, J. Geophys. Res. 108(C3), 3088 (2003).

(1999).

HIGHER ORDER BOUSSINESQ EQUATIONS FOR WATER WAVES ON UNEVEN BOTTOM*

HUA LIU Department of Engineering Mechanics, Shanghai Jiao Tong University

Shanghai, 200030, China

BENLONG WANG Department of Engineering Mechanics, Shanghai Jiao Tong University

Shanghai, 200030, China

Higher order Boussinesq-type equations for wave propagation over variable bathymetry are derived. The time stepping problem is based on the free surface boundary conditions. The free surface velocities and the bottom velocities are connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth are retained in the inverse series expansion of the exact solution, with which the problem is closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model is extended to the not so mild slope bottom. For linear properties, some validation computations of linear shoaling and Booij’s tests are carried out. The problems of wave-current interactions are also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results conform to the theoretical solution as well as other numerical solutions of the full potential problem available.

1. Introduction

Coastal engineering requires knowledge of the wave field over an area of 1- 10 lan2 in which the water depth may vary significantly. Generally, the wave data are not available at the site required. Predicting of wave transformation from offshore or nearby location to near shore locations needed is a main problem in coastal engineering. In this scope, the depth change is considerable. Waves propagating through shallow water are strongly influenced by the underlying bathymetry and currents.

To improve the dispersion and nonlinear characteristics and the distribution of the vertical velocity is the main problem on the developing of the Boussinesq

This work is supported by the National Natural Science Foundation of China (No. 10172058) and theDoctora1 Program Foundation for Higher Education from the Ministry of Education of China (No. 2000024817).

128

129

equations (e.g. Madsen et al. [l], Gobbi et al. [2]). In the same time, Hong [3] developed a hgh order Boussinesq model of nonlinear and dispersive wave in water of varying bottom and studied the linear properties of the model. Higher- order Boussinesq equations for the case of strong currents are studied through a perturbation method by Zou [4]. Both Hong and Zou’s work gave successful theoretical analysis of the Boussinesq models. The numerical results reported indicated that the theory needs to be extended.

The propagation of wave over current is another interesting topic. If there are submarine bars, the combination of the shoaling and current over the bars can induce significant choppiness on the sea surface and therefore present navigation hazards. Although Chen et al. [5] and Kristensen [6] extended the Boussinesq equations to improve the accuracy of the dispersion relation, the inherent properties of traditional Boussinesq equations could not give a satisfactory dispersion relation for large kh number, need not to say the combination of shoaling and the interaction with current. The model proposed by Chen et al. [5] could simulate the wave blocking on a submerged bar, but strong friction need to be added to stabilize the flow simulation. It’s difficult to evaluate the model equations. At this point, they did not give any explanation.

With the objective of improving the accuracy of the vertical velocity field as well as the linear and nonlinear properties, enhanced Boussinesq formulations are proposed by Madsen et al. [7]. During the derivation, mild slope assumption is introduced. At the mean time, only the leading order terms of the inverse series are retained for the velocity components. Consequently, the application range is limited to the case of mildly changed bottom. In their work, it was found that the linear shoaling characteristic is irrelevant to the bottom slope terms Vh in the series expansion for the specified model. This interesting conclusion is only limited to the linear shoaling. Although the nonlinear effects become significant when the wave travels from deep water to shallow water, a linear solution is still very useful. The main purpose of the present work is to extend the application range of Boussinesq-type wave model from mild slope bottom to rapidly varying bottom. This work completes the models given by Madsen et al. [7]. The hghlight of the present work is that we take the bottom slope terms into account in the wave models without deterioration of the excellent shoaling characteristics. The overall capacities on the problems with rapidly varying bottom are improved significantly.

In the paper, we generalize the procedure of the inverse series expansion given by Madsen et al. [7] to obtain good performance on uneven bottom, including the theoretical formulations, numerical scheme and several numerical results about linear shoaling, Booij ‘s test and wave-current interaction.

130

2. Mathematical Formulation

Aiming to provide an accurate vertical distribution of the velocity field, Madsen et al. [7] proposed the enhanced Boussinesq model. In this section, we will extend the model to include the effect of bottom slope. The start point follows the ones given by Madsen et al. [7], which might be derived using the projection of the Eulerian equations on the free surface developed by Wu [8], -

q, -G+Vq*i i = o (1) N U

7-7 G - - v, + g v q + V(- --(1+ v q V q ) ) = 0 (2) 2 2 - - where 7 = z + GVq , are the horizontal and vertical velocity components evaluated at the free surface, v = (gx ,d , ,O) is the gradient operator in the horizontal space. The bottom

is the free surface elevation, ii and

boundary condition reads: wb +Vh-iib = O (3)

Gb and wb are the velocity components at the bottom, h is the water depth. The origin is on the still-water level and z is positive upwards. However, to establish a connection between the vertical and horizontal velocity variables at the free surface and the bottom, the exact solution of Laplace equation in the interior domain is introduced. The obvious approach is to express the exact solution in terms of the velocity components at the still-water level, i.e.

i i(x,y,z;t) = cos(zV)iio + sin(zV)wo w(x,y,z;t) = cos(zV)wo -sin(zV)iio

(4)

in which the definitions of cos- and sin-operators are: m 22n+l

.Zzn vZn , sin(zV) = Z(-I)~+' ~ 2 n + l m

cos(zV) = Z(-1)" - n=O (2n)! n=O (2n + l)!

Consequently, from the definition we have N - 24 =C(x,y,q,t) , K=w(X,y,q,t) I

iib = ii(x,y,-h,t) and wb = w(x,y,-h,t) . *

To improve the vertical profile of the velocity field, u and %, which are velocity components taken at an arbitrary level z = 2 , are introduced. 2 is assumed to be a constant fraction of the undisturbed water depth. With the inverse series expansion, the velocity field could be expressed as

131

where

r, = (z - ;)(cos((z - ~)v)v .2 + sin((z - ~)v)vG) r, = (z - ;)(cos((z - ~)V)VG - sin((z - ;)v)v. 2)

From the numerical viewpoint, the i n f ~ t e sin- and cos- operator must be truncated. With the original coefficients of the Taylor coefficients, the performance of the model is not good when kh is large. To further improve the application range for kh number, L - operator is introduced following MBL. The principle of this operation is to optimize the coefficients of the M i t e sin- and cos- operator.

The system of equations to be solved consists of (1),(2),(3),(5) and the

defition of V , i.e. six coupled equations in six variables: the surface elevation, the horizontal gradient of the free-surface velocity potential, the horizontal and vertical velocities at the free surface and the cr = -112 of the water depth.

Madsen et al. [7] ignored the r, and r, terms because their shoaling

analysis shows that TVZ terms have no impact on the imbedded linear shoaling properties. This is true for infimte harmonic analysis. When the truncation of the sin- and cos- operator is introduced, this conclusion would not hold any more for arbitrary d level. Although r, and r, terms have no influence on linear shoaling of the wave amplitude, they do influence on the velocity profile leading to a phase shft relative to the surface elevation.

N

3. Numerical Method

The numerical solution procedure follows the one as Madsen et al. [7]. A finite-difference solution has been developed. At each time step, the variables

77" , v"" are known, when marchmg (1) and (2) in time, the variables %and u" are unknowns. From the formulae (5 ) , % and u" could be expressed by 6 and % as:

A@] +Bl[G] = [u"] , -B1[ti] + Al[G] = [%I.

v"=u"+%Vq=(A1 -[qx]Bl)[ti]+(B1 +[qx]A1)[G].

(5 '1 from the definition:

In the mean time the bottom boundary condition (3 ) could be rewritten as: A2 [ti] + B2 [GI = 0 . The system to be solved for , G are linear:

132

After ti , G have been solved, u" and i% are evaluated by the equation (5 ' ) . In order to compute the fifth-derivatives numerically, a seven point hfference stencil is introduced. For time integration, fifth-order Cash-Karp-Runge-Kutta scheme is used. Savitsky-Golay smoothing is applied to remove the hgh frequency instabilities for every 10-20 time steps.

4. Numerical Results

In this section we consider several applications of the model, all of which are severe tests normally beyond the reach of conventional Boussinesq-type models: (a) linear shoaling from rather deep water to shallow water, 30 2 kh 2 0.35 ; (b) reflection on plane shoal from mild slope to steep slope; (c) deep water wave blocking in adverse current. All of these tests concern the uneven bottom. Tests (a)-(b) deal with the linear properties, while (c) involves the nonlinear terms in the model equations.

(a) Linear shoaling In order to verify the linear shoaling properties of the new model, the

following test cases have been studied. An infinitely smoothmg bathymetry is defined by:

sin(?) )] -+l~l-$ (6)

1 - (F)2 h(x) = h, -- ho - h1 [1+ tanh( 2

At the seaward (west) boundary ( ~ = - 1 2 0 ) the water depth is h, = 9.55m. The bottom is flat for the first 20m. The length of the slope region is 200 meters. Finally the bottom is flat again with a water depth of h, = O.O36m, shown in Figure 1. The space and time step are chosen to be 0.04m and 0.03s, respectively. The period of the wave is 1.13s) and the kh number at the end of the flume is 30 and 0.35, respectively.

From the linear theory, the wave amplitude distribution along the flume could be resolved (solid line in Figure 2), and the dash line is the snapshot of the free surface elevation. Numerical results confirm the linear theory perfectly.

133

6 -

4 -

2-

-100 -50 0 1 0

I m,

Figurel. The topography of 1/20 slope

0,006 ........................... 1 ........................... 4 .................... 1) 0 0 4

- s 0002

- 0

5 - c

$ - o w 2

-0 o m

‘Z

...................... i ..................... i . . ............................ I

I I - 1 0 0 -50 0 SO I00

1 (,,,) Figure 2. Amplitude envelope of a linear wave shoaling on a slope

(b) Reflection from plane slope: Booij’s test Booij’s test is used to test the capabilities of the numerical model in case of

uneven bottom. The wave period in the Booij’s test [9] is 2s and the water depths on the up-wave and down-wave sides of the slope are 0.6m and 0 . 2 ~ respectively. The difference of the water depth between the two constant depth regions is 0.4m. The slope of the shoal depends on b, , the length of the slope region. Comparisons between the discussed model and other numerical models, such as the finte element solution of the Lapalace equation, are conducted. The numerical results show that including the h, terms in the inverse series expansion improves the perfonnance of the model significantly. Without considering r, and r, terms in the expressions of the velocity filed, there are larger discrepancies between the Boussinesq model and modified mild slope

134

models given by Suh et al. [lo], as shown in Figure 3. The oscillations character of the reflection curve does not exist. When r, and r, terms are added in the inverse series expansion, the model could predict the reflection rather well. It should be noted that the inverse series expansion bases on the Taylor series expansion, where h, is the small parameter and h, < 1 is assumed.

0.4 0.8 1.6 3.2 6.4 4, (d

Figure 3. Reflection coefficients versus horizontal length of a plane slope: Boussinesq model with and without terms in the inverse series expansion.

Computation domain

Figure 4. Set-up of the flume associated with fully coupled wavecurrent maker.

(c) Wave-current Interaction The set-up of the combined wave-current flume is illustrated in Figure 4.

Assuming the water depth at both ends of the flume has the same values. The extension to different depth is straightfonvard. Before the description of the setup of each zone, the following definitions are introduced: [.I" is the input

135

wave; uc is the input current; [.I, is the value to be given at each time step; 7 and u are the computed value. The updating rules of the variables at the specified zone are listed as following: 0 Wave maker + current sponger

where [Cr]xrn, = Oand [CrIxm, = 1 Left relaxation zone (the left-going reflecting waves are absorbed)

where [Cr], , = 0 and [CrIxrn, = 1 Right relaxation zone (the right-going waves are absorbed)

7, = [Crlq , urn = [Crlu + (1 - [Cr])Uc where [Cr],, = 1 and [Cr]

7, = [cr]qW1 , u, = [cr]uW1 + [C~IUC

0

7, = [ ~ r l q + (1 - [~r] ) f ' , u, = [ ~ r l u + (1 - [cr])(uW + UC) rnm

0

= 0 Xrnm

0 Current maker 7, = 0 , u, = [CrIUc

where [CrIxmi. = 1 and [Cr] = 0 xmlx

In each zone, [Cr] is a smooth function. For example, the following kind function is used in the current maker zone:

if O < x < L Cr = 0, i f x = L

here L is the length of the current maker zone and x is the local position in this zone and the origin is located at the left boundary of the zone.

Test of harmonic generation of wave train with and without current serves a check of the combined wave-current flume. It has long been known experimentally that it is extremely difficult to generate long, simple harmonic progressive waves of finite amplitude in the shallow tank. A harmonic analysis of the wave records at various stations indicates that all the harmonics of the input period vary periodically with respect to the distance from the wave-maker. This is the phenomenon of harmonic generation. The set-up and wave parameters are the same as the test by Chen et al. [5 ] . At the wave-maker zone, first-order Stokes waves with 2.5s for wave period and 0.084m for wave height are generated in the flume with water depth of 0.4m The nonlinear equations with the linear input waves generate spurious free high-order waves, which resulting in the modulation of the wave train along the wave flume. Steady, uniform flows with the Froude numbers Fr = 0.15 are generated through the coupled wave-current maker. Currents with the typical strength of Fr = 0.15 have significant effects on the harmonic generations. The numerical results are shown

136

in Figure 5. Both the beat lengths and amplitudes match the results given by Chen et al. [5] perfectly. Generally, a following current intensifies the extent of the energy exchange between harmonics, and vice versa for an opposing current. The numerical results indicate that the combined wave-current maker works well.

FOllDWlng Current

Quiescent Water 0.m, 1

ODDOS~Q current

"'""I I

0 5 10 15 20 25

Figure 5. Amplitudes of first three harmonic in case of following current, quiescent water and opposing current. Numerical results: - - is 1'' order, - -the 2"d order and ------the 3rd order

harmonic. Results of Chen et al. [S] are given in sample points: 0 represents the I s t order, o the

2 n d order and the 3rd order harmonic.

Various dispersion relations of the Boussinesq equations with currents were discussed in Chen et al. [l 11. The numerical simulations of wave-current interaction in the frame of PadC[4,4] Boussinesq model are conducted in the same paper as well. While some artificial friction must be added in the computation region to avoid the numerical instability for the numerical model in Chen et al. [ 5 ] .

137

x to4 F 51

x (m) Figure 6. Snapshots of the surface elevation along the flume. The top panel is the envelope of the free surface elevation in one period; The bottom panel shows the mean water level due to water set- down. T=l.2s and Uc=-O.l6m/s.

Figure 6 shows the envelope of the surface elevation near the blocking point during a wave period. The flume setup is the same as the current test. To avoid wave breaking, the small amplitude wave train is generated from the wave- maker. With the enhanced Boussinesq equations, the high wave number reflection waves could be simulated. Due to the resolution of the mesh and the increase of the wave number, the reflection waves are damped rapidly far from the blocking point. The kh number distribution of the incident wave and reflection wave is shown in Figure 7. Within the scope of the linear stokes theory, the wave numbers distribution of the incident and reflection waves could be obtained by solving the dispersion relation accounting the Doppler effects: w + lkUcl= ,/-, the local water depth h and Uc are calculated by the

nonlinear shallow water equation for the steady flow. At the point where the current UC is large enough, the two solutions that exist for the dispersion relation are the same. This is the blocking point. During the animation of the numerical results of the free surface elevation, the following phenomenon could be clearly seen: the wave front propagates along the wave flume before it reached the blocking point, after that, the reflection occurs and the front of the reflecting wave goes with the same direction of the current while the crests of the reflection wave still move upstream. The separation details of the incident and reflection waves are illustrated in figure 8, in which the solid line is the instance free surface elevation. With the Savitsky-Golay(48,48,8) filter, the incident and reflection wave components could be well identified in most region except the part close to the blocking point, where the wave numbers of the incident and reflection wave are very close.

138

Figure 7. Wave number distribution: H=l .Oe-4m, T=l.2s, Uc=-O.l6m/s.

x 10.' I

I 29 30 31 32 33 34

-5 I 28

J 29 30 31 32 33 34

-5 I 28

x (m)

Figure 8. Analysis of the wave elevation along the flume. The solid line in the top-panel is the computed wave surface elevation and the dash line is the separated incident wave; the bottom-panel shows the separated reflection wave.

5. Concluding Remarks

Boussinesq-type formulations with improved bottom slope terms are presented. With the mild slope terms in the inverse series expansion, the Boussinesq model could be applied to the water wave problems with uneven bottom. Although including the mild slope terms in the inverse series expansion

139

did not change the shoaling gradient, the performance of the reflection on the plane slope is improved significantly. Obviously, the mechanism of shoaling and reflection is different. The local velocity distribution affects the reflection more severe than shoaling. Although shoaling is one of the most important factors on the wave propagation from open sea to coastal line, reflection serves an important role when the bottom vanes rapidly. With the modification on the inverse series expansion, reflection over steep slope could be well predicted.

With the improved higher order Boussinesq model, the phenomena of wave-current interaction are simulated. A new fully coupled wave-current generator is developed to accomplish this purpose. All of these works provide an excellent basis for the further study of the wave current interaction and wave breaking.

Acknowledgments

HL is greatly indebted to Professor Theodore Y. Wu at Caltech. Professor Wu introduced him to the mathematical modeling of nonlinear water waves.

References

1.

2.

3. 4. 5.

6. 7. 8.

9. 10. K. D. Suh, C. Lee and W.S. Park, Coastal Engineering. 32,9 1 (1 997). 1 1. Q. Chen, P.A. Madsen, H.A. Schaffer and D.R. Basco, Coastal Engineering

12. Y. Agnon, P.A. Madsen and H.A. Schaffer, J. of Fluid Mechanics. 399,

P.A. Madsen, and H.A. Schaffer, Phil. Trans. Roy. Soc., London. A356, 3123 (1998). M.F. Gobbi, J.T. Kirby and G. Wei, J. of Fluid Mechanics. 405, 181

G. Hong, China Ocean Engineering. 11,243 (1997). Z. Zou, ACTA Oceanologica SINICA. 22(4), 41-50 (2000). Q. Chen, P.A. Madsen and D.R. Basco, J. of Waterway, Port, Coastal, and Ocean Engineering. 125(4), 176 (1999). M.K. Kristensen, Master’s thesis, Technical University of Denmark, 2000. P.A. Madsen, H.B. Bingham and H. Liu, J. FluidMech. 462, 1-30 (2002). T.Y. Wu, Advances in Applied Mechanics. Boston: Academic Press, 1

N. Booij, Coastal Engineering. 7, 191(1983).

(2000).

(2000).

, 33, 11 (1998).

319 (1999).

WAVES ON A LIQUID SHEET

S . P. LIN Mechanical and Aeronautical Engineering Department Clarkson University, Potsdam, N Y 13699-5725, USA

The flow in a liquid sheet with two free surfaces is unstable. The onset of instability in an inviscid liquid sheet in the absence of gravity and ambient gas is shown to result in two independent interfacial modes of wave motion. In the sinuous mode the two interfaces are displaced in the same direction, exactly in phase. In the other mode, called varicose mode, the two interfaces are displaced in opposite directions, exactly out of phase. It is shown that the sinuous wave can propagate only in the downstream direction if the Weber number We, is greater than one. The Weber number is defined as the ratio of the inertia force to the surface force. If W e < 1, the sinuous wave propagates in both upstream and downstream directions. Regardless of if W e 5 1 or W e < 1, it is shown that when a disturbance is introduced at a given location, it quickly disintegrates and forms two permanent wave forms which propagate along two characteristics without distortion. On the other hand the varicose waves are dispersive. It is shown how an impulsively introduced initial varicose hump disperses and decays. Two partial differential equations are obtained to describe the weakly nonlinear evolution of the sinuous and varicose waves. The qualitative properties of these equations reveal that the linearly independent modes of wave motion are nonlinearly unstable.

1. Governing Equations

Consider the capillary wave motion on the two free surfaces of an inviscid incompressible liquid sheet. The equations governing the motion and the mass conservation are respectively

pDV = -VP,

and

v .v=o, (2)

where p is the density, V , is the velocity vector field, V is the gradient operator, P is the pressure field, and D denotes substantial derivative, i.e.,

D = 8~ + V V. (3)

140

141

In Eq. (3) the first term on the right side denotes partial differentiation with time T . A flow with a constant velocity U and a constant pressure Po across the liquid sheet of uniform thickness 2H satisfies Eqs. (1) and (2) and the boundary conditions at the free surfaces defined by

Hk =Z-F&(X,Y,T)=O, (4)

where ( X , Y, 2) is the Cartesian coordinate and F is the distance measured perpendicularly from the centerline of the sheet to the free surface in the Z-direction at time T. The plus and minus subscripts in Eq. (4) as well as the subsequent equations refer respectively to the right and left free surfaces of a uniform sheet. The boundary conditions at the free surfaces are [c.f. w u (2001))

W = D F & , (5)

f S V . n & = P, (6)

where W is the Z-component of velocity, S is the surface tension and n is the unit normal vector pointing outward from the liquid at the free surfaces. Hence (Krishna and Lin, 1978)

n& = VH&/IVH&I.

The local free surface mean curvature is -3/2

V . n& = - [1+ HZ,x + q Y ] [H*,xx (1 + G , Y )

+H&,YY (1 + H&x) - 2H*,XYH*,XH*,Y] * (7)

The kinematic condition (5) states that the fluid particle must remain at the free surface. The dynamic boundary condition (6) states that the surface tension force must balance the pressure force at the massless free surface so that Newtons second law of motion applied to the free surface can be satisfied.

2. Sheet Waves

The uniform flow described in the previous section is unstable (Savart, 1983; Taylor, 1954). The uniform basic flow is irrotational, and so is the wave motion subsequent to the onset of instability. Since V x V = 0, there exists a velocity potential function @ related to V by

v = V@. (8)

142

Here we consider only twedimensional wave motion. It follows from (2) and (8) that @ satisfies the Laplace equation

@,xx + @,zz = 0. (9)

Integration of the Euler equation (1) with V given by (8) yields the Bernoulli equation

(10) 2 1

P = -@,t - - + . 2

In the basic flow = U , = 0, and thus the gauge pressure is

U2 P ---. 2 0 -

To bring out the relevant flow parameters, we non-dimensionalize the length, velocity, time, and pressure respectively with H , U , HIU, and pU2. The dimensionless Bernoulli's equation is then

(11) 1

' 2 p = - + t - - [+:z + q2 + W , z ] , where the lower case variables correspond to the upper case dimensional variables. In Eq. (ll), + is, the perturbation velocity potential, i.e.,

- = 2 + + , @ UH

and p is the perturbation pressure, the constant basic flow pressure po = -112 being subtracted out. The dimensionless boundary conditions corresponding to (5) and (6) at the free surface E = f l + f&(x, t ) are

where We is the Weber number defined by

PU2 We=-, S

and f is the dimensionless displacement from the unperturbed free surface. The subscript + or - again denotes the right or left free surface.

Substituting Eq. (11) into Eq. (14), we have

143

2.1. Linear Sheet Waves

Near the unperturbed free surface, the solution of the non-dimensional counter part of Eq. (9), i.e.,

4,m + 4,zz = 0, (17)

can be expanded in the Taylor series about z = f l for any given t and x as

1 4 ( f l +f*, 2, t ) = 4(fL 2, t)+4z(fl , 2, t)f& + ~ 4 r z ( f L Z , t)fZ +..., (18)

Substituting Eq. (18) into Eq. (13) and Eq. (16), and neglecting the nonlinear terms, we have

4z ( f l , x , t ) = ( a t+az ) f* ( f l , x , t ) , (19)

fWe-'f*,,, - (at + az) 4 (fl, 2, t ) = 0. (20)

The governing differential system Eqs. (17), (19) and (20) is linear. Hence, there are two linearly independent modes of interfacial wave motion; namely sinuous and varicose modes. The two free surfaces are displaced in the same direction and exactly in phase in sinuous mode. The two free surfaces are displaced exactly out of phase in the opposite direction, in the varicose mode. Hence 4 is anti-symmetric with respect to z, i.e. $(+z,z, t) = -4( -z, x, t ) for the sinuous mode, and symmetric for the varicose mode, i.e., $(+z,x,t) = 4(-z ,x , t ) .

2.2. Sinuous mode

Consider the sinuous mode first. If the sheet thickness is much smaller than the wavelength of the sinuous wave, the variation of the fluid velocity across the sheet thickness is negligibly small. We may put +,z =z (+l ,z , t ) = &(-l,x,t). Hence

4 ( f l + f*,x,t) =

144

Substituting Eq. (19) into (22), we have

*We-'f,, F (at + 8,)' f5 (*I, x, t ) = 0.

For sinuous waves f+ = f- = f . Thus the above equation can be simplified to

[& + (1 + We-'/2) a,] [at + (1 - We-'/2 > I 8, f = 0. (23)

Equation (23) states that the sinuous wave propagates without distortion along the characteristics

Hence if We > 1 there are two sinuous waves both propagating in the downstream direction; one faster than the other. If We < 1 the faster one propagates downstream, but the other one now propagates upstream. We have just extracted the properties of the sinuous waves without actually obtaining the explicit solution for 4.

It will be of interest to see how an arbitrary initial disturbance introduced at a given spatial location will sort itself out to propagate along the two distinct characteristics. Fig. 1 illustrates the sorting process. The results in this figure have been obtained (Lin, 2003) by use of a finite dif- ferention solution of Eq. (23).

2.3. Varicose mode

Consider next the varicose mode for which 4(+z ,x , t ) = +(-z ,x , t ) , +z(+z,x , t ) = -$z(-z,x, t ) , and f+ = -f- = f.

Hence Eqs. (19) and (20) can be rewritten as

+,z (Lx , t ) = (at + 8,) f(L x, t ) , (25)

We-lf,,, - (at + [4 + 4,& + +,zf,z]z=l (26)

Again, we wish to extract the properties of the varicose wave without obtaining the explicit even mode solution of 4. This can be achieved by differentiating Eq. (26) twice with respect to x and exploiting the symmetry of the solution. Differentiating Eq. (26) with respect to x, we have

We-lf,,,,, + (at + 8,) 4,zz (1, x , t ) = 0. (27)

It should be pointed out that the nonlinear term +,zzf,t + $,zf,z, from differentiation twice with respect to x of +( 1 + f, x, t ) is consistently neglected

145

t

Figure 1. Sinuous wave, W e = 0.2.

in (26). Moreover, c$zz has been replaced by -q5zzl since 4 must satisfy the Laplace equation. This latter function can be estimated by expanding & about z = 0 by use of Taylors series,

The first term of expansion, i.e., 4z(0, z, t ) vanishes since 4 is a symmetric function of z. Assume the sheet thickness is so thin that the magnitude of the z-component of the velocity increases linearly, i.e., 4zz(01zlt) = 4+zz(1 + f,z,t). It follows from Eq. (28) that

4 , z z (1 + fl z, t ) = (1 + f)-ld,z (1 + f, z, t ) *

We-lf,,,, + (at + az)'f = 0.

(29)

Substituting the linearized Eq. (29) into Eq. (27), and then applying Eq. (25), we have

(30)

The normal mode solution of Eq. (30) shows that the varicose wave is dispersive. The dispersion of the wave from a hump of disturbance introduced initially at a given spatial location is illustrated in Fig. 2.

Both sinuous and varicose waves have been observed experimentally on a planar liquid sheet by Taylor (1954). Lin and Roberts (1981), and de

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t

X

Figure 2. Varicous wave, W e = 1.002.

Luca and Costa (1997). Taylor (1959) demonstrated both theoretically and experimentally that the sinuous wave manifests itself as a Cardioid wave in an radially expanding liquid sheet. Lin and Jiang (2003) showed that the downstream propagating sinuous waves are the consequence of convective instability and that the upstream propagating waves are the consequence of absolute instability. They showed that the varicose wave is stable in the absence of surrounding air, but is convectively unstable in the presence of ambient air. Lin and Jiang also showed that the cardioid wave observed by Taylor must terminate at the edge of a radially expanding sheet where absolute instability leads the sheet to break. (C.f. Lin, 2003).

2.4. Nonlinear Sheet Waves

To investigate the nonlinear evolution of sinuous and varicose waves discussed in the previous section, we must retain the neglected nonlinear terms in (13) and (16).

For the sinuous wave, the solution of the Laplace equation (9) is anti- symmetric in z, and f+ = f- = f . The in phase sinuous wave motion is characterized by 4+ = -4- = 4, 4+,= = 4-,= = 4,= = 4+,z = $-+ = 4,z.

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Hence the kinematic boundary condition (13) can be written for the right and left free surface as

4 , z = (at + 8, + 4 7 ) f. (31)

The dynamic boundary condition (16) for the right and left free surfaces can be written respectively as

(32) 1

We-lf,,, - (at + 8,) 4 = 5 (4:, + 4:J 7

Although the kinematic boundary condition for both free surfaces remains the same as it should, the right sides of (32) and (33) have opposite signs. If the nonlinear term for the right side free surface acts as the source of energy, it will act as an energy sink for the left surface. Consequently the nonlinear evolution of the two free surfaces will follow different courses and destroy the in phase motion of the sinuous waves.

On the other hand 4+ = d-, 4+,, = &,,, c#J+,~ = -c$- ,~ , f+ = -f- in the symmetric varicose mode. Again the kinematic condition remains the same for both free surfaces, and is given by (31). Moreover the dynamic boundary condition for both free surfaces are given by the same equation.

(34) 1

We-lf,,, - (at + a,) 4 = 5 (4:z + d$) .

Hence contrary to the case of sinuous waves, the two free surfaces may evolve symmetrically. Whether the nonlinear evolution will allow the varicose wave to evanesce as predicted by the linear theory, or it will lead to stationary wave motion resulting from the balance between the surface tension force and the pressure force can only be determined from the solution of the Laplace equation with the boundary conditions (31) and (34).

3. Discussion

The linear theory shows that the long sinuous wave in the thin liquid sheet is non-dispersive and can follow two distinct characteristics without distortion. The emergence of characteristic lines from an initially single hump of disturbance is demonstrated. However, the nonlinear effect tends to destroy the in-phase motion of the two free surfaces in the sinuous mode. On the other hand the varicose waves are dispersive and evanescent according to the linear theory. The nonlinear effect tends to maintain the varicose

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structure. However, the precise nonlinear evolution of either mode has not been given.

The discussion given in the above paragraph is based on the assumption that the effect of the gas surrounding the liquid sheet is negligible. Jiang and Lin (2003) and de Luca and Costa (1997) show that the inertia of the ambient gas forces the neutral sinuous wave to become convectively unstable when We > 1. When We < 1, the sinuous waves become absolutely unstable. They grow and propagate simultaneously in the upstream and downstream directions. Convectively unstable waves can only grow in the downstream directions. It has also been demonstrated (Lin, 2003) that the inertia of the ambient gas forces the dispersive evanescent varicose wave to become convectively unstable, but cannot make it absolutely unstable for all Weber numbers.

Taylor (1959) showed with a linear theory that the upstream propagating sinuous wave in a uniform planar sheet in the absence of ambient gas can be arrested by the uniform stream. Two stationary wave fronts analogous to the Mach wave meet at a point where a disturbance is introduced at an upstream position. The half apex angle 8 between these two wave fronts is given by

e = sin-' ( ~ e - 1 / 2 )

Taylor verified his prediction with an experiment. Taylor also demonstrated how the dispersive varicose waves can be arrested to form a series of stationary waves. Lin and Robert (1981) found similar wave patterns for both modes in a viscous sheet. Taylor showed that the sinuous wave can be captured in a radially expanding liquid sheet to form a Cardioid wave. Lin and Jiang (2003) showed that the Cardioid wave cannot extend beyond a radius where the transition from convectively unstable waves to absolutely unstable ones commence. The observed finite amplitude wave patterns appear to agree with the predictions by the linear theory. The neglected nonlinear effects and viscosity seems to stabilize the linearly unstable waves. However, nonlinear theories are required to support this conjecture.

References 1. de Luca, L. and Costa, M. 1997. J. Fluid Mech. 331, 127. 2. Krishina, M.V.G. and Lin, S.P. 2005. Phys. Fluids 20. 3. Lin, S.P. and Jiang, W.Y. 2003. Phys. Fluids 15, 1745. 4. Lin, S.P. 2003. Breakup of Liquid Sheets and Jets. Cambridge U. Press, pp.

43-44.

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5. Savart, F. 1833. Ann. Chim. Phys. LIX, 55-87, 257-310. 6. Taylor, G.I. 1959. Proc. Roy. SOC. Lond. A 253, 296. 7. Wu, T.Y. 2001. Advances in Applied Mechanics edited by E. VanderGiessen

a d T.Y. WU, pp. 1-88.

A DIFFERENT VIEW ON DATA IN A NONLINEAR AND NONSTATIONARY WORLD

NORDEN E. HUANG NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA

The world we live in is neither stationary nor linear. Yet, the traditional view, based on established mathematical paradigm, is decisively linear and stationary. Such a linear view of the reality has impeded our understanding of the true physical processes. To break away from the inadequacy of the traditional approach, we have to adopt a totally new view with a new data analysis method, for data is the only connection we have with reality. The existing methods such as the probability theory and spectral analysis are all based on global properties of the data, and a priori defined basis and the stationary and linear assumptions. For example, spectral analysis is synonymous with the Fourier-based analysis. As Fourier spectra can only give a meaningful interpretation to linear and stationary processes, its application to data from nonlinear and nonstationary processes is problematical. To break away from this limitation, we should let the data speak for itself. We should develop adaptive data analysis techniques. The basics of the Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA) will be presented. This approach actually offers a different view of the nonlinear and nonstationary world.

1. Introduction

Traditional data analysis methods are all based on the linear and stationary assumptions. Only in recent years have new methods been introduced to analyze nonstationary and nonlinear data. For example, wavelet analysis and the Wagner-Ville distribution (Flandrin, 1995, Grochenig, 200 1) are designed for linear but nonstationary data. Meanwhile, various nonlinear time series analysis methods (see, for example, Tong, 1990; Kantz and Schreiber, 1997 and Diks, 1999) are designed for nonlinear but stationary and deterministic systems. Unfortunately, most real systems, either natural or even man-made ones, are most likely to be both nonlinear and nonstationary. To analyze data fiom such a system presents a daunting problem. Even the universally accepted mathematical paradigm of data expansion in terms of an a priori established basis would need to be eschewed, for the convolution computation of the a priori basis creates more problems than solutions. A necessary condition to represent nonlinear and nonstationary data is to have an adaptive basis. We cannot rely on an a priori defined fimction as a basis, no matter how sophistic the function can be. A few adaptive methods are available for signal analysis as

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summarized by Windrow and Steams (1985). The methods given in their book, however, are all designed for stationary processes. For nonstationary and nonlinear data, where adaptation is absolutely necessary, no available method can be found. How can we define such bases? What are the mathematical properties and problems of those basis functions? How should we approach the general topic of an adaptive method for data analysis? Being adaptive means a posteriori defined basis, an approach totally different from the established paradigm of mathematical analysis. Therefore, it presents a greater challenge to the mathematical community. We desperately need new methods to examine those data from the real world. A recently developed method, the Hilbert-Huang Transform (HHT, Huang et al. 1996, 1998, and 1999) seems to be able to meet some of the challenges.

HHT consists of two parts: Empirical Mode Decomposition (EMD) and Hilbert Spectral Analysis (HSA). It is a potentially viable method for nonlinear and nonstationary data analysis, especially for time-frequency-energy representations. It has been tested and validated exhaustively, however, but only empirically. In all the cases studied, HHT gives results much sharper than any of the traditional analysis methods in time-fiequency-energy representations. And it reveals true physical meanings in many of the data examined. Powerful as it is, the method is empirical. In order to make the method more robust and rigorous, many outstanding mathematical problems related to the HHT method need to be resolved. In this section, we will list some of the problems facing us now, hoping this list will call the attention of the mathematical community to this interesting and critical research area. Some of the problems are easy, and might be resolved in the next few years; others might last a long time. In view of the history of Fourier analysis, invented in 1807, with full proof established in 1935 (See, for example, Titchmarsh, 1948, Chapter 3), we should anticipate a long and arduous road ahead. Before discussing the mathematical problem, we will first outline the methodology of HHT in the next section.

2. The Hilbert-Huang Transform

For an arbitrary time series, X(0, we can always have its Hilbert Transform, Y(0, as

1 x ( t l ) & I , Y ( t ) = - P - ?r t - t'

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where P indicates the Cauchy principal value. This transform exists for all fimctions of class Lp (see, for example, Titchmarsh, 1948). With this defintion, X(t) and Y(t) form the complex conjugate pair, so we can have an analytic signal, m9, as

Z ( t ) = X ( t ) + i ~ ( t ) = a(t) eiSO ,

in which

a( t ) = [ X’(t) + Y’(t)] x ;

A description on the Hilbert transform with the emphasis on its many mathematical formalities can be found in Hahn (1996). Essentially, Equation (1) defines the Hilbert transform as the convolution of X(t) with I/t; therefore, it emphasizes the local properties of X(t). In Equation (2), the polar coordinate expression M e r clarifies the local nature of thls representation: it is the best local fit of an amplitude and phase varying trigonometric function to X(t). Even with the Hilbert Transform, there is still considerable controversy in defining the instantaneous frequency as

In fact, a sensible instantaneous frequency cannot be found through this method at all. A straightforward application as advocated by Hahn (1996) will only lead to the problem of having frequency values being as equally likely to be positive and negative for any given data set. The real advantage of the Hilbert transform only became obvious after Huang et a1 (1998) introduced what they called the Empirical Mode Decomposition method (EMD).

2.1. The Empirical Mode Decomposition Method: The Sifring Process

As discussed by Huang et al. (1996, 1998), the Empirical Mode Decomposition method is necessary to deal with both nonstationary and nonlinear data. Contrary to almost all the previous methods, this new method is intuitive, direct,

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a posteriori, and adaptive, with the basis of the decomposition based on and derived from the data. The decomposition is based on the simple assumption that any data is consisted of different simple intrinsic modes of oscillations. Each mode, which may or may not be linear, will have the same number of extrema and zero-crossings. Furthermore, the oscillation will also be symmetric with respect to the ‘local mean’. At any given time, the data may have many different coexisting modes of oscillation, each superimposed on the others. The result is the final complicated data. Each of these oscillatory modes is represented by an Intrinsic Mode Function (IMF) with the following definition: (a) in the whole data set, the number of extrema and the number of zero- crossings must either equal or differ at most by one, and (b) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

An IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but it is much more general. With this definition, one can decompose any function as follows: Take any data, such as that given in Figure la. Identify all the local extrema then connect all the local maxima by a cubic spline line as the upper envelope. Repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean, shown in Figure lb is designated as ml, and the difference between the data and ml is the first component, hl, i. e.,

X ( t ) - m, =.h, . ( 5 )

The procedure is illustrated in Huang et a1 (1998). Ideally, hl should be an IMF, for the construction of hl described above

should have made it satisfy all the requirements of an IMF. Yet, even if the fitting is perfect, a gentle hump on a slope can be amplified by the procedure to become a local extremum in changing the local zero from a rectangular to a curvilinear coordinate system. After the first round of sifting, the hump may become a local maximum. New extrema generated in this way actually recover the proper modes lost in the initial examination. In fact, the sifting process can recover signals representing low amplitude riding waves with repeated siftings.

The sifting process thus serves two purposes: to eliminate riding waves, and to make the wave profiles more symmetric. While the first condition is absolutely necessary for separating the intrinsic modes and for defining a meaningful instantaneous frequency, the second condition is also necessary in case the neighboring wave amplitudes have too large a disparity. Toward these

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ends, the sifting process has to be repeated as many times as is required to reduce the extracted signal an IMF. In the subsequent sifting process, hl is treated as the data, as shown in Figure lc, then

h, - m,, = h,, . (6)

After repeated sifting up to k times, h l k becomes an IMF, that is

only then is it designated as

the fust IMF component from the data, shown in Figure Id. Here we have a critical decision to make: the stoppage criterion. Historically, two different criteria have been used: The first one was used in Huang et al. (1998). The stoppage criterion is determined by a Cauchy convergence type of test. Specifically, the difference between two successive sifting operations is squared and normalized as

r=o If this squared difference, SDI, is smaller than a predetermined value, the sifting process is stopped. This definition seems to be rigorous, but it is very difficult to implement. Two critical questions need to be resolved First, the question on how small is small enough must be answered. Second, this definition does not depend on the definition of the IMFs. The squared difference might be small, but there is no guarantee that the h c t i o n will have the same numbers of zero- crossings and extrema, for example. These shortcomings prompted Huang et al. (1999 and 2003) to propose a second criterion based on the agreement of the numbers of zero-crossings and extrema. Specifically, a number, S, is pre- selected. The sifting process will stop only if after S consecutive times, the numbers of zero-crossings and extrema stay the same, and are equal or at most differ by one. This second choice is not without difficulty either: how to select

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the S number. Obviously, any selection is ad hoc, and a rigorous justification is needed.

Let us assume that a stoppage criterion is selected, and we can get to the first IMF, cl. Overall, cI should contain the finest scale or the shortest period component of the signal. We can, then, separate cI from the rest of the data by

Since the residue, rJ , shown in Figure ley still contains longer period components, it is treated as the new data and subjected to the same sifting process as described above. This procedure can be repeated to all the subsequent rj’s, and the result is

r, - c, = r, , ...

The sifting process can then be stopped finally by any of the following predetermined criteria: either when the component, c, , or the residue, r,, , becomes so small that it is less than the predetermined value of substantial consequence, or when the residue, r,, becomes a monotonic h c t i o n from which no more IMF can be extracted. Even for data with zero mean, the final residue still can be different from zero. If the data have a trend, the final residue should be that trend. By summing up Equations (10) and (1 l), we finally obtain

j=l

Thus, we achieve a decomposition of the data into n-empirical modes, and a residue, r,, , which can be either the mean trend or a constant. As discussed here, to apply the EMD method, a mean or zero reference is not required; EMD only needs the locations of the local extrema. The zero reference for each component will be generated by the sifting process. Without the need of the zero reference, EMD avoids the troublesome step of removing the mean values for the large DC term in data with non-zero mean, an unexpected benefit. The components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data.

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2.2. The Hilbert Spectral Analysis

Having obtained the Intrinsic Mode Function components, one will have no difficulties in applying the Hilbert transform to each IMF component, and compute the instantaneous frequency according to Eq. (2.4). After performing the Hilbert transform to each IMF component, the original data can be expressed as the real part, RP, in the following form:

j=l

Here we have left out the residue, r,, , on purpose, for it is either a monotonic function, or a constant. Although the Hilbert transform can treat the monotonic trend as part of a longer oscillation, the energy involved in the residual trend representing a mean offsetting could be overpowering. In consideration of the uncertainty of the longer trend, and in the interest of obtaining information contained in the other low energy but clearly oscillatory components, the final non-IMF component should be left out. It, however, could be included, if physical considerations justify its inclusion.

Equation (13) gives both amplitude and frequency of each component as functions of time. The same data if expanded in a Fourier representation would be

j=l

with both aj and q as constants. The contrast between Equations (13) and (14) is clear: The IMF represents a generalized Fourier expansion. The variable amplitude and the instantaneous frequency have not only greatly improved the efficiency of the expansion, but also enabled the expansion to accommodate nonlinear and nonstationary data. An example is given in Figure 2, to compare the results from Hilbert Spectral analysis with the traditional Fourier and more recent Wavelet analysis. With the IMF expansion, the amplitude and the frequency modulations are also clearly separated. Thus, we have broken through the restriction of the constant amplitude and fixed frequency Fourier expansion, and. arrived at a variable amplitude and frequency representation. This frequency-time distribution of the amplitude is designated as the Hilbert Amplitude Spectrum, H(4 4, or simply Hilbert Spectrum. If amplitude squared is more preferred to represent energy density, then the squared values of

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amplitude can be substituted to produce the Hilbert Energy Spectrum just as well.

The skeleton Hilbert Spectrum presentation is more desirable, for it gives more quantitative results. Actually, Bacry et a1 (1991) and Carmona et a1 (1998) have tried to extract the Wavelet skeleton as the local maximum of the wavelet coefficient. Even that approach is still encumbered by the harmonics. If more qualitative results are desired, a somewhat “fiuzy” or smeared view can also be derived from the skeleton presentation by using two-dimensional smoothing.

With the Hilbert Spectrum defined, we can also define the marginal spectrum, h(w), as

T

h(w) = jH(w,t)dt. (15) 0

The marginal spectrum offers a measure of total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the entire data span in a probabilistic sense.

The combination of the Empirical Mode Decomposition and the Hilbert Spectral Analysis is also known as the Hilbert-Huang Transform (HHT) for short. Empirically, all tests indicate that HHT is a superior tool for time- frequency analysis of nonlinear and nonstationary data. It is based on an adaptive basis, and the frequency is defined through a Hilbert transform. Consequently, there is no need for the spurious harmonics to represent nonlinear waveform deformations as in any of the a priori basis methods, and there is no uncertainty principle limitation on time or frequency resolution from the convolution pairs based also on a priori bases. A summary of comparison between Fourier, Wavelet and HHT analyses is given in Table 1. From thls table, we can see that HHT is indeed a powerful method for the analysis of data from nonlinear and nonstationary processes:

it is based on an adaptive basis the frequency is derived by differentiation rather than convolution therefore, it is unlimited by the uncertainty principle it is applicable to nonlinear and nonstationary data it presents the results in time-frequency-energy space for feature extraction

Table 1 : Comparisons of data analysis methods.

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Basis Frequency

Presentation

Nonlinear Non-stationary Feature extraction

Theoretical base

~~

Fourier Wavelet a priori a priori Convolution: Convolution: Global, Regional, Uncertainty Uncertainty Energy-frequency Energy-time-

No NO frequency

Discrete: No Continuous: Yes

Hilbert Adaptive Differentiation: Local, Certainty

Energy-time- frquency Yes Yes Yes

Empirical

3. Mathematical Problems related to HHT

Over the past few years, HHT has gained some following and recognition. Unfortunately, the full theoretical base has not been fully established. Up to this time, most of the progresses in HHT are in the application areas, while the underlying mathematical problems are mostly left untreated. All these results are case-by-case comparisons conducted empirically. We are approximately at the stage where wavelet analysis was historically in the early 1980s: producing great results but waiting for a mathematical foundation to rest our case. We are waiting for some one like Daubechies (1992) to lay the mathematical foundation for HHT. The outstanding mathematical problems, as we see them now, are listed as follows:

1 .) Adaptive data analysis methodology in general 2.) Nonlinear system identification methods 3 .) Prediction problem for nonstationary processes (end effects) 4.) Spline problem (best spline implement of HHT, convergence and 2-D) 5.) Optimization problem (the best IMF selection and uniqueness) 6.) Approximation problem (Hilbert transform and quadrature) 7.) Miscellaneous questions concerning the HHT

3.1. Adaptive data analysis methodology

Most data analysis methods are not adaptive. The established approach is to define a basis, such as using trigonometric functions in Fourier analysis, for example. Once the basis is determined, the analysis is reduced to a convolution computation. This well-established paradigm is specious, for there is no a priori

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reason to believe that the basis selected truly represents the underlying processes. Therefore, the results produced will not be informative. It, however, provides a definitive quantification with respect to a known metric for certain properties of the data based on the basis selected.

If one gives up this paradigm, there is no solid foundation to tread on. Yet data analysis methods need to be adaptive, for the goal of data analysis is to find out the underlying processes. Only adaptive methods can let the data reveal their underlying processes without any undue influence from the basis. Unfortunately, there is no mathematical model for such an approach. Recently, adaptive data processing has gained some attention. Some adaptive methods are being developed (Windrow, B and S. D. Steam 1985). Unfortunately, most of the methods available depend on feedback; therefore, they are limited to stationary processes. To generalize these available methods to nonstationary data is not an easy task.

3.2. Nonlinear system identification

System identification methods are usually based on being given both input and output data. For an ideally controlled system, such data sets are possible. Yet for most of the cases studied, natural or man-made, no such luxury on data is available. All we might have is a set of measured results. The question is whether it is possible to identify the nonlinear characteristics from the data? This might be an ill-posed problem, for this is very different from the traditional input vs. output comparison. Whether it is possible or not to identify the system through data only is an open question. Unfortunately, in most of the natural systems, we not only do not have control of the input, but also do not know what is the input or what is the system. The only data we have are the output from an unhown system. Can the system be identified? Or short of that, can we learn anythmg about the system? The only thing working to our advantage is that we might have some general knowledge of the underlying controlling processes. For example, the atmosphere and ocean are all controlled by the generalized equations for fluid dynamics and thermodynamics, which are nonlinear. The man made structures, though linear under design condition, will approach nonlinear under extreme loading conditions. Such a priori knowledge could guide our search for the characteristics of the signatures of nonlinearity. The task, however, is still daunting.

So far, most of the definitions or tests for nonlinearity from any data are only necessary conditions: for example, various probability distributions, higher order spectral analysis, harmonics analysis, instantaneous frequency, etc. (see,

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for example, Bendat, 1990, Priestly, 1988, Tong, 1990, Kantz and Schreiber, 1997). There are certain difficulties in making such identifications from observed data only. This difficulty has made some scientists talk about only nonlinear systems rather than nonlinear data. Such a reservation is understandable, but this choice of terms still does not resolve the basic problem: How to identify the system nonlinearity from its output alone. Is that possible? Or, is there a sure way to define a nonlinear system from the data (system output) at all? This problem is made even more difficult when the process is also stochastic and nonstationary. With a nonstationary process, the various probabilities and the Fourier-based spectral analyses are problematic, for those methods are based on global properties as well as linear and stationary assumptions.

Through our study of instantaneous frequency, we have proposed intra-wave frequency modulation as an indicator for nonlinearity. More recently, Huang (2003) has also identified the Teager Energy Operator (Kaiser, 1990) as an extremely local and sharp test for harmonic distortions within any IMF derived from data. The combination of these local methods offers some hope for system identification, but the problem is not solved, for this approach is based on the assumption that the input is linear. Furthermore, all these local methods also depend on local harmonic distortion; they cannot distinguish a quasi-linear system from a truly nonlinear system. A test or definition for nonlinear system identification based on only observed output is urgently needed.

3.3. Prediction problem for nonstationary processes, the end effects of EMD

End effects have plagued data analysis from the beginning of any known method. The accepted and timid way to deal with it is by using various kinds of windows as in Fourier analysis. Sound in theory, such practice inevitably sacrifices some precious data near the ends. Furthermore, the use of windows becomes a serious hindrance when the data is short. In HHT the extension of data beyond the existing range is necessary, for we use spline through the extrema to determine the IMF. Therefore, we need a method to determine the spline curve between the last available extremum and the end of the data range. Instead of windowing, Huang et a1 (1998) introduced the idea of using a ‘window frame’ as a way to extend the data beyond the existing range, in order to extract some information from all the data available.

The extension of data, or data prediction, is a risky procedure even for linear and stationary processes. The problem we are facing is how to make predictions

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for nonlinear and nonstationary stochastic processes. Here we have to abandon the cozy shelter of the linear, stationary, low dimension and deterministic assumptions and face the complicated real world. The data are mostly from high dimensional nonlinear and nonstationary stochastic systems. Are these systems predictable? What conditions do we have to impose on the problem to make it predictable? How well can we quantify the goodness of the predictions? In principle, data prediction cannot be made based on past data alone. The underlying processes have to be involved. Can we use the available data to extract enough information to make a prediction? l h s is an open question.

Again, there is an advantage in our favor: we do not need to make a prediction for the whole data, but only to make a prediction for the IMF, whch has a much narrower bandwidth, for all the IMF should have the same number of extrema and zero-crossings. Furthermore, all we need is the value and location of the next extrema, not all the data. Such a limited goal notwithstanding, the task is still challenging.

3.4. The Spline problem (best spline implement of HHT, convergence and 2-0)

EMD is a ‘Reynolds type’ decomposition: to extract variations from the data by separating the mean, in this case the local mean, from the fluctuations using a spline fit. Although this approach is totally adaptive, several unresolved problems arise from this approach.

First, among all the spline methods, which one is the best? This is critical for it can be shown easily that all the IMFs other than the first are a summation of spline functions, for from equations (5) to (8), we have

in which all m functions are generated by splines. Therefore, from equation (1 0)

5 = X ( t ) -cl = (mlk +ml(k-l) +...+ m,, +m,) , (17)

is totally determined by splines. Consequently, according to equation (1 l), all the rest of the IMFs are also totally determined by spline fimctions. What kind of spline is the best fit for the EMD? How can one quantify the selection of one spline vs. another? Based on our experience, we found the higher order spline functions need additional subjectively determined parameters, which violates the adaptive spirit of the approach. Furthermore, hgher order spline functions could

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also introduce additional length scales, and they are also more time consuming in computation. Such shortcomings are the reason that only the cubic spline was selected. But the possible advantages and disadvantages of higher order splines and even taut splines have not been definitively established and quantified.

Finally, there is also the critical question of convergence of the EMD: is there a guarantee that in finite steps, a function can always be reduced into a finite number of IMFs? All intuitive reasoning and our experience suggest that the procedure is converging. Under rather restrictive assumptions, we can even prove the convergence rigorously. The restricted and simplified case studied was sifting with middle-points only. We further restrict the middle-point sifting to linearly connected extrema, then the convergence proof can be established by reductio ad absurdurn, that the number of extrema of the residue function has to be less than or equal to that in the original function. The case of equality only exists when the oscillation amplitudes in the data are either monotonically increasing or decreasing. In this case, the sifting may never converge and forever have the same number as in the original data. The proof is not complete in another aspect: can one prove the convergence, once the linear connection is replaced by the cubic spline? Therefore, this approach to a proof is not complete.

Recently, Chen et a1 (2004) have used the B-spline to implement the sifting. If one uses the B-spline as the base for sifting, then one can invoke the Variation Diminishing Property of B-splines and show that the spline curve will have less extrema. The details of this proof still have to be established.

3.5. The Optimization Problem (the best IMF selection and uniqueness Mode mixing)

Does EMD generate a unique set of IMFs, or is EMD a tool to generate rnftnite sets of IMFs? From a theoretical point of view, there are infinitely many ways to decompose a given data set. Our experience indicates that the EMD can generate many different IMF sets through varying the adjustable parameters in the sifting procedure. How are these different sets of IMF related? What is a criterion or are there criteria to guide the sifting? What is the statistical distribution and significance of the different IMF sets? Therefore, a critical question is: How to optimize the sifting procedure to produce the best IMF set. The difficulty is in not sifting too many times to drain all the physical meaning out of each IMF component, while at the same time not sifting too few times and thereby failing to get clean IMFs. Recently, Huang et a1 (2003) has studied the

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problem of different sifting parameters and established a confidence limit for the resulting IMFs and Hilbert Spectrum. But the study was empirical and limited to cubic spline only. Optimization of the sifting process is still an open question.

This question of uniqueness of the IMF can be traced to the more fundamental one: how to define the IMF more rigorously? The definition given by Huang et a1 (1998, 1999) is hard to quantify. Fortunately, the results are quite forgiving: even with the somewhat fuzzy definition, the results produced are similar enough. Is it possible to give a rigorous mathematical definition and also find an algorithm that can be implemented automatically?

Finally, there is the problem of IMF Mode rectifications. Straightforward implement of the sifting procedure will produce mode mixing (Huang et a1 1999,2003), which will introduce aliasing in the IMFs. This mode mixing can be avoided if an “intermittence” test is invoked (see Huang et a1 2003). At this time, one can only implement the intermittence test through interactive steps. An automatic mode rectification program should be able to collect all the relevant segments together and avoid the unnecessary aliasing in the mode mixing. This step is not critical to the HHT, but it would be a highly desirable feature of the method.

3.6. Approximation problem (Hilbert transform and quadrature)

One of the conceptual breakthroughs in HHT is to define the instantaneous fiequency through Hilbert Transform. Traditionally, the Hilbert Transform has been considered as unusable by two well-known theorems: the Bedrosian theorem (Bedrosian, 1963), and the Nuttall theorem (Nuttall, 1966). The Bedrosian theorem states that the Hilbert transform for the product functions can only be expressed in terms of the product of the low frequency function with the Hilbert Transform of the high frequency one, if the spectra of the two functions are disjointed. This guarantees that the Hilbert transform of a(t) cos8(t) is given by a(t) sinqt). The Nuttall theorem (Nuttall, 1966), M e r stipulates that the Hilbert transform of cos8(t) is not necessarily sin6yt) for an arbitrary function e(t). In other words, there is a discrepancy between the Hilbert transform and the perfect quadrature of an arbitrary function e(t). Unfortunately, the error bound given by Nuttall (1966) is expressed in terms of the integral of the spectrum of the quadrature, an unknown quantity. Therefore, the single-valued error bound cannot be evaluated.

Through our research, we have overcome the restriction of the Bedrosian theorem through the EMD and the normalization of the resulting IMFs (Huang, 2003). With this new approach, we have also improved the error bound given by

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Nuttall by expressing the error bound as a function of time in terms of instantaneous energy. These are major breakthroughs for the Hilbert Transform and its applications. We have to quantie the influence of the normalization procedure. As the normalization procedure depends on a nonlinear amplification of the data, what is the influence of h s amplification on the final results? Even if we accept the normalization, for an arbitrary e(t) function, the instantaneous frequency is only an approximation. How can we improve this approximation?

Related to the normalization scheme, there are other questions concerning the Hilbert Transform: For example, what is the functional form of e(t) for the Hilbert Transform to be the perfect quadrature and also be analytic? If the quadrature is not identical to the Hilbert Transform, what is the error bound in the phase function (not in terms of energy as we have achieved now)?

One possible alternative is to abandon the Hilbert Transform, and to compute the phase function using the arc-cosine of the normalized data. Two complications arise, however, from this approach The first one is the high precision needed for computing the phase function when its value is near nd2. The second one is that the normalization scheme is only an approximation; therefore, the normalized functional value can occasionally exceed unity. Either way, some approximations are needed.

3.7. Miscellaneous statistical questions concerning HHT

The first question concerns the confidence limit of the HHT results. Traditionally, all spectral analysis results are bracketed by a confidence limit, which gives a measure of comfort to us, either truly or fallaciously. The traditional confidence limit is established from the ergodicity assumption; therefore, the processes are necessarily linear and stationary. If we give up the ergodic assumptions, can there still be a confidence limit without resorting to truly ensemble averaging, which is practically impossible for most of the natural phenomena? The answer seems to be affirmative for Fourier analysis. For HHT, however, we have tentatively established a confidence limit based on the exploitation of repeated applications of EMD with various adjustable parameters, whch thus produce an ensemble of IMF sets. How representative are these different IMFs? How can the definition be made more rigorous? Additionally, how the statistical measure for such a confidence limit be quantified? A recent result by Wu and Huang (2004) has addressed this very question through the study of white noise. They have established a method to assign the statistical significance of the IMFs and to certify whether the Ih4Fs are in the noise range or are truly representing a statistically significant signal. The

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second question concerns the degree of nonstationary. This is another conceptual breakthrough, for we change the qualitative definition of stationary to a quantitative definition of the degree of nonstationary. In Huang et a1 (1998), in addition to a degree of nonstationary, a degree of statistical nonstationarity was also given. For the degree of statistical nonstationarity, an averaging procedure is required. What is the time scale needed for the averaging?

4. Conclusion

These are some of the problems we have encountered in our research. Even without these issues settled, HHT is still a very useful tool. With these questions settled, however, the HHT approach will become much more rigorous, and the tool more robust. We are using HHT routinely now, as Heaviside famously said, when he encountered objections from the purists on h s step function: ‘I do not have to wait till I hlly understand all the biochemistry of digestion before I enjoy my dinner.’ Nevertheless, to understand all the ‘biochemistry of digestion for HHT’ is necessary for our “inquisitive and scientific health”. It is the steps that should be taken now.

Furthermore, the need for a unified framework for nonlinear and nonstationary data analysis is urgent and real. Right now, the field is fragmented between partisans belonging to one camp or the other. For example, researchers engaged in wavelet analysis will not mention Wagner-Ville distribution methods, as if it does not exist (see, for example, any wavelet book). On the other hand, researchers engaged in Wagner-Ville distribution methods will not mention wavelet analysis (see, for example, Cohen, 1995). Such an extreme position is unscientific, and unhealthy to the data analysis community. Now is the right time, and long overdue, for some support to unify the field and push forward. We should mount a concerted effort to attack the problem of nonlinear and nonstationary time series analysis. One of the suggestions is to organize an activity group within SIAM to address all the mathematical and application problems, and all the scientific issues related to nonlinear and nonstationary data analysis. This is underway now, and we eagerly await future progress.

Acknowledgements

This research is supported in part by a NASA RTOP grant from the Oceanic Processes Program, and in part by a grant from the Process and Prediction Division, Office of Naval Research, N00014-98-0412. Additional support by DARPA grant # 03-QSSS-00 is also gratefully acknowledged.

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References

1. Bacry E, AmCodo A, Frisch U, G a p e Y, Hopfinger E. 1991. Wavelet analysis of hllydeveloped turbulence data and measurement of scaling exponents. In Proc. Turbulence89: Organized Structuresand Turbulence in Fluid Mechanics, Grenoble, Sept. 1989, ed. M. Lesieur, 0. MCtais, pp. 203- 2 15, Dordrecht: Kluwer

2. Bedrosian, E., 1963: On the quadrature approximation to the Hilbert transform of modulated signals. Proc. IEEE, 5 1, 868-869.

3. Bendat, J. S., 1990: Nonlinear system analysis and identification fiom random data, Wiley Interscience, New York, NY, 267 PP.

4. Carmona, R., W. L. Hwang and B. TorrCsani, 1998: Practical Time- Frequency Analysis : Gabor and Wavelet Transforms with an Implementation in S. Academic Press, San Diego, CA.

5. Chen, Q., N. E. Huang, S. Riemenschneider and Y. Xu, 2004: A B-Spline Approach for Empirical Mode Decompositions. Adv. Math. Computations. (in Press).

6. Cohen, L., 1995: Time-fiequency Analysis, Prentice Hall, Englewood Cliffs, NJ

7. Diks, C., 1999: Nonlinear Time Series Analysis, World Scientific Press, Singapore.

8. Daubechies, I., 1992: Ten Lectures on Wavelets, Philadelphia SIAM. 9. Flandrin, P., 1995: Time-Frequency Time-Scale Analysis, Academic Press,

San Diego, CA. 10. Grochenig, K., 2001: Foundations of Time-Frequency Analysis.

Birkhauser, Moston, MA. 11. Hahn, S . L. 1996: Hilbert Transform in Signal Processing. Artech House,

Boston, MA. 12. Huang, N. E., S. R. Long, and Z. Shen, 1996: Frequency Downshift in

Nonlinear Water Wave Evolution, Advances in Appl. Mech., 32, 59-1 17. 13. Huang, et al. 1998: The empirical mode decomposition and the Hilbert

spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. SOC. Lond., 454, 903-993.

14. Huang, N. E., Z. Shen, R. S. Long, 1999: A New View of Nonlinear Water Waves - The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31,417-457.

15. Huang, N. E., M. L. Wu, S. R. Long, S. S. Shen, W. D. Qu, p. Gloersen, and K. L. Fan, 2003: A confidence limit for the Empirical Mode Decomposition and Hilbert Spectral Analysis, Proceedings Royal Society of London, A459, 2,317-2,345.

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16. Kaiser, J. F., 1990: On Teager’s energy algorithm and its generalization to continuous signals. Proc. 4h IEEE Signal Processing Workshop, Mohonk, NY.

17. Kantz, H. and T. Schreiber, 1997: Nonlinear Time Series Analysis, Cambridge University Press, Cambridge.

18. Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signals, Proceedings of IEEE, 54, 1458-1459.

19. Priestley, M. B., 1988, Nonlinear and nonstationary time series analysis, Academic Press, London, 237 pp.

20. Titchmarsh EC. 1948. Introduction to the Theory of Fourier Integrals, Oxford University Press, Oxford

21. Tong, H., 1990: Nonlinear Time Series Analysis, Oxford University Press, Oxford.

22. Windrow, B and S. D. Stearns, 1985: Adaptive Signal Processing, Prentice Hall, Upper Saddle River, NJ

23. Wu, Z. and Huang, N. E. 2004 A study of the characteristics of white noise using the empirical mode decomposition method, Proc. Roy, SOC. London, (in press).

Envelopes and the Mean data 10

8

6

4

2

-2

-4

-8

-8

-10 203 250 300 350 400 4% 500 550 600

Time : second

Figure la: Data (blue line) and the envelopes through the local extrema (green lines) and the local mean (red line).

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Data and hl

200 250 300 3M 400 450 500 550 600 Time : second

Figure 1 b: The first difference between the data and the local mean. Notice the numbers of zero-crossings are not the same as the local extrema.

Envelopes and the Mean : hi

-'SO0 250 300 350 400 450 500 550 600 nmc : second

Figure Ic: The same as in Figure la, but with h l as the data to repeat the sifting process.

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IMF h12=cl

-‘%U 250 300 350 400 450 500 550 600 Time second

Figure Id: The first Intrinsic Mode Function Component extracted from the data.

Data and residue r l 10

6

4

2

-2

-4

-6

i -101 I

200 250 300 350 400 450 500 550 600 Time ; second

Figure le: The residue (red line) after extracting the first IMF component, which bisects the data through all the local extrema. This residue is to be used as the data for the next round of sifting process.

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Comparison among Fourier, Hilbert, and Mode! Wavelet Spectra

-2 ' 1 0 200 400 600 800 1000

Hilbert

Fourier

1000

Wavelet

1 0 0 0 ~ 1000;

Figure 2: Comparisons between the Hilbert Spectrum with the Fourier and Morlet Wavelet spectra.

CHAPTER 2

NONLINEAR WAVES: EXPERIMENTS AND COMPUTATIONS


SOLITARY- WAVE COLLISIONS

JOSEPH HAMMACK Department of Mathematics, Penn State University

University Park PA I6802 USA

DIANE HENDERSON Department of Mathematics, Penn State University

University Park PA 16802 USA

PHILIPPE GUYENNE Department of Mathematics & Statistics, McMaster University

Hamilton, Ontario L8S 4Kl CANADA

MING YI State College High School

State College PA 16801 USA

Dedication

This paper is dedicated to Professor Theodore Yao-Tsu Wu, a gentleman, a scholar, and my teacher. His lectures on “Hydrodynamics of Free Surface Flows” provided inspiration for my fascination with, and study of, water waves. His research provided the standard of rigor and precision to which I strive. I am honored and grateful that our lives intersected.

Joe Hammack

Experimental and theoretical results are presented for binary collisions between co- propagating and counter-propagating solitary waves. The experiments provide high- resolution measurements of water surface profiles at fixed times, thereby enabling direct comparisons with predictions by a variety of mathematical models. These models include the 2-soliton solution of the Korteweg-deVries equation, numerical solutions of the Euler equations, and linear superposition of KdV solitons.

1. Introduction

The study of solitary-wave collisions has an old and venerable hstory that dates from the seminal experiments reported by John Scott Russell in 1845. His discovery of the solitary wave precipitated many mathematical investigations that provided a theoretical foundation and physical understanding for many of its interesting properties. In particular, Korteweg & deVries (1895) derived their

173

174

now famous equation for water waves propagating in one direction on shallow water. Moreover, they found an exact solution of the KdV equation for a single wave that is localized in space and propagates without change of form-the solitary wave.

Both the deeper mathematical and physical significance of the solitary wave was not realized until the subsequent development of “soliton theories” that was initiated by Gardner, Greene, Kruskal & Miura (1967). GGKM demonstrated a method to solve the KdV equation exactly on the real line for a wide class of localized initial data. Their results showed that initial data evolve into a finite number of co-propagating solitary waves, rank ordered by their amplitude (largest first), and a trailing train of dispersively decaying waves. Each of these solitary waves is referred to as a “soliton” based on the previous work of Zabusky & Kruskal (1965) who coined the name for particle-like waves that collide “elastically”, i.e., they emerge from a collision with no change in form. This collision property of solitary waves was made explicit by the exact, N- soliton solutions of the KdV equation found by Hirota (1971), who showed that the only lasting evidence of a co-propagating (following) collision is a phase shift in space. Weidman & Maxworthy (1978) provided much experimental collaboration of predictions based on Hirota’s exact solution for two solitons, e.g., phase shifts. In particular, they used photography to obtain spatial data at fixed times. These photographs provided qualitative results (only) for spatial profiles in consequence of the disparate vertical and horizontal wave scales of solitary waves.

Studies of binary collisions of counter-propagating solitary waves appear to have been initiated analytically by Mayer (1962). Byatt-Smith (1971) derived an explicit, approximate prediction for the maximum runup amplitude for the head- on collision of two equal-amplitude solitary waves. This special case is often used to model the reflection of a solitary wave by a vertical wall, and much of the literature concerns this special collision case. Cooker, Weidman & Bale (1 997) provide an excellent literature review as well as new numerical results for the special case of solitary-wave reflection by a vertical wall. (This special case is not the focus of the present study, and will not be reviewed here.) Maxworthy (1976) presented cinematic-based measurements of phase shifts and maximum runup amplitudes for two counter-propagating solitary waves. Maxworthy did not present detailed spatial profiles during the interaction, and the presented data showed considerable scatter. We note that, like the photographs of Weidman & Maxworthy (1978), the cinematic-based measurements did not resolve vertical wave structure with high resolution. Su & Mirie (1980) and Mirie & Su (1982) present approximate, analytical and numerical studies for the head-on collision of two solitary waves. They found that the collision was not elastic, i.e., in addition to small phase shifts a small amount of energy was lost by each of the waves to form secondary waves. This reduction in amplitudes leads to a reduction in wave speeds; hence, the phase shifts become spatially dependent.

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Byatt-Smith (1989) obtained higher-order, approximate results for the head-on collision of solitary waves with unequal amplitudes, and confi ied the results of Mine & Su (1982). Yih & Wu (1995) and Wu (1998) present analytical studies for head-on and following collisions of solitary waves of unequal amplitudes. In particular, Wu (1998) shows that there is an instant during both following and head-on collisions in which the spatial wave profile exhibits fore-and-aft symmetry.

Herein we investigate both the co-propagating (following) and counter- propagating (head-on) collisions of two solitary waves. Precise experimental data of spatial wave profiles at fixed times are presented and compared with the predictions of several mathematical models. For the head-on collision we use linear superposition of two KdV solitary waves and numerical solutions of the Euler equations. For the following collision we use the 2-soliton solution of the KdV equation and numerical solutions of the Euler equation. All of these mathematical models neglect viscous effects that are intrinsic in the experimental data. Obtaining hgh-resolution spatial measurements of experimental waves evolving in time and space is exceedingly dificult; hence, the emphasis of the discussions herein is on the experimental aspects of the study. The mathematical models are discussed briefly.

2. Experimental Program

In order to obtain quantitative experimental data for spatial wave profiles at fixed times that are needed for definitive comparisons with the mathematical models, it was necessary to develop special experimental facilities and procedures. The key idea is to use the most sophisticated electronic and mechanical systerix available and to develop experimental procedures that enable us to repeat the same experiment over and over as precisely as possible. Indeed, the use of repeatable experiments to obtain spatial data at fixed times was the basis of Russell’s pioneering work on the solitary wave. In order to understand both the strengths and limitations of the data that we obtained, it is necessary to provide a detailed accounting of this experimental program.

2.1 Wave Channel

Experiments were conducted at the W. G Pritchard Fluid Mechanics Laboratory in a horizontal wave channel that was 13.165m long, 25.4cm wide, and 30.0cm deep. Channel walls and bottom were made of glass that was precisely aligned. Stadess steel rails spanned the channel along the top of the two sidewalls. These rails supported an instrumentation carriage whose motion along the channel was provided by a linear belt drive and motor. The carriage supported four wave gages, spaced 40cm apart, and their associated electronics. A 10m long section of the channel was used for the experiments. This section

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was bounded by a vertical glass wall at its downstream end (x = lOm) and by a Teflon wave-maker piston at its upstream end (x = 0). A pressure gage was mounted at x = 7.155m in the center of, and flush with, the channel bottom.

2.2 Wave-maker

Solitary waves were generated by the horizontal, piston-like motion of a paddle made from a Teflon plate (0.5 inch thick) inserted in the channel cross- section. The paddle was machined to fit the channel precisely with a thin lip around its periphery that served as a wiper with the channel’s glass perimeter. This wiper prevented any measurable leakage around the paddle during an experiment. Paddle motion was driven directly by a state-of-the-art linear motor and integral carriage with up to 55cm of stroke and a position resolution of 20,000 countslcm The motor and paddle assembly were supported over the wave channel by a separate steel frame.

2.3 Wave & Depth Measurements

In all experiments waves were measured by a bottom-mounted pressure transducer and by four, non-contacting, capacitance-type gages, 40cm apart, and supported above the water surface by the instrumentation carriage. The sensing element of the wave gages was about 6mm wide and extended 12.7cm across the channel, thereby providing an average cross-channel measurement of instantaneous water surface elevations. The sensing element was 3cm above the water surface, and this maximum-possible height limited the maximum wave amplitudes that could be used in the experiments. Each gage was supported on a rack-and-pinion assembly with motor so that it could be calibrated under computer control. The pressure transducer measured the bottom water pressure (head) in the range of 0-10.16cm with an output voltage in the range of 0-5V. Both the wave gages and pressure transducer have remarkably repeatable and linear calibrations.

Precise control of the quiescent water depth (h = 5cm) was essential during these experiments. Although a traditional point gage was used, we found that the pressure transducer provided much greater resolution and control. In fact, we were able to monitor the depth to within about 0 . 2 5 ~ which corresponded to a water volume in the channel of one liter. This resolution enabled us to avoid significant depth changes during experimental series.

2.4 Data Acquisition & Control

Analog signals from the four wave gages and pressure transducer were low- pass filtered (30Hz) and digitized using a state-of-the-art (sigma delta technology) computer system that enables exactly simultaneous sampling among signal channels with 16-bit accuracy. The system runs under the (hard) real-time

177

operating system of VxWorks. Sampling was initiated by, and synchronized with, another real-time computer system (Programmable Multi-axis Controller by Delta Tau, Inc.) dedicated to control of the motors that generated the waves, calibrated the gages, and moved the instrumentation carriage. The integration of the data acquisition and control systems enabled an entire experiment to be performed under computer control.

2.5 Procedures

Since only four wave gages were available on the instrumentation carriage we could only measure waves at four spatial locations during a single experiment. To circumvent ths limitation we exploited the technological sophistication of both the mechanical and electronic systems that enable the (near) repeatability of an experiment. First, an initial carriage position was chosen and an experiment conducted. Then the carriage was shifted lcm downstream from its previous position and the experiment was repeated. Repeating this procedure 40 times provided a data set that spanned 1.6m in the x-direction (since the gages were spaced 40cm apart) with a resolution of lcm. This data set could then be interrogated to provide spatial profiles of the water surface beneath the instrumentation carriage at any fixed time. Specific procedures differed for the head-on and following collision experiments, and are described below.

2.6 Wave Generation

The motion of the wave-maker was programmed to generate a solitary wave by forcing a (horizontal) velocity field in the water that is ‘close’ to that occurring during passage of a solitary wave. We adopted a procedure similar to that introduced by Goring & Raichlen (1980), which accounts in part for the finte displacement of the wave-maker paddle and the propagation of the wave during generation. Generation was based on the KdV solitary wave whose horizontal velocity field is given by:

in which h, is the quiescent water depth, c, =a, g is gravitational acceleration, a, is the wave amplitude, and u, = aoc,/h, is the maximum horizontal velocity. The displacement xp of the wave-maker paddle from its initial position (x=O) is then found numerically by solving the differential equation:

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2.5

2

1.5

h cm 1

which gives the Lagrangian path of a water particle. Figure 1 shows the resulting paddle motion (solid line) for a solitary wave used in both the head-on and following collision experiments with a, = 2cm and h, = Scm. For convenience we also show (dashed line) the linear approximation of Equation

.z! * . . . . . . . . . 8 ;

8 / - -

/

0 0.2 0.4 0.6 0.8 I 1.2 t sec

Figure 1. Wave-maker displacement for a0=2cm. Solid line is solution of Equation (1). Dashed line is linear approximation.

20 40 60 80 100 120 140 160 x cm

Figure 2. Experimental solitary wave generated with a, = 2cm. Carriage window in the interval.

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(1). An experimental measurement of the wave generated by the paddle motion of Figure 1 is shown in Figure 2. In this measurement the wave is propagating to the right underneath the camage, which was positioned so that its 160cm measurement window was centered about n = 5m The amplitude of the solitary wave is about 1.9cm in consequence of viscous damping during propagation from the wave-maker. Note that there is a small shelf-like wave, with a maximum amplitude of about 0 . 3 ~ trailing the solitary wave, followed by even smaller, decaying, long-period oscillations that are barely perceptible.

2.6 Discussion

In spite of all efforts some small differences between two repeated experiments will occur. Three unavoidable sources of differences were recognized at the outset of the experimental program. The first of these sources is latency, which is lnherent in all electro-mechanical servo system. Latency is the small time interval between when an electro-mechanical system is commanded to move and when movement actually begins. A measure of this time is the servo update period, which is 0.885ms in these experiments.

A second source of experiment differences is water su$ace contamination. An exposed water surface accumulates surfactants with time (both from the air and fluid interior) that enhance wave damping during propagation. We conducted a series of experiments in which we measured damping of the solitary wave shown in Figure 2 at different times over a period of two days. We concluded that experiments would not be affected significantly by surfactant accumulation for up to six hours. After 6 hours, it was deemed necessary to drain and clean the channel and then refill it in order to begin the next series of experiments.

A thud source of experimental differences is residual boundary layer motions that are left behind as a solitary wave propagates in the channel. In both the co- and counter-propagating binary collision experiments solitary waves encounter the boundary-layer wakes of the other wave. Ths wake does have a small effect on wave speeds that can be significant in our data analysis. Detailed measurements of these boundary layer motions were not made; hence, it is not known how reproducible they are.

It is straightforward to cope, in part, with the three sources of difference described above in our set of 40 repeated experiments. The pressure-gage measurements, whch would be identical in exactly reproducible experiments, are used to time shift each experiment’s measurement to yield the maximum correlation with the first experiment in the set. Typically, these time shifts were about 0.01s resulting in correlation coefficients greater than 0.99. The worst case we encountered required a time shift of about 0.06s and had a correlation coefficient of 0.964. This case produced significantly poorer results in the data analysis, and appears to result from an unanticipated and unavoidable random

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fourth source of difference that occurs during the refection of a solitary wave from a vertical wall. We discuss this phenomenon in more detail below.

3. Mathematical Models

We compare the experimental results with theoretical predictions of several initial-value mathematical models that assume incompressible, inviscid, and irrotational wave motions. Numerical solutions of the Euler equations are presented for both the head-on and following collisions. In addition we employ explicit analytical predictions based on the Korteweg-deVries equation(s).

3.1 Euler Model

Consider a two-dimensional layer of water in a domain defined by n(q) = [ ( x , y ) : x E %,y E [ -h ,q]) , in which q(x, t ) denotes the free surface elevation referenced to the quiescent water level y = 0, and y = -h denotes a rigid bottom boundary. The velocity vector u ( x , y , t ) is given by u = Vq5 in which the velocity potential p(x . y . t ) satisfies

A p = O in R(7). (3)

On the bottom boundary, y = 4, the velocity potential satisfies the Neumann boundary condition:

The free surface boundary conditions on y = q(x,t) are:

(Surface tension effects are neglected.) Following (5a) we set c ( x , y ) = p(x, q(x,t),t) and define the Dirichlet-Neumann operator:

where n is the exterior unit normal of the water surface. The operator G ( q ) maps Dirichlet data to Neumann data on the free surface. It is linear in c but nonlinear, with explicit nonlocal dependence, on q which determines the fluid domain. In terms of the surface quantities q and 4 the free surface conditions of (5) become:

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These are Hamilton’s canonical equations in Zakharov’s (1 968) formulation of the water-wave problem as a Hamiltonian system, i.e.,

with the Hamiltonian:

Coifman & Meyer (1 985) showed that when 7 E Lip(%) the Dirichlet-Neumann operator can be written as a convergent Taylor series:

j = 0

and Craig & Sulem (1993) showed that explicit expressions for the G can be computed using a recursion formula.

The above system of equations are solved numerically for specified initial data using periodic boundary conditions in the x-direction and a pseudo-spectral method for the spatial discretization. The Dirichlet-Neumann operator is approximated by a finite number, M, of terms in (10). In practice, it is not necessary to use large values of M due to the fast convergence of the series expansion for G(v) . The two variables 7 and < are expanded in truncated Fourier series with the same number of modes. Applications of Fourier multipliers are performed in spectral space, while nonlinear products are calculated in physical space at a discrete set of equally spaced points. All operations are performed using the FFTW routines.

Time integration is performed in Fourier space. The linear terms in (7) are solved exactly by an integrating factor technique. The nonlinear terms are integrated using a fourth-order Adams-BashfordIMoulton predictor-corrector scheme with constant time step. In the computations it was observed that spurious oscillations developed in the wave profile after some time of integration due to onset of an instability initiated by growth of numerical errors at high wave-numbers. To circumvent this difficulty we applied an ideal low-pass filter to T,I and 5 at each time step.

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2.3 Kd V Models

An asymptotic approximation of the Euler equations in the limit of weak dispersion and weak nonlinearity is the well-known Korteweg-de Vries equation:

Equation (1 1) is for right-running waves only; a similar KdV equation applies to left-running waves. The solitary-wave solution of (1 1) is:

in which the speed of the wave is:

Wayne & Wright (2004) have shown formally that, to the KdV order of weak nonlinearity and dispersion, left-running and right-running solitary waves interact linearly during their collision. Hence, we will use linear superposition as an approximate model of head-on collisions for comparison with the hlly nonlinear, numerical solutions of the Euler equations.

In the case of collisions among N co-propagating solitary waves Hirota (197 1) found an exact solution of (1 1) when the reference frame translates to the right with the speed c,. For the case of a binary ( N = 2) collision (also see Whitham, 1974) the solution is:

r 1

in which:

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and (al ,a2) are the amplitudes of the two individual solitary waves and (xl, n,) are arbitrary (initial) shifts in wave positions.

4. Head-on (counter-propagating) collision

Specific procedures for the experiments on counter-propagating binary collisions of solitary waves are the following. Since only one wave-maker was available, it was necessary to generate a first solitary wave (ao=2.00cm) that propagated down the channel and reflected from the end-wall. Subsequently a second solitary wave (ao=l .25cm) was generated that collided with the reflected wave near the center of the channel test section. The instrument carriage was fixed during each experiment so as to provide a spatial window of the collision in the interval. Data were collected for about 64s so that the two solitary waves reflect and collide multiple times. Results for the first collision are reported herein.

Once the raw data (voltages) are converted to wave amplitudes using the calibration results, correlations are performed between the pressure measurement of the first experiment and that of each of other 39 experiments. In th~s manner we obtain the necessary time shifts to obtain the maximum correlation values among the 40 experiments. Typically, these shifts are about 0.01s. Initially we used the entire 64 seconds of pressure data to shift the records. The resulting spatial profiles were wholly unsatisfactory, exhibiting a lack of smoothness that was clearly an artifact of the data reduction algorithm. Second, we computed correlations using only an interval of the pressure data containing the first solitary wave before its’ reflection. The resulting time shfts yielded excellent results for right-running waves as shown in Figure 2. However, the results for the reflected, left-running wave were non-smooth. An example of these results is shown in Figure 3a, which shows the counter-propagating waves prior to collision. Note that the non-smooth, left-running wave (on the right in the figure) has a jump discontinuity at one location. Third, correlations were performed using an interval of the pressure data containing the reflected (left-running) solitary wave. These time shifts resulted in smooth left-running spatial profiles, but non-smooth, right-running waves as shown Figure 3b. We conclude that the reflection process (or, perhaps, propagation through the boundary-layer wake left by the incident wave) produces a small (about 0.05s) shift in the wave arrival times at our measurement sight. This small shift is random, i.e., it differs for each experiment. Hence, we are not able to resolve both the left-running and right-running waves with a single correlation procedure. The results presented below use correlations based on the reflected wave as in Figure 3b.

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18.401 3

1.5

2.5

2

1.5

h cm 1

0.5

0

I 1

20 40 60 80 100 120 140 160 x cm

18.40 13

I 20 40 60 80 100 120 140 160

x cm

Figure 3. Counter-propagating solitary waves before collision (at t=l8.4013s). Time shifts based on pressure data for first solitary wave before reflection. Time shifts based on pressure data for reflected solitary wave.

(a) (b)

A sequence of spatial profiles during the collision of counter-propagating solitary waves is shown in Figure 4. The experimental times (in seconds) are shown above each profile. Note that the total collision interval spanned in Figure 4 is about 1.7s. The experimental data at t = 18.2999s were fit theoretically with the linear sum of two solitary waves having initial amplitudes of (right-running) and (left-running), respectively. This theoretical fit (dashed line) served as the initial data for the Euler computations. Recall that the experimental spatial profiles in Figure 4 are based on correlations that resolve the left-running wave best.

The spatial profiles of Figure 4 between the times of t = 18.7024s and t = 19.9205s for the left-running solitary wave agree well with both the linear superposition and Euler predictions, which are nearly the same. The maximum wave amplitude of the collision occurs in the spatial profile at t = 19.0311s

185

2.5

2 :

1.5

3 r . . . . . . . - . . . . . . . . . . - . . . . . . . - . . . .

:

:

Figure 4. Spatial profiles of counter-propagating collision of two solitary waves at different times (listed above each frame). Solid points are experimental data. Solid line is Euler computations. Dashed line is linear superposition of KdV solitons.

186

18.92s

r i 20 40 60 80 100 120 140 160

x c m

19.0311

2.5

2

1.5

I

0.5

h m

r . . . . . . . . 20 40 60 80 I W I20 140 160

x a n

l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 40 60 80 100 120 140 160

x cm

Figure 4. Continued.

187

3

2.5

2

I .5

1

0.5

0

h cm

3

2.5

2

1.5

1

0.5

h cm

19.1939

20 40 60 80 LOO 120 140 160 x cm

19.9988

0

20 40 60 80 100 I20 140 160 x em


according to both the measurements and the theories. Both the Euler prediction and the measured data (for the remnant of the left-running wave) agree, and the maximum amplitude is about 2.45cm -- linear superposition predicts 2.27cm. Subsequent to the time of maximum amplitude, the Euler predictions and the measured data agree well; however, there are now quite large discrepancies with linear superposition. In other words, linear superposition is fairly accurate until the maximum amplitude is achieved, but much less accurate thereafter. The last spatial profile at t = 19.0311s shows the two solitary waves after the collision. According to linear superposition there should be no phase shifts in consequence of the collision. Clearly both the experimental data and Euler computations show that a small phase shift has occurred, i.e., the collision has delayed both waves. The observed shift is increasing with propagation distance; hence, the collision was not elastic. Interestingly, Euler theory predicts slightly more phase shift than observed in the last frame.

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5. Following (co-propagating) collision

Unllke the head-on collision described above, the collision of a larger solitary wave overtakmg a smaller solitary wave occurs over a very large distance down the wave channel. In order to measure a following collision, the instrument carriage must move with the waves and measure in a traveling reference frame. The co-propagating collision experiments were conducted in the following manner. The instrument carriage was positioned near the wave-maker so that its’ initial measurement window spanned. The wave-maker then generated a smaller solitary wave with a, = 5cm followed immediately by a larger solitary wave with a, = 2.00cm. Once these two waves reached the instrument-Carriage window, the carriage accelerated smoothly for 2.5s and at a distance of 1.125m to a constant speed of 90 c d s . The constant speed was maintained for 4.5s when the carriage decelerated for 2.5s and stopped. The total move time of motion for the carriage was 9.5s and the total move distance was 6.35m. Programmed and actual motions of the carriage are shown in Figure 5. It is important to note that the “actual” carriage displacement and velocity shown in Figure 5 is inferred from a rotary encoder on the back of the motor powering the belt drive attached to the carriage. This was necessary due to the long distance traversed by the carriage, whch prohibited the use of a feedback sensor for the carriage position. Since the belt between the motor and the camage is not rigid, there is necessarily uncertainty in carriage position, especially during the acceleration and deceleration intervals. Comparisons of carriage motions between two different experiments showed that the actual motion shown in Figure 5 is reproduced for each experiment.

Figure 6 shows the waves underneath the instrument-carriage window just before it begins moving. The dashed line is a 2-soliton solution (Equation 14) fit to the experimental data. The solid line, which agrees with the data much better, is the linear superposition of two solitary waves with a , = 2.15cm and a, = 0.68cm. This linear fit is used as the initial data for the Euler computations. (It should be noted that the wave-maker motion corresponded to the linear superposition of two solitary waves also.)

A sequence of spatial profiles during the collision of two co-propagating solitary waves is shown in Figure 7. Times (in seconds) during the collision (measured from the initial data of Figure 6 ) are shown above each profile. At all times the Euler predictions agree with the measurements better than the KdV predictions. At all times the maximum amplitudes predicted by KdV model exceed those by the Euler model, which exceed those in the measurements. In addition, the discrepancies in maximum-amplitude predictions increase with time and distance down the channel. This behavior is consistent with viscous damping that is significant in these experiments in consequence of the long distances of propagation during the collision. The small differences in phases between the

189

predicted and measured positions of wave peaks, e.g. at t = 8.79081s, is due, in part, to viscous effects and, in part, to experimental errors in the resolution of carriage position.

2 4 6 8 10 i s

(4

6

5

4

displacement m 3

2

100

80

60

Velocity c d s 40

20

2 4 6 8 LO

(b) t

Figure 5. Actual (solid line) and programmed (dashed line) carriage motion. (a) Displacement, @) Velocity.

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0

20 40 60 80 100 120 140 160 x cm

Figure 6. Initial data for co-propagating collision of two solitary waves. Solid points are data. Dashed line is a best fit of the 2-soliton solution. Solid line is best fit of a linear superposition of KdV solitons.

1.10285 2.5

2

1.5

h cm 1

0.5

0

20 40 60 80 100 120 140 160 x cm

2.09818

20 40 60 80 100 120 140 160 x cm

Figure 7. Spatial profiles of co-propagating collision of two solitary waves at different times (listed above each frame). Solid points are experimental data. Solid line is Euler computations. Figure Dashed line is 2-soliton solution of the KdV equation.

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3.1058 2.5

2

1.5

h crn 1

0.5

0

20 40 60 80 100 120 140 160 x crn

4.09806 2.5

2

1.5

h cm 1

0.5

0

20 40 60 80 100 120 140 160 x cm

5.09953 2.5

2

1.5

h cm I

0.5

0

20 40 60 80 100 120 140 160 x cm


192

h cm

2.5

2

1.5

1

0.5

0

6.09793

20 40 60 80 100 120 140 160 x cm

7.0994 1

20 40 60 80 100 120 140 160 x cm

8.09781 2.5

2

1.5

h cm 1

0.5

0

20 40 60 80 100 120 140 160 x cm

Figure 7. Confinued.

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As noted earlier, Wu (1998) shows that there is an instant in time during the collision of two co-propagating solitary waves for which the interaction profile exhibits fore-and-aft symmetry. This behavior is shown in Figure 7 by both the measured and Euler computations at t = 4.09806s. Interestingly, the KdV model does not show the fore-and-aft symmetry at this time.

6. Summary

High-resolution experimental data were presented for both co- and counter- propagating collisions between two solitary waves. The data are in the form of spatial wave profiles at fixed times during the collision. The data for the counter- propagating (head-on) collision are compared to numerical solutions of Euler equations and to linear superposition of two KdV solitary waves. Linear superposition is fairly accurate until the time at which maximum runup amplitude occurs. It is much less accurate thereafter. The Euler model predicts accurately the measured profiles and the resulting maximum runup amplitude and subsequent phase shifts. The maximum measured runup amplitude is 2 . 4 5 ~ ~ which is predicted by the Euler model. Linear superposition predicts a value of 2.27cm

The Euler model also agrees well with the measured data for the following collision; however, it over-predicts wave amplitudes. This disagreement is anticipated since viscous damping over the long distance spanned by the collision is significant in the experiments. Both the experiments and the Euler model exhibit a profile with fore-and-aft symmetry as predicted by Wu (1998) at an instant during the collision. The 2-soliton solution of the KdV equation agrees qualitatively with the measurements. However, it greatly over predicts amplitudes (more than the Euler model), and it does not show a profile with fore- and-aft symmetry at the instant measured and predicted by the Euler equations.

References

Byatt-Smith, J.GB. 197 1. An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49,625-633.

Byatt-Smith, J.GB. 1989. The interaction of two solitary waves of unequal amplitude. J. Fluid Mech. 205,573-579.

Coifman, R. & Y. Meyer 1985. Nonlinear harmonic analysis and analytic dependence, Pseudodifferential operators and Applications, Notre Dame IN (1984), Amer. Math. SOC., 71-78.

Craig, W. & C. Sulem 1993. Numerical simulation of gravity waves. J. Comput.

Gardner, C.S, J.M Greene, M.D.Kruska1, & R.M. Miura 1967. Method for Phys. 108,73-83.

solving the Korteweg-deVries eqation. Phys. Rev. Lett. 19, 1095-1097.

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Goring, D.G & F. Raichlen 1980. The generation of long waves in the laboratory. Proc. I 71h Zntl Con. Coastal Engrs, Sydney, Australia.

Hirota, R. 1971. Exact solutions of the Korteweg-deVries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192-1 194.

Korteweg, D.J. & G deVries 1895. On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves. Phil. Mag. (5) 39,422-443.

Maxworthy, T. 1976. Experiments on the collisions between solitary waves. J. Fluid Mech. 76, 177-185.

Mayer, R.E. 1962. Brown University Tech. Rept. Mirie, R.M. & C.H. Su 1982. Collisions between two solitary waves. Part 2. A

numerical study. J. Fluid Mech. 115,476-492. Russell, J.S. 1845. Report on waves. Brit. Assoc. Rept. Su, C.S. & R.M. Mirie 1980. On collisions between two solitary waves. J. Fluid

Wayne, C.E. & J.D. Wright 2004. Higher-order modulation equations for a

Weidman, P.D. & T. Maxworthy 1978. Experiments on strong interactions

Whitham, GB. 1974. Linear and Nonlinear Waves. John Wiley & Sons. Wu, Theodore Yaotsu 1998. Nonlinear waves and solitons in water. Physica D

Yih, C.S. & T.Y. Wu 1995. General solution for interaction of solitary waves including head-on collisions. Acta Mech. Sinica 11, 193-199.

Zabusky, N.J. & M.D. Kruskal 1965. Interactions of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15,240-243.

Zakharov, V.E. 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190-194.

Mech. 98, 509-525.

Boussinesq equation. Preprint.

between solitary waves. J. Fluid Mech. 85,417-431.

123,48-63.

COMPUTER SIMULATIONS OF' OVERTAKING INTERNAL SOLITARY WAVES

BRIAN C. BARR University of Florida, Gaineswille, FL 32611-6580, USA

DANIEL T. VALENTINE Clarkson University, Potsdam, NY 13699-5725, USA

clara@clarkson. edu

The results of computer simulations presented in this paper are numerical solutions of the two-dimensional equations of motion for an incompressible, viscous fluid. The computational results provide a relatively comprehensive description of the types of waves observed experimentally. The leading wave generated by a disturbance imposed upon an initial state of rest is a solitary wave. It is the fastest propagating mode. Much of its qualitative behavior is indeed describable by asymptotic theory. However, there are some significant differences including the fact the viscous waves are not actually waves of permanent form. In this paper we examine simulations of wave-wave overtaking interactions of viscous, internal solitary waves.

1. Introduction

The internal solitary waves described in this paper are numerical solutions of the Navier-Stokes, the convection-diffusion of density anomaly, and the continuity equations to within the Boussinesq approximation. The dimensionless parameters that characterize the flow field were selected to model waves observed in the laboratory. The waves generated are waves of the type observed by Kao, Pan & Renourd (1985). There are waves analogous to these waves that are generated each tidal cycle in coastal seas around the world; see, e.g. Liu (1988). Hence, the waves examined computationally are not only of academic interest, they are of practical importance in the characterization of the wave environment in coastal seas.

In an elegant and succinct paper Benney (1966) described, among other problems, the theory for weakly nonlinear, internal waves. The asymptotic approximation of the Euler equations for shallow water long waves in a density stratified fluid led to a Korteweg & de Vries (1895) (or KdV)

195

196

equation. This equation admits a well-known solitary wave solution. This weakly-nonlinear theory provides insight into scaling the computational results. Benney’s paper and several others around the time of its publication led to numerous theoretical investigations that extended the original theory in various directions. A recent paper of importance to the present investigation describes the fully-nonlinear theory of internal solitary waves on the interface of a two-layer fluid system. It is by Choi & Camassa (1999). The weakly nonlinear and the fully nonlinear theories help describe the solitary waves generated in the present investigation.

Wu (1998) investigated the overtaking interaction of a small leading solitary wave with a larger solitary wave following it in an inviscid fluid. The theory describes two types of interaction phenomena depending on the ratio of the amplitude of the two waves. In all cases, if the waves collide, the following wave must be larger than the leading wave (this is because, in general, larger amplitude waves travel faster than smaller amplitude waves). If the difference in amplitude is sufficiently small, a two-peak interaction occurs. If a certain critical difference in amplitude is reached, then the waves interact such that the two waves form a single peak at the center of the interaction. The waves emerge from the interaction unscathed, except for a phase shift. We show that the solitary waves described in this paper behave in a similar manner.

In this paper we describe internal, viscous solitary waves that we generated by solving the Navier-Stokes equations. In the next section we describe the computational analysis methodology. This is followed by a summary of both the weakly nonlinear and fully nonlinear theories used to compare with the properties of the viscous solitary waves. We then describe the results of simulations of wave-wave overtaking phenomena; the simulations reported were inspired by a talk given by Professor Wu at a special symposium in 1998 in memory of the late Professor Chia-Shun Yih held in Gainesville, Florida. The first author (B.C.B) began an investigation of overtaking internal wave-wave interaction immediately after this symposium; see Barr (2001).

2. Computational analysis methodology

2.1. Flow-field geometry and initial condition

The geometry of the two-dimensional, rectangular water tank in which the internal waves were simulated by computer is illustrated in Figure 1. The density structure of the water column investigated is that of a nearly two-

197

layered stably-stratified fluid with the density changing continuously across a region between the two layers, known as the pycnocline. The initial condition consists of a sech2 shaped pool of lighter fluid, with amplitude A, and horizontal extent W, suspended in the surrounding denser fluid. This initial condition was established by applying the following formula:

z + A , sech2(2z/W) - D

Simulations to examine the properties of the leading, viscous solitary waves were conducted by Barr (2001) and reported by Barr & Valentine (2005) for five different upper layer depths of h l / D = 0.075,0.1, 0.15, 0.2, and 0.9 as well as three different pycnocline widths, a / D = 30, 45, 60. The three pycnoclines are illustrated in Figure 2. The computational simulations in a rectangular domain mimics a laboratory wave tank like the one used by Kao, Pan & Renourd (1985). The simulations were conducted with a tank aspect ratio, LID, of thirty.

n t

Figure 1. Schematic of the initial condition and tank geometry.

Initially, there is no motion anywhere in the tank. When the displaced pool is released, the pressure variation and buoyancy force initiate the flow. The potential energy of the displaced pool is converted into a combination of kinetic and potential energy contained in the leading solitary wave and in the following train of waves, along with some losses due to localized mixing in the formation process.

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0.8 I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e

Figure 2. Illustration of pycnocline width.

2.2. Equations of motion

This flow field is modeled as the unsteady flow of a Newtonian, viscous, incompressible, Boussinesq fluid. The model equations for this flow are the Navier-Stokes, continuity and density-difference transport equations. These equations, in dimensionless form, are

1 U t + (uu), + ( W U ) , = -p, + -('zL,, Re + 4, (2)

where the subscripts denote differentiation with respect to that variable. The depth of tank, D = hl + hz, has been used as the characteristic length

199

scale and the linear interfacial wave speed,

as the characteristic velocity scale. The density difference, 8, and the reference density-difference stratification, e, have been scaled as:

The dimensionless parameters are defined as follows: v

s c = - hlh2 F2 = - Re= -, U D D2 Dab ' U

where Dab is the molecular diffusivity. Since the computations are in a two-dimensional domain, we can sim-

plify the governing equations as follows. The only finite component of the vorticity vector is I, the y component, which is perpendicular to the velocity vector. This component of the vorticity is defined by the equation c = u, -toz. Hence, substituting Equations (2) and (3) into this definition, we get the vorticity-transport equation, viz.:

From continuity, i.e., Equation (4), we can define the stream function $ such that u = gZ and 20 = -&. Substituting this into the definition of the vorticity, we get the Poisson equation for the stream function in terms of the vorticity, viz.:

$zz + $2, = c. (10)

With water as a working fluid and salt water being the denser lower fluid, the Schmidt number, Sc, is 833. The densimetric Froude number, F , varies according to the layer depths from 0.0693 to 0.16 for h l = h l / D = 0.075 and 0.2, respectively. The Reynolds number, Re, is 10,000, corresponding to a laboratory scale.

2.3. Computational-solution methods

The computational method applied is the ETUDE finite-difference method described by Valentine, Barr & K m (1999) to solve similar wave problems. The details of this method were published by Valentine (1995). This method solves the equations of motion in terms of the vorticity, stream function and

200

8 fields. The Poisson solver in the computer tool based on this method was modified to improve computational efficiency for the finer-grid simulations presented here.

The boundary conditions applied to the numerical wave tank are as follows. The side walls of the tanks are rigid, no slip, non-diffusive boundaries. The bottom of the tank is a rigid, no slip, constant density surface. The top surface is modeled as a pure slip, rigid lid with constant density.

The improvement to the ETUDE method applied in this investigation is the way we solved the elliptic problem, Equation (10). The boundary conditions for this problem are homogeneous Dirichlet boundary conditions. To solve this problem a geometric, V-cycle multi-grid solver was developed. Multigrid techniques are optimal solvers, which converge like O(N) . Gauss- Seidel relaxation was used as the smoothing operator. On the collocated grid of the vorticity stream function code, the restriction and prolongation operator pair weTe chosen to be full weighting and bilinear interpolation.

The solutions are reasonably well resolved. For wave propagation in a density stratified fluid, it is well known that the maximum wave frequency possible is the Brunt-VaisuZu (BV) frequency; see, e.g., Yih (1980). The frequency spectrum of a solitary wave is relatively narrow and significantly lower than the BV frequency. The horizontal resolution was selected such that the highest frequency waves are resolved with roughly five grid points. Since our interest lies in the solitary wave with significantly longer wavelength than the train of waves following the leading solitary wave (that are inevitably generated with the wave), this selection ensures that the solitary wave is well resolved. For quicker convergence in the Poisson equation solver, AZ was chosen to be equal to Ax = 1/128.

The temporal resolution was selected to satisfy the Courant-Freidrichs- Lewy (CFL) and Neumann (diffusive effects) stability criteria. For all the cases discussed here, At was set equal to 0.001. This time step satisfied the stability criteria and provided sufficient time accuracy for two numerical schemes; the second numerical scheme was developed by Barr (2001). It was applied to solve the primitive-variable formulation of the numerical wave tank of this investigation. The results of simulations of internal, viscous solitary waves reported by Barr & Valentine (2005) compared excellently with the ETUDE solutions. The results also compared favorably with the laboratory results of Kao, Pan & Renourd (1985). Hence, Barr’s investigation confirms the fact that the waves predicted are, indeed, waves that one can expect to observe in the laboratory.

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3. Wave Theory

3.1. Weakly nonlinear model

Benney (1966) showed that for relatively small, but finite amplitude internal long waves in a shallow basin of density stratified fluid, the horizontal and temporal dependence of motion satisfies the following equation:

At = -cOAx + ~ T A A , + sA,,,, (11)

where $(z, z , t ) = A$O(z) + . . . . The parameter $J is the stream function that is defined from Equation (4) such that u = $Jz and w = -$Jx. The equation for A = A ( z , t ) is a KdV equation. It admits the well-known solitary wave solution:

where

and

x - c t A = a sech2 [ ,

2 3

C = C, + - U T,

where a is the amplitude of the wave in terms of the stream function. If a is the displacement of a material surface within the density stratified fluid, i.e., the displacement of the 0 = 0.5 isopycnal (or the center of the pycnocline), then, to zeroth order, at = w = -$,. By substituting for $J the perturbation expansion that represents it, and using the KdV equation for A, we get, after intergrating once, a = c,a. We use the symbol a to represent the wave amplitude in terms of the displacement of the 0 = 0.5 isopycnal to be consistent with the same measure of wave amplitude used in examining the computational simulations and also used in the fully nonlinear theory described in the next subsection.

The parameters c,, T , s and 4' = @ ( z ) depend on the density structure of the water column. To determine c, and $O, we must solve the Sturm- Liouville eigenvalue problem given by

with 4 O ( O ) = 4'(1) = 0. The parameter c,, the eigenvalue, is the speed of the infinitesimal linear long wave on the pycnocline.

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The constants r and s are a consequence of integrability constraints. It can be shown that for a Boussinesq fluid

The eigenvalue problem must be solved for a specified $, i.e., for the specific density stratification of a particular case. The numerical values of T and s, that are required for the comparisons with the computational simulations discussed herein, are given in the Results and discussion section.

3.2. Fully nonlinear model

Choi & Camassa (1999) derived model equations, which also follow from Euler’s equations, that describe the motion of interfacial waves in a distinct two-fluid system that are long compared to the undisturbed depth of one of the layers. This differs from KdV theory in that there is no restriction on the smallness of the amplitude.

The equations are decomposed into depth mean quantities and fluctuations, allowing the vertical dependence to be “integrated away”, but necessitating the need for approximate closures to the equations.

The wave speed is given by:

c2 - (hl + a)(h2 - a) - _ c: hlh2 - (c?/g)a .

For the majority of the simulations, where

the wavelength is given by

- (a , - u-)lI2 (F(6, m) - E(6, m)) , (20) 1 1/2 2

where F and E are elliptic integrals of the first and third kind; see, e.g., Byrd & F’riedman (1971).

203

and hlh2(Plhl + P2h2)

plhq - p2hB ’ a, = -

and a- and a+ are the two roots of the quadratic equation

Also, 2 a* -a+ m =-

a, - a- I

and

] 1’2 . a+(.- - a,) .-(a+ - a,)

sin6 = (25)

It should be noted that in the previous expressions, the sign of the amplitude is important. However, the results as plotted in the following sections should be interpreted as la[ .


4.1. Internal solitary waves

The types of waves generated are internal solitary waves of depression on pycnoclines in a viscous fluid. They are described in detail in a recent paper by Barr & Valentine (2005); see, also, Barr (2001). The wave properties, i.e., wave speed and wavelength versus amplitude, were reported and compared with the KdV and Choi-Camassa theories. The range of wave amplitudes, as measured by the displacement of the 13 = 0.5 isopycnal located at the center of the pycnocline from its initial position of rest, is from 0.03 to 0.28 of the total depth of the tank. The width of the pycnocline is about 0.1 of the depth of the water column. The range of upper-layer depths examined is h l from 0.075 to 0.2 of the total depth of the water column. The KdV theory provides a means to scale the properties of the waves as predicted numerically and, thus, provide insight into the nature of the viscous solitary waves on finite-depth pycnoclines as compared with the theoretical, inviscid waves. We will examine the results of the wave speed next.

The increment of the wave speed, c, and the linear wave speed, c,, can be scaled as follows:

204

where T is defined by Equation (16) and co by Equation (15). The data for the a = 45 pycnocline over the entire range of upper-layer depths and the entire range of amplitudes investigated by Barr (2001) collapsed to a single curve. The formula for this correlation is

dc = 0.15 - 0.157e5a. (27)

This curve is plotted and compared with the corresponding result of the Choi-Camassa fully-nonlinear interfacial wave theory in Figure 3. For h l = 0.1, T = 5.836 and c, = 0.9433. Thus, the theoretical prediction based on Equation (18), to within the Boussinesq approximation, was scaled with these parameters. This is consistent with the viscous solitary wave predictions. Thus, this figure illustrates the fact that viscous waves on a finite- depth (yet relatively shallow) pycnocline travel slower than the inviscid waves. Yet, the viscous solitary wave speed does indeed follow the trend with amplitude as predicted by the fully-nonlinear theory. The scaling applied puts in evidence the viscous effects. For other details and comments on comparisons with experimentally determined wave speeds, wave shapes and wave lengths with amplitude, in addition, to the vorticity, pressure, and other fields predicted for viscous internal solitary waves in this numerical wave tank, the reader is referred to Barr & Valentine (2005). The main point for reviewing these results in this paper is to discuss the importance of the scaling based on theory and emphasize the difference between viscous and inviscid solitary waves. Finally, it is the interaction of these kinds of waves that are generated to investigate the over-taking encounters described in the next section.

4.2. Overtaking solitary waves

Wu (1998) described three regimes for overtaking surface solitary waves (where the larger, faster wave travels behind the slower, smaller wave), depending on the ratio of the interacting wave amplitudes: two distinct peaks (ularge/asmall < 3), a single peak with vanishing curvature at the crest (ularge/asmall = 3), and a single peak whose crest has curvature (alarge/asmall > 3). While an interesting analytical problem, it is experimentally and numerically challenging to investigate. Generating two solitary waves from any single disturbance leaves them rank ordered, with the largest solitary wave leading, just the reverse of what is required for an overtaking interaction. Numerically, our solution was to create a superposition of waves in the configuration required.

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0.12

-0.02 ‘ 0 0 . 0 5 0.1 0.15 0.2 0

Wave amplitude, a 25

Figure 3. Scaled increment of Wave speed versus amplitude: Solid line is fully-nonlinear theory. Dashed line is the speed determined from simulations of viscous solitary waves.

Each wave needed is generated using the vorticity-stream function code in a tank with an aspect ratio of thirty and allowed to propagate. Once it is well developed and has left the trailing tail, the wave is “captured”; the fields in an area around the wave are saved, eliminating the trailing oscillatory tail behind the wave. The two captured waves are then superimposed in a tank with an aspect ratio of eighty, propagating into quiescent fluid. This procedure is valid for a number of reasons. Cutting out and pasting together fields is valid as long as the respective fields are small enough at the edges. Second, the effect of “pasting” together the waves is smoothed over in one time step. The ellipticity of the stream function communicates the effect of the superposition of the solitary waves to the entire tank during that first time step.

For the figures illustrating the overtaking interactions, the waves travel from left to right, with the larger wave initially on the left overtaking the smaller wave on the right. The figures are made with a moving frame of reference that keeps the center of the figure aligned with the midpoint of the interaction.

The first case studied has two waves whose amplitude ratio is 1.71. The

206

ratio of their wave speeds is 1.07, so the waves are very nearly traveling at the same speed. The top frame in Figure 4 shows the density field for the two waves at time t = 8. The middle frame shows the waves interacted to the point that the larger wave has dumped enough mass into the smaller wave so that they are now roughly the same size and traveling at the same speed. Yet, the two waves are still distinct entities. Mass transfer continues, and eventually the wave on the right starts to pull away as it gains mass and momentum from the wave on the left. Notice in the third and final frame of Figure 4, the larger wave now appears on the right, and the smaller wave appears on the left.

The second overtaking case has two waves whose amplitude ratio is 3.95 and speed ratio is 1.23. Here the waves have formed a slightly concave, “dimpled” peak when they are most interacted in the middle frame of Figure 5. Once again in the last frame, the waves seem to have passed through each other.

For the final case, the amplitude ratio is 5.87, and the speed ratio is 1.36. Now at the point of maximum interaction, the peak no longer has any concavity to it. The interacting waves form one single definitive peak. The last frame in Figure 6 again illustrates that the small wave “pops” out t o the left of the larger wave, still retaining its identity.

Wu (1998) states that for an interaction that maintains two distinct peaks (his example has qarge/usmall = 1.09) the local flow velocity at the center of the interaction reaches a maximum when the two peaks are the shortest distance apart. This is confirmed in Figure 7, showing that the maximum velocity at the midplane of the interaction is an order of magnitude larger than at either the beginning or the end of the interaction process.

During the interaction, it is only possible to track both wave peaks for the case with an amplitude ratio of 1.71. For the other two cases, the wave peaks merge. Extrapolating the initial trajectories forward, first reveals that as the waves approach each other, the larger wave accelerates slightly, and the smaller wave decelerates. This is illustrated for the first and third cases in Figures 8 and 9, respectively. The results illustrated in this figure also reveal that the waves have undergone a phase shift. The larger wave appears ahead of the linear trajectory while the smaller wave lags its predicted linear trajectory. While the viscous decay makes the phase shift a function of measurement location, it is possible to give mean values for the computation time considered. For the case with amplitude ratio of 3.95, the large wave was shifted forward a distance of 1.08 while

207

1

0.95

0.9

N 0.85

0.8

0.75

0.7 ' I 12 13 14 15 16 17 18 19 20 21

0.75 O I

-_, 45 46 47 48 49 50 51 52 53 54

0.81

Figure 4. 36.4, and t = 54.

Density field for overtaking waves with qarge/asmall = 1.71 at t = 8, t =

the small wave was shifted back 0.83. For the case with amplitude ratio of 5.87, the large wave was shifted forward a distance of 0.18 while the small wave was shifted back 1.16.

208

0.6 I 9 10 11 12 13 14 15 16 17 18

1

0.95

0.9

0.85

N 0.8

0.65

0.6 28 29 30 31 32 33 34 35 36 37

0.6’ 48 49 50 51 52 53 54 55 56 57

X

Figure 5. 19.5, and t = 35.

Density field for overtaking waves with alaTge/asmall = 3.95 at t = 5 , t =

5. Conclusions

For the internal solitary waves on pycnoclines in a Newtonian-Boussinesq fluid (like water with density differences due to salinity variations), the following was found:

209

0.7

0.6

0.5 I 10 11 12 13 14 15 16 17 18 19

1

0.9

08 N

07

-." 26 27 28 29 30 31 32 33 34 35

-.- 42 43 44 45 46 47 48 49 50 51

X

Figure 6. 16.9, and t = 29.

Density field for overtaking waves with alarge.asma1l = 5.87 at t = 5 , t =

0 The internal solitary wave properties are similar to the fully- nonlinear, inviscid solitary waves on the interface of a two-layer fluid system. The internal waves traveling on a finite-depth pycnocline propagate more slowly than the interfacial waves of the same

210

U

Figure 7. Horizontal component of velocity at the midplane of the interaction for alarge/asmall = 1.71.

amplitude. The trends of celerity and wavelength versus wave amplitude follow the trends predicted by the theory. This is true for amplitudes up to 0.3 and for upper-layer depths from 0.075 to 0.2 and relatively shallow pycnoclines.

0 The overtaking interaction of two internal solitary waves traveling on a finite-depth pycnocline undergoes a two-peak collision for an amplitude ratio less than about 4. If the following wave amplitude is greater than 4 times the leading wave, the center of the overtaking event is single peaked. Although the critical amplitude ratio is not the same as the critical ratio predicted by Wu (1998) with Hirota 's two-soliton solution, the fact that a critical ratio exists for overtaking internal solitary waves in a viscous fluid was predicted by the computer simulations reported herein.

References 1. BARR, B . C. 2001 Internal Solitary Waves: From Weakly to Fully Nonlinear.

Ph.D. Dissertation, Clarkson University.

211

time

Figure 8. Trajectories of the wave peaks for alarge/asmall = 1.71.

2. BARR, B. C. & VALENTINE, D. T. 2005 Internal solitary waves on pycnoclines: Simulations of characteristics and interaction phenomena. Accepted for publication, Computers €9 Fluids.

3. BENNEY, D. J . 1966 Long non-linear waves in fluid flows. J. Math. €9 Phys.

4. BYRD, P. F. & FRIEDMAN, M. D. 1971 Handbook of Elliptic Integrals f o r Engineers and Scientists. Springer-Verlag.

5. CHOI, W. & CAMASSA, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1-36.

6. HIROTA, R. 1973 N-soliton solutions of the wave equations of long waves in shallow-water and in nonlinear lattices. J. Math. Phys. 14, 810-814.

7. KAO, T. W., PAO, H.-P. & RENOURD, D. 1985 Internal solitons on the pycnocline: generation, propagation, and shoaling on breaking over a slope. J. Fluid Mech. 159, 19-53.

8. KORTEWEG, D. J . & DE VRIES, G . 1895 On the change of the form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39, 422-43.

9. LIU, A. K. 1988 Analysis of nonlinear internal waves in the new york bight. J. Geophysical Research 10, 12,317-12,329.

10. VALENTINE, D. T. 1995 Decay of confined, two-dimensional, spatially- periodic arrays of vortices: A numerical investigation. International Journal

45, 52-63.

212

I I I I

12 14 16 18 20 22 time

Figure 9. Trajectories of the wave peaks for alarge/asmall = 5.87.

for Numerical Methods in Fluids 21, 155-180. 11. VALENTINE, D. T., BARR, B. C. & KAO, T. W. 1999 Large-Amplitude Soli-

tary Wave on a Pycnocline and Its Instability. Fluid Dynamics at Interfaces (edited by W. Shyy & R. Narayanan), Cambridge University Press, 198-210.

12. WU, T. Y. 1998 Nonlinear waves and solitons in water. Physica D 123,

13. YIH, C. S. 1980 Stratzfied Flows. Academic Press. 48-63.

THEORETICAL AND EXPERWIENTAL INVESTIGATION OF WAVES DUE TO A MOVING DIPOLE IN A STRATIFIED FLUID

SHI-QIANG DAI, GANG WEI, DONG-QIANG LU Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University.

Shanghai 200072, P. R. China

XIAO-BING SU Faculty of Science, University of Science & Technology,

Nanjing 21 1101, P. R. China

The waves induced by a moving dipole in a two-fluid system are theoretically and experimentally investigated. The velocity field of a dipole horizontally moving in the lower layer of a two-layer fluid with finite depth is obtained by using Green’s function of a pair of source and sink, and the far-field waves are studied by the method of stationary phase. The effects of two resulting modes, i.e., the surface- and internal-wave modes, on both the surface divergence field and the interfacial elevation are analyzed. A laboratory study on the internal waves generated by a moving sphere in a two-layer fluid is conducted in a towing tank under the same conditions as in the theoretical approach. The qualitative consistency between the theoretical and experimental results is confirmed.

1. Introduction

It is known that multimode waves can be generated by a body moving on or beneath the free surface of a stratified fluid. The relevant problems are of theoretical and practical significance. The pioneering work on waves generated by moving bodies in a two-layer fluid with infinite depth was conducted by Hudimac [I], who showed that, in addition to the surface mode, there is also an internal wave mode which has a pattern similar to the classical Kelvin wave [2]. Crapper [3] presented an analytical approach simpler than Hudimac’s and pointed out that the surface mode is only slightly affected by the stratification. For the case of continuous stratification, using Phillips’ theory [4], Shannan [5] arrived at the conclusion that there might appear one kind of the surface Kelvin wave mode and an infinite number of the internal Kelvin wave modes.

Applying ray theory in a continuously stratified fluid, Keller & Munk [6] derived an explicit expression for the far- and near-fields of kinematical wave patterns in the pycnocline. Yih & Zhu [7] further simplified the ray method and

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214

extended the results on the patterns to a strongly stratified fluid. Their results were only dependent on the dispersion relation, regardless of the types of waves. The ray methods, however, provide less information about the effects of various parameters on the amplitudes of induced waves.

In the aforementioned studies, although the kinematical properties of internal waves, including interfacial waves in a two-layer fluid, were discussed in detail, interactions between the two kinds of wave modes were not thoroughly examined. This might be of fundamental importance to gain a deep insight into the possible effects of the interaction of wave-modes.

The interaction between the internal- and surface-wave modes induced by a moving source in the upper layer of a two-layer ocean with finite depth was studied by Yeung 8z Nguyen [8] using the theory of potential flow. Recently their results were extended to a multi-layer model by Radko [9]. The effect of a source moving in the lower layer of a two-layer ocean with finite depth on the surface divergence was considered by Wei et a1 [ 101 with a similar approach. In addition, the problem of internal wave excited by a point source in a three-layer atmosphere was investigated in a linear formulation by Ter-Krikorov [ 1 13. However, these authors did not compare their results with any experiments, and the interfacial elevation calculated by the internal-wave mode under linearized interfacial conditions will tend to infinity as the density ratio of upper and lower layers of fluid approaches one.

The present study extends the above results to the case of a moving dipole in the lower layer of a two-layer fluid with finite depth. Our investigation is focused on the interaction of two resulting modes. An experimental study on the waves generated by a submerged sphere moving in a towing-tank is described and the qualitative agreement between our theoretical and experimental results is demonstrated.

2. Theoretical Approach

2.1. Formulation of Problem

A pair of a source and a sink moving horizontally at a constant speed U in a two-layer fluid of finite depth is considered. A rectangular coordinate system (0-xyz) attached to the source-sink pair is assumed, in which the x-y plane is put on the undisturbed interface between two fluid layers, the positive x-axis points in the moving direction and the positive z-axis points upward. The motion of ths pair of source-sink is confiied to the lower fluid layer and its coordinates are denoted by (r,, q,, <,) = ((-l)”+’x,, O,<,) , where s = 1 represents the source while s = 2 the sink (x, > 0 and<, < 0). The densities and depths of the upper and lower fluid layers are denoted by p, , h, and p2, h, , respectively.

215

It is assumed that the fluid is inviscid and incompressible and pl < p2 . Thus, the governing equations are

2

V2@(" = 0, and V2@(2' = ~ ( - l ) s ~ ' Q s S ( ~ - ~ s , y - ~ s , ~ - < s ) (1) s-1

where is the velocity potential of the upper ( m = 1 ) and lower ( m = 2 ) fluid layers, 6 ( x - gs , y - qs , z - 4,) the Dirac delta function, Qs the

intensity of the source or the sink, and Qs > 0 . It is assumed that the wave amplitudes are small in comparison with the

wavelength. Thus, the linearzied boundary conditions can be applied on the undisturbed free-surface ( z = h, ) and interface ( z = 0 ), that is,

,,@y +@? -@g) = 0 , ( 2 = I$ ),

0;) = @(2) z > ( z = 0 ),

@ Y ' = O , ( z = - h , ) , ( 5 )

(2)

(3)

(4)

y(oo@, (1) + @(I) -@t)) = oo@y + - pa?', ( z = 0 ),

where oo = g / U 2 , p is a positive constant and @?) represents the Rayleigh fictitious dissipative force [2,8].

2.2. Method of Green's Function

The solutions of Eq. (1) can be given as

(7) ~ ( 1 ) = and@(2) = 2 (-~)'Qs 06') , 4n.is 0

s=l

where Fs2 = (x - gs)2 + (y - vs)2 + ( z - <s)2 . The functions OF) (m=l , 2) satisfy the Lapace equation. By using the Fourier-transform, the following solutions are obtained:

By calculating the contour integrals in the k -plane, the expressions for @F) in the far field can be written as

where A' denotes the derivative of A with respect to k .

216

8-

2.3. Divergence field on the free surface

the free surface for the large parameter r,/h >> 1 Using the method of stationary phase, we can derive the divergence fields at

. . , ..-., -1 6 . . , . . , , . I . . .

It follows that there exists simultaneously the action of two lunds of modes, i.e., the surface-wave mode ( n = 1 ) and internal-wave mode ( n = 2 ), on the free surface divergence field. Each mode possesses two kinds of wave systems, i.e., the divergent waves ( 1 = 1.) and the transverse waves ( 1 = 2 ).

A horizontal dipole is a conventional example for the comparison with the laboratory study. Let Q, = Q2 = Q , and R denotes the radius of a sphere, we can easily arrive at the above conclusion with the same argument, i.e., the dipole moment M = lim (2xsQ) = k U R 3

2x, -+O,Q+m

A simple mathematical treatment in the expression (10) yields

n=l I=l

where the non-dimensional form of i $ ( r , v ) is

N

where co = 0 for A$ > 0, and co = 4 for i,$ < 0 .

The amplitude functions A$) represent the effects of the two resulting

wave modes on the free surface and interface. A{,\) and 4j are usually not of the same order of magnitude as shown by Wei et a1 [lo], but will have an equivalent influence on the free surface under certain conditions that y is not

close to one, the dipole approaches to the pycnocline and FrlFr, + 1. The effect of parameter Fr on the surface divergence field is shown in Fig.

field at y = 18” , y=O.4 , h,/h =0.286 , surface divergence of divergent wave at

and R/h = 0.1785. G0/h = -0.429 and R/h = 0.1785. y = 18” , h,/h = 0.286 , <,/h = -0.429

217

1 , in which the intersection between two solid (or dash) curves represents the exact magnitude of the equivalent influence of the two modes on the surface divergence field of the divergent (or transverse) wave, while the intersection between one solid and one dash curves represents the exact magnitude of the equivalent influence of the same mode on the surface divergence of the transverse and the divergent waves. The Froude number Fr, corresponding to

the former intersection is always near the critical value Frz = 0.378 . It can also be seen that the maximum for the surface-wave mode ( II = 1) is always larger than that for the internal-wave mode ( n = 2 ) for either the divergent wave or the transverse wave at the free surface.

Furthermore, the effect of different density ratio y on the amplitude A,$

of the divergent wave is examined. Fig. 2 shows that the value of A::! corresponding to Fr, decreases with increasing y . As y > 0.7 , Fr < 0.275 , A;:; for internal-wave mode is responsible completely for the divergence field at the free surface. Hereby it can be deduced that as y + 1 the intersection of

curves vanishes and also A$ gradually tends to zero, whle there exists only the surface-wave mode.

2.4.

written as

Wave elevation at the interface

The internal wave patterns due to the motion of a source-sink pair can be

Through the mathematical treatment similar to Sub-section 2.3, for the case of a horizontally moving dipole in the lower layer of two-layer fluid, we can easily deduce the interfacial elevation in the far-field ( r/h >> 1)

with c,=O for i,",(:)>O,and c,=4 for i $ ) < O . The numerical evaluation of Eq. (15) shows that the effect of wave modes

on the interfacial elevation is mainly contributed by the internal one, i.e.,

4:) >> A!,;), but is dependent simultaneously on the resulting modes for the

218

special case that y is not close to one, the dipole approaches the pycnocline and FrlFr, + 1.

The effect of Fr on the two modes for the divergent and transverse waves is illustrated in Fig. 3, in which the intersection between two solid (or dash) curves represents the point of the equivalent influence of the two different modes on the interfacial elevation of the transverse wave (or the divergent wave), while the intersection between one solid and one dash curves represents the point of the equivalent influence of the same mode on the interfacial elevation of the transverse and the divergent waves. The Froude number Fr, corresponding to the former intersection is also near the critical value Fr2 = 0.4924. It can also be deduced that for the greater value of y , the

s; y[ 3 2 I

0 0.0

2-1 f t 1

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fr

Fig. 3 Effects of Fr on the interfacial Fig. 4 Effects of Fr with different y on the elevation at v/ = 18" , y = 0.1 , interfacial elevation of divergent wave at h , / h =0.286 , c 0 / h = -0.429 , v/ = 18' , h, /h = 0.286 , R/h=0.1785. <o/h =-0.429, R/h = 0.1785.

maximum for the internal-wave mode (n=2) is always larger than that for the surface-wave mode (n=1) €or either the divergent wave or the transverse wave at the interface.

The effect of y on the amplitude A::! of the divergent wave is illustrated

in Fig. 4. It can be seen that the value of A::! corresponding to Fr, decreases

with the increase of y and for smaller y , the maximum A,(,:) for the surface-

wave mode (n=1) is larger than the maximum A;:! for the internal-wave mode

( n = 2 ) .

3. Comparison with experimental results

Experiments were carried out in a towing tank of 600cm in longth, 50cm in width and 50cm in depth. The density stratification of a two-fluid system was

219

formed by fresh water and saline water or by fresh (or saline) water and kerosene. The densities of fresh water and kerosene are l.OOg~m-~ and 0.82g~m-~ respectively, and the range of the density of saline water is in 1.03 1-1.127 gem- ’. The range of towing speed was restricted in 0.00-19.01cms~’. The sheet-light techmque and the continuously shooting by two CCD cameras were employed to carry out flow visualization and records the results from both top and side at the same time.

The “rigid lid” effect at the free surface is so strong that the elevation of the surface wave is hardly observed. Effects of two wave modes on the interface are only recorded.

Fig.5 The top-view images of ship-wave at the interface Fr = 0.2295 > Fr2

Fig.6 The top-view images of ship-wave at the interface. Fr = 0.1887 < Fr2

Fig.7 The top-view images of ship-wave at the interface. Fr = 0.1332 < Fr2

220

Figures 5-7 show the interfacial image produced by a moving sphere in the fresh water and kerosene media with h, = 2.0cm, h, = 5.0cm. The two-fluid system corresponds to the critical Froude numbers Fr, = 0.9808 and Fr, = 0.1952 . The sphere is located at the distance 3.Ocm blow the interface and moves at the constant velocity. The velocity is 19.0lcms-' in Fig. 5 , 15.63cms-' in Fig. 6 and 11.034cms-I in Fig. 7, and the corresponding the Froude numbers are 0.2295,0.1887 and 0.1332, respectively.

Figure 5(a) shows that when Fr = 0.2295 , stronger divergent wave with the internal-wave mode and weaker transverse wave with the surface-wave mode can be found, which agrees qualitatively with the theoretical result presented in Fig. 5(b). Figure 6(a) shows that when Fr < Fr2 there exist the divergent and transverse waves from both the internal-wave mode and the surface-wave mode, and the former one plays a more important role since y = 0.82. The qualitative consistency between the experimental and theoretical results is presented in Figs. 6 (a) and (b). As Fr becomes small, the transverse waves with the internal-wave mode dominate over the interface, which is shown by the experimental and theoretical results in Figs. 7(a) and (b).

4. Conclusions

By means of Green's function for single source (or sink), the velocity potentials of a dipole moving in the lower layer of a two-layer fluid with finite depth have been derived. The singularity of integral terms in these velocity potentials has proved to be independent of either the motion of the dipole in the upper or lower layers and of the numbers of point sources (or smks). The asymptotic far-field behavior of the divergence field at the free surface and the wave elevation at the interface has been deduced by applying the method of stationary phase. The calculated results show that if y is not close to one, the dipole approaches to the pycnocline and FrlFr, + 1, the equivalent influence of the surface- and internal-wave modes on both the divergence field at the surface and the wave elevation at the interface will be produced.

A series of experiments for the lower Froude number Fr has been conducted in a towing-tank at the Shanghai Institute of Applied Mathematics and Mechanics (SIAMM). In the two-layer fluid of the fresh waterkerosene medias (y=O.82) the experimental results near the lower critical Fr, are consistent qualitatively with the theoretical ones, though for Fr much smaller than Fr, the interfacial patterns by the experiments do not agree with those by the theory.

221

5. Acknowledgment

A great help for preparing a series of experiments from Mr. Zhong Baochang is sincerely acknowledged.

6. References

1. A.A. Hudimac, J. Fluid Mech., 11: 229-243 (1961). 2. H. Lamb, Hydrodynamics (Sixth Edition), Cambridge University Press

3. G.D. Crapper, J. Fluid Mech., 29: 667-672 (1967). 4. O.M. Phillips, Dynamics of the Upper Ocean (Second Edition). Cambridge

5. R.D. Shaman and M.D. Wurtele, J. Atmos. Sci., 40(2): 396-427 (1983). 6. J.B. Keller and W.H. Munk, Phys. Fluids, 13(6): 1425-1431 (1970). 7. C.H. Yih and S.P. Zhu, Q. Appl. Math., 47(1): 17-33 (1989). 8. R.W. Yeung and T.C. Nguyen, J. Eng. Math., 35: 85-107 (1999). 9. T. Radko, J. Ship Res., 45(1): 1-12 (2001).

(1932).

University Press (1977).

10. G. Wei, J.C. Le and S.Q. Dai, Appl. Math. Mech., 24(9): 20-36 (2003). 1 1. A.M. Ter-JWcorov, PMM J. Appl. Math. Mech., 66( 1): 59-64 (2002).

THIN FILM DYNAMICS IN A LIQUID LINED CIRCULAR PIPE

ROBERTO CAMASSA Department of Mathematics, University of North Carolina at Chapel Hill

Chapel Hill, NC 27599, USA

LONG LEE Department of Mathematics, University of North Carolina at Chapel Hall

Chapel Hill, N C 27599, USA

A two-phase core annular flow in a cylindrical pipe is considered. The inner core is assumed to be a pressure driven gas flow. The other phase is highly viscous fluid lining the inner wall of the pipe. Several models are presented, including the classic Poiseuille solution for two-phase flows, to predict the mean thickness of the liquid layer in the experiment by Kim et al. (1986), where a given fixed gas-flow rate drags the liquid injected into the pipe at a fixed feed rate. In particular, a nonlinear evolution equation based on the lubrication approximation is derived. The strong pressuredriven gas flow is incorporated as a forcing term into the equation for the liquid, with an effective viscosity for turbulent flow replacing the molecular viscosity of the gas. We study numerically the interface evolution of an initially axisymmetric disturbance of the annular film of viscous liquid. The mean height of the liquid layer in the experiment can be accurately predicted using this model, and the existence of the ring-like waves reported in the experiments is confirmed by the interfacial dynamics of the model.

1. Problem formulation and Poiseuille solutions

Kim, Greene, Sankaran, and M. A. Sackner conducted an experiment to investigate the effects of air drag in the dynamics of the mucus layer that coats the airways of human respiratory systems. The schematic diagram of the experiment is shown as in Figure l(a). A pair of identical glass tubes in series are connected via a cylindrical chamber. Liquid is forced at a fixed rate into an annular space formed between the chamber and the tube walls. The liquid is then carried away along the wall of the upper tube by an airflow that continuously passes upward through the tubes. When the liquid layer reaches the top of the upper tube, it is allowed to overflow naturally into the collection cup. Analysis of such a two-layer gas-liquid

222

223

flow within a tube is important for understanding the mobility of thin layers of biofluid in wetted membranes, especially when the core flow is a high- Reynolds-number, (strong) pressure driven gas flow. We remark that Kim et al. actually considered several fluids with viscoelastic properties, in addition to Newtonian fluids of same viscosity which were used as control. In this paper we consider the Newtonian case only.

Assuming axisymmetry for either the liquid or the gas mean flow, the incompressible Navier-Stokes equations in cylindrical coordinates govern the dynamics of both fluids,

P("t + uu, + wu,) = - pz + CL

1 r -d,(rUl) + 212 =o,

where the coordinates are (z,O, r ) , with associated velocity components (u, v, w). Here p is pressure, p is density, p is molecular viscosity, and g is gravity. For a flat interface, the boundary and interfacial conditions are: (i) no-slip boundary condition at the wall, (ii) the continuity of velocity across the interface, (iii) the continuity of normal and tangential stress across the interface. In searching for the Poiseuille solutions for this system, the only nonzero velocity component is the axial velocity u, which is a function depending solely on T . Let the axial velocity for the gas be u ( g ) , and let the liquid velocity be dZ). The radius of the pipe is r = a, and the interface is located at r = R. While applying conditions (i) - (iii) to equations (l), the axial velocity for each layer is

(2)

where u(') is kinematic viscosity and G(g) is defined by

224

COLLECTION C

L L

CONNECTR\IG

:UP

2 j 1 1 j 2

h R

g

r

Figure 1. geometry of the concentric two-phase flow system.

(a) Schematic diagram of experimental system of Kim et a]. 5 . (b) The

The gas flux can be obtained from the velocity u ( g ) ,

225

Similarly the liquid flux is

nR'g 1 = -- 7rG(') (a' - R')' + -[-(a - R') - a'log(a/R)](p(') - P ( ~ ) ) . 8 d Z ) 2p(') 2

(6) Given the fluxes Q(g) , Q(') , the three equations (4), (5), and (6) can be solved together for the three unknowns G ( g ) , G('), and R. The theoretical prediction for the liquid layer thickness is then given by the difference a- R.

1.1. Comparison with experimental data

In order to compare with the experimental data, the parameters used in the Poiseuille solutions are as follows: the density of air is 1.205 x g/m13, the density of the liquid is 0.96 g/m13, the molecular viscosity of air is 1.81 x cm'/sec, and gravity is 980.665 cm/sec'. Three different viscous fluids were tested in Kim et al.'s experiments, with viscosities 80,200, and 600 Poise, respectively. Four airflow rates 330,500,830, and 1170 ml/sec, and two liquid feed rates 0.5 and 1 ml/min were used.

A comparison of the experimental measurement of the mean liquid thickness with the Poiseuille solutions is shown in figure 2(a). The data show that for each fluid tested the mean liquid thickness decays as the airflow rate increases, and the Poiseuille solutions do capture this trend, while grossly over-estimating the liquid layer thickness.

1.2. Eflective viscosity

To explain the discrepancy between the experimental measurement and the theoretical prediction, we estimate the Reynolds numbers for pipe flows in the experiments. Using

ReD uave D / " , where D is the diameter of the airway, U,,, is the cross-sectional average of the air velocity, and v is the kinematic viscosity, the (fast) airflow rate of 1170 ml/s yields for Reynolds number ReD x (1170 x 0.8)/(n x (0.4)' x 0.15) M 12500, while for an airflow rate of 330 ml/sec (slowest in the experiment) the Reynolds number is ReD x 3500. It is known that for monophasic pipe flows the flow is said to be undergoing transition to turbulence at Reo x 2000. Above ReD M 3000, the pipe flow is fully turbulent

(page 117). Hence we expect the core airflow to be turbulent.

226

3.5

; ; - ,,,;;;*;----- =...=;;;;;a -

3-

E - E 2.5. - H t 2 - z

0.4 0.8 0.8 1 1.2 gas flow rate (Vsec) (4

1.2 0.5

0.4 0.6 0.8 1 gas flow rate (Vsec)

Figure 2. (a) Comparison of the Poiseuille solution with experimental data, where the liquid feed rate of the liquid is 0.5 ml/min. (b) The same as (a), but using effective viscosity.

Suppose that the liquid is laminar, and the core is fully turbulent with no entrained droplets, the molecular viscosity of the gas can be replaced by an effective (eddy) viscosity peff = ~ T , G , where ~ T , G is the turbulent viscosity of the gas flow. In evaluating ~ T , G , a conservative estimate of eddy viscosity for pipe flows can be derived from Blasius formula, neglecting any influence of rough or wavy surface (pp 422)

where p is molecular viscosity of gas, and ReD is the Reynolds number defined above. After replacing the molecular viscosity by the effective vis-

227

cosity in the Poiseuille solutions, we compare the experimental data with the resulting predictions. Figure 2(b) shows that the predictions using this effective viscosity formula (7) are an improvement with respect to those from the exact Poiseuille solution, at least quantitatively. However, the predictions no longer capture the qualitative trend of the decay rate in liquid thickness for increasing gas flow rate. Notice that we estimated the effective viscosity based on the assumption that the core airflow is turbulent. The estimate is less conservative when the airflow rate is high, which brings the prediction closer to the experimental data than those of low airflow rates. This results in steeper slopes than those using the exact Poiseuille solution. We remark that the turbulence closure we used here is the so-called zero-equation model 7, whereby the molecular viscosity is simply replaced by an effective viscosity in the Poiseuille solution.

2. A two-phase gas-liquid model

Surface tension has long been known to play an important destabilizing role for two-phase liquid layer flows in cylindrical geometries, and the instability is responsible for wave generation 2 7 4 . The theoretical prediction by means of Poiseuille solutions in the previous section neglects surface tension. Thus, these solutions fall short in two ways: (i) the predicted liquid layer is too thick, which implies that the liquid transport rate is lower than that of the experiment, and (ii) the z-independent interface does not agree with the undistorted smooth ring-shape waves observed in the experiment '. To investigate the role of surface tension effects in the rate of liquid transport, we derive a nonlinear evolution equation for the liquid thickness, based on lubrication or long-wave approximation. The derivation is similar to that in 6 , but carried out here in cylindrical coordinates.

2.1. The evolution equation for a bounded film

Consider a viscous-liquid flow lining the inner wall of a vertical circular pipe, surrounding a pressure-driven gas flow. The side-views of the concentric two-phase pipe flow is shown in Figure l(b). The governing equations for the liquid film are the incompressible axisymmetric Navier-Stokes equations in cylindrical coordinates (1). Equations (1) need to be nondimensionalized with respect to a set of scales. The length scale in the z direction is set by a typical wavelength A, while the scale in the radial direction is R1, the typical radius of the cross section of gas-column. The distortions is said to be of long-wave scale if E = Rl/A << 1. Other scales are set by an

228

axial velocity UO and a radial velocity WO. Continuity equation requires WO = EUO. The list of all dimensionless variables is as follows:

r* = r/R1, z* = ./A, u* = u/Uo, w* = w/Wo, (8) t* = t Uo/A, p* = ~p R~/ ( /A( ' ) Uo), T* = T R~/ ( /A( ' ) Uo),

where t* is the dimensionless time, T* is the dimensionless stress, and p* is the dimensionless pressure. If the dimensionless variables are substituted into the system ( 1 ) and gravity is absorbed in the pressure term, then a scaled system for the liquid is (drop stars for dimensionless variables hereafter):

1 r E Re(')(ut + uu, + wu,.) = - p , + - a,.(ruT) + c2u,,

1 -a,.(r w ) + u, =O, r

with Reynolds number Re(') = UO Rl/v('). Boundary conditions are

u = o , w = o , (10) at the inner wall of the pipe, r = a. Assuming that the surface tension of the liquid is constant in z, the conditions at the interface, r = R, are (i) continuity of tangential stress

(u,. + c2 w, ) ( l - c2 R,) + 2e2(w,. - u,) = dg)(l + ( E R2)2), (11) where ~ ( 9 ) is the dimensionless tangential stress of gas, and (ii) jump of normal component of stress is given by Laplace's formula

-p(l+ ( E Ri)) + 2e2(w, + U ~ ( E Rz)2) + E ~ ( u , + EW~)RZ

(12)

where d g ) is the dimensionless normal stress of gas, and C = Uop(')/y is the capillary number '. Finally, the dimensionless kinematic condition at the interface is

w = R~+uR, . (13) Integrating the continuity equation across the annular-sectional area of

the liquid and using (13) yields the layer-mean equation

229

for the interface location. With an approximate solution for u, the system can be closed and reduced to a single evolution equation for the interface location.

The leading order approximation of u can be obtained by taking the limit E --f 0 in equations (9)-(12). Since the liquid is highly viscous, Re(') is at most O( 1) and hence E Re(l) 0. While the higher-order terms vanish in the system of equations, similar to the planar case in 6 , it is essential to retain the surface tension terms at the leading order, for which one term is of O ( E ) , and the other is of O(e3). The O(E) term is responsible to the onset of instability of an initial disturbance, while the O(e3)-terrn stabilizes the evolution. Hence it is important to retain both of them, regardless of the order of magnitude. A more careful asymptotic analysis would be required to justify this, however it will not be pursued here. It is also worth pointing out that because of the geometry, surface tension in the planar case consists of only the O(e3) term. The long-wave regime in the planar case does not have any instability.

At leading order in E , the momentum and continuity equations, in dimensional form, become

-a,.(ru,.) = P, (t ) ( I )

R t - z a Z L 1 8 a urdr=O.

The boundary condition at r = a is

u = 0. (18) At the interface r = R we have

/J"),. = +),

From equations (15), (16), (18)-(20), the boundary value problem has a solution of the form

( 1 ) P Z u(r) = - (r2 - a2) + Aln(r/a), 441)

where A = R(dg) - 1/2pi1) R). From (20), pi') is known through pig), hence the terms that need to be determined in formula (21) are pig) and dg), the z-component of the pressure gradient and the shear stress from the gas flow.

230

To model the gas flow, we adopt the so-called zero equation turbulence closure 7. We proceed to compute the mean flow using again a Poiseuille solution, modulated to take into account the slow variation in z of the gas-liquid interface. More refined closure models can certainly be used, especially taking into account the wavy surface, but we leave this study to future work. The solution u for Poiseuille flow is of the form

Pz 2 u(r) = --T +c(z ) . 4 P

We need a boundary (interfacial) condition to determine c(z) . Since the dynamic viscosity ratio of liquid to gas is of order lo6, by continuity of velocity, it may be appropriate to assume that the velocity at the interface r = R ( z , t ) , is of order E or smaller. Hence we assume for the mean velocity field of the gas

With this velocity, the flux function for gas is

If the flux, Q(g) , is kept constant, the z-component of the gas pressure gradient is

Similarly, the tangential stress at the interface is

Substituting these two terms, (25) and (26), into (21) results in the approximation for the liquid velocity field. The velocity field is then put into the continuity equation (14) to obtain the nonlinear evolution equation for the interface location R. The dimensional evolution equation can finally be

231

written as: 2a2Q(g)p(g)

P R t + TR4 [ (i)2 - 11 R, - [ (i)4 - 1 + 4 ln(R/a)] R Z 2

- 3R + 4Rln(r/a) R,, 1 ln(R/a) R,,,R, 1

Normal-mode analysis can be applied to the above equation to investigate the behavior of small perturbations of a uniform film T = &. In particular, the behavior reflecting how (small) wavenumbers surface tension parameter , and mean radius &, are related to instability growth can be analyzed.

The experiment with which we compare the mean thickness of the liquid layer was conducted with a given fixed liquid feed rate. This implies that the mean liquid flux in our model must be kept constant. The flux function at any point z and time t is given by

2 p(z)Q(l) = p(9)Q(9) [ ( .%)2 - 11

1 a4 8 R2

+ 3 [-- + 4a2 - 3R2 + 4R2 ln(R/a) R,

+ 4a2 - 3R2 + 4R2 ln(R/a) R,,,. 1 It is worth pointing out that equation (27) does not make an assumption

of a small ratio of liquid thickness to pipe radius. Although this assumption makes a number of simplifying approximations possible, in particular to promote the surface tension effect to be of the same order as pressure, it might not be suitable for our problem. For instance, in 2 1 the ratio of liquid thickness to pipe radius must be assumed of order or smaller, unless the radius is fairly small, e.g. cm or smaller. For our problem the radius of pipe is 0.5 cm, and the mean liquid thickness ranges from 0.05 cm to 0.1 cm for different gas flow rates, provided the mean liquid flux is 0.5 ml/min.

For the initial-boundary-value problem (27) , the boundary conditions have to be determined from those of the experiment under the same scaling and approximations that went into deriving the model. It is not clear how

x

232

to pursue this, but convenient choice suitable for conditions away from the ends of (sufficiently long) pipes is offered by periodic boundary conditions. In the next section, we solve numerically this evolution equation with periodic boundary conditions, while maintaining a fixed mean liquid flux as required by the experiment. The resulting mean thickness of the liquid is our theoretical prediction.

2.2. Numerical solutions and comparison

Equation (27) is solved by the standard method of lines: first, spatial derivatives are discretized with the second-order center differencing, generating a system of coupled ordinary differential equations (ODE) in time for the variable R, then the ODE is solved by an ODE solver. Due to high spatial derivatives in the equation, time-step restriction to the resulting ODE is severe, and it is necessary to employ an implicit time-stepping method for the solution. Gear’s method as implemented in the ODE solver DLSODE from Netlib is used for solving our system of ODE, and has proved to be efficient and stable. Since equation (27) is solved with periodic boundary conditions, we look at the initial condition of the form

(29) 2TZ L R(0, 2 ) = RrJ - pcos -,

where L is the period, an the periodical domain is a multiple of L. The surface tension value of the liquid is required for solving equation

(27). Since the values of the silicone oils used in the experiment are not documented, we choose the values suggested by 3, page 147: the surface tension values for silicone oils whose viscosities range from 4.86 to 122 Poise vary from 21.1 to 21.5 dyn/cm. The density of the silicone oil listed in is 0.971-0.975 g/cm3 which is comparable to that of the experiment. The viscosity values of the silicone oils in the experiment are 80, 200, 600 Poise, respectively. Extrapolating linearly, we use the values 21.4, 21.75, and 22 dyn/cm for our calculations.

The experiment was conducted with a fixed gas flow rate and a fixed liquid feed rate. The fixed gas flow is incorporated into the model, but the fixed liquid feed rate is not satisfied automatically with periodic boundary conditions. Our strategy is to choose an initial mean height perturbed by a small-amplitude periodical wave as shown in equation (29). We run the code and monitor the liquid flux computed by formula (28), until the flux reaches a “quasi-steady” value consistent with that of the experiment. Then the averaged mean thickness over a time period is used as our theoretical

233

3.5

3 -

prediction. Figure 3 (a) is a plot of theoretical predictions against experimental

data using the model described in this section. The periodical domain is 2n, and the effective viscosity (7) is used in the calculation. We see the trend of the experiment is captured by the model, expect for group B with flow rates between 330 and 500 ml/sec.

- . r

-

2 -

.^

- - experiment - - prediction OroupA

0 group6 groupC

groupA o group B + groupC

+

- -. axnarimental valiipc

0.61

- . .. . . +

o *

0.4' ' ' ' ' ' ' ' ' ' ' ' I 0 2 4 8 8 10, 12 14 16 ,l8 20 22

multiple of the effective viscostty (equation (7))

Figure 3. (a) The comparison of the predicted liquid mean thickness with the experimental data. (b) Projected viscosity values for predicted mean thicknesses to match the experimental data. The gas flow rate is 830 ml/sec.

234

2.3. Modified effective viscosity

The effective viscosity (7) is derived from the Blasius formula for smooth- wall data of monophasic pipe flows. It is likely to be a conservative estimate for wavy surfaces, just as it is for rough surfaces. To modify the effective viscosity for our problem, we perform a sequence of simulations for one particular gas flow rate, with different effective viscosities, in the unit of ~ T , G defined in (7), until our predictions match experimental data. Fig- ure 4(a) shows the projected viscosity values for predicted mean thicknesses to match the experimental data. The gas flow rate is 830 ml/sec. By examining these values, we notice that the effective gas viscosity that matches the experiments may depend on the liquid viscosities. We hence propose a modified effective viscosity for our problem

where m = p(’ ) /p(g) , the ratio of liquid viscosity p(’) to the molecular viscosity of gas ~ ( 9 ) . The power law of m approximates the projected values shown in Figure 3(b), namely the associated x-coordinate when each curve touches the horizontal dash line.

Using this modified effective viscosity, we repeat the simulations for predicting the mean liquid thickness. Figure 4 shows the predictions match the experimental data. The liquid feed rate is 0.5 ml/min in (a) and 1 ml/min in (b). The mean liquid flux in the simulation is kept at 0.5 f 0.005 ml/min in Figure 4(b) and 1.0 f 0.005 ml/min in (b).

2.4. Surface tension parameter

If the power-law modification for the effective viscosity is not used in our calculations, the predicted mean liquid thicknesses are thicker than those reported in the experiment, as shown in Figure 3(a). To investigate the discrepancy, we plot the waveforms of the interfacial waves. Figure 5(a) shows snap shots of the interfacial waves that develops from the initial perturbation after a sufficiently long time has elapsed to saturate the initial instability. At time t = 168, wave snapshots are shown for each of the three groups, with liquid viscosity, from the top, of 80, 200, and 600 Poise respectively. The effective viscosity without the power-law modification is used for the calculation. The gas flow rate is 330 ml/sec. For all three groups, the wavelength is about 30 mm, the mean height is less than 2.7 mm, and the amplitude is less than 0.2 mm. It is clear that the waves

235

3.5

3-

E E 2.5

- experiment

0 grwpB

-

-

- experiment - - orediclicn

Figure 4. (a) A comparison of predicted mean liquid thickness with the experiment using the wave model and the modified effective viscosity. The liquid feed rate is 0.5 ml/min. (b) The same as (a) but with double liquid feed rate 1 ml/min

are within the long-wave regime, whereas the wave amplitudes are small compared with the layer means. A remedy for the thickness discrepancy can be offered by large amplitude waves. These waves could provide the additional liquid flux needed for the model, and the key for generating large amplitude waves is larger surface tension values.

suggests that for certain fluids a dynamic surface tension, in contrast to the static surface tension measured under equilibrium conditions, needs to be taken into account. It remains to be seen whether silicon oils of the kind used in the experiment are fluids for which this quantity plays a role, but the wavy motion in the experiment might

The paper by Bechtel et al.

236

belong to the class of flows where dynamic surface tension appears. As a preliminary attempt to investigate this dynamic effect and to parametrize it, we collect the different values of constant surface tensions needed to match the experimental data by performing a sequence of computations.

Figure 5(b) shows the interfacial dynamics for liquid with viscosity 80, 200, and 600 Poises respectively. The surface tension values are chosen to be 390, 400, and 600. These values are the ones for which the mean liquid fluxes are 0.5 f 0.01 ml/min after waves are saturated and propagate in and out of the periodic domain several times, while the mean liquid thicknesses match the experimental values. The gas flow in the calculation is 330 ml/sec and the effective viscosity without power-law modification is used. We see that, from the initial axisymmetric perturbation of the interface, large amplitude waves evolve and saturate into a wave-train propagating to the right. These waves are consistent with observation of undistorted smooth ring-shape waves reported in the paper of Kim et al. 5.

A more thorough study is necessary for us to characterize the surface tension effect in this problem. Nonetheless, the evolution equation derived in this paper might be used as a model to measure the dynamic surface tension in the experimental data of these highly viscous liquid in two-phase pipe flows.

3. Discussion and conclusion

We have presented several models for predicting the mean liquid thickness in the experiment by Kim et al. '. The models do capture the main behavior of the newtonian liquids reported in this reference. With the help of a modified effective viscosity, fitted on just one datum for each fluid group, the model evolution equation is able to predict with some quantitative agreement the experimental data. Nevertheless the model, to some extent, fares worse with respect to the data relative to the largest viscosity fluid, and this mismatch worsen for the fastest liquid feed rate 1 ml/min.

Improving the agreement and prediction capability of the model should be achievable by modelling more accurately the stress at the interface beyond the zero-equation turbulence model. In fact, the lack of sufficiently strong tangential stress at the interface results in weaker momentum transfer from the gas to the liquid, and hence in a reduced liquid transport rate. Thus, a thinner liquid layer cannot support a fixed liquid feed. Increasing momentum transfer requires either a larger surface tension, which increases the pressure contribution to this transfer, and/or an effective viscosity for

237

1.8 1.6

0 10 20 30 40 50 60

2 20 30 40

2.5

2.3 0 10 20 30 40 50 60

(b) mm (4

10 20 30 40 50 60

1

OO

'0 10 20 30 40 50 60 5 A

1

OO 10 20 30 40 50 60 mm

Figure 5. (a) Interface dynamics at t = 168. From the top, the liquid viscosity is 80, 200, and 600 Poise respectively. The effective viscosity without power-law modification is used. The gas flow rate is 330 ml/sec. (b) Same as (a), but with surface tension values chosen so that the predicted mean thicknesses match the experimental data.

the gas that increases the tangential stress contribution. A model similar to the wind-over-waves coupling for pipe flows could generate enhanced wave-induced shear stress, with turbulent gas Aow over non-breaking waves. Whether this approach can make up for the inadequate momentum transfer at the interface remains to be seen and is under investigation.

Treating surface tension as a parameter offers an alternative to the route of modified effective viscosity. In this respect, we notice that dynamical (as opposed to static) surface tension variation by order of magnitudes have been reported in the literature for certain liquid mixtures. The applicabil-

238

ity t o the present experiment is another direction of investigation we are currently pursuing.

References

1. S. E. Bechtel, J. A. Cooper, M. G. Forest, N. A. Petersson, D. L. Reichard, A. Saleh, and V. Venkataramanan. A new model to determine dynamic surface tension and elongational viscosity using oscillating jet measurements J. Fluid Mech., 293:379-403, 1995

2. P. S. Hammond. Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe J. Fluid Mech., 137:363-384, 1983.

3. D. D. Joseph and Y . Y . Renardy. Fundamentals of two-fluid dynamics: part I: mathematical theory and applications Springer-Verlag, 1992.

4. V. Kerchman. Strongly nonlinear interfacial dynamics in core-annular flows J . Fluid Mech., 290:131-166, 1995.

5. C. S. Kim, M. A. Greene, S. Sankaran, and M. A. Sackner. Mucus transport in the airway by twephase gas-liquid flow mechanism - continuous-flow model Journal of Applied Physiology, 60:908-917, 1986.

6. A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films Reviews of Modern Physics, 69:931-980, 1997.

7. F. M. White. Viscous Fluid Flow 2nd edition, McGraw-Hill, New York, 1991.

TRANSVERSE WAVES IN A CHANNEL WITH RECTANGULAR CROSS SECTION

L.M. DENG AND A.T. CHWANG Department of Mechanical Engineering,

The University of Hong Kong, Pokfulam Road, Hong Kong

E-mail: [email protected]; atchwang@hkucc. hku.hk

Transverse waves are interesting phenomena that occur in the direction parallel to the cross section of a channel when incident waves propagate along the water channel. They are studied in a channel with rectangular cross section 12 m long, 0.73 m wide and 0.90 m deep. Incident regular waves are generated by a computer-controlled piston-type wavemaker operating at one end of the channel and propagate away along the longitudinal direction of the channel. Transverse waves are found in the cross section when the wavelength of the incident wave is smaller than twice the channel width. Transverse waves, however, do not occur in the continuous mode but at some specific discrete frequencies. Different wave mode appears only when the channel width is an integer multiple of the half wavelength of the incident wave. By means of capacitance-type wave gauges placed in the water channel, the evolution of waves at different positions is recorded. Under the assumption of linear waves, the transfer function method based on the linear wave theory is employed to obtain the amplitude of transverse waves. Transverse waves with different mode number are observed in the channel and the mechanism and characteristics of transverse waves are discussed. Finally, the importance of transverse waves in the laboratory and in the natural environment as well as the relationship between original waves and transverse waves are discussed.

1. Introduction

Transverse waves were observed by Zhu in a small, fixed water flume where the incident waves propagated along the flume. Zhu and Chwang conducted a series of experiments in a 30 cm wide wave flume and took the initial effort to give a theoretical explanation on transverse waves. They found that a transverse wave has the same frequency as that of the incident wave and it took place when the wavelength of the incident wave is less than or equal to twice of the width of the wave flume. The interaction between incident waves and their oblique reflection of relatively short waves was suggested to be the reason for the occurrence of transverse waves2.

239

240

Similar phenomena include cross-waves and sloshing waves. They are all standing waves oscillating between the sidewalls of a channel. Cross-waves are generated when a plane wavemaker operates in a rectangular channel and produces propagating plane waves. Cross-waves have been proved to have a frequency that is one-half that of the wavemaker 3 9 4 9 5 t 6 , 7 . Sloshing waves, however, are created when the container is subjected to a periodic horizontal oscillation. Sloshing waves have the same frequency as that of the container oscillation 899t10711. Crosswaves and sloshing waves are comparable with transverse waves. Cross-waves are similar to transverse waves in lacking a container oscillation, but differ in having a different wave frequency. On the other hand, sloshing waves resemble transverse waves in having the same wave frequency as the excitation frequency. However, the source of excitation is different: a container periodical oscillation for sloshing waves and an incident wave for transverse waves. The present study aims to further investigate the characteristics and occurrence conditions of transverse waves in a wide rectangular channel. To generate more modes of transverse waves, we use a large wave channel than the 30 cm wide channel used by Zhu and Chwang '. In the present experiments, transverse waves are successfully generated and carefully described. The possible reasons for the generation of transverse waves are discussed.

2. Experimental Configuration

G1

122 142

4-

Figure 1. The location of five wave gages in the water channel.

Experiments were performed in a water channel 12 m long, 0.73 m wide and 0.90 m deep. Absorbing facilities were not fixed at the end of the channel in order to make convenient observation. At another end of the channel, a piston-type wavemaker with only one pushing plate was used to generate incident waves. The movement of the wavemaker was controlled by a computer system. Regular waves with reflection absorbing feature were applied in the present study. In this program, the wavemaker can partially absorb reflected waves.

241

Surface elevations were measured by the capacitance-type wave gages. Five gages labeled as G1 to G5 were placed at different locations as shown in Fig. 1. G1, G2 and G3 are on the centerline of the channel. The distance from the wavemaker to G1 is about 8 m, while G4 and G5 are in the same cross section with G3 at a distance of 1.42 m away from the end wall of the channel. G4 and G5 were near the sidewalls and the span between these two gages is 23 cm. All gages were calibrated before the experiments. Data were collected at a rate of 50 times per second. A stopwatch was used to obtain the frequency of transverse waves.

3. Experimental Results

The tested period of incident waves ranges from 0.50 sec to 1.0 sec. At every tested period, the wave height of incident waves was set at five different values of 1 cm, 2 cm, 3 cm, 4 cm and 5 cm. For longer wavelengths, the measurement was also performed at a larger wave height of 6 cm. The water depth is kept at a level of 40 cm during all tests.

Incident waves, the plane waves, propagated away from the wavemaker and were reflected by the endwall of the channel. Consequently, transverse waves, with the crests perpendicular to the wavemaker occurred in the channel after certain time t. Time t is different with different periods, which is generally after standing waves were formed by the incident waves and the reflected waves. When transverse waves occurred, the time series recordings of wave gages spanned across the channel width were different from those before the occurrence of transverse waves. The data of two wave gages G4 and G5 are shown in Figures 2, 3 and 4. Before the generation of transverse waves, the recorded values of two gages were approximately the same. The values were obviously different after transverse waves occurred.

With the occurrence of transverse waves, the wave pattern on the free surface was accordingly changed. For some cases, the original steady state was broken and wave breaking took place. In our experiments, three steady modes of transverse waves were captured in the cross section as shown in Figures 5, 6 and 7. The mode number n is an integer of the half-wavelength across the channel. In these figures, the pictures on the left-hand side were captured by a digital camera and the diagrams on the right-hand side were drawn according to the relations between the channel width and the wavelength. These figures schematically show the mode differences.

The frequency of transverse waves was found to be the same as that of the wavemaker motion. The occurrence frequency of transverse waves is not

242

I i 8s

Iw: Fo tf (10 90 165 i t 0 17s 8s 411

(a) before the occurrence of transverse waves (b) after the occurrence of transverse waves

The surface elevation versus the time at T = 1.0 sec and H = 6 cm recorded Figure 2. by G4 (dash line) and G5 (solid line).

- 8 1 . - I M TL 34 71; ?8 m L65 tB? wo lflt L18 t7$

111: *n





continuous in the tested frequency range. When the frequencies are 1.0 Hz, 1.25 Hz, 1.46 Hz, and 1.79 Hz, transverse waves could be observed clearly in

243

Figure 5. Photograph of the first-mode (N = 1) oscillation.

Figure 6. Photograph of the second-mode (N = 2) oscillation.

Figure 7. Photograph of the third-mode (N = 3) oscillation.

the cross section. However, for frequencies different from the above values, no clear transverse waves were observed. At the occurrence frequency, not

244

all the amplitudes of incident waves would produce transverse waves. When the wave frequency is 1.0 Hz, transverse waves were not found when the incident-wave amplitude A was smaller than 3 cm. However, transverse waves occurred at amplitude of 1.5 cm when the frequency is 1.46 Hz. The frequencies and amplitudes of incident waves when clear transverse waves could be observed are shown in Table 1. As the frequency increases, the necessary amplitude of incident waves to excite transverse waves decreases.

Based on the linear wave theory, the amplitude of transverse waves was obtained by revising the transfer function method of Zhu l2 and Zhu and Chwang 13, in which transverse waves were assumed to be regular waves in the cross section and the recording data from the wave gages consisted of three parts: the progressive plane waves, the corresponding reflected waves and transverse waves. Table 1 also shows the nondimensional amplitudes of transverse waves scaled by the mean water depth h when they occurred at given periods and no serious breaking took place.

Table 1. incident waves at different periods.

Amplitudes of transverse and

T(sec) f ( H z ) a / h A / h

1.000

0.800

0.700

0.700

0.685

0.685

0.680

0.680

0.650

0.650

0.600

0.600

0.559

0.559

0.550

0.550

1.00

1.25

1.43

1.43

1.46

1.46

1.47

1.47

1.54

1.54

1.67

1.67

1.79

1.79

1.82

1.82

~~

0.0648

0.0371

0.0239

0.0391

0.0265

0.0300

0.0384

0.0428

0.0266

0.0348

0.0130

0.0093

0.0249

0.0150

0.0149

0.0113

0.0750

0.0375

0.0250

0.0375

0.0375

0.0500

0.0375

0.0500

0.0250

0.0375

0.0250

0.0375

0.0250

0.0375

0.0250

0.0375

From Table 1, it is found that amplitudes a of transverse waves are very close to those of incident waves A . The maximum wave amplitudes

245

of transverse waves mostly appear near the mode frequencies, which are indicated as fi= 1.0 Hz (the first mode), f2=1.46 Hz (the second mode), and f3=1.79 Hz (the third mode). At the frequency of 1.25 Hz, which is in between the first and the second mode, large amplitude of transverse waves was also found. This is probably due to the interaction of the first mode and the fifth mode.

4. Discussion and Conclusions

From the experiments, transverse waves were observed at some discrete frequencies. They first appeared near the wavemaker and then gradually moved into the whole channel. The amplitude and frequency of incident waves are found to play an important role in the excitation of transverse waves. As the frequency of incident waves decreases, larger amplitude to excite transverse waves is required.

In comparison with cross-waves and sloshing waves, transverse waves are more complicated. If the transverse waves were caused by an asymmetric wavemaker oscillation, they would grow in the whole channel at all frequencies. However, they did not happen in this way. The occurrence frequencies suggest that transverse waves are probably resonance phenomena. The first three modes were observed clearly in the experiments. It is noted from the experimental results that the maximum amplitude of transverse waves occurs when the flume width is nearly an integer multiple of the half-wavelength of incident waves. This can be explained from the dispersion relation w2=(2rg/X) tanh(2~h/X),where X is the wavelength, w the angular frequency, g the gravitational acceleration and h the water depth. Based on the mode analysis, the wavelength of transverse waves should be X=(2W/n), where W is the channel width and n the mode number. Therefore, the n-th mode of transverse waves occurs a t frequency w,, wi=(nrg/W) tanh(nrh/W). Table 1 shows that transverse waves also occur at frequency 1.25 Hz, which is not a mode frequency. This is probably due to the interaction of the first mode and the fifth mode.

The following conclusions can be drawn from the present experimental study:

(1) Transverse waves occur when the wavelength of incident waves is

(2) Transverse waves have discrete frequencies, which are near the mode

(3) The incident waves with large enough amplitude are necessary to

less than or equal to twice the channel width.

frequencies.

246

excite transverse waves at the occurrence frequencies.

Since transverse waves may occur in natural environment, for example, in a harbor or along a narrow sea-route, they are worthy to be investigated further.

Acknowledgments

The authors would like to thank Dr. Yuhai Chen and Mr. Toman Mok for their technical assistance. This research was sponsored by the Hong Kong Research Grants Council under Grant Number HKU 7076/023.

References 1. S. T. Zhu, PhD thesis, The University of Hong Kong, (1999). 2. S . T. Zhu and A. T. Chwang, Proc. 4th Int. Conf. Hydro. Yokohama, 497

3. C. J. R. Garret, J. Fluid Mech. 41, 837 (1970). 4. B. J. S. Barnard and W. G. Pritchard, J.Fluid Mech. 55, 245 (1972). 5. J . J. Mahony, J. Fluid Mech. 55, 229 (1972). 6. A. F. Jones, J. Fluid Mech. 138, 53 (1984). 7. J. W. Miles, J. Fluid Mech. 186, 119 (1988). 8. B. J. S. Barnard, J. J. Mahony and W. G. Pritchard, Phil. Trans. R. SOC.

London A286, 87 (1977). 9. L. Shemer, E. Kit and T. Miloh, I n nonlinear water waves, IUTAM Symposium

Tokyo/Janpan, 103 (1987). 10. E. Kit, L. Shemer and T. Miloh, J. Fluid Mech., 181, 265 (1987). 11. L. Shemer and E. Kit, Fluid Dyn. Res. 4, 89 (1988). 12. S. T. Zhu, Ocean Eng. 26, 1435 (1999). 13. S. T. Zhu and A. T. Chwnag, J. Eng. Mech. 127, 300 (2001).

(2000).

LONG TIME EVOLUTION OF NONLRJEAR WAVE TRAINS IN DEEP WATER

HWUNG-HWENG HWUNG

Dep f. of Hydraulics and Ocean Engineering, National Cheng-Kung University Tainan Hydraulics Lab., National Cheng Kung Universiv, Tainan ,Taiwan

h h hwung@mail. ncku. edu. tw

WEN-SON CHIANG

Dept. of Hydraulics and Ocean Engineering, National Cheng-Kung University Tainan Hydraulics Lab., National Cheng Kung Universiv, Tainan ,Taiwan

chws@mail. ncku.edu. tw

A series of experimental studies on nonlinear wave modulation in deep water are conducted in a super tank (300m*5.2m*5m) at Tainan Hydraulics Laboratory (THL). In particular, the effects of sideband space (dimensionless frequency difference between the carrier wave and the imposed sideband) on long time evolution of nonlinear wave trains are investigated. For initially smaller sideband space, the multiple downshift of the spectrum is observed through a series of wave breaking. The permanent downshift is observed for the sideband space slightly smaller than the most unstable mode. On the contrary, when the sideband space is larger than the most unstable mode, the transient downshift is observed. In the case that the sideband space is greatly larger than the most unstable mode, no downshift is observed even after a series of wave breaking.

1. Introduction

Evolution of nonlinear wave train has provided researchers with a wide variety of interesting phenomena. Benjamin and Feir (1967) examined experimentally and analytically, that the slowly modulated Stokes wave is unstable. Their analysis predicted that the unstable sideband components would grow exponentially with a time rate, whch depends on dimensionless sideband space (6) and initial wave steepness ( E ) . For weakly nonlinear and narrow banded wave trains, the leading order modulated surface displacement satisfies the nonlinear Schrodinger equation (NLS). On the basis of NLS theory, it is expected to have the recurzence of initial state and the envelope of surface elevation which is symmetric with respect to the peak. Meanwhile, the

247

248

numerical simulation of NLS equation has revealed that a wave field with initially narrow-band could breakdown due to the energy leakage to high wave number modes which violates its narrow bandwidth constrain. Lo and Mei (1985) solved numerically the extended NLS equation, which was first derived by Dysthe (1979) and showed the asymmetric growth of an initially symmetric wave train, but, the permanent downshift was not seen. Trulsen and Dysthe (1996) further developed a mo el in which the restriction of spectrum bandwidth was relaxed to 0 ('b) while retaining the same accuracy in nonlinearity. Trulsen and Dysthe (1997) investigated numerically the conservative evolution of weakly nonlinear narrow-band waves. No permanent frequency downshift was observed in the two-dimensional simulation.

Pertinent experiments have lagged behind theoretical and numerical studies. Lake et al. (1977) conducted experiments on the evolution of nonlinear wave trains, in which confurnation of Benjamin and Feir's analysis was claimed. Melville (1982) found that the spectrum evolution of nonlinear wave train is not only restricted to a few discrete frequencies but also involves a growing continuous spectrum. The systematic tests of well-controlled experiments were conducted by Tulin and Waseda (1999). The near recurrence of wave train without frequency downshift was observed for non-breaking case. The further evolution after frequency downshift revealed that the amplitude of the lower sideband nearly coincides with the amplitude of carrier wave at post breaking stage.

The long time evolution of initially modulated wave train is examined experimentally in a large tank. The results for wave with constant wave steepness and different sideband spaces are discussed in this paper.

2. Experiments

The experiments were performed in a large wave flume at THL, Taiwan, which is 300 m long, 5.0 m wide and 5.2 m deep. Figure 1 shows the schematic diagram of experimental setup. A programmable, high resolution wave maker is located at one end of the tank with an effective wave-absorbing beach at the opposite end. The wave maker is a piston type paddle activated by a hydraulic cylinder. The motion of hydraulic cylinder is commanded by a programmable controller which takes an external input signal for the generating wave motion. The evolution of surface wave trains are recorded as they propagated through

249

0.625

0.625

the tank, using capacitance-type wave amplitude gauges which were made at THL, total 66 wires, located at 15 m to 240 m downstream from the wave maker. The data were acquired using the PC based Multi-Nodes-Data-Acquisition- System ( W A S ) developed by THL. The time series of water surface elevation are acquired simultaneously by 25 Hz data rate and stored for further processes. The time interval between two successive experiments is about 35 min.

0.1 0.80,0.90,0.95, 1.0 0.05,0.10,0.15,0.2,0.25

0.14 0.3,0.5,0.7 0.9, 1.0, 1.2 0.05,0.10,0.15,0.2,0.25

46.1111 142111 L a 82m 7- 1-

Fig. 1. Experimental setup and configuration of the wave tank.

The experiments were performed using computer-generated wave forms, given by equation (l), given below, as input to the wave-maker servo-system. In

Table 1 Experimental wave conditions

I 0.625 I 0.17 I 0.3,0.5,0.70.9, 1.0, 1.2 I 0.05,0.10,0.15,0.2,0.25 I order to minimize the transient wave front from initial wave generation, a ramp function is used to the wavemaking system during the starting and ending of waveboard motion. The measured wave amplitude from gauge at 15 m station is used to define the initial wave amplitude and steepness. The experimental conditions are tabulated in Table 1. Equation (1) is given next:

250

~ ( t ) = a, sin(@$) + a, sin(o,t + 4,) W* = W, f Am 4, = - 4 4

where 7 is surface displacement, a, , a, are the amplitudes of carrier wave and sidebands, W, is angular frequency, Am is sideband space, x is horizontal coordinate, t is time. The piston stroke amplitude is related to the desired wave amplitude by linear theory.

The amplitude spectrum appears to be a useful method in analyzing the wave evolution phenomena such as sideband amplitude levels, initial growth rate of sideband. The spectra are obtained using a discrete Fourier transfonn with the Hanning window applied. Typical calculations are done for a frequency range from 0 to 12.5 Hz and a resolution bandwidth of 0.012 Hz which is small enough to distinguish different wave modes; frequency difference between the carrier and imposed sideband waves ranges from 0.03 to 0.125 Hz in the experiments. From the spectrum, the amplitudes of wave modes can be calculated as the square root of total energy of spectral peaks.

3. Evolution of non-breaking type

In this section, the near recurrence evolution of a nonlinear wave train with a long carrier wavelength (-4m) is presented. The spatial development of the wave train is shown in Fig 2. The initially generated wave train exhibits a liner superposition of one carrier wave and two small sidebands. During the initial stage of evolution, the modulation of surface elevation exposes a slow oscillation with an increasing trend accompanied with the symmetric growth of the imposed sidebands components. As the wave train further evolves, the modulation becomes stronger and the envelope of the wave group leans forward which the corresponding energy spreads over the frequencies higher than the upper sideband shown in Fig 3. A maximum modulation is then reached where the frequency downshift of the spectrum peak is observed temporarily. At later stage, the wave train demodulates and eventually the greatest portion of the energy still resides in the original three wave components. However, the energy spreads over the lower and higher frequencies, which is broadening out the spectrum. This prevents the full recurrence because the earlier spread of energy

251

is not perfectly transferred back to the original wave components. The number of wave in a group is temporarily reduced near the location of the peak modulation that is related to the observed “crest-lost” phenomenon before the peak modulation. However, the initial number of waves is recovered after the “crest-gained” phenomenon observed during the demodulation process. Fig. 4 shows the evolution of the dimensionless amplitudes including one carrier wave and a-pair imposed sidebands.

4. Evolution of breaking type

The modulated wave train eventually evolved into breaking wave for initial wave steepness greater than 0.12. A set of experimental results are presented here with constant wave steepness but different sideband spaces. The results illustrate the effect of sideband space on the long time evolution of nonlinear wave trains. Several different types of evolution including the multiple downshift, the permanent downshift and the transient downshift are discussed in the following.

For the initial stage, the continuous spectrum is investigated on the high frequency side of the spectra peak at a sideband space smaller than the most unstable mode according to the calculation of Tulin and Waseda (1999) based on Krasitskii’s (1994) theory. The continuous spectrum is developing with fetch as shown in Fig. 5 . The multiple downshift of the wave spectrum for wave train with S=0.5 is illustrated in Fig. 6, which shows the snapshot of amplitude contours. The results indicate that the wave train evolves gradually to lower frequency, which is accompanied with the growth of continuous spectrum. Eventually, the spectrum peak resides roughly at the most unstable lower sideband. The evolution of the continuous spectrum is accelerated for initially smaller sideband space.

The evolution of the dimensionless amplitudes is shown in Fig. 7 and 8 for different sideband space, respectively. Generally, the important features of sidebands instability such as exponential growth, asymmetry of the upper and lower sidebands near the maximum modulation, and the frequency downshift after breaking, are observed in our experiments which confirm the experimental results of Tulin and Waseda (1999). Due to the limitation of tank length, they displayed the evolution of the sidebands for a series of measurements in which sidebands of different amplitudes were imposed at the paddle. Then, the

252

measurements were patched together to obtain the evolution in an effectively longer channel. Fortunately, in this paper, the simultaneous measurement of the sidebands evolution is investigated. Two evidently different evolutions at post breaking stage are observed. In Fig. 7, permanent downshift of the wave spectra is displayed. In Fig. 8, the camer wave eventually recovers its amplitude accompanied with the destruction of the lower sideband. The frequency downshift is transient in this case. This phenomenon is also observed in the evolution of naturally evolving sidebands in regular wave experiments.

-0.2 I I I I I

0 5 10 15 20 25 time(sec)

I.

-0.2 I I I I I

0 5 10 15 20 25 time(sec)

Fig. 2. The surface elevations at several selected locations for wave w, = 0.625 Hz, E = 0.1 , 6 = 1.0 , a,/a, w 0.25 (time coordinate is moving with the linear group velocity of waves in deep water).

253

ch6 kcx=74 0.1

chi kcx=36 0.0 I\ A - I

0.0 0.5 1 .o 1.5 frequency(Hz)

?

L E 2 8 m 3 .f

i3 d z 4

-1 -3 -5

E,

0.0 1 .o 2.0 3.0 frequency(Hz)

Fig. 3. The spectrum evolution of initially modulated wave train. w, = 0.625 Hz, E = 0.1 , 6 = 1 .O , a,/a, = 0.25 . (a) linear-linear plot of the spectrum evolution (b) semi-log plot of the spectrum evolution.

V.V

0 50 100 150 200 250 300 350 400 450 500 550 600 kx

Fig. 4. The normalized amplitudes evolution of the imposed sidebands and the carrier wave versus dimensionless fetch.

254

1 03" lo3"

1 02' I 0''

- 4

S l O 2 O $32"

.z 4 - d

;=

g l O 1 5 E

lo1" lo1"

lo5 10'

lo" 10"

0.0 1 .o 2.0 3.0 0.0 1 .o 2.0 3.0 frequency Wz) frequency(Hz)

Fig. 5. The evolution of amplitude spectra for different sideband spaces for wave , wc=0.625 Hz, ~ = 0 . 1 7 . ( a ) S=O.5 (b) 6 = 0 . 3 .

. . . . . . . .......... .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ , ........

. . . . . . .

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (m-mJ'mp

~

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Fig. 6. The snapshot of amplitude contours for wave train with w, = 0.625 Hz, E = 0.17, 6 = 0.5 as a function of dimensionless frequency difference and fetch.

255

1.2 I

0 50 100 150 200 250 300 350 400 kr

Fig. 7. The normalized amplitudes evolution of the carrier and imposed sidebands versus dimensionless fetch. The initial wave condition is w, = 0.625 Hz, E = 0.17, 6 = 0.7 .

0.4

0.1

nn

oD000000000000

-.- 0 40 80 120 160 2W 240 180 320 360 400

kr

Fig. 8. The normalized amplitudes evolution of the carrier and imposed sidebands versus. dimensionless fetch. The initial wave condition is w, = 0.625 Hz, E = 0.17 , 6 = 0.95 .

5. Conclusions

The long time evolution of modulated wave trains is investigated experimentally. For non-breaking cases, the wave recurzence of wave train and the asymmetrical growth of sideband amplitudes are observed in our experiments. For breaking cases, the evolution of wave spectrum strongly depends on the sideband space. In particular, for initially smaller sideband space, the multiple downshift of the spectrum appears in the wave flume. The permanent downshift is observed for the sideband space slightly smaller than the most unstable mode. On the other hand, when the sideband space is slightly larger than the most unstable mode, the transient downshift is evolved in the experiments.

256

Acknowledgments

The authors gratefully acknowledge funding from the Ministry of Education, Taiwan, Grant No. A-9 1 -E-FA09-7-3.

References

1.

2.

3.

4.

5.

6.

7.

8.

9.

Benjamin, T. B. and Feir, J.E.(1967), “The disintegration of wave trains on deep water. Part 1. Theory.”, J. Fluid Mech., Vol. 27, pp.417-430 Dysthe K. B. (1979), ‘Note on a modification to the nonlinear Schrodinger equation for application to deep water waves”, Proc. R. SOC. Lond., A. 369, pp.105-114. Krasitskii, V. P. (1994), “On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves”, J. Fluid Mech., Vol. 272, pp.1-20 Lake , B. M., Yuen, H. C., Rungaldier, H. and Ferguson, W.E. (1977) “Nonlinear deep-water waves: theory and experiment. Pat 2. Evolution of a continuous train.”, J. Fluid Mech., Vo1.83, part 1, pp.49-74. Lo, E. and Mei, C. C. (1985), “A numerical study of water-wave modulation based on a higher-order nonlinear Schrodinger equation”, J. Fluid Mech., Vol. 150, pp.

Melville, W. K. (1982) “The instability and breaking of deep-water waves.”, J. Fluid Mech., Vol. 115, pp.165-185. Trulsen, K. and Dysthe, K. B. (1996), “A modified nonlinear Schrodinger equation for broader bandwidth gravity waves on deep water.”, Wave Motion, Vol. 24,

Trulsen, K. and Dysthe, K. B. (1997), “Frequency downshift in three-dimensional wave trains in a deep basin”, J. Fluid Mech., Vol. 352, pp. 359-373. Tulin, M. P. and Waseda, T. (1999) “Laboratory observations of wave group evolution, including breaking effects.”, J. Fluid Mech., Vol. 378, pp. 197-232.

395-416.

pp.281-289

ON THE ZHANGWU RUN-UP MODEL

HONGQIANG ZHOU Dept. of Civil and Environmental Engrg., Univ. of Hawaii at Manoa

Honolulu, Hawaii 96822, USA

MICHELLE H. TENG Dept. of Civil and Environmental Engrg., Univ. of Hawaii at Manoa

Honolulu, Hawaii 96822, USA

KELIE FENG M& E Pacific Inc., Honolulu, Hawaii 96813, USA

Experiments were carried out to measure solitary wave run-up on smooth plane slopes of 15 and 20 degrees. The innovative Eulerian-Lagrangian run-up model developed by Zhang and Wu [8, 91 was applied to simulate the wave run-up. The predicted results showed excellent agreement with the experimental data further validating the accuracy of the Zhang-Wu run-up model. In addition, the Manning's term is added to the existing equations to attempt to model the frictional effect during long wave run-up in the present study.

1. Introduction

When a large ocean long wave such as a tsunami or a storm surge reaches the shore, it can run up on coastal land and cause severe flooding to the coastal community. It is important for scientists to develop numerical models that can predict how far inland a long wave can inundate. In Figure 1, R represents the run-up and X is the inundation. Thus far, several models based on the depth- averaged shallow water equations have been developed for predicting tsunami run-up including the Japanese model by Imamura et al. [3], the Cornell COMCOT model by Liu et al. [4] (applied in Gica [ 11 for the Pacific Ocean) and the USC/PMEL MOST model by Titov and Synolalus [6 ] , among others.

In the aforementioned models, the wave front is treated as a moving boundary whch is tracked and updated at each time step. Recently, Zhang [8] and Zhang, Wu and Hou [9] developed a new Eulerian-Lagrangian run-up model. This model is unique in that by introducing a proper transformation, the

257

258

moving wave front is transformed into a fixed location, which makes the numerical simulation easier. In addition, since the model is a hybrid Eulerian- Lagrangian model, the transformed equations are shorter than the equations under the fully Lagrangian description (Zelt [7]).

In the present study, the Zhang-Wu run-up model is re-visited from Zhang [8] and Zhang, Wu and Hou [9]’s studies. Specifically, more experiments on wave run-up were carried out to further validate the Zhang-Wu model. To account for the frictional effect during long wave run-up, an attempt is made to add an additional term based on the Manning’s equation to the Zhang-Wu run-up model. The effectiveness of the Manning’s term in modeling the frictional effect is discussed.

2. The Zhang-Wu Run-Up Model

A sketch of a plane beach with slope angle p , and connected to the open ocean of uniform still water depth h is shown in Figure 1.

Figure 1. Sketch of a solitary wave running up a plane beach.

The governing equations for describing the propagation and run-up of long waves are the normalized shallow water wave equations:

a77 a - + -[(A + q ) u ] = 0 , at ax

259

where q is the water surface elevation, h the still water depth, u the longitudinal depth-averaged velocity, f = 8gn2 I h’” the friction factor, and n is Manning’s roughness coefficient. The last term in the momentum equation (2) is based on the Manning’s equation to model the effect of bottom friction and is added in the present study. In tsunami modeling, this term is the commonly used term to describe the bottom friction. For the motion of the waterline, the boundary conditions were given by Bang [8] and Zhang, Wu and Hou [9] as follows:

h ( X ) + V ( X , t ) = 0 9

- = u ( X , t ) = U ( t ) ,

(3)

(4) dx dt

dU d q dt dx ’ -- - --

where X ( t ) and U ( t ) denote the position and moving velocity of the waterline at time t , respectively. Then an innovative transformation was introduced by Zhang [8] and Zhang, Wu and Hou [9]:

x = ( 1 + X I L)X’+X , (6)

(7) t = t ‘

where L is the initial length of the computational domain (i.e., from the open ocean at the left end to the initial waterline position at the right end). After this transformation, the governing equations in ( x’, t‘ ) become:

where,

-- % a a’ c,U-+ c, -[(h + q ) u ] = 0 , at & a x

au au au aq f u 2 - -C,U-+c, (u-+-) +- at ax aX ax 8 ( h + ~ ) ~ ” =”

l + x l L c, =-

1 + X / L ’

(9)

( 1 1 ) 1

c, =- 1 + X / L *

In the transformed coordinate system ( x ’ , t ‘ ) , the position of the moving waterline becomes a fixed location. It should be noted that the primes have been omitted in equations (8)-( 1 1 ) for simplicity.

260

The initial condition in the present study is a solitary wave at x = xo propagating towards the sloping beach as shown in Figure 1. The surface elevation and velocity for the initial solitary wave are given by:

11 = asech*k(x-x,), (12)

whereais the wave amplitude, k = m the wave number, x,the initial position of the wave peak, and c is the wave celerity.

The transformed shallow water equations (8)-(9) can be solved by applying the Richtmyer two-step Lax-Wendroff scheme (Zhang, [S]).

3. Numerical and Experimental Results

Laboratory experiments were carried out to measure solitary wave run-up on smooth plane beaches of 15' and 20' slopes in a wave tank in the hydraulics lab (Feng [2]). Since the frictional effect is negligible for wave run-up on steep slopes, the Zhang-Wu run-up model without friction was applied to simulate the experimental runs. All wave run-ups on these two slopes in the experiments were non-brealung. The comparison between the experimental results and the numerical results based on the Zhang-Wu run-up model is shown in Figures 2 and 3. In these figures, the solid line represents the asymptotic solution of solitary wave run-up for non-breaking waves solved by Synolalus [5] based on the inviscid nonlinear and non-dispersive shallow water equations (1)-(2) with f = 0. Our present results show that the numerical predictions based on the Zhang- Wu run-up model agree with the experimental measurement excellently.

261

0.7 -

o Experimental - - - - - -Numerical -

0.5 -

0.05 0.10 0.15 0.20 wave amplitude (a)

Figure 2. Wave run-up on smooth 20Oslope.

0.8 0.7 - -Synolakis (1987)

o Experimental - - - - - -Numerical

- 0.6 - (1 0.5 - c5

0.0 1 0.05 0.10 0.15 0.20

wave amplitude (a)

Figure 3. Wave run-up on smooth 15"slope.

In the present study, we also applied the Zhang-Wu model to simulate the experimental non-breaking wave run-up on a smooth 2.88' slope in Synolakis' 1987 study [ 5 ] . Th~s slope is a mild slope. Even though the artificial beach surface in the experiment was determined as hydrodynamically smooth, the fictional effect still existed due to viscous shear at the bottom of the water layer at the solid surface. For this reason, we added a term in the momentum equation based on the Manning's approach to attempt to model the fictional effect on long wave run-up. In determining the dimensionless frictional factor f; the equation f = 8gn2 I h"' is used. Here all the variables on the right hand side are in dimensional form and each has a unit. In the formula, g = 9.81 d s 2 is the gravitational acceleration, Manning's roughness coefficient n is given for hydrodynamically smooth surface (such as glass, plexiglass, smooth metal

262

surface, etc) as n = 0.01 s/m'", and the h values are based on the actual water depths used in Synolakis' [5] experiments. The comparison between the measured run-up and the predicted results based on the Zhang-Wu run-up model are presented in Figure 4 and Table 1.

0.250

0.200 - % 4 0.150

0.050

0.000

Synolakis (1 987) 0 Experimental

-Numerical (with friction) ....- --.- Numerical (without friction, Zhang, 1996)

0.005 0.010 0.015 0.020 0.025 wave amplitude (a)

Figure 4. Wave run-up on 2.88'slope ( n = 0.01 for hydrodynamically smooth surface).

Table 1. Run-up of non-breaking solitary waves on 2.88' slope

(m) Experimental Numerical (1) Numerical (2) Synolakis

0.335 0.335 0.337 0.298 0.342 0.338 0.291 0.338

0.0052 0.0065 0.0080 0.0092 0.0129 0.0170 0.0210 0.0230

0.0190 0.0220 0.0290 0.0360 0.0480 0.0630 0.0760 0.0870

0.0186 0.0242 0.03 11 0.0367 0.0546 0.0742 0.0929 0.1027

0.01 87 0.0245 0.03 15 0.0373 0.0564 0.0792 0.1029 0.1155

151 0.0176 0.0233 0.0302 0.0359 0.0543 0.0774 0.1008 0.1130

0.337 0.0280 0.1230 0.1344 0.1514 0.1440 (1) with friction, calculated by present model; (2) without friction, from Zhang [8].

From these results, we can see that the Zhang-Wu model agrees very well with Synolakis' theoretical solution for inviscid wave run-up. In the experiments, the measured run-up is smaller than the inviscid predictions. This is an expected result as the travel distance of wave run-up on mild slopes is usually much longer, therefore, the fictional effect plays a more noticeable role compared with the cases with steep slopes. By adding the Manning's term in the

263

momentum equation, we are able to predict a lower run-up value closer to the experimental data, however, the results show that the Manning’s term may not account for the full frictional effect adequately as the predicted results are still larger than the experimental measurement. The possible explanation is that the Manning’s term is derived in traditional hydraulics for uniform open channel flows whereas wave run-up is a highly unsteady and rapidly varied flow over an extremely thin water layer. Under these conditions, the validity of the Manning’s approach for describing the bottom friction becomes questionable.

4. Discussions and Conclusions

The numerical and experimental results from the present study further validated the excellent accuracy of the Zhang-Wu run-up model in predicting long wave run-up on sloping beaches. This model can be applied as an effective model for tsunami predictions.

We also found that the Manning’s term in the momentum equation can predict the bottom frictional effect to a certain degree but not completely. There may be a need to develop better theoretical terms to describe the frictional effect. We are currently carrying out a study to further examine this issue by investigating long wave run-up on both smooth and rough beaches. The results will be presented in a future separate paper.

The present study is focused on non-breaking wave run-up. In reality, both breaking and non-breaking wave run-up may occur. Recently, we carried out a preliminary study on predicting breaking wave run-up (Zhou et al. [lo]) and plan to conduct further studies on this issue.

5. Acknowledgement

The second author wishes to express her deep gratitude to Professor Theodore Y. Wu for educating and training her at Caltech, and for continuing to provide her with guidance and support for the past thirteen years since she left Caltech in 1992. The authors are very grateful for helpful discussions with Professor Wu and Professor Jin E. Zhang of the University of Hong Kong. The study was partially h d e d by the NOAA Sea Grant College Program (IUEP 13) and the Joint Institute for Marine and Atmospheric Research (JIMAR) at the University of Hawaii.

264

References

1. G. Edison, Risk analysis of coastal flooding due to distant tsunamis, Ph.D. thesis, Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, Hawaii (2005).

2. K. Feng, Experimental study of the effect of coastal terrain roughness on ocean long wave runup. M.S. thesis, Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, Hawaii (2001).

3. F. Imamura, N. Shuto and C. Goto, Proceedings of the 6Ih Congress of Asian and Pacific Regional Division, IAHR, Japan, 265 (1988).

4. P.L.-F. Liu, Y.-S. Cho, M. Briggs, U. Kanoglu, and C.E. Synolakis, Journal of Fluid Mechanics, 302,259 (1995).

5 . C.E. Synolahs, Journal of Fluid Mechanics, 185,523 (1987). 6. V.V. Titov and C.E. Synolakis, J. Waterway, Port, Coastal and Ocean Engineering,

7. J.A. Zelt, Coast. Engrg., 15,205 (1991). 8. J.E. Zhang, I. Run-up of ocean waves on beaches and 11. Nonlinear waves in

a fluid-filled elastic tube. PbD. thesis, California Institute of Technology, Pasadena, California (1996).

9. J.E. Zhang, T.Y. Wu, and T.Y. Hou, Advances in Applied Mechanics, 37, 89 (2001).

10. H.Q. Zhou, M.H. Teng and P. Lin, Proceedings of the 2005 Joint ASME/ASCE/SES Conference on Mechanics and Material (McMat2005), Baton Rouge, Louisiana, U.S.A., to appear (2005).

124 (4), 157 (1998).

A NUMERICAL STUDY OF BORE RUNUP A SLOPE

QINGHAI ZHANG & PHILIP L.-F. LIU School of Civil and Environmental Engineering, Cornell Univerity, Ithaca NY 14853,

USA

The runup process of a fully developed turbulent bore propagating over a uniform slope is studied numerically using a 2-D Reynolds Averaged Navier-Stokes (RANS) solver, coupled with a Volume of Fluid (VOF) method and a non-linear k - & turbulence closure. Numerical results for the assemble-averaged flow field show much better agreement with existing experimental data than those obtained from theoretical and numerical results based on shallow water equations (SWE). The numerical simulations are also able to illustrate the processes of bore collapse and other small scale features, such as the 'mini-breaking'. Numerical results of the turbulence field demonstrate that the maximum turbulent kinetic energy (TKE) intensity occurs near the still water shoreline where the bore collapses. The subsequent decay of TKE! reveals two distinct stages in terms of the decay rate. In the region near the still water shoreline, the dominant process is the advection of the strong TKE intensity produced by bore collapse; in the region landwards from the shoreline, the turbulence field has forgotten the influence of bore collapse and the TKE decay rate is very close to that of homogenous grid turbulence.

1. Introduction

In the last half century, several researchers have performed extensive investigations on bore runup over a beach using the shallow water equations:

-+;[(v drl + h)u] = 0 at ax au au d q - + u - + g - = 0 at ax' ax1

where X I points horizontally in the onshore direction, u(x' , t ) the depth- averaged water particle velocity, ~ ( x ', t ) the water surface elevation from the still water level, h(x') the water depth, and g the gravitational acceleration. For most of existing analytical approaches, the bore has been treated as a

265

266

moving discontinuity at x,, such that u jumps from 0 to U ( X , I) and water surface elevation from h(xb ') to h(x, ') + q(xb ') across the bore (Stoker 1957).

The bore front velocity, defined as

u=- 4 7 ' dt '

can be calculated from Eq. (1) and Eq. (2) as:

(3)

Whitham (1958) found that on a positive characteristic of bore propagation,

dx' I dt = u + c

c = J g ( h + 77)

du + 2 d ~ -- gdh = 0 U + C

By substituting Eq. (4) and Eq. (5) into Eq. (8), the following Whitham's rule is obtained:

1 1 -4(M + 1)(M - -)'(M3 + M 2 - M - -) - 2 2 1 dh

(9) 1 3 2 2

h d M (M-1)(M2--)(M4+3M3+M2--M-1)

inwhich M = U I c

Eq. (9) shows that the height of an initially weak bore over constant-depth water will increase when it climbs up a beach and will then eventually decrease to zero as it approaches the shoreline where the water depth is zero (Whitham 1958); i.e.,

q l h cc1 s 77 xh-'I4 (10)

267

q/h>>l 3 q a h ” 2 (1 1)

Keller, Levine & Whtham (1960) demonstrated numerically that Eq. (9) is an accurate approximation to Eq. (1) and Eq. (2). From their results, the bore front velocity U and the fluid particle velocity, u , converged to a common value U * at the still water shoreline. We remark here that these results are obtained based on the conservation of mass, Eq. (4), and momentum, Eq. (5 ) , across the bore. The bottom frictional effects are totally ignored, while the turbulent dissipation is implicitly included by allowing the jump across the bore.

Ho & Meyer (1962) derived a more general result than Eq. (9), using the advancing and receding characteristic parameters as independent variables. Meyer and his colleagues also studied shoaling of a bore on non-uniform slopes (Shen & Meyer 1963a) and subsequent run-up (Shen & Meyer 1963b). They found that the behavior of a bore at the shoreline only depends on the slope at the shoreline and the bore front velocity in the constant water depth region. In other words, different wave shapes and beach configurations along the bore path tend to be forgotten as the bore reaches the shoreline. This ‘forgetfulness’ is due to the dominant singularity of Eq. (1) and Eq. (2) by analogous to the Euler- Poisson-Darboux equation (Ho & Meyer 1962). After the collapse of the bore, their theory predicts that the fluid forms a thin layer and its run-up is totally governed by the gravity force:

u = (U *2 - 2 g p x , y 2

R = U *2 l2g (13)

where X , is the distance from the location of the run-up bore to the still

water shoreline, p the slope of the beach and R the maximum runup height.

Miller (1968) performed extensive experiments on bore run-up over a uniform slope and showed that some of the conclusions based on the SWE are not strictly correct. Miller demonstrated that the bore does not collapse abruptly to zero at the shoreline. Instead, there is a gradual transition from bore mode to runup mode. Later experiments by Yeh (1989) further identified the interaction between the bore and the initially still water wedge in front of it (The bore tends to push up the water-wedge.). This behavior is very different from what is suggested by Eq. (1 1). Moreover, the runup heights of the experiments are considerably lower than those predicted by Eq. (13). Miller’s results suggested that runup heights are also a h c t i o n of the roughness of the slope bottom.

268

Realizing the drawbacks of the characteristic methods, Hibberd & Peregrine (1979) numerically solved Eq. (1) and Eq. (2), using a finite difference method with the Lax-Wendroff scheme (Lax & Wendroff 1960). Packwood & Peregrine (1981) extended this numerical model by incorporating bottom friction effect with a Chezy term. However, the calculated runup heights are still much larger than those of Miller’s experimental results, particularly in the cases of mild slopes.

Turbulence due to breaking is another important aspect in bore propagation and runup. Madsen & Svendsen (1983) studied quasi-steady bore propagation on constant water depth with an algebraic k - E turbulence closure. They assumed that the horizontal length scale is much larger than the vertical length scale and, therefore, the pressure field is hydrostatic. This leads to the canonical boundary-layer equations with a longitudinal pressure gradient. The turbulence was assumed to be concentrated in a wedge that originates at the toe of the front and spreads towards the bottom. Their model was able to provide the free surface profiles, velocity and shear stress variations and dissipations withm the bore. It was also shown that the surface profile is insensitive to the precise form of the initial velocity profile. In a later paper (Svendsen & Madsen 1984)) they included SWE in the governing equations and extended their turbulence model to simultaneous hyperbolic partial differential equations in order to study a turbulent bore climbing a beach. While their model shed some lights on the turbulence field before the bore reaches the shoreline, it was not applicable to the bore collapse and the ensuing runup.

In this paper, we extend a breaking-wave model, COBRAS (Lin & Liu 1998), to simulate a fully developed turbulent bore interacting with a steep slope. With a VOF method tracking the free surface, this model uses a two-step projection method to solve 2-D RANS equations with a k - & model and nonlinear Reynolds stress closure. Our primary interest is to examine the ensemble averaged flow field and the TKE evolution during the bore collapse and runup stages. With this general 2-D model, we will show the gradual transition between different stages and reveal physical processes that cannot be appreciated by SWE due to their intrinsic limitations.

2. Governing Equations and Boundary Conditions

The ensemble averaged velocity field is governed by the RANS equations:

269

where uj is the j-th component of the total velocity, ( ) denotes the

ensemble average operator and u the corresponding fluctuating velocity component. In the momentum equation Eq. (15), p is the pressure and v the kinematic viscosity. The two-step projection method (Chorin 1968; 1969) is used to solve Eq. (14) and Eq. (15). The first step introduces an intermediate velocity that results from forward-in-time explicit calculation of Eq. (1 5), without considering pressure terms; the second step projects the intermediate velocity field onto a divergence free plane Eq. (14) to obtain the velocity for the next time step.

The Reynolds stress terms in Eq. (15), -p(u iu j ) , are closed by the k - &

model (Jones & Launder 1972):

where k represents the TKE and & the rate of turbulence energy dissipation. The flux terms have been modeled by gradient diffusion hypothesis (Rodi 1980). The turbulent viscosity is determined locally by:

k2 Vf = c -

" E

The values of the coefficients in Eqs. (1 6)-( 18), due to Launder & Sharma (1974), are:

Cp =0.09, C,, =1.44, C,, =1.92, ok =1.0, o& =1.3. (19)

For the specific assumption of turbulent viscosity hypothesis, we use the quadratic one of Shih et al. (1996):

270

The empirical coefficients appearing in Eq. (20) are given as follows (Lin & Liu 1998):

C, = 0.0054, C, = -0.0171, C, = 0.0027 (21)

Tracking free surface is another important component of the numerical simulation. On one hand, the free surface serves as a part of flow boundaries and thus demands appropriate boundary conditions; on the other hand, the location of the free surface is coupled with the velocity field through mass conservation. Therefore, numerical error in tracking the free surface location will propagate to the ensemble averaged velocity field as well as the turbulence field. This implies that the treatment of the free surface is crucial in obtaining accurate solutions.

The VOF method defines a volume-of-fraction for every computational cell as

T I

where Vc is the total volume of a cell and vf is the corresponding volume of the fluid in this cell. Assuming that the interface is non-diffusive and neglecting the turbulence effect on F field, the F values are conservative following the flow motions. Thus,

dF aF -+pi)- = 0 at axi

The VOF method consists of an advection step and a reconstruction step. New F values are first calculated from previous F values and the interface configuration. Then the new interface is reconstructed fiom the new F field. In

271

this paper, we follow the SOLA-VOF method proposed by Hirt & Nichols (1981). This method represents the interface within each cell as either a vertical or a horizontal surface, the direction of which only depends on the spatial gradient of F field. As for the advection step, a donor-acceptor method is used.

On the free surface, we neglect the density fluctuation and the effect of the air flow so that zero stress (both shear and normal) condition is imposed. The surface is also treated as a flux barrier for turbulence field, i.e., ak / a n = 0 , a& / a n = 0 , where n is the local outward-normal direction of the free surface.

Along the solid wall, we impose no-slip boundary condition for the ensemble averaged velocity. As for the turbulence field, a shear velocity, U, , is first calculated from the law of wall with the logarithmic velocity profile and it is then used as the velocity scale to calculate TKE and dissipation near the wall, i.e., :

2

kw=&,

3 u W &, =-

KYW

where E = 9.0 for a smooth wall, K = 0.41 , and subscript ‘ I ” denotes a quantity evaluated at one grid away from the wall.

Any obstacle inside the domain is treated as a special case of a flow with an infmite density.

This partial cell treatment partially blocks the cell face and the cell itself according to boundary geometries; the variables in the partial cells or on cell faces are then weighted by an openness coefficient 6 . The governing equations Eqs. (14) and (1 5 ) can then be written as:

272

where 8 = 0 , if the cell is entirely occupied by obstacle; 8 = 1 , if the cell is occupied entirely by fluid; and 0 < 8 C 1, if the cell is partially occupied by obstacle. Although this treatment is easy to implement, it does not retain the information of the orientation of the obstacle boundaries and smears out the sharp interface between the fluid and the obstacle.

3. Numerical Setup

Yeh (1989) studied experimentally the runup of a turbulent bore over a uniform slope. The bore was generated by a dam break mechanism. As shown in Fig. 1, in one of Yeh's experiments, the water in front of the dam (gate) was h, = 9.75cm , while the water depth behind the dam was h, = 22.52cm . The gate at X I = 2.97m was pulled instantaneously and a fully developed turbulent bore (FDTB) evolved and ran up a slope of p = 7.5" with its toe at x' = 3.37m . Although Yeh performed several other experiments in terms of different values for h, and h, , we only investigate this particular case because the bore generated by this configuration is a FDTB and the drastic bore collapse and the resulting turbulence field are of our primary interest. The setup of numerical simulations is the same as that of the physical experiments of Yeh (1989), except that the coordinates of the numerical model have been rotated so that x-axis coincides with the slope face with its origin at the still water shoreline. With this arrangement the slope coincides with the numerical boundary and, thus, it facilitates the accurate application of boundary conditions. Although this method still has partial cells (non-boundary-conforming cells) in the flat-bed region, the effect of these partial cells is confined in a relatively unimportant and uninterested region.

Fig.1 Setup of numerical simulations for a turbulent bore running up a slope

We remark here that before studylng the bore runup on a slope, it is important to make sure that the numerical model can generate the bore correctly. The verification of the bore generation can easily be done by simulating dam- break waves on a flat bed, which has been shown in our previous work (Shigematsu et al. 2004). The details will not be repeated here. However, to

273

0.5 0.44s -

n 1 I I I I

benchmark the accuracy of the numerical results, we show in Fig. 2 the comparison between numerical results and experimental data taken by Stansby et al. (1 998). In this case, the water depth in front of the dam is hu = 36cm and the depth is hd = 0. lh, behind the dam.

0.74s -

I I I 1 I

0.32s 0.5

0 I ” I I I I I I I I

0 1 2 3 4 5 6 7 8 9 10

1.24s -

I I

I I I , I I I I I

-0- - 1 2 3 4 5 6 7 8 9 10

I I I I I ! I I I I

n I I I I I I I I 1 -

-0 1 2 3 4 5 6 7 8 9 10

x ’/ h, Fig.2 Comparison of free surface elevation for dam-break generated waves at five different times (“dots” = experimental data by Stansby et al. (1998) and “solid line” = numerical results). The gate is pulled at t = 0.

In Fig. 2 the experimental and numerical results differ quite a bit with respects to the transient processes of bore-generation, particularly at the bore front where the free surface deforms violently. However, this is not unexpected. The numerical results are based on the ensemble averaged RANS equations and the free surface location shown in this figure is identified as the contour line for F = 0.5. On the other hand, the laboratory measurements are instantaneous, including the turbulent fluctuations and the violent air-water mixture. Nevertheless, the position of the bore front at each time frame is predicted reasonably accurately and the bore front velocity, consequently, is also well predicted. In addition, the final bore heights of these two results agree with each other. Following the argument of Ho & Meyer (1962) that the behavior of the bore at the shoreline only depends on its velocity in the constant depth region

274

and the slope at the still water shoreline, we are confident that generation of the bore is satisfactory.

4. Numerical Results

In our numerical simulations, structured, non-uniform rectangular grids with a total number of 2240-by-144 are used for computational efficiency and economy. Grids near and on the beach are Imm-by-lmm squares; grids far away from the beach are about 10 times coarser. To ensure numerical stability, the grid size in the transitional region changes slowly (hi /Axi+, < 1.05) and the size of time steps is dynamically adjusted to satisfy both advection and diffusion stability requirements:

aAY } (u>- ’ (v>, a.Ax

At I min{- -

1 1 ~ Ax2Ay2 At I

2(v, + v) Ax2 + Ay’

where a=0.3 and (U),, , (v), are the maximum horizontal and vertical ensemble averaged velocity in the computational domain. The initial condition for the mean, flow is zero velocity everywhere; as for turbulence field, it is clear from (1 6 ) that the model will not produce any TKE, if there is no TKE initially. Thus, a small amount of k = (6c0)* /2 , serving as ‘seed’, is specified

everywhereattime t=O,where S = 2 . 5 ~ 1 0 - ~ and q,=Jgh,. Before discussing the numerical results, we will first show the comparisons

between the numerical results and the experimental data in terms of the runup velocity, which is the same as the bore front velocity defined as U = dx, l d t , where xb is the location of the runup front. We define a non-dimensional beach-wise coordinate:

x, = x g s i n p I U * ’ . (31)

In our numerical simulations, the runup velocity is obtained by first recording the positions of the bore front on the slope at each computational time step and then differentiating the resulting time history of bore front positions. The comparison between numerical results and experimental data (Yeh, 1989) is shown in Fig. 3.

275

0 4 1 0

I I P I I Y I Q' 0 1 02 ' a 3 OL a5 2,

Fig. 3. FDTB runup velocity comparison. Straight line: theoretical results by (12); '+': numerical results of Hibberd & Peregrine 1979; '*': experimental results of Yeh (1989); diamonds: numerical results in this paper. U* = 2.43rn /s (Yeh 1989).

The experimental data, being very scattered, imply high uncertainties of the measured bore front velocity, particularly at the beginning of the runup stage when turbulence intensity is high. However, as the bore front progresses, the influence of breaking becomes less significant and the dynamic process is mostly governed by the gravity and viscous stresses. Accordingly, in Fig. 3, the scattering in experimental measurements becomes smaller and the numerical and experimental results agree with each other better at the end of the run-up stage. Numerical and theoretical results based on the SWE approach are considerably higher than those of experimental results.

Since there are no other detailed experimental data for this case, we shall discuss the physical processes based on the numerical results in the remainder of this paper. The entire interaction process of a FDTB with an impermeable beach can be divided into three stages: (a) bore collapse, (b) runup and (c) down rush. We shall discuss each stage in the following sections.

4.1. Bore Collapse

During the bore collapse phase the bore front collapses onto the slope and the bore loses its unique feature: vertical free surface profile and strong turbulence at the bore front, as shown in Fig. 4. This process can be viewed as the interaction between the bore, the slope and the small wedge of initially quiescent water near the still water shoreline. From Frame 2 of Fig.4, we can see clearly that the wedge is being pushed up by the bore and thus, the bore is slowed down. By reducing the momentum of the incoming bore, this wedge of fluid serves as a 'buffer' between the bore and the slope. Also because of this wedge, bore

276

collapse becomes less abrupt (although it still occurs in a relatively short time), which according to the SWE theory is a mathematical singularity and discontinuity. The simulated results confirm the experimental observations by Miller (1968) and Yeh (1 989).

Frame 1. t = 0.80s

Frame 3. t = 0.90s Frame 4. t = 0.95s ___- ---

. /--

//--ac- 9' Q- a1

Fig.4 Processes of bore collapse at four consecutive time steps. The first subplot has the contour lines of free surface and also serves as an overview. The gray bar indicates F values. The other two subplots are zoom-ins of the small-window shown in the first subplot. Plotted in the second subplot is & (m / s ), the value of turbulent velocity, also plotted is one single contour line of F = 0.5. The ensemble averaged velocity profiles for every 25 grids in X direction and every 2 grids in the y direction are plotted in the third subplot. t = 0 corresponds to the moment of dam-break.

The velocity profile in a water column (normal to the beach) is more or less uniform in the rear of the bore front. However, the bottom boundary structure is quite visible. Near the bore front, the fluid particle velocities near the free surface are much faster than those close to the beach face. Significant vertical (downward) velocity components appear during the bore collapse (Frame 2 and 3 of Fig. 4). We also remark here that the air bubble trapped inside the

277

collapsing water shown in Frame 3 of Fig.4 represents an air tube in the spanwise direction. In the field condition, the bore collapse is three-dimensional and the air-tube could be easily broken. Although we have not considered the surface tension in the current model, we argue that the dominant process in bore collapse is the violent momentum exchange of breaking surface and surface tension effect is relatively insignificant w i h n the time scale involved.

Before the bore reaches the still water shoreline, the highly concentrated turbulence is confined in the frontal zone of the bore (Frame 1 of Fig. 4). During the bore collapse, more turbulence is generated in the bore front and near the bottom boundary layer (Frame 2 of Fig. 4). At the end of the collapse, the turbulence is spread to a larger region (Frame 4 of Fig.4) and a water tongue is formed, which is ready to run up the slope.

4.2. Runup

In Fig. 5, several snap shots of surface profile, mean velocity and turbulence at six time frames, t = 1.00s, 1.40s, 1.90s, 2.45s, 2.70s, 2.85s, respectively, are displayed.

The runup phase starts (t = 1.00s) after the collapse of the bore front and the formation of the water tongue. It lasts until the maximum run-up height is reached ( t = 2.85s). During ths phase, the momentum flux is balanced by the gravity force and viscous force. The turbulence intensity is weakened as the T I E is diffused and dissipated. The TKE series in Fig.5 shows the translating, stretching and diluting of the intense turbulence field resulting from the bore collapse.

Shortly after the runup phase begins, the effects of bore collapse upon the ensemble averaged velocity field quickly diminish and velocity of the runup water tongue forms a nearly uniform profile ( F l . 4 0 ~ ) in a vertical water column. As the water body continues its runup, the effects of bottom stress accumulates and the water particles near the bottom travels noticeably slower than those near the surface (t =1.90s). On the other hand, because of longer exposure to bottom friction and gravity, fluid particles near the tip of runup tongue travel at a slower speed than those in the rear of the water wedge. A typical illustration can be found in Frame 3 of Fig. 5. The combination of these two features results in an interesting local phenomenon, “mini-breaking”. As shown in Frame 4 of Fig. 5 ( t = 2.45s), a vertical surface appears near the tip of runup tongue.

278

Frame 1: t = 1.00 s

Fig.5 FDTB Run-up Process (Caption same as Fig

Frame 2: t = 1.40s

To examine the “mini breaking” closely, in Fig. 6, we show the free surface profiles as well as the velocity field near the tip of the runup tongue for t = 2.50s, 2.55s, 2.60s, 2.65s and 1.05m < x <: 1.35m. The window size is only 0.3m wide in the on-offshore direction, allowing us to observe the effect of spatial variability of the velocity field. As shown in Frames 1 and 2, the tip of the

279

water tongue has come to a stop, while the water body behind it is still pushing shoreward. Water particles near the surfaces travels much faster than those near the bottom ( t = 2.55s), this resembles a bore. Although the height of the "mini" bore is only in the order of magnitude of lcm, it behaves in the similar way as the original bore, i.e., it collapses, generates turbulence locally (t = 2.60s) and forms a new runup water wedge (t = 2.65s).

Mini-breaking also has an effect on the bore front velocity. In Fig. 3, we observe a small hump in the bore front velocity nearX* =o.3, where the bore front velocity becomes almost zero, jumps up and then drops down again. This indicates that the small scale free surface discontinuity has caught up with the tip of the water tongue, broken there and ran up again. The effects of mini breaking on the mean velocity profile can be seen in Frame 5 of Fig.5. The similarity between Fig. 4 and Fig. 6 is striking.

Frame 1: t = 2.50s

.-

om

Fig.6 Mini breaking (Caption is the same as Fig.4, except that velocity profiles are plotted every 6 grids in a-direction and shown in y-direction is the actual computational resolution)

280

Since there are 10 numerical grids to resolve the mini bore height, we are confident that the mini breahng is not a numerical artifact. Furthermore, results for mini breaking are repeatable with fine enough resolutions. We remark here that mini breaking can be observed on natural beaches. It should be noted here that the bottom friction causes the spatial gradients in the velocity field, which is necessary for generating mini breaking. It is clear that SWE theory cannot capture the process of mini-breaking because of its overlook of vertical variances and bottom friction. This is a big advantage of the methods used in this paper over depth-averaged method.

4.3. Rundown

During the rundown phase the water tongue is pulled downwards by gravity force and the downward movements are resisted by bottom viscous stress. In Fig. 7 the free surface profile and the velocity fields at two instances, t = 2.90s and 3.30s, are shown. The free surface remains more or less horizontal during the rundown phase. The velocity profiles far away from the tip of the tongue are similar to those of a strong wall-jet. A very thm transient boundary layer is developing underneath the wall-jet like flow.

_/--

__./---

-..

I

Fig. 7 - Rundown stage (caption same as Fig.4-2)

281

4.4. Turbulent Kinetic Energy Evolution during Runup

To study the evolution of the TKE in the runup phase, we shall examine the depth-averaged TKE:

where t is limited to t € t, , and t, denotes the end of run-up phase.

At a fixed location on the slope, k(x,, t) only varies with time. We define

k,,,(x,) as the maximum value of k ( x , , t ) and t , (x,) as the corresponding - -

time of k,(x,) .

In Fig. 8, we plot Lm(x,) against xs . The maximum of depth-average TKE,

KO = max(k,(x,)} = 0.067m2/sZ , occurred at the still water shoreline ( x , = 0) when the bore collapsed ( t 0 = o . 8 g s ). The fact that Lm(x,) decreases monotonously all the way up the slope has an implication that there is no strong TKE production after the collapse of the bore. However, there does exist a region (0.056 < x, < 0.143) where dissipation of TKE is roughly balanced by production of TKE due to the bottom viscous effect, as discussed in section 4.2. Otherwise, the turbulence field of whole run-up process is dominated by dissipation, particularly at the beginning of the runup.

0.03-

0.02-

0.01 -

I sb, 0 a s 0.1 0 f5 0 2 0.25 0 3 .C*

Fig. 8 Spatial distribution of the maximum depth-averaged TKE.

The numbers near ‘*’ are t m ( x s ) in seconds. (x,) has the unit of m1 / s * .

282

In Fig.9 we plot ~ J ~ , ) against t , ( ~ , ) and a two-stage power log fitting to the

numerical results yields:

* (33) ln(k,lKo)=-0.3785*ln[(tm -to)lT]-1.46 ( 0 . 0 2 < ~ , <0.14)

ln(km /KO) = -1.2003*ln[(tm -t,,)lT] -2.29 (0.14 < x, < 0.26)

I

6"'- ----

Fig. 9 TKE decay with respect to time. *': numerical results; lines: power-law fits to the numerical results

T = re -to

The decaying rate of TKE drops abruptly at a certain point ( xs = 0.143 ), after which the turbulence decay rate (-1.2) is close to that (-1.3) of homogeneous grid turbulence. Interestingly, the experimental works of Cowen et al. (2003) also show a grid turbulence decay rate even for a totally different setup (periodic waves with a milder slope). This is, however, not unexpected. At the beginning of the runup stage, the effect of bore collapse still affect the turbulence field in terms of the produced anisotropy of Reynolds stresses. After a certain amount of time, this anisotropy dies out and the turbulence field becomes almost homogenous; meanwhile the production of TKE drops out and dissipation becomes the dominant process. Under these two conditions, the turbulence decay has all the reasons to be similar to that of grid turbulence.

This argument is significant on turbulence decay: within a range of beach slope and initial turbulence strength, the runup water tongue will eventually reach a point where all the previous effects have been forgotten and turbulence becomes almost homogenous. It then decays like the grid turbulence does.

283

5. Conclusions

Based on the numerical results shown in this paper, as well as previous experimental observations, the process of bore propagation over a slope can be described as follows. As the bore draws near the shoreline, the water wedge in front of the bore does is pushed up by the bore and serves as a buffer to slow down the bore. The height of the bore does not go to zero suddenly at the shoreline as predicted by the SWE theory. Although the bore collapse lasts a short period of time, it is still gradual and is best viewed as the adaptation of the bore to the slope. The initially still water wedge at the shoreline also participates in the violent momentum exchange. In the runup phase, in additional to the gravity force, the bottom stress plays an important role. Consequently, the water particle near the bottom moves much slower than those near the free surface. Furthermore, water particles near the front travels slower than those at the rear of the bore. All these eventually lead to the occurrence of a small-scale process: ‘mini-breaking ’, which remarkably resembles bore collapse qualitatively.

As for the turbulence field, our numerical results show that the TKE are confined in the frontal zone of the bore before it collapses. Bore collapse produces high TKE at the front and bottom and the maximum turbulence intensity occurs at the shoreline. In the runup phase, TKE produced by bore collapse is stretched, diluted and dissipated. There are two distinct regions with respect to turbulence decay rate: in the region near the shoreline, effects of the bore collapse on the turbulence field persist, while in the second region the run- up flow becomes almost homogenous and decays like the grid turbulence does.

Acknowledgment

This research has been supported by grants from National Science Foundation (Fluid Dynamics Program, Physical Oceanography Program and ITR program) to Cornell University.

284

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563-578.

STUDIES OF INTENSE INTERNAL GRAVITY WAVES: FIELD MEASUREMENTS AND NUMERICAL MODELING

HSIEN P. P A 0

Washington, D. C. 20064, USA Department of Civil Engineering, The Catholic University of America

ANDREY N. SEREBRYANY N.N. Andreyev Acoustic Institute

Moscow. Russia

The oceanic shelf is an area of significant internal wave intensification and transformation. Long internal waves (predominantly intemal tidal waves) almost regularly propagate across shelf shoreward evolving in the process of nonlinear transformation with generation of solitary and soliton-like internal waves. Intense internal waves are also generated when surface intrusion of lighter waters into coastal waters was taking place. One case study of intense internal waves on shelves is presented here. Comparisons of measurement data with numerical modeling were made whenever possible. Good agreement has been achieved.

1. Introduction

One interesting case of intense internal waves on shelf will be presented in detail. This case is a study of intense intemal waves generated by surface intrusion of warmer and fresher water on a shelf of the Black Sea. Observations were made from a stationary platform in the northwest part of the Black Sea, located 60 km from the nearest shore. A change of water masses occurred in the study area, leading to a corresponding change in the thermocline structure of the upper layer of the sea. At this time, a long train of intense internal waves was recorded. All data indicated the passage of a local front. The intrusion occurred at the surface and lasted several days. The freshened waters moved in the direction from the shore regions outward toward the sea. The process of surface intrusion propagating above sharp thermocline was also investigated by a numerical modeling. Results from the numerical study are in good agreement with observed data.

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2. Study 1: Intense Internal Waves on Shelf

Large-amplitude internal waves propagating on shelf of Pacific Coast of Kamchatka were observed in field experiments during August of 1990 [l]. Solitary internal depression wave with symmetrical profile transforms its profile into a steepened back face in the vicinity of turning point (see Fig. 1). Moreover, two depression waves of lesser amplitudes in the wave train follow the leading solitary wave. During and after the passage of turning point, the leading depression wave gradually transforms into elevation wave.

I

SOLITARY WAVE

Figure 1. A 13-h record of vertical displacements of the thermocline made by line sensor fiom an anchored vessel (top). Dotted line shows horizon of midpoint. Record of profile of solitary internal wave passing overturning point (bottom). Sampling is at 12-second interval.

2.1. Numerical Modeling

Results of numerical modeling of the process made on the basis of solving 111 Navier-Stockes and diffision equations are presented. An animation of the time- dependent solitary wave transformation based on the numerical results will also be presented. Fig. 2 shows plots of lines of constant density (or isopycnals) and streamlines of an intense internal wave propagating shoreward from a numerical simulation. The bottom slope is 1/60, which is the same as the field experiment. The results shown here for dimensionless time t = 20 and 40. The strain rate dddx at the surface is also plotted as function of horizontal distance x. The peak strain rate of 19.86 corresponds to a dimensional value of 1.92 x 10” s-’. This value is found to be in the range, which can be detected by S A R satellite imagery. It is seen that measurement data are in good agreement with numerical modeling results.

Figure 2. Plots of lines of constant density (or isopycnals) and streamlines of an intense internal wave propagating shoreward from a numerical simulation. The bottom slope is 1/60 which is the same as the field experiment. The results shown here are for dimensionless time t = 20 and 40. The strain rate dddx at the surface is also plot as function of horizontal distance x.

289

3. Study 2: Generation of Intense Internal Waves by Surface Intrusions on a Shelf

A study is made of intense internal waves generated by surface intrusion of warmer and fresher water on a shelf of the Black Sea. Observations were made from a stationary platform in the northwest part of the Black Sea, located 60 km from the nearest shore. A change of water masses occurred in the study area, leading to a corresponding change in the thermocline structure of the upper layer of the sea. Figure 3 shows a schematic diagram of flow configuration and density profile. Inflow enters through a vertical opening he into a channel of depth d and semi-infinite length.

At this time, a long train of intense internal waves was recorded. All data indicated the passage of a local front: a mass of freshened warm water intruded into the portion of the sea having a uniform salinity throughout the depth. The intrusion occurred at the surface and lasted several days; the salinity during this time decreased by 2.1 promille. The freshened waters moved in the direction from the shore regions outward toward the sea (Fig. 4). In the process, it evolved into a corresponding change in the thermocline structure of the upper layer of the sea, altered by the surface intrusion water. Figure 5 shows the density profiles corresponding to the five density soundings indicated in Fig. 4. It is seen that the density of the upper layer of the sea was decreasing with time as the surface intrusion continued.

P c , Ue

0- x

de si rofile

Qe - inflow discharge per unit width; Ue - inflow velocity; pe - inflow density; po - density of ambient fluid ; x*-- dimensional horizontal coordinate; x = x* / d z* -- dimensional vertical coordinate; z = z * / d

Figure 3. Schematic diagram showing flow configuration and density profile. Inflow through a vertical opening he into a channel of depth d and semi-infiite length.

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Generation of internal waves by moving surface intrusion (observation on N W shelf of the Black sea).

1 2 3 4 5

Figure 4. Observation of generation of internal waves by moving surface intrusion on NW shelf of the Black Sea.

Train of intense internal waves was observed on 22 July in the time of passage of sharp density front of surface intrusion of warmer and fresher water from coastal regions as shown in Fig. 6.

3.1. Numerical Modeling

We consider the initial configuration as shown in Fig. 3, which represents a continuous stratified fluid with a free surface in a channel of depth d. The goveming equations for an incompressible, viscous, diffusive, two-dimensional Boussinesq fluid may be written as

291

E

-20

-25

9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 1

U

0

Figure 5. The density profiles corresponding to the five density soundings as indicated in Fig. 4. o represents the departure of density from the fresh water value (kg m-3).

Figure 6. Observation of internal waves: train of intense internal waves observed on 22 July in the time of passage of sharp density front of surface intrusion of warmer and fresher water from coastal regions.

292

V2Y aY a a 1 -+-(Uy)+-(Wy)=- at dx az Sc Re

V2ry = 6 where < is the dimensionless vorticity, y is the dimensionless density difference, w is the streamdunction and t is the dimensionless time.

The process of surface intrusion propagating above sharp thermocline was investigated by a numerical modeling. The boundary conditions for the stream bction, density and vorticity are given as follows:

a2Y Y=O, y=o, <=-

az2 for z=l, O<x<w ;

for O<z<(I-hJd), x=O ;

for O<z<l , x j w ;

Y*m(z), y=Q c=o

293

The numerical method applied in thls study is a one-step explicit finite difference scheme that possesses both the transportive and conservative properties described by Roache [2]. Central differencing in space and forward differencing in time are used, except for the nonlinear terms of the governing equations, for which a special upwind method was applied. The upwind scheme originally introduced by Torrance and Rockett [3] has truncation errors that appear as false viscosity and false diffusion. These computational damping and diffusive effects begin to mask the real damping and diffusion as the Reynolds number increases (see, for example, Ref [4]). In order to limit the false diffusion, an explicit scheme with zero numerical diffusion called ETUDE (Explicit in Time Up-stream Difference Estimate) was adopted in ths study [5, 61 *

3.2. Numerical Results

Two calculation cases are chosen according to the observation density profiles, with U = 4 c d s , U, = 40 c d s , &Id = 0.1 in both cases. Isopycnic lines from numerical simulation at three different dimensionless times, t = 10, 20, 30, respectively, are shown in Fig. 7 for Case 1 (corresponding to the case with the density profile 2 as shown in Fig. 5) . The head wave of the intrusion plume is clearly seen to advance in the offshore direction. This actually initiates a surface density current. Intense internal waves are induced at the thermocline with the largest wave immediately below the head wave. The advancing speed of the leading internal waves is the same as that of the head wave (or front) of the surface density current.

For Case 1, streamlines at three dimensionless times, t = 10, 20, 30, respectively, are shown in Fig. 8. It is seen that the flow at the far right is uniform (barotropic), while the flow at the left is baroclinic in nature, accompanied by the generation of internal waves due to the forcing of the advancing of the head wave at the surface. Similarly, isopycnic lines at three different dimensionless times, t = 10, 20, 30, respectively, for Case 2, are shown in Fig. 9. In Table 1, a comparison is made between the field observation data and results from numerical computation for Case 2 (corresponding to the case with the density profile 3 as shown in Fig. 5). It is seen that the agreement for wave periods, speeds and heights are excellent.

294

4 0

3 0

2 0

1 0

4 0

3 0

2 0

1 0

I I ( 0 1 6 3 0 4 0 5 0 6 0 1 0 6 0

Figure 7. Isopycnic lines from numerical simulation at three different dimensionless times, t = 10,20,30, respectively; Case 1 corresponds to the case with the density profile 2 shown in Fig. 5 .

4 0

31)

2 0

1 0

4 0

S O

1 0

1 0

1 0 2 0 5 0 7 0 8 0

Figure 8. Streamlines at three dimensionless times, t = 10,20,30, respectively, Case 1.

295

4 0

3 0

2 0

1 0

I J $ 0 2 0 a o 4 0 6 0 6 0 7 0 8 0

1 0

a o

2 0

1 0

I 1 0 7'0 J O 4 1 6 0 5 0 7 0 a 0

4 0

3 6

2 0

1 0

1 0 2 0 3 0 4 0 6 0 6 U 7 0 110

Figure 9. Isopycnic lines from numerical simulation at three different dimensionless times, t = 10, 20, 30, respectively; Case 2 corresponds to the case with the density profile 3 shown in Fig. 5 .

Table 1. Comparison between the field observation data and results from numerical computation for Case 2.

Observation data Calculation result (case 2)

Internal wave period, (min) 4-5

Internal wave speed, ( d s ) 0.22-0.30

4

0.245

Internal wave height, (m) 2.5-3.0 3.0

296

References

1 . Serebryany A.N. and Pao H.P., 2002. The process of breaking of internal solitons on shelf: transition of internal solitary wave through turning point. Proceedings of American Geophysical Union 2002 Ocean Science Meeting, Honolulu, HI, February 2002, p. 296.

2. Roache P.J., 1972. Computational Fluid Dynamics. Hermosa Publishers, N.M.

3. Torrance K.E. and Rockett J.A., 1969. Nemerical study of natural convection in an enclosure with localized heating from below: creeping flow to the onset of laminar instability. J. Fluid Mech. 36,33-54.

4. Kao T.W., Park C and Pao H.P., 1978. Inflow, density currents and fronts. Phys. Fluids 21, 1912-1922.

5. Valentine D.T., 1987. Comparison of finte differences methods to predict passive contaminant transport. Computers in Eng., ASME, 263-269.

6. Saffarinia K. and Kao T.W., 1996. A numerical study of the breaking of an internal soliton and its interaction with a slope. Dynamics Atmos. Oceans, 23,379-391.

NONLINEAR INTERNAL WAVE STUDY IN THE SOUTH CHINA SEA

ANTONY K. LIU', YUNHE ZHAO Oceans and Ice Branch, NASA Goddard Space Flight Center

Greenbelt, Maryland 20771, USA

MING- KUANG HSU Department of Oceanography, National Taiwan Ocean University

Keelung, Taiwan

The internal wave distribution map in the northeast part of South China Sea (SCS) has been compiled from hundreds of ERS-112, RADARSAT and Space Shuttle Synthetic Aperture Radar (SAR) images. Based on the map compiled from satellite data, the wave crest can be as long as 200 km with amplitude of 100 m. In recent Asian Seas International Acoustics Experiment (ASIAEX), extensive moorings have been deployed around the continental shelf break area in the northeast of South China Sea. Simultaneous RADARSAT SAR images have been collected during the field test to integrate with the in-situ measurementS from moorings, ship-board sensors, and CTD casts. Besides providing synoptic information, satellite imagery is very useful for tracking the internal waves, and locating surface fronts and mesoscale features. Environmental parameters have been calculated based on extensive CTD casts data near the ASIAEX area. Nonlinear internal wave models have been applied to integrate and assimilate both SAR and mooring data. The shoaling, turning, and dissipation of large internal waves on the shelf break, elevation solitons, and wave-wave interaction have been studied.

1. Introduction

The ocean current over topographic features such as a sill or continental shelf in a stratified flow can produce nonlinear internal waves of tidal frequency and has been studied by many researchers [l], [2], [3]. Their observations provide insight into the internal wave generation process and explain the role they play in the transfer of energy from tide to ocean mixing. It has been demonstrated that surface signatures of these nonlinear internal waves are observable in the Synthetic Aperture Radar (SAR) images [4] from Russian Alma-1 and from the First and Second European Remote sensing Satellite ERS-1R [ 5 ] . Recently, the internal wave distribution maps in the northeast of South China Sea (SCS) and

' Office of Naval Research Global - Asia, Tokyo, Japan. E-mail: [email protected]

297

298

near Hainan Island have been compiled fiom hundreds of ERS-1/2, RADARSAT and Space Shuttle SAR images from 1993 to 1998 by [6] as shown in Figure 1 . Based on internal wave distribution map, most of internal waves in the northeast part of South China Sea are propagating westward. The wave crest can be as long as 200 km with amplitude of 100 m, due to strong current from the Kuroshio branching out into the South China Sea [7]. From the observations at drilling rigs near DongSha Island by Amoco Production Co. [8], the solitons may be generated in a 4 km wide channel between Batan and Sabtang islands in Luzon Strait. However, recent work [7], [9] has suggested that solitons in SCS may also be generated locally near the shelf break. Furthermore, based on satellite imagery the long crest internal waves near Luzon Strait are produced by the connection along the crest of many individual wave packets generated from different sources or sills in the strait [lo].

Y

0 3

..I

.I ... 3

22

20

18

I O R I10 112 I14 116 l i8 120

Longitude ('E)

Figure 1. Bathymetry and internal wave distribution map in the South China Sea

The Kuroshio moving north from Philippine Basin branches out near the south tip of Taiwan and part of the Kuroshio intrudes into the South China Sea through the Luzon Strait. Surface signature of huge internal wave packets has been observed in the ERS-I S A R images [7] in the South China Sea. The crest of soliton is more than 200 km long and each packet contains more than ten rank-ordered solitons with a packet width of 25 km. Within a wave packet, the wavelengths appear to be monotonically decreasing, front to rear, from 5 km to 500 m. This is the biggest internal waves ever been observed in this area. The internal wave amplitude is larger than 100 m based on the CTD casts. These huge wave packets propagate and evolve into the South China Sea and finally reach the continental shelf of southern China. The effects of water depth on the evolution of solitons have been modeled by Kortweg-deVries (KdV)-type equation [2] and linked to satellite image observations [7].

299

Since 1997, U.S. and Asian Pacific Rim scientists have initiated a joint effort to design and conduct an experiment to study shallow-water acoustics, physical oceanography, and bottom structure in the South and East China Seas. This experiment has been named the Asia Seas International Acoustics Experiment (ASIAEX). Started from 1999 a series of pre-tests in SCS for area survey [ 1 13, in conjunction with modeling and remote sensing studies. In recent South China Sea internal wave study, as a part of the ASIAEX pre-test program, five moorings had been deployed in April 2000 for a month. The moorings consisted of a chain of thermistors and Acoustic Doppler Current Profiler (ADCP). Simultaneous SAR coverage from ERS-2 and RADARSAT had been collected and processed. This preliminary survey from year 2000 definitely helped on the planning for the major field experiment in May 2001. During ASIAEX-2001 in SCS, three ships from Taiwan were used [ l l ] . The moorings had been deployed by the end of April, and SEASOAR survey was carried out from April 29 to May 14. The intensive period of measurements was from May 3 to May 18. It’s found that the internal wave field was dominant in the ASIAEX area and its magnitudes were also larger than the prediction from planning estimates.

2. Recent SAR observations

Besides it provides synoptic information, satellite remote sensing is critical to several aspects of ASIAEX, including tracking the internal waves, and locating surface fronts and mesoscale features. In recent South China Sea internal wave study, simultaneous SAR coverage from ERS-2 and RADARSAT have been collected and processed from Taiwan ground station in near real-time [12]. As an example, Figure 2 shows an ERS-2 SAR image (100 km * 100 km) of huge internal solitons collected on April 26, 2000 north of DongSha Island in SCS. The crest of solitons is longer than 200 km with amplitude of more than 100 m. Based on mooring data, the maximum current in the upper layer was over 2 d s , and 1 d s in the opposite direction in the bottom layer, which is typically induced by a mode-one wave. In four hours, the SAR images were interpreted and a delineated map was then transferred to the ocean research ship in SCS by fax. Based on the satellite information, then the Chief Scientist on board may coordinate the survey strategy for internal wave observation.

300

Figure 2. RADARSAT ScanSAR image collected northeast of the South China Sea on April 26, 2000, showing internal wave packet.

YB

ZI

10

411al9s 05;m 4lioIBB 06:w 4fID@S IXW

Figure 3. Temperature time series collected from thermistors at different depths on April 10, 1999 near Dong-Sha Island. The mode-two waves are lagging behind the mode-one waves by 4 hours.

301

A new wave process on the shelf break has been observed during ASIAEX pre-test from a mooring near Dong-Sha Island on April 10, 1999. The thermistor chain data (Figure 3) show and indicate the mode-two internal waves with negative temperature fluctuation in the mixed layer and positive value in the bottom layer. These mode-two waves are lagging behind the diurnal tide (with mode-one waves) by about 4-hours since mode-two waves have slower wave speed than mode-one waves. The ADCP data from mooring also confirm the mode-two internal solitons on April 10, 1999 with two-zero crossings in current profile and are consistent with the thermistor chain data. The mixed layer depth was about 110 m (at location of zero-crossing in temperature profile) from thermistor data, and ADCP data show a mixed layer depth of 120 m (at location of maximum current). The maximum current induced by these mode-two waves was over 1 d s . Due to their different wave speeds mode-one and mode-two waves will separate after evolving on the shelf. Based on the observation of 4- hours lagging or separation between mode-one and mode-two waves, the location of generation for mode-two waves can be traced back from mooring and was located at shelf break of 180 m water depth approximately.

1 %

**

Figure 4. Subscene of RADARSAT ScanSAR image collected over ASIAEX area on May 18,2001, showing wave-wave interaction of two internal wave packet systems.

302

However, the internal wave field in ASIAEX area sometimes can be quite complicated. As shown in Figure 4 on May 18, besides the regular soliton packet propagating in the west direction, there are the second wave packets refracted and generated by Dong-Sha Island propagating to the north in ASIAEX area. These two wave systems, an east-west propagating soliton system from the Luzon Strait and another local generated north-south linear wave packet from shelf break, have been merged as a circle by the nonlinear wave-wave interaction. The intersection areas show an interrupted front with a kink as a result of wave interaction. Figure 5 show three wave spectra from (a) east-west soliton system, (b) north-south linear wave, and (c) merged wave packet, respectively. The solitons have an averaged wavelengthkeparation of 1.2 km and the wavelength of linear waves is about 900 m, while the merged wave packet has a wavelength of 750 m only with a northwestern propagation direction. This nonlinear wave-wave interaction is quite similar to the observation in the Yellow Sea [6] . Not only has the direction of wave train shifted after interaction, but the number of waves in the wave packet and their wave lengths, amplitudes have also changed.

-- - a m ** -4m2 5tQI &ma (Ilgi o w

x @M

Figure 5. Two-dimensional wave number spectra of Subscenes of SAR images (a) East-West travelling solitons, (b) North-South travelling wave packet, and (c) waves in the interaction zone.

303

3. Internal Wave Data Analysis

All moorings have consisted of ADCP during ASIAEX. As an example, Figure 6 shows ADCP current time series data at different depths from a mooring near DongSha Island collected on April 9-18, 2000 showing large internal solitons passing through the area after April 15. The mixed layer depth was about 200 m (at location of zero-crossing in current profile) from ADCP data. As shown in Figure 6, the maximum current in the mixed layer was about 2 m/s in the western direction, and was more than 1 m/s in the eastern direction in the bottom layer, which is typically induced by a mode:one wave. In this study, mooring data from ASIAEX-2001 test area (S2, S4, S5, S6, S7), and S8 (in 800m water) are used. For example, there are two wave packets in SAR image on May 9, 2001. First the soliton packets are identified from ADCP data based on SAR image, so the shiWlagging time between moorings can be estimated from the arriving time of each identified packet. Then ADCP velocity components have been re-combined or rotated to find the direction of maximum current induced by internal waves that is also the propagation direction of local internal waves.

Using the shiR distance and time along wave propagation direction, the internal wave speed between moorings for each packet has been computed to compare with S A R data result. In deep water, wave speed of 1.7 m/s agrees well with estimate of 1.77 m/s near S8. On the shelf in ASIAEX area, the wave speeds (in m/s) are slowing down quickly from 1.77 (between S7 & S8) to 1.14 (between S6 & S7), 1.08 (between S5 & S6), 0.76 (between S4 & SS), 0.72 (between S2 & S4). Also, waves are shifting direction from 25 degree from the West to 25, 45, 45, 75 degree. Local wave speeds near mooring stations have also been estimated based on S A R (spatial) and mooring (temporal) observations as discussed before. These speeds of 1.67 m/s at S8, and 1.54 m/s at S7 are quite consistent with the results of mooring data.

In addition, number of solitons in packets, soliton widths, and wavelengths can be estimated based on KdV type model. Three solitons in packets are identified at moorings S5, S6, and S7 as also observed from SAR data. For the first wave packet, the soliton widths are found to be 0.7 km at S4, 1.5 km at S5, 0.9 km at S6, and 1.4 km at S7, which is consistent with SAR observation of 1.4 km. The separation distances (or wavelengths) are 2.4-1.6 km at S5, 4.2-1.6 km at S6, 4.4-2.1 km at S7, and 9.7-2.4 km at S8. The S A R estimates of separation distances are 4.3-2.8 km that is also consistent with observations at moorings S6 and S7. The number of solitons in each packet and the separation distance between solitons (wavelength) can be also estimated from mooring measurements once the local wave speed is known. Then, the characteristic

304

width can also be determined theoretically from the S A R data based on KdV- type formulations [13]. The result was quite good for the first packet (1.4 vs. 0.9 - 1.5 km). The current induced by internal waves can be obtained from ADCP data, and it ranges from 0.5 to 1.2 m/s for the first packet. In summary, the comparison of internal wave speed, direction, width, and length derived from ADCP data at moorings (S2 to S8) and SAR data in ASIAEX region show good general agreement and consistent results on May 9,2001.

u (cmlssc)

&gr r e g . d ? g r IWP. 1&r la% z&r I G T IwPr lE% 1M)B

Figure 6. ADCP current time series data at different depths from a mooring near DongSha Island collected on April 9-18, 2000 showing Mode-I internal waves passing through the area after April 15.

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4. Nonlinear Internal Wave Evolution on a Shelf

The evolution of nonlinear internal wave packet on a continental shelf has been studied and formulated [3], [7]. The dissipation effects on solitary wave evolution are considered to be important in the shallow water owing to internal wave breaking and strong turbulent mixing. The evolution equation of nonlinear internal waves in water of depth H(x,t) and amplitude A(x,t) with variable coefficients is given by

A, +COAX + aAA, + f l 0 A , - "1, = 0 (1)

where C, is the linear wave speed for very long waves (k+ 0), and the wave- induced velocity is

U(Z> = CoAo4'(z) (2)

The eigenvalue problem for the vertical modes is

4 " + [ N 2 -rn2]4/C0 = O ,

@(O)=#(-H)=O, c, = O / K (3)

where o and K and are wave frequency and wavenumber, and the Brunt-Vaisala frequency or stability frequency.

=I . N(z) is

The environmental parameters for the nonlinear term is

0 0

a = 3C0 JqS 'dz /Z JqS2dz - H - H

and for the dispersion term is

0 0

y = j 4 2 d z /2 J4'"dz I -H -H

and E is the eddy viscosity for the dissipation term.

(4)

(5)

An effective horizontal eddy viscosity of E = 1 to 10 m2/s for solitons was used [2], [3]. Because eddy viscosity is not a property of a fluid, its value may vary with location and water depth. It is possible that local, incipient shear flow instability or wave breaking could be a cause leading to an eddy viscosity of

306

such value. For shallow water in the ASIAEX area, the bottom friction and induced mixing can be another dissipation mechanism.

Figure 7. (a) Density and (b) Brunt-Vaisala frequency profiles at mooring S6 during ASIAEX-2001.

Figure 8. (a) Mode-I vertical eigenfunction profile and (b) its derivative for current induced by internal waves. The current data indicated by solid circles and triangles are ADCP measurements for two solitons at mooring S6 on May 9,2001.

Density profiles are provided from CTD casts data collected during ASIAEX [ 113. As an example, Figure 7a, and 7b show the density and B-V frequency profiles of CTD cast on April 20, 2001 at mooring S6. The peak of B-V frequency profile is at 50 m approximately. The vertical mode-1 eigenfunction and its derivative are shown in Figure 8a, and 8b. Notice that the peak of mode-1 eigenfunction is at 100 m, which is approximately at the bottom of mixed layer. The linear wave speed is determined by the eigenvalue and is found to be Co = 1.06 m/s for mode-1 wave (K = 0.0001 m-'). From Equation 2, then the wave amplitude can be estimated based on ADCP current measurements of solitons. Figure 8b also shows the best fit of ADCP current

307

measurements in two wave packets (as triangles and circles) to mode-1 eigenfunction derivative and results in the wave amplitude A, of 63 m and 32 m respectively. The same procedure has also applied to moorings $7 and S5 and results in the wave amplitudes of 70 m and 80 m, respectively.

In order to demonstrate the effects of varying depth on the environmental parameters, all CTD casts data collected extensively in April and May around ASIAEX area are used to solve the eigenvalue problem (Equation 3) for various depths. Figure 9a shows the resultant environmental parameters a and y as a function of depths. Notice that the nonlinear parameter a is across zero at depth of 100 m and peaked at 300 m approximately. The linear wave speed C, (from model) and wave speed C (from data) as a function of depths is shown in Figure 9b, and the difference between wave speed C and linear wave speed C, is mostly because of the nonlinear effects. These results for a wave propagating from a deep water of 800 m near the edge of the continental shelf to a nearshore water depth of 70 m can be summarized as follows: (1) the wave speed is reduced to less than half of its initial value, which shows wave speed retardation due to shoaling in shallow water; (2) the parameter y for the dispersive effect decreases exponentially, which indicates the compression of the wave; (3) the parameter a for the nonlinear effect is relatively flat in deepwater, then peaks up at 300 m depth, decreases rapidly in shallow water, and it is across a location of critical depth of 100 m where a = 0. Because a changes sign, the wave of depression will not survive after critical location and may disintegrate into a dispersive wave train, then evolve into an elevation packet [7].

Figure 9. Internal wave (a) environmental parameters, and (b) speeds as a function of depth based on CTD data.

308

5. Numerical Simulation and Comparison

The effects of varying depth on the numerical simulation of wave evolution can be considered to fall into three categories: (1) The appearance of variable coefficients in the K-dV type evolution equation represents the first-order effect; (2) the shoaling effects due to variation of wave speed and eigenfunction along the path of wave propagation; (3) the dissipation effects vary with location and water depth. Near ASIAEX area approaching shelf break, the wave amplitude initially will increase significantly due to shoaling effects. Once the solitons become highly nonlinear on shelf break, the eddy viscosity may increase and could be eroding the sharp peaks of the large solitons, reducing their amplitudes and increasing their apparent widths at the same time. Therefore the shoaling effect eventually could be suppressed by the dissipation effect.

A numerical approach using Fomberg’s pseudo-spectral method [2], [3], [7] has been developed to solve the evolution Equation (1). A Fast-Fourier Transform (FFT) algorithm is used in the spatial coordinate and the split-step method is used for time derivatives. The choice of time step and mesh size has to be made with care in order to obtain an accurate numerical solution. The time step was chosen in order to maintain numerical stability; the computational reference frame was chosen to move in the direction of wave propagation at a certain constant speed such that the wave train remains in the computational domain. Thus, changes in wave speed as well as shape will become apparent in a space-time evolution plot. Also, the Hanning window is used to filter out any waves entering the computational domain from the adjacent domain.

A numerical simulation is performed by solving the initial value problem described by Equation (1) with SAR observation of a soliton near S8 on May 9, 2001 as an initial condition. The initial condition of a well-developed soliton in deep water (about depth of 1 km) is given by

A = A, sech’ ( x / L ) (6)

where A, = 80 m, L = 1.5 km which corresponds to a width of 3 km, are used.

Figure 10a shows the simulated space-time evolution of nonlinear internal waves in the ASIAEX area from deep water (depth of 1 km) to S5 (depth of 180 m) with variable environmental parameters and dissipation effects. Because eddy viscosity is not a property of a fluid, its value may vary with location and water depth. For this case study, the effective horizontal eddy viscosity E = 1 m2/s in deep water linearly increased to 6 m2/s in shallow water at S5 is used. In

309

this sequence, an initial well-developed soliton in deep water develops to a large wave packet due to bottom effect. The wave amplitude initially will increase significantly and its width decrease at the same time due to shoaling effects. The simulation window is moving with a reference speed of 1.5 m / s . Once the solitons become highly nonlinear on shelf break, the eddy viscosity increase and erode the sharp peak of the large soliton, reducing its amplitudes. Near S5 in shallow water it evolves into a large wave packet of three rank-ordered solitons after 12 hours. The numerical simulated wave amplitudes are 60, 40, and 30 m with peak-to-peak distances of 7.1, and 1.3 km. As compared with mooring data at S5 in Figure lob, the wave amplitudes are 80,40, and 32 m with peak-to-peak distances of 7.6, and 1.8 km; the comparison is reasonably good.

-roo 0 1OO

Distance (loom)

Figure 10. (a) Numerical simulation of a nonlinear soliton propagating from SAR observation in deep water evolved into three solitons in a packet at mooring S5 on shelf and compared with (b) S5 ADCP current measurement.

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Figure 11 shows the numerical evolution of the internal soliton packet propagating from mooring S5 (Figure lOa) to mooring S3 (depth of 85 m) in shallow water with a decreasing dissipation from E = 6 m2/s to 2 m2/s. The change of polarization from depression waves to elevation waves across the critical depth has been demonstrated in this simulation. Since the nonlinear parameter is across zero at a location of critical depth of 100 m, the waves of depression can not survive after critical location (about 2.5 hr) and may disintegrate into a dispersive wave train (after 4 hr), then evolve into an elevation wave packet (after 6.25 hr). Since eddy viscosity may vary with location and water depth, therefore, the change of polarization from depression waves to elevation waves across the critical depth is always possible as observed [ 141 when the environment condition is favorable.

I

I i I

Blsmca (toan) *m .r200 -100 0 NXI 2M m

Figure 11. Numerical calculation of the internal soliton packet propagating from mooring S5 (Figure 10) to mooring S3 in shallow water for decreasing dissipation. The change of polarization from depression waves to elevation waves across the critical depth has been demonstrated in this simulation.

6. Summary and Discussion

In this paper, nonlinear internal wave evolution in the northeast of South China Sea is studies based on the ASIAEX SAR, CTD, and mooring data. Selective sets of mooring data concurrent with SAR observations are presented and

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analyzed for the parametric study of internal wave characteristics. Based on these results from data analysis, the typical internal wave characteristics evolved from deep water onto the shelf can be summarized as follows:

a) water depth is from 800 m (at S8) to 70 m (at S2) in shallow water; b) internal wave propagation direction is shifting from 25" to 75"; c) internal wave speed ranges from 1.8 m/s to 0.72 m/s; d) soliton width is changing approximately from 1.7 km to 0.7 km, e) soliton number in packet is from 5 to 3 to wave train in shallow water; f ) soliton separation ranges from 9.7 km and 2.4 km to wavelength of 1.2

km; g) wave amplitude in average is from 95 m to 63 m and less in shallow

water.

These observations have provided a calibration on SAR data and inputs for the numerical simulation of nonlinear wave evolution on the continental shelf.

The mesoscale variability, mean horizontal and vertical shears and varying stratification near the shelf-break are highly transient in AprilMay during the spring transition from winter monsoon to summer typhoon season. Therefore, the evolution of internal solitons in the ASIAEX test area at shelf-break is especially complicated in ApriVMay with many interested features such as mode-two solitons [14]. The generation of these mode-two waves is most likely due to the intrusion of seasonal thermocline at the shelf break. When the mode- one soliton from the open ocean with a deep permanent thermocline encounters the fresh shelf water with additional shallow seasonal thermocline (to form a three-layer system), the mode-one wave energy will then be redistributed into the mode-one and mode-two waves at shelf break. The solitons are in transient with continuous evolution and dissipation along the shelf. During ASIAEX, nonlinear internal waves propagating up the slope have been tracked by research ship [14]. The location of the initiation of the depression to elevation conversion has been identified in their high-resolution acoustic data.

It is clear that these internal wave observations near the ASIAEX area provide a unique resource for addressing a wide range of processes on soliton propagating up the slope. Among these the following are included: 1) the disintegration of solitons into internal wave packets, breaking, and dissipation; 2) the shoaling effects of variable bottom topography on wave evolution, generation of mode-two waves, and internal wave-wave interaction; 3) the evolution of nonlinear depression waves through the critical depth to convert to elevation waves. Numerical simulations have been performed by using SAR

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observed internal wave field in the shelf break region as an initial condition to produce the wave evolution. The numerical simulation compares reasonably well with the measurements at the downstream mooring S5 on shelf for three solitons in a large wave packet as regards their amplitudes and wavelengths and wave speeds. This type of numerical simulation demonstrates the method of data assimilation to integrate all data from SAR, moorings, CTD casts, and ship- board marine radar. The inclusion of these physical processes is essential to improve quantitative understanding of the coastal dynamics and the sensitivity for acoustic propagation.

Acknowledgement

The authors wish to thank ASIAEX science team for their valuable discussions and suggestions, especially to Steve Ramp of the Naval Postgraduate School, T.- Y. Tang and J. Wang of the National Taiwan University.’ Help provided by Y.- J. Yang of the Taiwan Navy Academy to collect mooring data is also acknowledged. ASIAEX work has been supported by U.S. Office of Naval Research (ONR) and also funded by Taiwan’s National Science Council (Hsu).

References

1. J. R. Apel, J. .R. Holbrook, A. K. Liu, and J. Tsai, “The Sulu Sea internal soliton experiment,”J. Phys. Oceanogr., Vol. 15, pp. 1625-1651, 1985.

2. A. K. Liu, J. R. Apel, and J . R. Holbrook, “Nonlinear internal wave evolution in the Sulu Sea, ” J. Phys. Oceanogr., Vol. 15, pp. 1613-1624, 1985.

3. A. K. Liu, “Analysis of nonlinear internal waves in the New York Bight,” J. Geophys. Res. Vol. 93, pp. 12317-12329, 1988.

4. A. K. Liu, and S. Y. Wu, “Satellite remote sensing: SAR,” Encyclopedia of Ocean Sciences, London: Academic Press, Edited by J. H. Steele, S. A. Thorpe, and K.K Turekian, vol. 5, pp. 2563-2573,2001.

5. N. K. Liang, A. K. Liu, and C. Y. Peng, “A preliminary study of S A R imagery on Taiwan coastal water,” Acta Oceanogr. Taiwanica., vol. 34, pp.

6. M. K. Hsu, A. K. Liu and C. Liu, “A study of internal waves in the China Seas and Yellow Sea using SAR,” Continental ShelfRes., vol. 20, pp. 389- 4 10,2000.

7. A. K. Liu, Y. S. Chang, M. K. Hsu and N. K. Liang, “Evolution of nonlinear internal waves in the East and South China Seas,” J. Geophys. Res. Vol. 103, pp. 7995-8008, 1998.

8. J. B. Bole, C. C. Ebbesmeyer, and R. D. Romea, “Soliton currents in the South China Sea: measurements and theoretical modeling,” Offshore Technology Conference, OTC 7417,367-376, 1994.

17-28, 1995.

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9. M. K. Hsu, and A. K. Liu, “Nonlinear internal waves in the South China Sea,” Canadian J. Rem. Sens., vol. 26, pp. 72-81,2000.

10. S. R. Ramp, J. F. Lynch, P. H. Dahl, C.-S. Chiu, and J. A. Simmen, “ ASIAEX fosters advances in shallow-water acoustics”, EOS, AGU Transaction, Sept. 2003.

11. A. K. Liu, and M. K. Hsu, “Nonlinear internal wave study in the South China Sea Using SAR,”Znt. J. Remote Sens., 25, 1261-1264,2004.

12. Q. Zheng, Y. Yuan, V. Klemas, and X.-H. Yan, “Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width,” J. Geophys. Res., vol. 106, pp.

13. M. K. Hsu, A. K. Liu, and C.-H. Lee, “Using SAR Images to Study Internal Waves in the Sulu Sea”, J. Photogrammetry and Remote Sensing, 8, 1-14, 2003.

14. M. H. Om, and P. C. Mignerey, “Nonlinear internal waves in the South China Sea: observation of the conversion of depression internal waves to elevation internal awves”, J. Geophys. Res., 108, C3, 3064, doi: 10.1029/2001 JCOOll63,2003.

3 14 15-3 1423,2001.

A NUMERICAL PREDICTIVE MODEL OF TIDES AROUND TAIWAN

HSIEN-WEN LI Department of Civil Engineering, Ming Shin University of Science and Technology, Shin

Feng, Shin-Chu County, Taiwan 304. E-mail: lihw@,must.edu.tw

YUNG-CHING WU Department of Oceanography, National Taiwan Ocean University, Keelung, Taiwan 202.

E-mail: william.w52062 1 @,msa.hinet.net

In this study we developed a two-dimensional numerical model to predict tidal currents around Taiwan. The finite difference method is used to solve the control equations. The model area is 900 Km x 900 Km, and Taiwan lies in the central part of model. The tides at open boundaries of model are the driving forces for tidal flow inside the model. The tides at model boundaries are calculated through harmonic method making use of the principal tidal constants of M2 , Sz, K1 and 01 partial tides. The computed tides are compared to the observed ones at some tidal stations around Taiwan. In the offshore area between Botsuliao and Danshui of westem Taiwan the strongest tidal current in a cycle is more than 65 c d s , some places are about 2 d s . The tidal currents are weak off eastern Taiwan, the magnitude are normally lower than 10 c d s .

1. IntrQdUCtiQn

The East China Sea has a broad continental shelf in the west Pacific. Taiwan Strait is situated in the southern part of East China Sea. The tidal current is strong and plays a major role of flow system in the Taiwan Strait. The deep ocean basin lies east of Taiwan, and Kuroshio is the dominant ocean current in this area. Many ships are cruising through Taiwan Strait and along the offshore area east of Taiwan. A cargo ship AMORGOS registered in Greece was wrecked off southern Taiwan on 14 January 2001. Her fuel oil spilled in the offshore area south of Taiwan. To monitor the oil spill spreading we must understand the flow pattern in this area, especially the tidal currents. There are many industrial areas along the west coast of Taiwan, which faces the Taiwan Strait. The study of tidal current in Taiwan Strait is an infrastructure of coastal and offshore engineering west of Taiwan. The tidal phenomena in the Taiwan Strait are mostly excited by the propagation of tidal waves from west Pacific. The tidal waves come into the

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strait from north and south boundaries. These two waves meet in the central part of the strait. Thus the tidal range is quite large (- 3 m) and tidal current is very strong in the central Taiwan Strait.

The tide prediction at selected locations can be successfully carried out by means of the harmonic method provided the tidal harmonic constants are known at those stations. The harmonic analysis of tidal potential has been well developed by Doodson’. The analysis of tidal spectroscopy developed by Munk and Cartwrighe is widely used for tidal prediction as well. The calculation of tides for the hture date at tidal gauge stations can also successfully be accomplished by means of harmonic method. The tidal table of most harbors in the world can be thus purchased. However these methods can only be used for calculation of tides at observed gauge stations. It is difficult to calculate tidal currents using the above methods. It is also impossible to predict the tides in a coastal area without long historical tidal records.

The numerical method for tide calculation was initiated by Hansen3 . Since that time many numerical methods for solving tidal problems have been developed4. In order to predict the sea levels directly from the astronomically derived tide generating forces, some attempts to solve tidal dynamics equations for the global ocean were made. However the results of these tidal models of the world oceans are inadequate to predict sea levels for practical use. Cartwright’ , Hendershod , Hendershott’ and Marchuk and Kagan’ gave detailed reviews of numerical models of global ocean tides.

Nevertheless knowledge of tides and tidal currents in marginal and adjacent seas plays an important role in coastal engineering. The tidal phenomena in a marginal sea, such as the Taiwan Strait, are mostly excited by the propagation of tidal waves from the open ocean. The tidal phenomena of Taiwan Strait are extremely complex. The semi-diurnal waves come into the strait from both ends. The wave coming in from the East China Sea propagates southwestwards and that coming in from South China Sea northwards. These two waves meet near the central part of Taiwan strait.

During the early 1930’s a considerable number of tide measurements along the coast of East China Sea and South China Sea were made by Ogura’. These tidal measurements are used as the basis to construct co-tidal charts in the seas adjacent to Taiwan. Thus we can calculate the tidal constants at model boundaries. In this study the principal tidal constants of MZ, Sz, K, and O1 at each grid point of the model boundaries are inferred based on these charts. The combination of the 4 principal partial tides constitutes more than 90 per cent of the tidal phenomenon. Then the tidal elevations at model boundaries are specified according to the harmonic theory.

316

The hydrodynamical-numerical method is employed to solve tidal equations, which govern the tidal flow. In this study we use a two-dimensional tidal model for calculating tidal currents around Taiwan. To verify the developed model the computed tides are compared to the observed ones at some tidal stations around Taiwan.

2. Governing Equations

The following system of hydrodynamical equations is used for development of a numerical model of tides.

ap 1 a a< rbx au - =fv - -(< + H)(- + pg-) - -

at P ax ax P

Here x, y, z are components of a Cartesian coordinate system, with x, y, z axes in eastward, northward and upward directions respectively; u and v are velocity components in x and y directions respectively; p is the density of seawater, which is assumed here as a constant. g is the acceleration due to gravity and f the Coriolis parameter. A is the horizontal eddy viscosity coefticient. The tide- generating forces are neglected because of the small area of h s model. (U, V ) is the volume transport vector. is the free surface displacement from the undisturbed state and positive upward. The atmospheric pressure at sea surface, Pa, will be neglected in this model. rbx and rb,, are bottom stresses in x and y components, which are exerted by the sea water upon the sea bottom.

317

Equations (1) and (2) are vertical integration of the equations of motion in x and y components respectively. Equation (3) is vertical integration of the continuity equation for an incompressible fluid. In the procedure of vertical integration we have used the hydrostatic equation to determine the pressure at depth z and incorporated the corresponding boundary conditions.

A measure of water velocity is then obtained by averaging this volume transport over depth, thus

Because the surface displacement ( is much smaller than the water depth H, we will take ((+ H) =H hereafter.

Assuming the horizontal eddy viscosity to be constant and the velocities to be uniform with the depth, the expressions for the turbulent fiiction of x- component can be approximated as follows, ,

The computation for y-component is similar as above. For the present calculation A is assumed to be of the order of lo6 cm2/sec.

On assuming a uniform velocity distribution in the vertical, the bottom friction can be approximated as follows,

where k is a nondimensional skin hction coefficient of the order of 2.5 x lo", which was used by Hansen" and Ueno".

The nonlinear terms may be approximated as follows,

j u u d z = - , j u v d z = - uv , j v v d z = - vv 5 uu H

5

H -H

H -H -H

With the above approximations equations (l), (2) and (3) can be solved numerically by means of finte difference method. Here we use the conjugate Richardson lattices'* for space-differencing.

The derivatives in time and space are approximated by the central difference scheme, where the divergence term, the pressure gradient terms, and the Coriolis terms are evaluated at a time-step centered between the old and new time. The dissipation terms and nonlinear terms are evaluated at the old time step. The stability criterion is the familiar Courant-Friednch-Lewy condition,

318

At 1/& -<- As c,

where C,, is the spe d f Prop e tion of the fastest waves in the model. Thus C,, = ,/gHmx , which is the fastest speed of shallow water wave, and H,, is the maximum depth in this area. Here H,, is 6361 m. In this model AS(= Ax = Ay) equals to fi Km. Therefore we choose At = 4sec for calculation.

The equations (l), (2) and (3) are then approximated as follows,

a2u a2u + A(2 ax + 2) (4)

2 l l

1 1 1

Ax 2 2 T ( x , y , t ) = < ( x , y , t - At) - - U(x + - A x , y,t --At) - U(x - - Ax, y,t --At)

(6) 2 ' 1 1

A t { 2

1 1 - ~ { v ( x , y + -Ay , t --At) - v ( x , y - - A y , t - - A t )

AY 2 2 2

For the horizontal viscosity to couple the conjugate Richardson lattices, the Laplace operator should be evaluated by a rotation of the coordinate axes over

319

45 degrees in order to accomplish a virtual smoothing to reduce the grid di~persion'~. The last derivatives in (4) and ( 5 ) are also approximated by central differences.

The lateral boundary points are composed of U, and V points along the coastline, where the non-slip condition are used, i.e. U = 0, V = 0. At the open boundaries the empirically determined tide by the common harmonic method will be used.

To substitute the predicted tides into open boundaries by means of harmonic method, we need the tidal constants. The tidal charts of M2, S1, KI and O1 constituents in the seas around Taiwan are constructed by Li14. These charts are based on the harmonic constants along the coast of Japan, Korea, Taiwan and the Chinese Mainlandg. Thus the amplitude and phase lag of M2, S2, K1 and 0, at each boundary point is calculated by interpolation fiom the constructed co-tidal charts.

The tidal height at boundaries is computed by use of the harmonic theory, m

<(ti = U, + C ~~a~ cos(mit + E~ - ai) i=l

where a0 is the mean sea level. Ni, ai, mi , Ei, and Ai are the node factor, amplitude, angular frequency, equilibrium argument and phase lag of the i-th constituent tide respectively. The node factor (Ni) and equilibrium argument (Ei) can be calculated by means of the related astronomical f~rmula '~. In this study we take m = 4, i.e. i = 1, 2, 3,4 which corresponds to M2, S2, K1 and O1 partial tide. In this model uo is taken to be zero.

3. Results

To verify this model we calculate the tides fiom January 1 to February 1, 2001. The bottom topography of the study area is shown in Fig. 1. Some tidal stations around Taiwan are taken for verification. These stations are shown in Fig. 2. The computed and observed tidal heights at these stations around Taiwan are shown in Fig. 3. The mean value in the observed tidal curve has been subtracted out.

In these figures we can see that most of the calculated tides are in agreement with the observed ones. Especially the phases of both the calculated and the measured tides are nearly matched. At some stations the calculated tidal ranges are different from the measured ones. The deviation of tidal range between the calculated and the measured tides may be due to the difference of model bottom depth fiom the actual topography. The bottom depth of the model is derived from the navigation chart, which is constructed by use of some limited survey. In particular the survey near the coast is quite few. The population density is very

320

high in the plain area of western Taiwan. Many constructions were built along the coast in the past years, including harbors, factories and aquaculture ponds. Thus these human activities have constantly changed the bottom topography in the offshore area west of Taiwan.

The vertically averaged tidal currents are also computed. Most calculated tidal currents in the sea off eastern Taiwan are very weak. The tidal stream in a cycle is smaller than 10 c d s in offshore area between Suao and Chenkung. In offshore southwest of Lanyu the strongest tidal current in a cycle is about 20cds. In the sea south of Taiwan, i.e., offshore south of Hobihu, the tidal current is stronger than that of eastern Taiwan. The strongest tidal current in a cycle is about 45cds. In the offshore area between Ryuchoyu and Kaohsung the tidal current is about 15 c d s , which is also weaker than that of southern Taiwan. The tidal current is about 25 c d s off Tungsu. In the offshore area between Botsuliao and Danshui the tidal current in spring tide is more than 65 c d s . The strongest tidal current is found in the Penghu channel between Taiwan and Penghu iIsland. The maximurn velocity is about 2 d s . The tidal current off Lensanbi, i.e., in the area of northern Taiwan, is about 45cds, which is similar as that of southern Taiwan. The tidal current off Gengfang is about 30 c d s .

4. Discussion and Conclusion

The model of this study is similar to Li’s model14. But here the grid distance is Km, which is only one tenth of the grid distance in Li’s model. Thus the

precision is improved and we get more accurate result. The comparison of calculated and measured tides shows better result than the previous study. The tidal current near coastal area can be depicted in detail with fine gnd. The emphasis of this research was put on practical aspects with the intention to provide a predictive scheme of tides and tidal currents. For improvement of the calculated results, we need more accurate data of bottom depth in the shallow sea, especially in coastal area near western Taiwan.

Acknowledgements

The authors give special thanks to the Central Weather Bureau for providing tidal data. This research is supported by the National Science Council of the Republic of China under contract NSC 92-2625-2-1 59-001.

References 1. A. T. Doodson, Proc. Roy. SOC., A, 305 (1921). 2. W. H. Munk and D. E. Cartwright, Philos. Trans. Roy. Soc.. A, 533(1966). 3. W. Hansen, Deut .Hydrog. Zeits., Erg. 1, (1952).

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4. W. Hansen, in the Sea, Vol. I , Ed. M. N. Hill, (Willey, New York , 1962), p. 764. 5 . D. E. Cartwright, Rep. Prog. Phys., 665(1977). 6. M. C. Hendershott, in the Sea, Vil. 6, Eds. E. D. Goldberg, et al., (Willy, New York,

7. M. C. Hendershott, in Evolution of Physical Oceanography- Scientic Surveys in Honor of Henry Stommel, Eds. B. A. Warren and C. Wunsch, (MIT Press, Cambridge,l981), p. 292.

8. G I. Marchuk and B. A. Kagan, Ocean tides-mathematical models and numerical experiments, (Pergamon Press, Oxford, 1984).

9. S. Ogura, Bull. Hydrog. Dept., 7 , (Imperial JapaneseNavy, 1933).

1977), p. 47.

10. W. Hansen, Tellus, 287( 1956). 11. T. Ueno, Oceanog. Mag. 53(Japan Oceanog. SOC., 1964). 12. G W. Platzman, Meteor. Monog., VoZ. 4, No. 26, (America1 Meteor. SOC., 1963). 13. T. J. Simons, Canada Centre for Inland Waters, Sci. Ser., No 12, (1973). 14. H.-W. Li,Proc. Natl. Sci. Counc.ROC(A), 11-1, 74(1987). 15. P. Schureman, Coast Geod. Survey, Spec. Pub., 98, (US. Dept. of Commerce,

1958).

3200000

3000000

2800000

2600000 *

2400000

2200000

X

Fig. 1 The bottom topography of the area for calculation of tidal current around Taiwan. The numbers in the axes denote distance in meter With respect to the reference point.

322

'Oenfang

Buao

Fig. 2. The selected tidal stations for comparison between the measured and the omputed tides around Taiwan.

Fig. 3.

Figure 3.1 Tide at Station Lenaanbi

-Irn1: 1i.5 1; 1i.5 1; 1G.5 14 l i . 5 1; l i . 5 16 2001/01/11 - zooimi/i5

Comparison of measured and calculated tidal elevation at fourteen stations (3. I).

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Figure 3.2 Tide sl Station Genrrng 80

60

40

20

B o 0 ._ = m L -20

-40

-W

-80

5 I

10 15 20 25 33 56 m 1 m i m 2 - n n i m i n i

150

l![ -50

- 1 0 0

-150

mim1n3 - aminin1 Figurs 3.5 Tide a Slstion Chenkung

m i m m z - a m m i n i

Fig 3. Continued (3.2, 3.3, 3.4, 3.5).

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16 70 2mimimz - 2w1mini

Figure 3.7 Tids at Station HobihU

Msasured .~aIua - 1w

80 .... Cslsulntsd value

5 I

10 15 20 25 30 35 m i m i m z - zmimini

Figure 3.8 T i d m -1 Station R'pChOyU

5 I

10 15 20 25 30 35 2m1m1m2 - 2wimin1

Figure 3.9 Tide at Station Kaohsung

2 ~ 1 1 0 1 x 1 2 - 2001mir31

Fig 3. Continued (3.6,3.7,3.8,3.9).

w

Elevat

ion lcm

l

II

II

U

J

326

Figure 3.13 Tide at Station C h u m

Measured value I i - 200

I 5 10 15 a3 25 30 35

-250 I 2001101102 - m1/01/31

202

150

100

-100

-150

Figure 3.14 Tide at Station Danahui

- Measured y a l u ~ ._._ Calculated value

, I . . .

5 10 15 20 25 a 2001101102 - 202lmlBl

Fig 3. Continued (3.13,3.14).

NEW CONCEPTS IN IMAGE ANALYSIS APPLIED TO THE STUDY OF NONLINEAR WAVE INTERACTIONS

STEVEN R. LONGt NASA GSFC / Wallops Flight Facility

Wallops Island, VA 23337. USA

Many earlier methods in image analysis are based on techniques that are best suited to linear and stationary processes. Here an application of the Empirical Mode DecompositiodHilbert-Huang Transform (EMDIHHT) techniques is presented as applied to water surface image analysis. The images represent nonlinear and non-steady wave interactions obtained by a vertical-looking, high speed, high resolution digital camera. Recent advances in this new and robust technique have been reported by [l], [2], [3], (41, [5], [6], and [7]. This work was originally developed for the analysis of nonlinear and non-stationary data as a function of time. These new methods have been extended here to include the analysis of image data, and have been used just as outlined in the numerous foundation articles listed above and several application articles (such as [7], [S], and [9]). The image data acquired can be expressed in terms of a numerical array of rows and columns, allowing this new concept developed for one-dimensional data (measured values recorded vs. time) to be now applied to these arrays row by row. Each slice of the data image, either row or column-wise, represents local variations of the image being analyzed. Just as much of the data from natural phenomena are either nonlinear or non-stationary, or both, so it is also with the data that form images of natural processes. Because of the nonlinear and non-stationary nature of natural processes, the EMD/HHT approach is especially well-suited for image data, giving frequencies, inverse distances or wave numbers as a function of time or distance, along with the amplitudes or energy values associated with these, as well as a sharp identification of imbedded structures. The various possibilities and products of this new analysis approach are presented, such as joint and marginal distributions which can be viewed as isosurfaces, contour plots, and surfaces that contain information on frequency, inverse wave length, amplitude, energy and location in time, space, or both. Additionally, the concept of component images representing the intrinsic scales and structures imbedded in the data will be described, with examples shown, as well as a technique for obtaining frequency variations of structures within the images.

1. Introduction

The world around us has always been appreciated visually. Our eyes constantly input a stream of data into our minds from the many processes occurring around us, images we process mentally to obtain distance, size, color, orientation, along

~~ ~

This work is supported by the NASA Oceanic Processes Program.

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with art and beauty, and even warnings of danger. Vision and images are so important to us, that we consider a lack of them to be a true handicap.

In recent times, it has become technically possible to obtain images that are more than just pictures in the usual sense, images that are actually an array of numbers of high precision that represent point-wise measurements over an area, not just a gray scale value in a photograph, but a detailed measure of electromagnetic wave length and intensity representing color, heat, or x-ray intensity, to name just a few, all with underlying physical sigmficance. Images are also routinely acquired from other processes other than electromagnetic waves as well, such as magnetic resonance images, for example. These arrays of numbers are managed easily by computer, and can thus be subsequently displayed, printed, and viewed by us as an image, while representing a reality that our own eyes could never see directly. In this sense, modern imaging technology and techniques have expanded our vision, allowing us to “see” new concepts never before observed directly. This will also occur when applying a totally different method to image analysis, a method such as the EMD/HHT techniques. New types of results can be expected, allowing our minds to see the reality around us in totally new directions and with completely new concepts. To date, some of these new concepts using applications of EMD/HHT on one dimensional data include new results from earthquakes, ocean waves, rogue water waves, sound analysis, length of day measurements, etc. The approach has been so successful with nonlinear and non-steady data due to the concept of its basis functions being time varying and adaptive.

In general, let any data be written as X(d). Then the first step in this new approach is to “sift” or decompose the data into n empirical components such that

n

X(d) = C cj + rn (1) j=l

where cj is the jm component, r,, is the residue, and d denotes the axis over which the data varies, such as time or spatial distance, for example. The sifting stops when either the last component c,, or the residue r, reaches a value lower than a predetermined level, which is a value small enough to be of no consequence, or when r, becomes a monotonic function from which no more components can be extracted. Even if the data has a zero mean, the final residue can be different from zero. If the data contains a trend (a slow drift in the instrument calibration or the tide changing during ocean wave measurements, for example), then when the sifting is completed, the residue r, will be that trend.

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Just as in previous work using EMDMHT analysis, this sifting will be employed here on data that make up each row or column of an image array. Thus the initial processing used here follows the identical steps used in the earlier reports. However, the application to images requires that this new sifting processing and subsequent analysis to be repeated perhaps hundreds even thousands of times, depending on image resolution and size. Consider the example of a square image array of 512 pixels on a side, such as will be presented here. It would require 512 repetitions of the sifting process, where each repetition would produce a complete component set and residue as outlined in Eq. 1. One image of 5 12 x 5 12 would produce 5 12 component sets, each set with several distinct components. Once the component sets are obtained, they become the input to fUrther processing steps that can produce unique products from the initial image. After a brief overview, these various products will be discussed in turn, and examples used to illustrate the new possibilities opened up by the application of this robust technique to images.

2. General Concepts of Image Processing

Image processing in general has always made full use of available computer capabilities. Because of the shear size of the data to be analyzed, the needs of image processing have often “raised the bar” on hardware requirements, such as more memory, increased computational speed, increased internal bus rates, greater and better storage, etc. Just as the images to be processed come from widely varied sources such as satellite imagery, aerial photography, microscope imagery, industrial imaging for quality control, CT scan data for medical applications, etc., so do the methods vary in how the image is processed. It is almost always driven by a need to reveal or detect some feature within the image, to clarify the feature or resolution, or to make possible the measurement of the feature from the image, to name but a few. Because of this, image processing encompasses a wide range of mathematical techniques with a proven track record of effectiveness and established mathematical foundations.

The approach taken here is to refer the reader to well-established texts such as [lo] or [11] as a starting point in a vast literature already built up on the subject, while outlining the new and important methods here that can now be added to the available tools, for producing new and unique image products. The descriptive and mathematical groundwork for this new approach has already been established in the series of articles, denoted here as foundation articles, by [I], PI , P I , P I , PI , P I , P I , PI , and W I .

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3. New Concepts in Image Analysis

3.1. The Laboratory That Produced the Nonlinear Waves

The laboratory used for producing the nonlinear waves studied here is the NASA Air-Sea Interaction Research Facility (NASIRF) located at the NASA Goddard Space Flight Center I Wallops Flight Facility, at Wallops Island, VA. The test section is 18.3 m long and .9 m wide, filled to a depth of .76 m of water, leaving a height of .45 m over the water for air flow, if needed. The facility can produce wind andor paddle generated waves over a water current in either direction, and its capabilities, instruments, and software have been described in detail by [9], [ 131, and [ 141. The basic description is shown as Figurel, with an additional new feature indicated as new coils in Figure 1. These were recently installed to provide cold air of controlled temperature and humidity for experiments using cold air over heated water.

5 05 m New Coil

T

Figure 1. NASIRF, the NASA Air-Sea Interaction Research Facility main wave tank at Wallops

Island, VA.

3.2. The Digital Camera and Setup

The camera used to acquire the laboratory images presented here is a Silicon Mountain Design M-60 camera, capable of acquiring images of up to 1024x1024 pixels at a resolution of 4096 intensity levels, and at a rate of up to 60 imagesls. For the examples shown here, the resolution was set at 512x512 pixels, at a rate of 60 imagesls. The camera was mounted to look vertically down at the water surface, so that its 512x512 pixel image area covered a physical square on the water surface of 26.54 cm x 26.54 cm. Each pixel thus covered a square of about 0.5 18 mm each throughout the image area. To obtain an image array of surface slope values from each 5 12x5 12 image, the configuration illustrated in Figure 2 was used. The light source was mounted at the bottom of the tank and produced a uniform field of light initially. A thin film that varied in transparency between

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clear and black along the down wind direction was placed on top of the light source, resulting in a light intensity output that varied linearly with fetch. Because of this variation and the laws of refraction, each image acquired by the camera had the down wind surface slope at each pixel location on the surface recorded in the intensity level of that pixel in the image array. Using calibration lookup tables obtained through simple geometry and the application of Schnell's law of refraction, this intensity level was then converted to down channel surface slope for each pixel withm the entire array of 5 12 x 5 12 numerical values.

3.3. Experimental Images of Nonlinear Waves

With this imaging system in place, steps were taken to acquire interesting images of wave blocking due to an opposing water current. This is also illustrated in Figure 2, where the presence of a false bottom is shown.

SMD Camera

I

F

Light Intensity

Figure 2. The experimental arrangement to capture images of nonlinear and non-steady waves

A transparent window was placed over the light source that allowed the light source to function, while maintaining the slope of the false bottom for the water flow. When water flowed against the wave direction, an area of current shear was set up over the light box. As discussed in [ 141, waves can become trapped in such a shear zone, and shift to higher frequencies in the absence of wind. To illustrate what happens, Figs. 3a-d records this phenomena in a series of images. Although the camera acquired images at 6O/s, here a selection from the sequence is shown. To help our eyes visualize this, the 4096 levels of gray have been used to display the slope variation. A horizontal line down channel is also included to

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mark an area of interest, where a very rapid development occurs during the time covered by these images. Even though 1/60 second variations may seem to be rapid for a water surface, one gets the impression from these images that something is developing faster than the camera acquisition rate. The images obtained do, however, capture significant stages in this rapid development.

Figure 3a. Surface slope image 80. Wave packet moves right to left against the opposing current. Intensity bar gives the surface slope.

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Figure3b Surface slope image 86. Wave packet continues to moves right to left against the opposing current. This is 6/60 or 0.1 s after the image of Figure 3a. Note the sudden steepening of the capillary waves on the front face of the incoming wave.

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Figure 3c. Surface slope image 88. Capillary packet now changing dramatically after only 2/60 or 1/30 s.

Figure 3d. Rapid change continues within capillary packet, 1/60 s after the image of Figure 3c. Note the standing wave patterns that the capillary wave packet is meeting by examining the 4 images shown in Figure 3. Standing waves are evident in the image center and in the upper and lower left comer areas.

Using the horizontal line through the phenomena in image 80, Figure 4a illustrates the slope detail contained in the actual array of data values. A complex slope structure is shown as the approaching wave packet is getting ready to produce a capillary wave packet on the front face of the wave at the area of maximum steepness.

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Figure 4a. Slice through image 80 of Figure 3a, taken as a horizontal row of pixel values at pixel line 285 (14.77 cm). Note the standing wave structure near the center, and the capillary packet about to form approaching from the right.

Figure 4b shows a comparison of image slices taken from images 80,86,88, and 89. These are horizontal slices through the region of an interesting phenomena, and show the complex changes that can occur in surface slope over a time period covering only 9/60 second. As can be seen, the capillary burst appears suddenly and changes rapidly.

9 - E y 5 & I - P 1

10 1s 20 r(O_O"UI

Figure4b. Horizontal slices at pixel row 285 (14.77 cm) from the images of Figure 3a-d. The bottom slice is image 80 (Figure 3a). Above this, images 86, 88, and 89 are offset by 2, 4, and 6 respectively (from Figure 3b-d).

3.4. EMDH$HT Analysis on Images of Nonlinear Waves

3.4.1. Components, Contour and Surface Visualization

From Figure 4b, the slice from image 86 (image Figure 3b) was processed following the EMD sifting procedures, and the results are shown in Figure 5, where it can be seen that the complex surface may be represented by

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comparatively few components. The sifting was done via the extrema approach discussed in the foundation articles, and produced a total of 7 components. Figure 5 shows the original data at the top, offset by +8. The components from EMD processing are plotted under the data, component C1 being offset vertically by +7, followed by C2 through C7 offset vertically by one less for each component. Cl describes the slope of the capillary burst, whde C2 and C3 outline the standing wave structure. C4 through C6 are connected with the incoming carrier wave. C7 shown at the bottom here is more a representation of the slight underlying slope produced by the current flow.

Image 88 Slope Slice w%b Component. ORDel Dmm by 1 Each 10

01 I 5 10 15 20 25

nodzonml Maan- (-)

Figure 5. Image 86 (Figure 3b) horizontal slope slice offset vertically by +8. Under the data is the results of EMD processing, giving component C1 offset vertically by +7, followed by C2 through C7 offset vertically by one less for each component. CI describes the slope of the capillary burst, while C2 and C3 outline the standing wave structure. C4 through C6 are connected with the incoming carrier wave. C7 shown at the bottom here is more a representation of the slight underlying slope produced by the current flow.

Using the components illustrated in Figure 5, the next step with Hilbert- Huang Transform (HHT) analysis produces a result that can be visualized as in Figure 6a showing the contour plot of the result.

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Une2B5. Image Number 88

0 2

0 15

0 1

0 05

10 15 20 25 Howontal Distsno~ (cm)

Figure 6a. Image 86 (Figure 3b) showing the results from EMD/HHT computation on line 285 (14.77 cm) on the across-tank scale, a line passing through the sudden development of the capillary packet on the incoming wave’s front face. Note the rapid occurrence of the shorter capillary waves (larger number on the l/cm scale) around the horizontal distance of about 17 to 18cm. The other features near the 12 and 13 cm horizontal location at an inverse wave length of about 1 (l/cm) are from the standing waves.

In Figure 6a, a standing wave pattern can be seen between about 12 cm and 13 cm on the horizontal scale of the image slice. This pattern persists through many images preceding and following this moment in the sequence, and within the group of standing waves, different peaks took a turn at being the largest. Because they persist in time, though, they can be thought of as standing waves, trapped in the bloclung conditions of the current shear flow when the images were acquired. The shorter capillary wave group that appears on the leading edge of the crest, and suddenly bursts out rapidly (around the 18 cm area on the horizontal scale) are represented in the hgher values on the l/cm scale. To produce a true wave number, one only has to convert using

k = 2x11,

where k is wave number (in l/cm), and h is wave length (in cm). The peaks occur where certain wave lengths are dominating in the data, and are entered by the HHT process in the resulting array by location (horizontal location on the slice in cm), by wave length inversed on the vertical (I/cm) scale, and by amplitude or intensity of slope on the intensity bar scales.

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To visualize how rapidly the capillary wave burst occurs, a comparison of what has changed a short time later is presented i Figs. 6b. Image 89 was taken 3/60 or 0.05 s later than image 86(Figure 3b). During this time, the wave represented by its inverse wave length near 4 on the l/cm scale has moved about 1.24 cm against the rapid opposing current (compare Figure 6a and 6b). Ths represents a rate of about 24.8 c d s .

Uns 285. Image Number: 88

4 5

4

3 5

3

E 2 5 - 2

1 5

1

0 5

0 ' I 5 10 15 20 25

Horironbl Dlstanoe (cm)

Figure 6b. As in Figure 6a, but now image 89 taken 3/60 or 0.05 s later. During this time, the wave represented by its inverse wave length near 4 on the l/cm scale has moved about 1.24 cm against the rapid opposing current (compare Figure 6a and 6b). This represents a rate of about 24.8 c d s .

3.4.2. Volume Computations and Isosurface Visualization

Many interesting phenomena happen in the flow of time, and thus it is of interest to include how changes occur with time in the images. In order to include time in the analysis, comparisons may be done as was illustrated in Figs. 6a-b, but for a detailed analysis that includes time, something more is needed.

By starting with a single horizontal line from the image, a contour plot as was shown in Figs. 6a can be computed from the EMD/HHT analysis. Using a set of images, such as from image 1 to image 109 (109 images covering 108/60 or 1.8 second), and a slice through the capillary burst location, a set of 109 numerical arrays can be obtained from the EMD/HHT analysis. Each array can be visualized by means of a contour plot as shown before. The entire set of 109 arrays can now be combined in sequence to form an array volume, or an array of

338

dimension 3. Within the volume, each element of the array contains the amplitude or intensity of the surface slope from the image sequence. The individual element location within the 3D array specifies values associated with the stored slope intensity. One axis (call it x) of the volume represents horizontal distance down the slice as before, in cm. Another axis (call it y) represents the resulting inverse length scale (lkm) that signifies the inverse wave length of the slope values associated with waves in the data. The additional axis (call it z), produced by laminating the 109 arrays together, represents time, because each image was acquired in steps of 1/60 second. Thus the position of the element in the volume gives location x (cm) along the horizontal slice, inverse wave length (l/cm) along the y axis, and time (seconds) along the z axis.

To visualize this, isosurface techniques are needed. Th~s could be compared to peeling an onion, except that the different layers, or spatial contour values, are not bound in spherical shells. After a value of slope amplitude or intensity is specified, the isosurface visualization will make transparent all array elements outside of the level of the value chosen, while shading in the chosen value so that the elements inside that level (or behmd it) can not be seen.

Some examples of this are seen beginning in Figure 7a. Here, the most energetic level is seen by choosing a peel level of 0.1. Note the general up and left trend, denoting that as the waves move left into the current shear, increasing time causes the waves seen to move up along the time axis. Along this general up and left trend, capillary waves form suddenly, shifting to shorter wavelengths and thus larger l/cm values. The progression ends when the waves merge with the standing waves near the 12 to 13 cm area horizontally. These are the most energetic occurrences, and are seen in the absence of lower levels. To illustrate the effect of the peeling level and viewing angles, a slow variation in peeling value is presented as Figs. 7a-f.

Figure 7a. The most energetic level, seen by choosing a peel level of 0.1. Note the general up and left trend, denoting that as the waves move left into the current shear, increasing time causes the waves seen to move up along the time axis. Along this general up and left trend, capillary waves form suddenly, shifting to shorter wavelengths and thus larger l/cm values.

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At the peeling level of 0.05 shown in Figure 7b, more structure is seen, but these are hidden somewhat in the view presented in Figure 7b.

Figure 7b. Peel level now at 0.05. The trend denoting movement is still visible, but now many more structures are visible. Not easily seen in this view are haw these structures are subdivided.

By rotating slightly to another view, as in Figure 7c, different features are seen. From this, it appears that with the passage of time, shorter waves are created suddenly around the 18 cm horizontal distance area. These in turn appear at larger values of inverse wave length, or l/cm.

Figure 7c. Figure 7b rotated to reveal the structure at the peel level of 0.05.

By looking at a lower level, 0.02, Figure 7d reveals fbrther details. Note the finger-llke branches forming around the 20 cm level at various times, but all

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merging with the general left and up trend of the carrier wave at longer wave length and lower llcm values as it moves into the standing wave area.

Figure 7d. Peel level now at 0.02. Note the finger-like branches forming around the 20 to 25 cm level at various times, but all merging with the general left and up trend of the carrier wave at longer wave length and lower l/cm values as it moves into the standing wave area.

Figure 7e. Rotated view of Figure 7d. Note the standing wave vertical structure in the bottom left comer, that rises to the 0.02 level at times, probably due to incoming energy from the arriving waves. Each of the different structures to the right representing the incoming capillary wave burst point to these standing wave clusters visible at the 0.02 level.

By rotating the view, Figure 7e reveals more. Note the standing wave vertical structure in the bottom left comer, that rises to be visible at the 0.02 level at times, probably due to incoming energy from the arriving waves. Each of

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the different structures to the right representing the incoming capillary wave burst all point to one of these standing wave clusters visible at the 0.02 level.

Figure 7f. Peel level now at 0.01, hiding the trends in the details.

With further decreases in the peeling level, a point is reached where the outer surface grows so large and complex that it starts to hide the underlying features, as seen in Figure 7f. Yet, the sequence shown in Figs. 7a-f demonstrates what is now possible by applying EMD/HHT to images, and considering the peel levels and viewing angles.

The examples shown here were developed using a single line from each image in the time sequence of images. Other approaches may be used as well, such as averaging over a subset of adjacent lines withm each image, if it is justifiable based on the content of the image and the range of lines chosen over which to average. The processing described here could also be repeated for each of the 512 lines across the wave tank of the present images, from top to bottom in each image, through the 109 images in the time sequence shown, or even longer by including more images. This would produce an array with dimension beyond 3, which would be no,problem for the computers, but difficult for us to visualize. Each slice along the across tank axis from the higher dimensional array would be a volume result similar to the ones shown here, where isosurface techniques could reveal interesting processes and features.

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Another approach with the analysis of images is to re-assemble the image data using a different format. Again starting with a single horizontal line, at line 285 horizontally through the image, each line then is laminated to its predecessor to build up an array that is 5 12 along one edge, with units of cm, and the number of images along the other axis, in units of time (seconds). Once complete, this two-dimensional array can be split into 512 slices along the time axis. Each of these time slices, representing the variation in slope with time at a single pixel location, can then be processed with EMD/HHT techniques. As an example of this, consider Figs. 8a-b. Figure 8a represents the change in slope values at pixel 250 on line 353, or 12.96 cm over a time period of just over 7 seconds.

Time Change on Llne 363 at Plxel 260 (I2.Becm) 0.0

0.S

0.4

0 0

0.2

go.* -0.l

-0.2

-0.3

-0.4,

Figure 8a. through the burst at pixel 250, or 12.96 cm

Change in slope over a time period of just over 7 seconds at a location on line 353

Time Change on Llne 353 at Pixel 300 (15.56cm)

Figure 8b. through the burst at pixel 300, or 15.55 cm

Change in slope over a time period of just over 7 seconds at a location on line 353

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Figure 8b is similar data, but at a location on line 353 at pixel location 300, or 15.55 cm. Both show the passage of wave groups of longer wavelength through the chosen pixel as a function of time. Using this data, the EMD/HHT techniques reveal variations in frequency of the waves passing through thls location. Consider Figure 9a, which shows the change of frequency with time using line 353 and pixel 250. Note the bursts to higher frequency occuning very rapidly around the 1 second area, and also the persistent 2 Hz wave with fluctuating frequency. The sequence of images examined in earlier figures, images 1 through 109, occur in time between 0 and 1.82 seconds, and the shorter 4 image sequence between images 80 and 89 occur on this time scale between 1.33 and 1.48 seconds,

Figure 9a. Change of frequency with time using line 353 and pixel 250 from Figure 8a. Note the bursts to higher frequency occurring very rapidly around the 1 second area, and also the persistent 2 Hz wave with fluctuating frequency.

A similar record of variations is seen in Figure 9b, which was processed identically to Figure 9a, but at another pixel location, pixel 300. Thus detailed frequency variation information can also be obtained from a sequence of images.

Figure 9b; Similar to Figure 9a, but now using pixel 300.

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2 -

1.5

Changes in curvature may also be examined by selecting two pixels along the same line. The spatial distance along this line between the pixels selected can be adjusted to vary the resolution and sensitivity to wave length. Curvature along the direction of wave motion is then

-

A(s1ope) / A(separation)

where the change of slope is the difference between the two measured values at the two selected pixels, and the change in separation is the length difference along the chosen horizontal line between the two selected pixels. By repeating this procedure for each image in a sequence, the time variation of curvature at the selected location can be obtained and studied. l h s is another avenue opened up by the application of the EMD/HHT methods to images. An example of this is found in Figs. 10a-b. In Figure 10a, the curvature is shown plotted against time. The curvature was computed between pixel 225 and 227 along horizontal line 285, from a sequence of 500 images taken at 601s.

O W 7 Images 1 lo 5M) 2.5

1

0.5

a

-0.5

1

-1.5 < 3 4

Time (sac)

Figure 10a. Variation of surface curvature with time over a sequence of 500 images taken at 60/s. The curvature was computed from the difference in slope values between pixels 221 and 227 along horizontal pixel line 285.

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OMP7 Image8 1 to 5w 2.51

0 1 2 5 4 5 8 7 8 -1.5'

Time (SSC)

Figure 10a. Variation of surface curvature with time over a sequence of 500 images taken at 60/s. The curvature was computed from the difference in slope values between pixels 221 and 227 along horizontal pixel line 285.

0 45

0 4

0 35

0 3 m

0 25 0

0 2

8 15 B

0 15 10

0 1 5

0 05

0 0 1 2 3 4 5 6 7 8

Figure lob. Frequency variation of curvature shown in Figure l la . Note the variation occumng during the first three seconds, the time period during the arrival of a wave packet.

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From the data of Figure IOa, the EMD/HHT techniques were used to produce Figure lob, showing the frequency variations for the curvature. Note the curvature variations during the first three seconds in Figs. 10a-b, during which time the incoming packet arrives and is stopped by the opposing current.

3.4.3. Use of EMLYHHT in Image Decomposition

Another application of EMD/HHT is a process similar to the filtering of images. A more accurate description of this would be a separation of the complete image into component images, from the shortest scales to the longer scales. Again, as in the earlier analysis, the summation of these components produces the original data, in this case an image, with minimal numerical differences fiom the original.

Starting with Figure 3c, image 88 is again processed line by line, so that each horizontal slice produces a set of components. The difference in this approach is that now the first component is taken from each component set, or 512 different first components from the 512 complete sets. These 512 first components are laminated back together in the correct order, producing a 512 x 512 array, which can then be viewed as an image, the first component image. Figure l l a shows the first component image of Figure 3c. Note that only the shortest scales are seen, and in this case, it is the capillary wave burst seen in the earlier images.

Figure 1 la. First component image from image 88 shown earlier (Figure 3a), and assembled from the first components obtained from each of the 512 horizontal lines of slope data making up the image. Note that only the shortest scale is present, and may be studied in the absence of all other scales.

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The decomposition (EMD) continues with Figure l lb , illustrating that the slightly longer scale now resides with the incoming wave packet that carried and produced the capillary burst.

Figure 1 lb. The next longer scale, the second component image from image 88, assembled as before, but now using the second component only.

Figure 1 lc. The third component image developed as before from image 88. Note the standing wave patterns now evident, and that the incoming camer wave has now reached the area of the standing waves.

Figure l lc , the third component image, reveals the scales associated with the standing wave field. The incoming wave has reached the area associated with the standing waves. This raises the possibility of the standing wave pattern being

348

the trigger for the burst that arose out of a momentary but extreme steepness (slope > 1) on the incoming wave packet’s front face near the crest.

Figure I ld. The fourth component image, revealing the camer wave scale and some of the persistent standing wave pattern. Lamination effects are becoming evident here (some horizontal mismatch), illustrating the need for some matching computations between the slices at this and higher component levels.

Figure l l d shows the fourth component image containing the longer scale associated with the incoming wave, and some smaller contributions from the standing waves. At this level, the lamination effects are becoming evident (some horizontal mismatch), illustrating the need for some matching computations between the slices at th~s and higher component levels. Up to now, the raw result of processing has just been assembled back together to form the image array.

The fifth and sixth component images are illustrated in Figs. 1 le-f. Even with the increasing problem due to the lamination effects, the component images have something to reveal, such as the standing wave pattern in Figure 1 If. The greatest slope of the central wave pattern running vertically through the image center is just about where the capillary bursts occurred, indicating a possible trigger source for their generation on extreme slopes.

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0 3

02

0 1

0

11

1 2

Danvmd DNm 1-1

Figure 1 le. the left edge and just right of the image center. The lamination effects continue to increase.

The fifth component image. Note the persistent standing wave patterns, especially at

Figure 1 If. The sixth component image, seen through an increasing blur of lamination effects. Even with that, standing wave patterns are evident. Note the central one, where the capillary bursts were initiated.

4. Summary

As has been illustrated here, the application of the EMD/HHT techniques to image processing opens up new and exciting frontiers in image analysis. It is hoped that this brief review of some of these new possibilities will raise still others in the minds of its readers, as well as point to new and interesting applications.

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Images of water waves were used here because that happens to be the research field of interest of the author. However, this in no way limits the wide application of the steps and results illustrated here to interesting images of other processes. The new views into the complex interactions occurring routinely at the interface between the atmosphere and earth's oceans was made possible entirely due to the power and versatility of the EMD/HHT breakthrough technology. If data from irregular heart beats, brain wave pattern during epileptic seizures, images from CT, MFU, or x-ray images of patients with a medical problem were analyzed by researchers in that field, it is certainly possible that new and useful results and techniques would result. That work has indeed already begun, and not only in the medical fields, but in science and engineering applications as well. Such is the case with usefkl tools. They can simplify existing tasks, and help do new ones we thought weren't even possible.

Acknowledgements

The author wishes to express his sincere thanks and gratitude to Prof. T. Y. Wu of the California Institute of Technology for his pivotal role in guiding and encouraging the development of the EMD/HHT techniques, as well as for his willing spirit and assistance on many occasions. The author wishes to especially express his continuing gratitude and thanks to his colleague Dr. Norden E. Huang, Senior Fellow at NASA Goddard Space Flight Center, Director of the Goddard Institute for Data Analysis, and inventor of the EMD/HHT techniques for his decades of constant help and discussions. Support ftom NASA Headquarters is also gratefully acknowledged, specifically Dr. Eric Lindstrom and Dr. William Emery, for their encouragement and gracious support of the work. The author wishes to also thank the reader for their interest in something new, and in addition, issue an invitation to those wishing to see the results in color to contact the author at Steven.R.Long@,nasa.gov.

References

1. Huang, N. E., Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-steady time series analysis. Proc. R. SOC. Lond., A,

2. Huang, N. E., Z. Shen, and S . R. Long, A new view of water waves - The Hilbert spectrum. Ann. Rev. FluidMech., 31,417-457 (1999).

3. Huang, N. E., H. H. Shih, Z. Shen, S. R. Long, and K. L. Fan, The ages of large amplitude coastal seiches on the Caribbean coast of Puerto Rico. J . Phys. Oceanogr., 30,2001-2012 (2000).

4. Huang, N. E., M. C. Wu, S. R. Long, S. S. P. Shen, W. Qu, P. Gloersen, and K. L. Fan, A confidence limit for empirical mode decomposition and Hilbert spectral analysis. Proc. R. SOC. Lond., A, 459,2317-2345 (2003a).

454,903-995 (1 998).

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5 . Huang, N. E., M.-L. C. Wu, W. Qu, S. R. Long, S. S. P. Shen, and J. E. Zhang, Applications of Hilbert-Huang transform to non-stationary financial time series analysis. Appl. Stochastic Models Bus. Ind., 19,245-268 (2003b).

6. Huang, N. E., Z. Wu, S. R. Long, K. C. Amold, K. Blank, and T. W. Liu, On instantaneous frequency, Proc. R. SOC. Lond., A, submitted (2004).

7. Wu, Z., and N. E. Huang, A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. SOC. Lond., A460, 1597- 16 1 1 (2004).

8. Huang, N. E., C. C. Chern, K. Huang, L. W. Salvino, S. R. Long, and K. L. Fan, A new spectral representation of earthquake data: Hilbert spectral analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seism. SOC. Amer., 91, 5 ,

9. Long, S. R. and J. Klinke, A closer look at short waves generated by wave interactions with adverse currents, Gas Transjer at Water Sutjfiaces, Geophysical Monograph 127, American Geophysical Union, 121-128 (2002).

13 10-1338 (2001).

10. Castleman, K. R., Digital Image Processing, Prentice Hall, 667 pp. (1996). 11. Russ, J. C., The Image Processing Handbook, 4Ih Edition, CRC Press, 732 pp.

(2002). 12. Long, S. R., N. E. Huang, C. C. Tung, M.-L. C. Wu, R.-Q. Lin, E. Mollo-

Christensen, and Y. Yuan, The Hilbert techniques: An alternate approach for non- steady time series analysis, IEEE GRSS, 3,6-11 (1995).

13. Long, S. R., NASA Wallops Flight Facility Air-Sea Interaction Research Facility, NASA Reference Publication, No. 1277,29 pp. (1992).

14. Long, S. R., R. J. Lai, N. E. Huang, and G. R. Spedding, Blocking and trapping of waves in an inhomogeneous flow. Dynam. Atmos. Oceans, 20,79- 106 ( 1 993).


CHAPTER 3

WAVES STRUCTURE INTERACTION


NONLINEAR WAVE LOADS ACTING ON A BODY WITH A LOW-FREQUENCY OSCILLATION

MOTOKI YOSHIDA, TAKESHI KINOSHITA AND WEIGUANG BAO Institute of Industrial Science, The University of Tokyo

4-6-1Komaba, Meguro-ku, Tokyo 153-8505, Japan E-mail : [email protected]. ac.jp

Wave-drift added mass results from nonlinear interactions between waves and low- frequency oscillatory motions of a floating body, in the presence of incident waves. First, wave-drift added mass is derived directly from a perturbation analysis with two small parameters and based on two time scales, using a Cartesian coordinate system that follows the low-frequency oscillations. Next, wave-drift added mass of floating bodies has been systematically measured from a slowly forced oscillation test or a free decay test in waves. The model is either a floating circular cylinder or an array of four cylinders. Experimental results are compared with calculated results, which do not include higher-order potentials yet. Well, it is necessary for evaluation of wave-drift added mass to solve higher-order potentials. This problem is solved for a uniform circular cylinder of which draft is same as the water depth for the convenience of analytical calculation, by means of Green's theorem. Analytical solutions and calculated results of wave-drift added mass are presented. Far field conditions, radiation conditions for each order of potenials are obtained to ensure the existence of unique solution.

1. Introduction

As well known, moored ocean structures and vessels subject to slowly varying non-linear wave loads. Under this non-linear wave excitation, structures or vessels might oscillate at a low frequency in the horizontal plane, i.e. in surge, sway and yaw. These low-frequency oscillations will in turn affect hydrodynamic forces acting on the body. It is now commonly accepted that wave-drift damping, which is the portion of wave-drift force in phase with the velocity of low-frequency oscillations, plays a key role in determining the amplitude of low-frequency drift oscillations at resonance. On the other hand, wave-drift added mass, the portion in phase with acceleration, attracted less attention. However it has been found that effects of wave-drift added mass on low-frequency drift motions, e.g. the resonant frequency, are not negligible'.

355

356

In the present work, the interaction of low-frequency oscillations with both diffraction and radiation wave fields of a floating body is considered. Wave-drift added mass will be calculated together with other non-linear hydrodynamic forces to make a better understand of physical mechanism of low-frequency drift motions, within the frame of potential theory. Further more, experimental measurements are carried out in a wave tank to validate calculated results. Next, the most important problem for evaluation of wave-drift added mass is to solve higher-order potentials. The discussion to solve these is concentrated on the interaction problem of slow surge oscillation of a uniform circular cylinder, whose draft is the same as the water depth, with the ambient diffraction wave field, there are no essential obstacles to extend this approach to more complicated problems and body configurations, such as a freely floating cylinder or a cylinder array slowly oscillating in all horizontal motion modes.

2. Introduction

Theoretical formulation can be found in our previous work', based on the potential theory. The frequency of low-frequency oscillations u is assumed to be much smaller than the incident wave frequency w (u << w ) . Linear responses to incident waves are not restrained, i.e. the floating body is free to oscillate a t the wave frequency. Expanding Newman's approach3, a perturbation analysis with two small parameters and based on two time scales is applied to evaluate wave-drift added mass.

2.1. Moving Frame

The low-frequency oscillation is restricted in the horizontal plane, i.e. in the mode of surge, sway or yaw designated by j=1, 2 or 6, respectively. Its displacement is expressed as &(t) = Re{i&e-i"t}, where f j represents the amplitude of slow oscillation and the symbol Re denotes the real part of comlex expression.

A Cartesian coordinate system 0-xyz following low-frequency oscillations, but not high-frequency responses to incident waves, is adopted to describe the problem.

The definition of phase function Sj( t ) comes from the incident wave potential @lo(z , t ) , which is the only specified component in the first order potential. On the moving frame, the incident wave potential is given by

where la is the wave amplitude, ko is the wave number of incident waves

cash ko(z+h) ei[ko((z cos@+y sin@)] @l~(z, t ) = Re{4lo e-isj(t)} with 410 = i", cash koh 7

357

and p is the propagating angle of incident waves, h is the water depth and g is gravitational acceleration. Here, the phase function is defined as Sj( t ) = w t - ( S j 1 + S j Z ) & ( t ) ~ j , where ~1 = kocosp, ( j = 1) , 6 2 = ko sin@, ( j = 2) , 4 3 = z m , ( j = 6) and 6j1,2 denotes Kronecker's delta. . a

2.2. Perturbation Expansion of Velocity Potential

The fluid is assumed to be inviscid and the flow to be irrotational. The velocity potential @(z, t ) can be expressed by a perturbation expansion up to the quadratic order in the wave amplitude ca as follows:

Here, in the subscript, the number 0 , l or 2 referes to the order of magnitude in <a while the symbol j referes to the mode of low-frequency oscillations, and superscripts are used if needed to denote time dependence on the wave frequency.

High-frequency oscillations, i.e., linear responses of body, are denoted by q,(t), (s = 1 N 6). Since those responses will be affected by low-frequency oscillations, they can be expanded in a similar way as the potential:

where, in the subscript, the number 0 refers to the part not related to low- frequency oscillations, the symbol s referes to the mode of high-frequency motions, and qos, 753;) is the amplitude of qoS,qjf) motion, respectively.

The potential $\:I, together with the corresponding part of linear body responses 753;) , are further expanded into series of 0. Considering boundary value problems, especially body surface and free surface conditions, they are divided into diffraction parts and radiation parts in a similar way as

358

the linear potential $1 but the form is more complicated:

( 5 )

W 2 with u = - 9 (7)

where $is is an auxiliary potential satisfying the inhomogeneous free surface condition of 4 j .

2.3. Calculation of Hydrodynamic Forces

Once potentials and linear body motions are solved in accordance with each order boundary value problems and equations of motions, the hydrodynamic pressure can be obtained by the Bernoulli equation and then hydrodynamic forces are evaluated by the integration of hydrodynamic pressure along the instantaneous wetted body surface 30.

Hydrodynamic forces are then expanded in the same way as the velocity potential with two time scales, i.e.

Fo + Fle-isj(t) + F r ) + C T ~ [ F ~ j e - z " ~ ]

The last term in Equation (8) F8' is a force component in quadratic order of wave amplitude, which can be separated into two parts that are in phase with acceleration and velocity of low-frequency oscillation, respectively, i.e.

F.$ = ioA2ji - B2ji (9)

The real part of it is involved in the calculation of wave-drift damping coefficient B 2 j i . On the other hand, the imaginary part of force component A2ji gives wave-drift added mass, where i denotes the direction of force.

359

3. Experimental and Calculated Results

3.1. Experiments

Experiments to measure wave-drift added mass are carried out in a towing tank, which is 54m long and 10m wide with a depth of 2m. A single circular cylinder and an array of four circular cylinders are used as the models. The radius of cylinder a is equal to 0.125m and the draft d changes from a to 3a. When the cylinder array is used, cylinders are located at the corners of a rectangular, which is L=lOu long and B=5a wide. As shown in Fig. 1, the models are hung up by four wires, which have an average length of about 4.5m. Both free decay (denoted as FD) tests and forced oscillation (designated as FO) tests are performed for a systematic combination of experimental parameters, i.e., the wave amplitude and frequency, the amplitude and frequency of forced oscillations. In FD tests, the models are disconnected from the carriage and forces acting on the models are not measured. Only the displacement of the models is measured by an optical position sensor system. When FO tests are performed, the models are connected to the carriage through soft springs and a pair of cantilever load cells, which axe used to measure the force acting on the models. The carriage is driven to move along rails by a servomotor so that it leads the models to oscillate slowly in the surge direction. Both the displacement and force are measured in FO tests. FO tests in the present work have two special features. First, a linear wave exciting force can be parried and the linear response motion can be permitted through soft springs. Secondly, the frequency of forced low-frequency oscillation is set to be a little bit higher than the natural frequency of whole test system. Hence hydrodynamic forces can be extracted precisely.

3.2. Analysis of Experimental Results

To analyze measured data of FO tests, the method used by Kinoshita et al.’ is adopted. Added mass in still water M, and wave-drift added mass M, are determined by

where Mm and Mb are mass of model and ballast, respectively, C is the restoring coefficient. ala,w, b ls ,w and u2a,w, b2a,w are Fourier coefficients

360

position sensor

tor

Figure 1. Set up of forced oscillation tests

of displacement and force obtained from the Fourier analysis, respectively. The subscript s or w refers to the value in still water or in waves, respectively.

On the other hand, the method to analyze FD tests is a routine activity.

3.3. Results and Discussion

Experimental results are presented and compared with calculated results in this subsection.

First, effects of wave amplitude are examined. The ratio of wave-drift added mass to added mass in still waver % is plotted against the nondimensional wave amplitude % in Fig.:! for the cylinder array. In this figure, parabolic curves have been drawn to the corresponding experimental data using the least square method. It is observed that the ratio is generally proportional to the square of wave amplitude. This confirms that wave-drift added mass is a quadratic quantity of wave amplitude.

Secondly, effects of wave number are considered. Wave-drift added mass M, normalized by Lpo.rrac2 is plotted against the wave number ko in Fig.3 for the single cylinder and Fig.4 for the cylinder array. Here L is the total number of cylinders. The wave number is nondimensionalized by the radius of cylinder a (single cylinder) while the longitudinal distance L (cylinder array). Calculated results, represented by lines, are presented in these figures to compare with experimental ones. The dashed line (cal. dif.only)

361

A d=2a, k0L=5. 0 0 d=la, koL=5. 0 A d=2a,koL=6.0

.. .,A'

-0.4-

Figure 3. Wave-drift added mass vs. wave number (single cylinder)

x. =+.

ii.., \

L..

represents the case of diffraction problem only, and solid line (cal. dif.&rad.) represents the case including linear radiation problem. The contribution of higher-order potentials 4 j s and 4g) are not included in the calculation of both cases. It can be seen that calculated and experimental results agree fairly well with each other in general tendency.

3

2 -

1 - N

L 3

g o . 3

x -1

-2

-3

0 d=3a,c =O. 030m 0 d=3a,T =0.040m

- cal. dif.&rad. - - - cal. dif.only

- - - - - -

-

362

FO, d-a, <,-O. 020m F0,d=a,<a=0.030m

0 FD,d=a,~a-=0.030rn

cal. dif.6rad. cal. dif.only

Figure 4. Wave-drift added mass vs. wave number (cylinder array)

12

10

8

6 *I." Q- iP 3 2 . I:

0

-2

-4

-6 2 4 6 8 10

k"L

Figure 5. Effects of draft on wave-drift added mass (cylinder array)

Thirdly, effects of draft of model are discussed. Wave-drift added mass *z is plotted against the wave number koL for various drafts (d = a,2a,3a) of cylinder array in Fig.5. It can be seen that there is no difference when the draft changes in both of calculated and experimental results. Accordingly, it can be concluded that wave-drift added mass is a

363

phenomenon near the free surface.

ment is good. Finally, results of FD and FO tests are compared in Fig.4, their agree-

4. Solutions of Higher-order Potentials

It is necessary for evaluation of wave-drift added mass to solve higher-order potentials 4 j S and $!$. In this section, solutions of higher-order potentials are going to be obtained, extending Matsui et al.'s approach4.

Now, the discussion is concentrated on the interaction problem of slow surge(j = 1) oscillation of a uniform circular cylinder, of which draft is the same as the water depth, with the ambient diffraction wave field, without linear radiation wave fields for the sake of simplicity. It is not an essential difficulty to extend the present approach to the interaction with radiation wave fields. The radius of cylinder is a and the draft is d. The depth of still water is h=d. The order of potential in Ca and cr is described as (order of Ca , order of cr).

4.1. B. V.P. for Higher-order Potentials

Boundary value problems for higher-order potentials are arranged as follows according to our previous work2. That is, for (1,2)order potential $ j s :

A&7=0 at r > a , - h < z < O

-- - 0 at ~ = a a417

aT = O at z = - h a417 -

az proper far field condition as T -+ 00

for (2,l)order potential +El:

A+')=O at r > a , - h < z < O

364

proper far field condition as r -i 00 (21)

where $17 and fiy) are inhomogeneous terms of free surface conditions.

4.2. Solution of (1,2)order potential

In general, the potential 4 can be expressed in an integral form according to the Green’s theorem.

with

aV = S = SO U SF U SB U SR

where V is the fluid domain, SO is the body surface, SF is the free surface, S, is the sea bottom and SR is the control surface, where SR is the surface of a large circular cylinder with radius R (i.e., R -+ CXJ). n denotes the normal direction of surface S pointing out of fluid domain, and G is the Green function.

In the case of a uniform circular cylinder, 4, G, the inhomogeneous term f in the free surface condition are expanded into the Fourier series of azimuth angle 29 as follows:

M

n=-ca

n=-ca

Making use of boundary conditions and substituting these to Equation(22), then the integral with respect to the azimuth angle can be performed to yield

365

where P(r, 0, z ) represents the field point and Q(p, 8,c) represents the source point. The Green function g, is the well known Fenton solution. The integral over SR, the fourth term on the right-hand side of Equation(26), can then be written as

where J, represents the first kind Bessel function and H, denotes the first kind Hankel function. On the other hand, the integral over SF, the third term on the right-hand side of Equation(26), may be expressed as

I F = i2TCo &@or) Jn(kOP)fn(P)PdP [ lr + ~ n ( k o r ) ( ~ n ( ~ 1 - ~n(r))] zo(z>

where I , and Kn are modified Bessel functions, and

m=O:

cos km(z + h) cos k , h

cm = k& v = -km tank&, Zm(Z) = h(k& + v') - V'

(30)

Here F,(p) is a function defined by the following indefinite integral:

Fn(P) = Hn(kOP)fn(P)PdP (31) s In the case that F,(R) does not tend to zero, the contribution from the control surface IR should cancel the value of F,(R) to ensure that unique solution exists. This will yield the following condition to be imposed on the potential for large R:

366

Then, the velocity potential for a field point inside V may then be expressed as

m=l

4.3. Solution of (2,l)order potential

The approach described in the previous subsection is in general applicable to solve for the (2,l)order potential. Nevertheless, the Green function has to be changed and governed by the following boundary value problem:

AG(P, Q ) = -4nb(P, Q ) Q E V

A solution for the above problem is suggested by Isshiki5 as follows:

gn(r, 2; P, C )

mn mn =I

367

The result of integral on SR may be written as

where

(42) mr h Z m ( Z ) = cos -(z + h) , (m = 0,1,2,. * - )

On SF, a function defined by the following indefinite integral is used:

As well as the previous subsection, the function Fn(R) should cancel the integral IR over the control surface to ensure the existence of unique solution. It gives the far field condition satisfied by the potential such that

(44)

The velocity potential for a field point inside V may be expressed as:

368

4.4. Eigenfunction Solutions of Potentials

Having potentials (1,O)order $1 , (0,l)order $oj and (1,l)order &a solved, the (1,a)order potential 4js is going to be solved for example in this subsection. The forcing term in its free surface condition is divided into three parts.

f 1 7 n = .fgn + f g n + (46) where

The solution of potential 4 1 7 contains three parts correspondingly.

Here, in superscripts, the symbol U referes to the part not related to the potential $oj while the symbol S referes to the part related to $oj. As well known,

00

$1 = $10 + $17 = AZO(z) &(Jn(kOr) + BnHn(kOr))eine (5l) n=-w

(52) a2 r

+oj = $ol = --case

and potentials 4% and 4f7 are known. Accordingly, eigen function solutions of potentials 4g and @a are ob-

tained from substitution of Equations(48)-(52) to Equations(33) and (45). Furthermore, it is derived that the (1,2)order potential has the far field

behaviour 4jS N RqeikoR as compared with the (1,l)order potential 4jS ~ 4 ~ i k o R

369

4.5. Analytical Solution of Wave-drift Added Mass

In the case of this section, the analytical solution of wave-drift added mass is expressed as the following formula:

where po is the density of fluid, SO is the mean position of wetted body surface, co is the mean water line. The asterisk in the superscript denotes the complex conjugate being taken and the symbol Im designates the imaginary part of complex expression. Calculated results of wave-drift added mass are plotted against the incident wave number in Fig.6. First, the dashed line shows the result without the contribution from $js and 4;). Next, the dotted line gives the result including the contribution from $js. Finally, the solid line is the result with the contributions of all potentials. It can be seen from Fig.6 that contributions from $js and are not negligible.

Figure 6. Analytical solutions of wave-drift added mass (uniform cylinder)

370

5. Conclusions

An equation for wave-drift added mass was derived directly from the perturbation analysis with two small parameters and based on two time scales. The wave-drift added mass of floating bodies was measured from slowly forced oscillation tests in waves to compare with the calculated result and validate the theory. The calculated results agree fairly well with experimental results in spite of the fact that higher-order potentials are not included.

Next, the method to solve higher-order potentials was described and eigenfunction expansion solutions were obtained. The simplest case considered was the case of a body with geometry of a uniform circular cylinder. In this case only the diffraction problem was considered.

References 1. Kinoshita, T. and Takaiwa, K. 1990 Time domain simulation of slow drift

motion of a moored floating structure in irregular waves including time varying slow motion hydrodynamic forces. ASME Offshore Mechanics and Arctic Engineering Conference paper , volumel-partA, 199-204.

2. Kinoshita, T., Bm, W., Yoshida, M. and Ishibashi, K. 2002 Wave drift added mass of a cylinder array free to respond t o the incident waves. ASME Offshore Mechanics and Arctic Engineering Conference paper , No.28442.

3. Newman, J. N. 1993 Wave-drift damping of floating bodies. J. Fluid Mech.

4. Matsui T., Lee S. and Sano K. 1991 Hydrodynamic forces on a vertical cylinder in current and waves. Journal of SNAJ 170, 277-287.

5 . Isshiki, H. 2003 Private communication.

249, 241-259.

ANALYTICAL FEATURES OF UNSTEADY SHIP WAVES

XIAO-BO CHEN Research Department of Bureau Veritas

1 %is, Place des Reflets, 92077 Paris La Dkfense 2, Fkance E-mail: xiao-bo.chenObureauueritas. corn

The unsteady waves generated on the free surface by a source pulsating sinusoidally and moving with constant horizontal velocity are treated by an asymptotic analysis. A direct relationship between the dispersion curves in the Fourier plane associated with the linear boundary condition at the free surface and the corresponding wave system is established. Considerable information about far-field features of ship waves is revealed, especially, the wave crestlines and related wavelength, directions of propagation, cusp angles, and phase & group velocities which are determined explicitly from the dispersion function. Furthermore, some preliminary results on the effect of surface tension and discussions on fluid viscosity are presented.

1. Introduction

The unsteady waves generated by time-harmonic motions of a ship animated with constant forward speed can be represented by those created by a distribution of singularities. The potential flow generated by a source pulsating sinusoidally and moving with constant horizontal velocity is then fundamental to the description of the flow past a moving ship and to the prediction of its motions in waves. The time-harmonic source potentials, including time-harmonic flows with forward speed and the special cases of Neumann-Kelvin steady flow (zero frequency) and time-harmonic flows without forward speed, play a critical role of theoretical basis due to their intrinsic properties such as proper far-field wavy behavior and radiation condition. However, the unsteady waves associated with a pulsating and translating source have been treated in very few studies as Eggers (1957), Becker (1958) and Wehausen & Laitone (1960). Some confusion and controversy exist in describing the kinematics of unsteady wave patterns.

The main objective of the present paper is then to perform a new analysis and to establish a new basis for analyzing time-harmonic wave kinematics. Associated with the linearized free-surface boundary condition, the velocity potential of time-harmonic ship waves is expressed as the sum

371

372

of simple (Rankine) singularities satisfying the zero-flow free-surface condition, and the double Fourier integral accounting for the free-surface effects. Following the analysis performed in Noblesse & Chen (1995), the free-surface component expressed by the double Fourier integral is decomposed into a wave component which propagates into the far field, and a local component which is only significant in the near field. This formal decomposition gives a useful expression of the wave component written as a single integral along the dispersion curves.

Based on using the stationary-phase analysis of the wave-component line integral, considerable information about far-field features of ship waves is revealed. Especially, the constant-phase curve (e.g. crestlines) and related wavelength, directions of wave propagation, cusp angles, and phase and group velocities can be determined explicitly from the dispersion function. Indeed, the direct relationship between the dispersion curves in the Fourier plane and the corresponding wave systems on the free surface is established in Chen & Noblesse (1997). Furthermore, analytical expressions of far-field ship waves are obtained in Chen & Diebold (1999).

An important and complex feature of the ship-motion Green function is its rapid oscillations with singular amplitudes when a field point approaches to the track of the source point at the free surface. This peculiar behavior of the ship-motion Green function is analyzed in Chen & Wu (2001) by using the wave-component integral along the open dispersion curves. The asymptotic calculation of the integral along the portion of the open dispersion curves at large values of wavenumber yields an analytical expression which captures the behavior of highly oscillations. This result covers the special case of steady flow studied by Ursell (1960) who showed that the wave elevation near the track of steady-moving pressure point oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength.

These gravity waves of very short wavelength cannot be ignored and cause substantial difficulties in numerically modeling them. The singular and highly oscillatory properties being manifestly non-physical, it is expected that the surface tension and fluid viscosity play an important role in modeling short ship waves. The effect of these small parameters is then studied and some preliminary results are summarized here.

2. Time-harmonic ship waves

The reference system moving with the ship at the mean forward speed U along the positive x-axis, is defined by letting (x, y) plane coincide with the

3 73

mean free surface and z-axis be positive upward. The velocity potential of the flow generated at a field point (= (c , 7, C ) by a source of unit strength located at the point Z3 = ( x 3 , y3, z,) is expressed by

@(x, y, z , t ) = %{4(x, y, z ) e-Zwt} and 4 = $R + $ J ~ (1)

with w the wave encounter frequency and (2, y, z I 0) = (<-x3, m3, C+z3). Here c $ ~ is the Rankine part

q5R = (-l/r + l / r ' ) / ( 4 ~ ) (2)

with r and r' the distances between the field point (and the source point Ic3 , and between f and the mirror image of Z3 with respect to the mean free surface plane z = 0, respectively. The free-surface part 4F originally defined by a double Fourier integral, following Noblesse & Chen (1995), is decomposed into

q5F=4W+4L (3)

where q5w is called the wave component 4F = q5w in the far field x2+y2 4

00 while q5L is non-oscillatory and only significant in the near field, and then called local component.

The wave component is obtained by an asymptotic analysis of $F and expressed by a single integral

along the so-called dispersion curve determined by D = 0. In (4),

C1 = sign(Df) and C2 = sign(zD, + yDp) (5)

and IIVDI) = ,,/DS + DZ with D, = dD/da and Dp = d D / d p .

2.1. Dispersion curves

The dispersion function involved in (4) is given by

D = ( F c Y - ~ ) ~ - k = ( F c o s ~ ) ~ ( ~ - k+) (k - k - ) (6)

in which (a , p) = k (cos 6 , sin 0) are Fourier variables, the wavenumber k = d w and polar angle 0 = arctan(P/a), with k*(0) given by

k* = ( 1 / 2 f d m ) 2 / ( F c o s 0 ) 2 (7)

374

The dispersion function (6) is associated with the classical free-surface boundary condition linearized about the uniform stream opposing the forward speed of the ship. The nondimensional frequency f , the Froude number F , and the Brard number r are defined as

f = u r n , F = U / @ , r = F f =Uw/g

where w is the encounter frequency of the regular ambient waves exciting the ship motions, L and g are the ship length and the acceleration of gravity.

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0 2.0 , , I I

-4.0 -2.0 0.0 2.0

r = 0.25

1 .o "-.. k-

0.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0

.. ." 0.01 I ?? I I I I I ' # I

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0

Figure 1. Dispersion curves in ( c Y , ~ ) or ( k , B ) plane

From the expression (6) of the dispersion function, the dispersion relation D = 0 defines several dispersion curves which are symmetric with respect to p = 0 so that only those in the upper half p 2 0 of the Fourier plane is now considered. k = k* with k*(8) given by (7) represents two types of dispersion curves : open dispersion curves k = k+ and closed dispersion curve k = k-. In fact, there exist three dispersion curves for r < 1/4 which intersect the axis p = 0 at four values of a, which are denoted a: and a:, and given by

such that two open dispersion curves are located in the regions -00 < a 5 a; corresponding to k = k + ( 8 ) within 7r/2 < 8 5 7r and a$ 5 a < 00

375

corresponding to k = k+(O) within 0 I 8 < n/2, and a closed dispersion curve in the region a; 5 a I a t corresponding to k = k - ( 8 ) within 0 I 8 5 n.

At the special value of r =1/4, a; =a; from (8) so that the dispersion curves in the region -00 <a 5 a; and a; 5 a

For r > 1/4, we have one open dispersion curve located in the left region -GO <a I a’ corresponding to k = k+(O) within n/2 < 8 In-8, where 8, is defined by

a+ are connected.

8, = arctan d 1 6 ~ ~ - 1

connecting with k = k - (8) within 0 5 8 5 n - 8, at 8 = n - BC, and another open dispersion curve located in the right region a: 5 a < 00 corresponding to k = k + ( 8 ) within 0 5 8 < n / 2 .

Therefore, four distinct dispersion curves exist : the right open dispersion curve for F > 0, the left open dispersion curve for r < 1/4, the close dispersion curve for r < 1/4 and the left open dispersion curve for r > 1/4, associated with which four corresponding wave systems are baptized in No- blesse, Chen & Yang (1996) as : inner-V waves, outer-V waves, ring waves and ring-fan waves, respectively. The four dispersion curves may be further classified into two types : a closed dispersion curve for r < 114 and two open dispersion curves for F > 0. They are depicted for T = 0.2, 0.25 and 0.3 in Fig. 1 where the Fourier plane is scaled with respect to f /F .

2.2. Dispersion relation and far-field waves According to (4), the wave component associated with one of dispersion curves can be written

4ni 4w = ds (El + &)A e-ihp I D = ,

(9)

with El =sign(Df), E2=sign(xDa+yDp) and h2=z2+y2, A=ekz/JJVDJI and the phase function cp=aZ+pg with

(z, 0) = (2, y)/h = (cosy, sin y)

Both (3, g) and (cosy, sin y) are used in the following analysis. The far-field features of q5w are determined by the stationary points

of the phase function cp along the dispersion curves which are defined by (p‘ = Za’+jjp’ = 0 and satisfy

?D,-gD, = 0 = hllVDll sin(6-y) (10) Here 6 is defined by (cosd, sind) = (0, , Dp)/IIVDII and thus represents the angle between the unit vector normal to a dispersion curve and the a-axis.

376

The wavelength of the waves corresponding to a stationary point (10) is given by X=2n/k where k is the wavenumber at the stationary point.

Expression (10) shows that a point of stationary phase on a given dispersion curve is defined by 6 = y or 6 = y+n . Thus, a point of a dispersion curve generates waves in the physical space in a direction normal to the dispersion curve. The sign function sign(zD,+yDp) in (9) is equal to 1 if 19 = y or -1 if 6 = y+n . Expression (4) therefore indicates that a point of a dispersion curve generates waves in the direction of the normal vector llVDll to the dispersion curve if sign(Df) = 1, or in the opposite direction if sign(Df) = -1 . Furthermore, at the stationary point ‘p‘ = 0, the second derivative of the phase function is expressed as :

cpt’ = c&FTiF(atcosy-ptsiny)/ sin27 (11)

where a’ and p’ are differentiation of a and p with respect to the integral variable along the dispersion curves, and the curvature c is given by :

c= (2DaD&P - D p p p - ~;~a,)/llv~l13 As other terms in the expression (11) cannot be zero, p”=O occurs only at the point of inflection where c = 0. Using (6) in above expression, the wavenumber at the inflection point is determined by the relation

F4k; - (312) F2 kc+ E ~ ~ T F & - 3r2 = 0 (12)

Two points on both sides of the inflection point may have the same unit normal and then two groups of waves may propagate in the same direction but with different wavenumber. In fact, an inflection point (ac , pc) of a dispersion curve, determined by c= 0 , defines a cusp line along which two distinct wave systems are found with a unique wavelength A, = 2n/kc and

Table 1. Cusp angles of wave systems

7

Ring-fan waves - 2 125O16’

90000’

@ 35O26’

f l 19O28’

377

8.0

6.0

4.0

2.0

0.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0

Figure 2. Crestlines of wave systems at T = f l

the corresponding angle 6, is defined by

6, = arctan(Dp/D,),

where the subscript c indicates evaluation at the inflection point (a,, p,).

the source point is obtained by using (6) and (12) The cusp angle defined as the angle between cusp line and the track of

7, = n-6, = arctan J1/(6F2k, - 1) (13)

for both inner-V and outer-V waves, and for the ring-fan waves at r > m, and

7, = .?r-arctan v/l/ (6F2 k, - 1)

for the ring-fan waves in 1/4 < I- 5 m. In fact, 7, = n/2 at r = G, i.e. strictly no waves propagate upstream for r 2 @, an interesting exact result. Another interesting value of r is a t which the unsteady waves (ring-fan waves plus inner-V waves) axe contained within the wedges of steady waves. Approximate values of I- = @% M 0.27 and I- = M

1.63 can also be found in Lighthill (1978). Other values of cusp angle with respect to r are given in Tab. 1 for inner-V waves, outer-V waves and ring-fan waves.

The curves along which the phase cp is constant, equal to C,” = f 2 r n - sign(cp”)n/4 with n = l , 2 , . . . are given by

with a supplementary condition

378

-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 5 .O

Figure 3. Right open dispersion curve and inner-V waves for F > 0

The formulations (15) and (16) are applied to the inner-V waves associated with the right open dispersion curve and the ring-fan waves associated with the left open dispersion curve at the value of r = &@?. The constant- phase curves (crest lines) are depicted by Fig. 2.

Following Lighthill’s work, the wave phase and group velocities are considered now. The phase velocity Gf, determined by the stationary-phase relation (lo), is given by

Gf = -(a, P ) f / k 2 (17)

which is orthogonal to constant-phase curves (15) and different, both in magnitude and in direction, from the group velocity 3, at which wave energy is transported, defined by

fl = -(af/aa, a f l m = (Day DP)/Df

( z ,y ) . iP > 0

(18)

Expressions (18) and (15) together with condition (16) yield

which shows that wave energy is propagated away from a wave generator in accordance with the radiation condition. Using (6), the group velocity (18) is written as

p = - [F + ~ ~ ~ / ( 2 k ~ / ~ ) , C ~ P / ( ~ ~ ~ / ~ ) I (19)

379

in the system of coordinates moving with the mean forward motion of the ship, and

dg = v'9 + (F , 0) = -c&, p ) / ( 2 1 ~ ~ / ~ ) (20)

in the absolute system of coordinates. The absolute velocity ?g (20) is orthogonal to the constant-phase curves, whereas the relative velocity iP (19) is not.

The foregoing simple analytical relationships between the dispersion curves in the Fourier plane and important 'features of the corresponding far-field waves in the physical plane are illustrated by Fig. 3 for the inner-V waves, Fig. 4 for the outer-V waves, Fig. 5 for the ring waves and Fig. 6 for the ring-fan waves.

On Fig. 3, the right half represents the Fourier (c.,,B) plane (scaled with 10F2) and the left half represents the physical (z,y) plane (scaled with 1/F2). Along the right open dispersion curve on the right half plane, there is an inflection point at which the normal is parallel to the cusp line of inner-V waves on the left half plane, in the opposite direction since sign(zD,+yDp) = -1 according to the foregoing analysis. The transverse waves are associated with the segment of dispersion curve comprised between the intersection point and the inflection point while the divergent waves are associated with the segment from the inflection point to infinity. The normal direction along the dispersion curve is parallel to the propagation direction of the wave group velocity iP which is always in the radial

B (108) Fourier plane \ 20.0 1 10.0

0.0

-10.0

-20.0

-30.0 -60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0

Figure 4. Left open dispersion curve and outer-V waves for r < 1/4

380

-30.01 I I I I I I I I I

-50.0 -40.0 -30.0 -20.0 -10.0 (

B f lo$) Fourier plane

Physical plane n (f2)

1 ' 1 ' 1 '

Figure 5. Closed dispersion curve and ring waves for r < 1/4

direction in the physical (5, y) plane. The direction of wave phase velocity v'f is perpendicular to the wave crestlines and parallel to the direction of wavenumber vector, as shown by the orthogonality between the wavenumber vector and the lines tangent to crestlines. Furthermore, it is shown that the phase velocity has a component in the negative-y direction, i.e. the wave crests appear to propagate downstream and to the centerline,

30.0

20.0

10.0

0.0

-10.0

-20.0

-30.0 -60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 4 .O

Figure 6. Left open dispersion curve and ring-fan waves for r > 1/4

38 1

different from the group velocity which is oriented radially. Finally, The wavelength (e.g. Xi or Xiv) measured between two successive crestlines is associated with the wavenumber (e.g. a' or k,) at the corresponding point along the dispersion curve.

The outer-V wave system is presented on the lower half of Fig. 4 and the left open dispersion curve on the upper half. The same correspondence as the inner-V wave system is established. Since sign(zD,+yDp) =+1, the normal at a point of dispersion curve is coincided with that of wave propagation (group velocity). The phase velocity is shown to have a positive-2 component and a negative-y component in the lower half plane, i.e. the wave crests appear to propagate upstream and outward from the centerline. At T + 0, the outer-V and inner-V wave systems tend to occupy the same space but their phase velocities are exactly opposite so that the sum is stationary - wave crests don't move while the group velocity keeps well its the radial direction.

The ring wave system and the closed dispersion curve are presented on Fig. 5 where the Doppler effect is evident by the shorter wavelength in the upstream and longer wavelength in the downstream. On Fig. 6, the ring-fan wave system and the left open dispersion curve are presented respectively on the lower half and upper half planes. The partial ring waves are associated with the segment between the point a: and the inflection point and the fan waves with the segment from the inflection point to infinity. There is one interesting point k = 4f2 > k, at which the wavenumber vector and the normal are orthogonal, i.e. the corresponding fan waves propagate (group velocity) in the direction parallel to the wave crests.

In summary, the four wave systems above-described exist in a group of three for T < 114 as the inner-V, outer-V and ring waves or in a group of two for T > 1/4 as the inner-V and ring-fan waves. The inner-V and outer-V waves for T < 1/4 or the inner-V and ring-fan waves for T > 1/4 are associated with the open dispersion curves. Although the integrand of the single integral is not singular, care is needed for the two indefinite integral as the unbounded limit can lead to highly oscillatory and singular behavior. Indeed, this analysis has established the link between the singular and highly-oscillatory properties of the wave component and the open dispersion curves along which two indefinite integrals are performed.

2.3. Singular and highly-oscillatory terms Previous studies presented in Ursell (1,960, 1988) and Clarisse & Newman (1994) on the Neumann-Kelvin steady flow show that the Green function is

382

singular and highly-oscillatory near the singular axis - the track of source point located at the free surface. These studies were mostly based on asymptotic analyzes of the single integral similar to (4) for steady flows along the complex path of steepest descent. However, it seems more complex to extend this analysis to general cases such as time-harmonic ship- motion Green functions.

Very recently, an asymptotic analysis based on an asymptotic expansion of open dispersion curves at large wavenumber and properties of the complex error function, has been realized in Chen & Wu (2001). The singular and highly-oscillatory properties of the ship-motion Green functions are then expressed in an analytical closed form :

where ?t= ,/- and p=arctan[y/(-z)]. The expression (21) being valid for large values of ~ ~ / ( 4 ? t ) ~ , captures the behavior of highly oscillations with indefinitely increasing amplitude and indefinitely decreasing wavelength of ow for y + 0 at z = 0, and the finite limit for 7-l -+ 0 at z < 0 and a finite value of x < 0 as assumed. Thus, the limits are non- uniform depending on p = 1r/2 ( z = 0) or p < 7r/2 ( z < 0). At z = 0, the term (21) is depicted in Fig. 7 by solid and dashed lines for the real and imaginary parts, respectively.

0 0.1 0.2 0.3 0.4 0.5

Figure 7. Singular and highly-oscillatory term for T = 1/5 at (z = -5 and 0 5 31 5 1/2)

The singular and highly-oscillatory term (21) of ship-motion Green functions is valid for any values of T including T = 0 and T = 1/4. Indeed, if we take T = O , the expression (21) reduces to the simpler form which is in total agreement with previous results given in Ursell (1960, 1988) and Clarisse & Newman (1994) for the special case of Neumann-Kelvin steady flow.

383

The complex nature of the wave component captured by (21) reveals insight of the difficulty in the numerical prediction of the motion of a ship advancing in waves. A commonly used method is to convert the Laplace equation in the fluid domain into an integral equation over the boundary of the fluid domain, by using the Green function. The integration over the free surface is further transformed into two line integrals, one at infinity and another along the waterline of the ship. The former often ignored in the analysis is shown to be recently by Doutreleau & Chen (1999). However, the waterline integral imposing both source and field points are on the free surface has to be justified since a stable numerical solution is not easy to achieve due to the difficulty associated with these singular and highly- oscillatory properties.

3. Effect of surface tension

These singular and highly oscillatory properties is manifestly non-physical as they show that the wave elevation near the track of pressure point oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength. It is expected that the surface tension plays an important role here. The dispersion function (6) associated with the free-surface boundary condition including the surface tension becomes

F 2 D ( a , p ) = (a -~ ) ' - k - a2k3 (22)

in which (a,p) are the speed-scaled Fourier variables associated with the Froude number F and the parameter a is defined by

representing the ratio between the characteristic wavenumber of capillary waves and that of gravity ship waves. In (23), T is the surface tension (T=0.074 N/m for the air-water interface at 20°C).

The dispersion curves defined by D = 0 are symmetrical with respect to p = 0 as those associated with (6). There exist also three or two distinct dispersion curves for T < 1/4 or T > 1/4, respectively. The dispersion curves are nearly identical in the zone a2I3k << 1. However, the most striking difference is that the two open dispersion curves on the left and right half parts of the Fourier plane (Fig. 1) are now closed. In fact, the dispersion curves intersect the axis p=O at the values of a denoted 5: and 5: :

(24) 5; = F2a: - + O(a4) ~ = F2a,f + a 2 ( F 2 a , f ) 3 / d m + O(a4)

384

in which a: and a,’ are given by (8). Furthermore, the left and right dispersion curves are closed at the intersection points on the axis p=O :

E$ = - (1f2T) - (T2(1fT)(1&3T)] + 0(CJ4) (25)

This closure of dispersion curves bears two major significations. The wavenumbers are finite not infinite as for pure gravity waves and the divergent waves including in the inner-V, outer-V and ring-fan wave systems must be largely modified according to the relationship between the dispersion relation and far-field waves as foregoing described.

For the sake of simplicity, we consider the case of steady flow T = 0 in which only two dispersion curves exist and are symmetrical with respect to both axes a=O and p=O. In the quadrant a 2 0 and /3 2 0, the dispersion curve is defined explicitly in polar coordinates (k, 0 )

(26) k, ( e ) = 2/ (co~2 e + J C O S ~ e - 4 4 Ic~(6 ) = (COS~ 6 + v/c0s4 6 - 4u2)/(2o2)

k 5 ku k 2 k,

k ( 6 ) =

with k, = l/u. The curve described by (26) is a closed one limited in the region

0 5 0 56, with 8 , = arctan[d(l- 20)/(2u)]

At 0=6, we have k = k,. At 6 = 0 , we define

k;=2/(1+4=) and k $ = ( 1 + 4 s ) / ( 2 u 2 ) (28)

so that the dispersion curve intersects the a-axis at a = kg” and a = k!$. The dispersion curves given by (26) are depicted on the left part of Fig. 8

at a Froude number F = 0.1 (using L = 1 m) for u = 0 when the surface tension is ignored and u=0.275 when the surface tension is included. The dispersion curve without the surface tension (u = 0) represented by the dashed line is given by k = 1/ cos2 6 and corresponds to the case usually called Neumann-Kelvin ship waves. It is an open curve as k --+ 00 when 0 --+ ~ / 2 . The dispersion curve with the surface tension (u # 0 ) is a closed one with a maximum wavenumber k!$ defined by (28). The point (kc, 0,) divides the dispersion curve into two portions : one (k, < k,, thick solid line) along which the effect of gravity is dominant and another ( k ~ > k,, thin solid line) along which the capillarity is dominant.

On the right part of Fig. 8, we depict the crestlines following the formulation (15) for n = (1,2, . - , 5 ) associated with the dispersion curves plotted on the left part of the figure. The Neumann-Kelvin ship waves represented by dashed lines composed of transverse and divergent waves are present

385

10

8 -

6 -

I , I I I I I

: u a

,... ,,./

/"

0 2 4 6 8 1 0 1 2 1 4 -30 -25 -20 -15 -10 -5 0 5

Figure 8. Dispersion curves (left) and crestlines (right) of capillary-gravity ship waves

only in the downstream and limited by a cusp line (dot-dashed line). The ship waves including the surface tension are present in both upstream and downstream. The upstream crestlines associated with the part of dispersion curve at kT > k, are capillary waves and plotted by thin solid lines. The wavelength of upstream capillary waves is of order 2xF2/k&.

The downstream crestlines (thick solid lines) associated with the part of dispersion curve at k, < k, are gravity-dominant waves. Comparing to the pure-gravity waves (dashed lines), the transverse waves keep the same profile with a slight shorter wavelength 2xF2/k i instead of 2xF2. The most striking feature concerns the divergent waves which disappear completely at this value of u (in fact for u > uo given in the following) due to the effect of surface tension. In their place, the transverse waves are extended smoothly outward to a region limited by the ray (dotted line) forming an angle y, with the negative-x axis defined in

y = arctan[y/(-s)] I y, = x / 2 - 0, (29)

The crestlines for n = ( 1 , 2 , . . . ,5) are depicted on Fig. 9 for u = 0.02 (left part). Only those of downstream waves are drawn for the sake of clarity. The transverse waves are represented by thick solid lines and the divergent waves by thin solid lines, while the rest of capillary-gravity waves by dashed lines limited by the dotted ray (y = yo). yu = 0 at u = 0 means that no capillary waves exist since the effect of surface tension is ignored. At u=u,=1/2 , the dispersion curve reduces to a point ( 2 , O ) and y, = x / 2 which means that all steady waves disappear (no wavy deformation of the free surface) since ship's speed is less than the minimum velocity of capillary-gravity waves so that waves propagating at ship's speed cannot be generated.

There are two other important rays, more evident on the right part of Fig. 9 on which only crestlines of divergent waves are kept. One represented

386

by the thin dot-dashed line is close to the cusp line of Kelvin ship waves, and another by thick dot-dashed line. We denote the two rays respectively by y = yc and y = yo the angles forming with the negative-a: axis. Same as yo, the ray-angles and yo are function of the parameter u. Following (13), the value of -yc is associated with the normal direction at the first point of inflection along the dispersion curve, which is quite close to that for the Neumann-Kelvin ship waves. There exists a second inflection point along the dispersion curve of capillary-gravity ship waves at low values of CT. The value of yo is given by the normal direction at this second point of inflection. The ray-angle yc becomes the cusp angle 7,“ =rc(a=0) x 19’28’ of pure-gravity ship waves when u -+ 0 while 70 tends to zero. It is shown that the divergent waves can be found only in the region (70 <rl T ~ ) where transverse waves appear as well. In the region near the ship’s track (0 <y< +yo), only transverse waves are present. Since 70 increases significantly with increasing u (corresponding to the decrease of forward speed), the region (70 < y < 7c) where divergent waves appear is more and more reduced. At u=uo x 0.133 (corresponding to U=Uo x 0.450 m/s), there does not exist any divergent wave.

These results concerning divergent waves are welcome in the modeling of ship waves, since by excluding the surface tension, the part of divergent waves becomes extremely oscillatory and singular so that substantial difficulties arise in their numerical computations. According to (4), the wave component of capillary-gravity steady flow is written as the sum of two parts :

in which the gravity-dominant waves 4 r and the capillarity-dominant waves @ are expressed by a single integral along the part of dispersion

15 15 I I I I I I

10

5

0 -35 -30 -25 -20 -15 -10 -5 0 -35 -30 -25 -20 -15 -10 -5 0

Figure 9. Crestlines of capillary-gravity ship waves (left) and definition of rays (right)

387

curve k, < k,

Icg de ezkg sin(zk, cos 0) cos(yk, sin 0) (31)

and along the part kT>k,

respectively, and with

S, = sign(zD, + yD~) lk=k, ; ST = sign(zD, + yDp)lk=kT

and

lDk1g=12kgCOS2e- l -3u2k921 ;

The single integrals (31) and (32) are convergent even for z=O, i.e., when both the source and field points are located at the free surface, since the amplitude of oscillatory integrands is of order O(l/k) instead of O(k) if the surface tension is excluded. Furthermore, both kg and k~ have finite value for u > 0 so that 4r and @' are not singular.

In summary, the steady ship waves including the surface tension are analyzed using directly the relationship between the dispersion relation and far-field waves. It is shown that the role of surface tension in modeling ship waves is twofold. Firstly, including the effect of surface tension yields more realistic description of ship waves. Especially for low forward speed, the divergent waves are largely compressed and appear only in a zone between two rays : the line (y = yo) and the cusp (y = yc). At lower speed of U 5 UO ~ 0 . 4 5 0 m/s (corresponding to a=uox0.133), no divergent waves exist. When U < Urn x 0.232 m/s (corresponding to u =urn = 0.5), no wave can be generated. Secondly, introducing of surface tension in the formulation of ship waves eliminates the singularity of the Green function when both the source and field points are at the free surface. These benefits will be much more enjoyed in the numerical development of practical computation methods.

4. Discussions and conclusions

The single integral representation of the wave component of the free-surface effect following the decomposition of Noblesse & Chen (1995), is used in the analysis. By using the method of stationary phase, it reveals that a simple and direct relationship between the geometrical properties of a dispersion

388

curve and important aspects of the corresponding far-field waves, including wavelength, directions of wave propagation, phase and group velocities, and cusp angles.

Application of this relationship to the time-harmonic ship waves yields very detailed descriptions of secalled inner-V waves, outer-V waves, ring waves and ring-fan waves. This work has also established the link between the singular and highly-oscillatory properties of the wave component and the open dispersion curves along which two indefinite integrals are performed. Indeed, by analyzing the leading terms of the wave component associated with the open dispersion curves and making use of asymptotic properties of the complex error function, it is shown that the time-harmonic potential flow is highly oscillatory with indefinitely increasing amplitude and indefinitely decreasing wavelength when the field point approaches the track of the source point located at the free surface.

These singular and highly oscillatory properties is manifestly non- physical and it is expected that the surface tension and fluid viscosity play an important role here. The analysis of time-harmonic ship waves including the effect of surface tension shows that the open dispersion curves are closed and the wave patterns of steady ship waves are largely modified especially when the forward speed is low. Fhrthermore, the single integral along the closed dispersion curves is no longer singular even when the field point and source point are both located at the free surface. Still the wavelength of capillary-gravity ship waves is very small and some of them propagate upstream. These small capillary-gravity waves must be very sensitive to damping effect of fluid viscosity.

If the fluid viscosity is introduced and the equation system of Stokes or Oseen type is established as in Chan & Chwang (2000) and Lu (2002), the free-surface waves are expressed by a double Fourier integral which involves a dispersion function including an additional imaginary part like :

(33) F2D(cx,p) = (a-T) 2 - k - a2k3 - i46(cx-7)k2

in which the parameter E is associated with the fluid viscosity. It is this imaginary term in the dispersion function that introduces the damping effect. The damping factor must be of exponential type with an argument proportional to the absolute value of the imaginary part which is high for capillary-gravity waves of small length. The detail analysis of free-surface waves associated with a complex dispersion function of type (33) is being pursued presently.

389

References

1. K. Eggers, Schifl and Hufen 11, 3-7 (1957). 2. E. Becher, 2. angev Math. Mech. Bd 38, 9/10, 391-99 (1958). 3. J.V. Wehausen & E.V. Laiton, Handbach des Physik 9, 446-778 (1958). 4. F. Noblesse & X.B. Chen, Ship Technol. Res. 42, 167-185 (1995). 5. X.B. Chen & F. Noblesse, Proc. 12th Intl WWWFB, 31-35 (1997). 6. X.B. Chen & L. Diebold, Proc. 14th Intl WWWFB, 25-28 (1999). 7. X.B. Chen & G.X. Wu, J. Fluid Mech., 445, 77-91 (2001). 8. F. Ursell, J. Fluid Mech. 8, 418-31 (1960). 9. F. Noblesse, X.B. Chen & C. Yang, Proc. 2lst Symp. Naval Hydrodyn. 120-

35 (1996). 10. M.J. Lighthill, Cambridge Univ. Press (1978). 11. F. Ursell, Proc. Royal Society of London A 418, 81-93 (1988). 12. J.M. Clarisse & J.N. Newman, J. Ship Res. 38, 1, 1-8 (1994) 13. Y. Doutreleau & X.B. Chen, Proc. 14th Intl WWWFB, 33-36 (1999). 14. H. Lamb, Cambridge Univ. Press (1932). 15. G.D. Crapper, Proc. Royal SOC. London. Series A, 282 547-58 (1964). 16. A.T. Chan & A.T. Chwang, Proc Instn Mech Engrs 214, 175-79 (2000). 17. D.Q. Lu, Univ. Hong Kong (2002).

A NOTE ON THE CLASSICAL FREE SURFACE HYDRODYNAMIC IMPACT PROBLEM

CELSO P. PESCE Department ofMechanica1 Engineering, Escola Politkcnica

University of SLio Paulo, SLio Paulo, Brazil

This work treats the classical problem of determining the hydrodynamic impact force acting upon a rigid body, during the water entry phenomenon. Under the usual Wagner‘s approach, the paper discusses a well-known controversy regarding discrepancies obtained in impact forces calculations, if either integrating pressures methods or energy approaches are used. By addressing previous analyses, including a proof given, by G. X. Wu’ this work confirms conclusions drawn by other authors. According to those authors a proper explanation for the apparent discrepancy can be found the kinetic energy transferred to the jets.

1. Introduction

Hydrodynamic impact of a solid body against a liquid surface is a classical subject in applied mechanics, responsible for an important core of relevant problems in engineering. Firstly addressed by Von Karman’ and Wagner3, motivated by loading on seaplane floaters during “landing”, the problem has received attention of many investigators. In the naval architecture and ocean engineering fields the impact problem extends from structural applications concerning “slamming” of ship bows, “impulsive wave-loads” on offshore structures to “design of planning crafts” (see, e.g., Faltinsen4). The complete treatment of the problem includes the elasticity of the impact body as well as compressibility effects. A comprehensive and mathematically thorough review can be found in Korobkin and Pukhnachov’.

The simplest problem is the rigid-body impact against a liquid fiee-surface, in which compressibility effects are not take into account and the flow is considered inviscid and irrotational. This is valid in a subsonic stage. Within potential theory, Wagner’s approach is probably the most popular one. It treats the interaction problem as the ‘impact of a floating plate’ whose size 3changes in time. The usual free-surface condition is usually replaced by an equipotential boundary condition, say I$ = 0 , corresponding to the limit of infinity frequency

390

391

in the sense of the wave radiation problem. Taking into account the so-called wetted-surface correction in the added mass quantity, the impact force acting upon the body, for a purely vertical impact, is usually written F, = - d ( M , ~ ) / d t (see, e.g., Faltinsen4, chapter 9), being W the vertical velocity and Mzz the corresponding added-mass. However, if +t = 0 is considered valid everywhere on an equipotential control surface that replaces the actual free surface, the impact force would be derived as F ~ , = -1/2 x wMoa - M~,W ; see G.X. Wu'.

This apparent discrepancy was noticed by many authors; see, e.g., Miloh6 and Shifhan and Spencer'. It would exist when methods based on integration of the pressure field or, alternatively, energy approaches were used instead. Nevertheless, the discrepancy would not exist if energy arguments were properly used, namely: part of the kinetic energy is transferred to the jets and part to the bulk of fluid. This conclusion, inferred in the 30's by Wagne?, was drawn by many other authors, as Cooker and Peregrine', through a pressure- impulse theory approach, Molin, Cointe and Fontaineg, based on matched asymptotic expansion solutions (previously obtained by Cointe and Armand", after Cointe") and by Faltinsen and Zhao'*, for some particular impacting bodies as cylinders and spheres.

This simple work re-addresses such a discussion, aiming to contribute to enlighten some aspects of the problem.

2. Basics on Momentum and Kinetic Energy

2.1. Dejinitions

Consider a body aB impacting the water free surface. The whole volume of fluid will be denoted, in an arbitrary instant of time t, by R. The initial instant t = 0 is defined as the instant when the body first touches the interface. Let R be bounded, at a given instant t > 0, by dn = S = S, US, US, US,, a piecewise, regular and continuous surface. S, is the part in contact with the instantaneous wetted body surface, S, the free surface, S, a distant control surface and S, the bottom surface. One shall take S, as a fixed vertical cylindrical surface of radius R, with no loss of generality. The positively outward normal unit vector n gives orientation to the closure surfaces.

Let x = (x, y, z ) denote a particular point in a Cartesian coordinate system orienting the inertial frame of reference, being the initially quiescent free surface given by z = 0, z assumed positive outward the fluid. The origin 0 is taken at the first point of contact between the body and free surface. The flow is here assumed inviscid and irrotational, such that a potential scalar function

392

$(x ,y , z , t ) defines the velocity field. For the sake of simplicity, consider the fluid at rest before the first contact. The fluid is supposed to be ideal, such that no compressibility effect exists, Laplace equation V2$ = 0 being the field equation.

SQ I

n=U, on

I. I

Figure 1. Whole volume of fluid R in an arbitrary instant of time I, and its closure surface.

It is worth recalling that the fluid surfaces composing S can be classified according to: (a) if material or non-material; (b) if fuced or moveable. Material surfaces can be of two types: permeable and non-permeable. According to criterion (a), surfaces S,, S,, and S , are said to be non-permeable material surfaces. According to criterion (b), S , is classified as a fuced surface whereas s, and s, are moveable sui$aces. On the other hand S , is afixed, non- material surface, or a fured control surface. Usual kinematic conditions apply; see Figure 1.

At the very start stage of body-fluid interaction, inertia forces dominate gravitational ones. In this sense, the dynamic free-surface boundary condition may be written,

-+-V$.V$=O a 4 1 on S , . at 2

393

2.2. Momentum and force

From momentum considerations, the force acting on the body may be written (see, e.g., NewmanI3, page 133).

or, observing that

(3)

may also be given in an alternative form,

(4) 1 F = - p - d j$ndS+p I ( v$ - - - (Ve) 'n a$ 1 d S . an 2

SB +SF SB +SF dt

Considering $ ( x , y, z , t ) as conveniently evanescent, the integral over S, vanishes as R -+ 00, and then, from (3),

( 5 ) F = - p - d I$ndS+--p 1 f (V$) 2 ndS.

di 2 SB +SF SH

Equation ( 5 ) recovers the result given in Molin, Cointe and Fontaineg. In fact, regarding the water entry problem on the basis of a body moving inside a fluid domain limited by solid boundaries, Molin, et a1 state: "The previous analysis based on momentum conservation remains valid if we replace S, with S, u S, , since the free surface S, is a material surface at zero pressure". For a semi-finite fluid domain and taking $ = 0 at S,, those authors then concluded that, in a simple translational case the force upon the body would be given by F = -d(M,U)/dt , being U and M a , velocity and corresponding added mass respectively.

Notice, also, that the flux of momentum through the fluid boundary is given by

Q = p fv$( - -c i , )dS= @ p p $ y d S @ an n S SR

394

There is no flux through S, and S, or even through S, , since all three are non-permeable material surfaces. The same may not be true through S , , of course, a non-material, permeable and fixed surface.

2.3. Energy time rate

The kinetic energy introduced in the fluid simply reads

In general, the parcel of the integral on the free surface should not be ignored, unless the kinematic boundary condition on the free surface is replaced by the Dirichlet condition: 4 = 0 .

Another well-known result refers to the kinetic energy time rate,

If condition (1) is applied at S, , equation (8) simplifies as

where p is the dynamic pressure. As S, has been properly taken as a non- permeable material surface, there is no rate of energy flux through it. If the potential is taken to be conveniently evanescent, the second contribution vanishes and (9) takes a well-known form,

dT - = - [pU,,dS dt

SB

395

3. Discussion

3.1. Subsidiary discussion on boundary conditions

A usual hypothesis in the theory of hydrodynamic impact is to consider the free surface to be known at t=O', such that ~ ( x , y , t = O ' ) = O . It may observed that, far enough from the body suyface, 6 , = C y = 0 at t = 0' is consistent with the assumption 6 (x , y, t = 0') = 0 , thus leading to

at; - 8 4 onz=O at t = 0 + at az

This is also consistent with conventional mathematical modeling of impact problems in classical mechanics. Indeed, if the perturbation caused by the body is considered impulsive, a jump in momentum and velocity has to occur though not, of course, a jump in position. A Dirichlet type boundary condition is then usually assumed on the free surface z = 6 (x , y,t = 0') = 0 , namely $(x, y, t = 0') = 0 , leading to the conclusion that, at t = 0' , the kinetic energy transferred to the fluid could be well represented through an integral over the body surface. Note that this last condition would satisfy the linearized free surface condition for the harmonic problem as the frequency goes to infinity and could suggest that $, = 0 on z = 6 (x , y,t = 0') = 0 .

However on the free surface,

Clearly, the condition 4, = 0 on z = t; (x,y, t = 0') = 0 may not be generally valid; at least not in the neighborhood of the body, where certainly very high velocities would exist. Studying a similar problem regarding the initial pressure distribution due to a jet impact on a rigid body, G.X. W U ' ~ discusses the singularity at the intersection. That author points out that in the impulsive stage the matter of practical interest is the pressure-impulse, which turns out to be finite, varying in space according to the potential itself.

On the other hand, a non-flat impact would present a time jump on normal velocity at every point that would gradually start being in contact with the body. In general, for regular convex shapes, there is a velocity-jumping frontier at a

396

given instant, where the spray is formed”. The continuous front of points where normal velocity jumps occur will march as the body penetrates the water, defining the intersection r ( t ) = S, n S, in a given instant. On theji-ee surface, any jumps [V$ $1;- , [$,JlIi- and I&, J;z- should be mathematically consistent with each other, according to (1 1) and (12). Time jumps as \& J,li- and 14 J=;- would indicate, strictly speaking, time discontinuities in the geometrical configuration space.

If an impulse idealization is taken, the contact of the body with the free surface is considered as ‘instantaneous’, so that the whole body surface will be, suddenly and ‘simultaneously’, in contact with the free surface. A way to model the impact as a mathematical impulse idealization, enabling a time jump in the velocity potential to exist, would be to represent the body as an equivalent ‘time-varying floating plate’. This is, essentially, Wagner’s approach. This procedure is quite common and useful. Nevertheless, as pointed out by Korobkin and Pukhnachov’: “. . . it is not clear how Wagner method may be generalized to essentially three-dimensional problems, Besides, the Wagner method leads to an incorrect description of the flow picture in the vicinity of the three-phase contact lineb, r ( t ) . In particular, it leads to predictions of an unlimited increase in pressure as r(t) is approached from the wetted part of the body surface”. Actually such a singularity is found in many other analyses as pointed out by G.X. Wu14 and, also, by Peng and Peregrine15, within a pressure- impulse theory for a plate impact. These latter authors observe: “Cooker’s approach leads to difficulties when the velocities after impact have a singularity, as they do in this case. Some effort has been put into overcoming this problem with no success to date, It seems likely that a local solution for jet formation at the edges of the plate may be needed to exclude the singularities from the problem”.

3.2. Discussion on 4, at z = 0

The analysis is here restricted to a control volume Q,, bounded by So = S,, LJ S,, LJ S,, LJ S, . For clarity, refer to the surface & (x,y,O+) = z = 0 as S,, and the velocity potential as 4, . The normal velocity U , of S,, is zero. Strictly speaking, S,, is a fixed control surface, an equipotential surface and not a free surface. For consistency, the body surface is designated S,, , and the

a In the case of a parallel and flat impact, a finite surface of fluid is instantaneously made in contact with the body, producing a time velocity-jump over a finite area.

The intersection of the free and the body surfaces; the marching jet-root frontier.

397

distant control surface as S,, = S, . High-frequency asymptotics in free surface waves might involve the following boundary conditions

Taking (14) as true everywhere at the control surface S,, , the pressure would be given by

(16) 1 2

2 p o = - - P $ ~ ~ onz=O.

Discussion of condltion (14) is the essential issue of the next section. As it will be shown, assuming condition (14) true from the onset, may lead, in a sense, to a circular reasoning (a tautology).

3.2.1. Preliminary discussion

Without assuming conditions (1 3-1 5 ) to hold, and observing that S,, is a fixed control surface, such that

one might easily conclude, from momentum considerations, that the force acting on the body may be written as

or, using (3), by

398

This recovers G.X. Wu's' equation (6), there derived for vertical impact. That author claimed to have enforced $, = 0 on S,, . Note that, in the present derivation conditions (13-15) where not used, i.e., $, = 0 on S,, , has not been claimed as true. Consistently, there is, now, flux of linear momentum not only through S,, but also through SFo since both are non-material, permeable and fixed surfaces,

Looking now at the energy time rate and recalling that: U , = O on S,,,S,,,S,; a ) , / a n = O on S,; a$,/an=U, on S,, and taking the potential to be evanescent, one obtains,

where p o = -p($,, +1/2(VI$, .V$,) is the dynamic pressure. Clearly, the flux of energy rate through the control surface S,, would be zero if $,, = 0 everywhere on that surface.

3.2.2. Vertical impact

The problem is, for simplicity, restricted to a simple vertical impact with varying velocity W, such that 4, = W w , . The usual heave added mass definition gives

Mo, = P J b o n J S (22) S.90

Take (13) and (15) both valid, but not necessarily (14) (and consequently not (16), either). Following, otherwise, G.X. Wu's' steps, except for not considering w t = 0 everywhere on S,, , one obtains

399

In the corresponding G.X. Wu's' equation (15) there is, of course, no integral on S,, . It also follows, from Wu and Eatock Taylor16, that the first term in (23) may be written

jVorn,dS=W fko,". -(Vv0)2n,)?S- ~VOr(WO,-n,)dS. (24) sBO+sF 0 sE 0 SF,

Therefore,

Equation (25) does reduce to the result obtained by G.X. Wu', if (14) is enforced, thus eliminating the last term. From (25) and recalling that, on S,, , n, = 1 and +,, = W y o , +yo dW/d t = W v o t , it follows that

Therefore, from (1 9), the vertical force reads,

where the subscript M indicates that momentum approach has been taken.

Alternatively,

If an energy approach is used, instead, the kinetic energy reads

400

and

From (30) and (21), with U, = Wn, , it follows

where the subscript E means an energy approach has now been adopted.

By enforcing (28) and (3 1) to produce equal answers, it follows

Apart the constraining imposed by the integral identity itself, there are two straightforward solutions to (32), on S,, , at t = 0' .

The first one is y o z E -1, everywhere on S,, . It must be disregarded since implies a constant and negative vertical velocity everywhere on S,, , what is obviously physically inconsistent.

The second one is yot = 0 everywhere on S,, . This condition, together with y o = 0 on SF0 , would lead to I$,( = 0 on that surface. As stated before, it is clear from (21) that this solution would imply no flux of kinetic energy through S,, . Recall, however, that S,, is, at t = 0' , an (equipotential) control surface, i.e., a fixed permeable surface across which fluxes of energy and momentum occur. Therefore, this second solution should be disregarded, as well. Note from (31) that, if yo, E 0 were taken as true, the force F,, would be given by,

as given by G.X. Wu'. In summary: with the analysis restricted to the control volume R, , the condition = 0 on S,, could be interpreted as a condition resulting from taking momentum and energy approaches equivalent to each

401

other. In other words, this would justify G.X. Wu’s’ claim: “...the forces obtained by integrating the pressure over the body surface and by an energy argument are the same.. .”. However, that author imposed, from the onset, the hypothesis $ot = 0 as valid everywhere on SFo , at t = 0’. Such reasoning could be regarded as a tautology. In fact, the claim is true. However, to be properly justified, the general mathematical and physical constraints given by the integral identity (32) should be re-written and properly interpreted, as a singularity certainly exists at ro ( t ) = SFo n S,, .

It should be noted that, in Batchelor’s” analysis on a flat-nosed projectile impact, section 6.10, page 473, no restriction is made on bOt , though there is a singularity in the velocity at the comer. Moreover, quoting Batchelor, page 474: “It should be noted that the motion produced by impact of the flat-nosed body is identical only instantaneously with that in (one half of) the flow field of a flat plate moving through infinite fluid”. It is also worth quoting a comment (on the apparent discrepancy), by Molin, Cointe and Fontaine’: “The problem actually results from the fact that the correct free surface condition (gravity playing no role at the initial instant) has been replaced with $ = $, = 0 , which is in fact valid only in an outer domain (away from the body and free surface intersection)”. At the intersection, the constraining integral identity, equation (32), must be consistently treated, both physically and mathematically, as, e g , suggested and even done by many authors through matched asymptotic expansions.

An alternative and more general approach, from the analytic mechanics point of view, may be found in some notes on the application of Lagrange Equation to mechanical systems with mass explicitly varying with respect to position; Pesce”. This will be summarized in section 3.4.

3.3. The pressure-impulse approach

As mentioned, Cooker and Peregrine’ studied the hydrodynamic impact problem through the concept of pressure impulse; see Batchelor”, art. 6.10

n+

P(x) = jp (x , t )d t . (34) 0-

The recurrent argument, regarding an apparent loss of energy resulting from the impact, is readily deducible from the pressure-impulse approach. In fact, Cooker and Peregrine’ show that the change of kinetic energy in the bulk of the fluid can be written,

402

Y 3

Figure 2. Pressure-impulse approachc.

As an integral approach, it obviously filters out any sudden variations that may occur in the integrand. In the present case, it misses the description of the flow close to the impacting body. Since the flow is assumed incompressible, inviscid and irrotational, all lost energy has to be drained through the boundary of R' ; see Figure 2. In the present case, through the free surface Sk , near the impacting body, i.e., through the sprays or jets.

In this sense, the spray can be viewed as a local relief of a very large pressure field, developed in the neighborhood of the impacting body. This is the way to relieve energy, since the body is supposed rigid and the fluid compressibility has been taken as null. In a sense, such a local nature of energy relief could be regarded as a result of a variational principle. Remember that

' The superscript I indicates that the body and free surfaces are to be understood from an impulsive model p in t of view.

403

under the optics of Variational Principles in Fluid Mechanics, Hamilton Principle turns out to be “one of stationaty pressure“; see, e.g., Seliger and Whithamlg. Particularly for the gravitational waves, as shown by Luke”, the pressure is the Lagrangian density. For the impact problem, one could think of the Lagrangian as a quantity related to the pressure-impulse field.

Part of the energy may indeed be thought as drained through the jets. This conclusion is consistent with that previously drawn by many authors. In fact, Molin, Cointe and Fontaineg stated: “...it may be concluded that, in all impact situations, at the initial instant, the kinetic energy is equally transferred to the jets and to the bulk of the fluid”. Though mass and momentum fluxes are negligible through the jets, energy flux is not, and represents part of the energy transferred to the whole fluid (bulk and jets).

and aR = S, v S, = S denote the spray and respective enclosure surface. S, is the external part of the jet surface and S, , a moving control surface, is the internal part, at the jet root. The body and free surface intersection is dC = S, nS, . Figure 3 below shows these definitions. Both, (S, -S,) and S, are supposed to be surfaces at zero pressure. Neglecting variation of pressure across the jet, S, may be said at zero pressure on the jet side, what means the whole jet volume R, is itself at zero pressure. Approaching S, from the bulk of the fluid side, a very high pressure exists. Such a model is consistent with considering a significant part of the kinetic energy relieved through the jet root, i.e., through the control surface S, .

Yet, let R

ac = s,’ n s,

s: Figure 3. Jets or sprays and closure surfaces.

Let 6 be the thickness of the section at the jet root (or spray root) and VJ be the average velocity across the jet root. Let also V, be the velocity of the jet root (the velocity of the control surface S, or, equivalently, the velocity of the boundary aC). The rate of kinetic energy through the jet root may then be approximated, as

404

3.4. The Lagrange equation approach

It is usual practice to treat potential hydrodynamic problems involving motion of solid bodies within the frame of system dynamics. This is done whenever a finite number of generalized coordinates can be used as a proper representation for the motion of the whole fluid. Terming this approach as ‘hydromechanical’, the present impact problem can be formulated under the Lagrangian formalism, recalling the added mass dependence on the position of the body; PesceI8.

For the sake of clarity, consider first a very simple and hypothetical problem of a particle of mass m(x) , explicitly dependent on position x , acted on by an external force F(x,X,t) , mass being expelled at null velocity. The equation of motion is simply rn‘(x)XL + m(x)x = F(x , X , t ) . However, if a somewhat naive application of the usual Lagrange equation, d (Ti ) /df - T, = F(x , X , t ) , were made, one would obtain m’(x)XL / 2 + m(x) l = F(x, i, t ) , in an obvious disagreement with respect to the first and correct equation of motion derived from Newton’s Law. What is the reason for such a somewhat unexpected discrepancy? The answer to this question could be easily guessed: the usual form of Lagrange Equation is not the most general form that could be conceived, concerning a system presenting variation of mass, explicitly dependent on position. In this simple one degree-of- freedom example, we could guess that the correct ‘Lagrange’ equation should be written d (Ti )/dt - T, = F(x, X , t ) - m’(x)XL/2 .

The extended Lagrange equation of motion can be derived in a general case of a system of particles, for which mass is explicitly dependent on position (and velocity), mi = mi (q ; q ; t ) . Such equation reads as; see, e.g., PesceIgd,

j = 1, ...., M d dT aT - S j ; dt aQi aqj

i

being, v, = v,(qi;qj; t ) ; j = 1, ..., M , where qj denotes a generalized coordinate and Q j , the respective non-conservative generalized force of a system composed by N particles, including all active forces f, and reactive forces mivo,, due to addition or expelling of mass, with ‘absolute’ velocity vOi ’.

This independent derivation recovers another by Cveticanin2’, for the simpler case of mass only dependent on position.

Sometimes, the reactive force is expressed in terms of relative velocity in the form mi(voi - vi) , known as Metchersky force, as presented in Cveticanin”.

405

Taking now the purely vertical impact case of a rigid body against a free surface, the kinetic energy defined in the bulk of the liquid may be written as

1 2

T = --M,W2

M , = Mzz(Z) . t

Z = IWdt O+

The true added mass is consistently defined in the bulk of the liquid, at each instant of time, by taking into account the so-called wetted correction, due to the marching of the jet root. The correct Lagrange equation approach is to use (37), such that the total vertical force applied by the body and the jets on the hydromechanical model of the bulk of the fluid is given by

The first three terms correspond to the force applied by the body on the bulk of the liquid. The fourth termf corresponds to the reactive force applied on the bulk of the liquid by the jets; ri? is the flux of mass through the jets and vJ the absolute velocity of the fluid particles at the jet root; a is the instantaneous angle of the jets with respect to the horizontal.

Therefore, the force applied by the bulk of fluid on the body is simply,

The third term appearing on the right hand side, if not considered, would lead to an erroneous assertive, given by equation (33). In fact, equation (40) transforms as

d dMzz dMzz W 2 =--(M,,W). d (41) dt 2 dZ 2 dZ dt

F, = -- (Mn W ) + - W - - --

This term is in fact small. For the particular and important case of a circular cylinder of radius R, e.g., it can be proved, from the asymptotic analysis by Molin et al?, after Cointe and Armand”, that the vertical force, per unit length, applied by the jets on the bulk of fluid is of order OFnpRW’ sina) , where E = , /Wf/K is a small parameter measuring a short scale of time. Contrarily, the energy flux is of order G = OkpRW’) and d(M,W)/dr = O ( E - ’ ~ ~ R W ~ ) .

406

Equation (41) recovers the expected result. Actually, the present note was inspired by the recognition of a lacking term, -1/2x W L d M , /dZ , in the analysis presented by G.X. Wu’. Note also that in the present analysis the changing in the added mass, explicitly dependent on position, is due to an actual changing of size and shape of the surface of the body in contact with the bulk of the liquid. This should not be confused with usual cases where the body has the size and shape invariant and the added mass varies according to its proximity to material surfaces; see, e.g., the analysis in Lambz2, art. 137.

4. Concluding Remarks

This work addressed the classical problem of determining the hydrodynamic impact force acting upon a rigid body, during the water entry phenomenon. The author discussed boundary conditions on the free surface, suggesting appropriate ways of looking at it. An apparent controversy regarding apparent discrepancies obtained in impact forces calculations, if either integrating pressures methods or energy approaches are used, was re-discussed. The presented reasoning recovered a previous and proper explanation given by many authors: a considerable part of the kinetic energy is transferred to the jets. It should be pointed out-that is very intriguing the existence of an apparent discrepancy in a potential problem in classical mechanics persisting as late as in the ~ O ’ S !

Acknowledgments

The author acknowledges research grants, no. 304062185 and 30245012002-5, from CNPq - Brazilian National Research Council. The author is specially grateful to Professor Armin W. Troesch, University of Michigan, for introducing him to the impact problem, indicating a bench of valuable references and for his encouragement and time, spent in very interesting discussions.

My deepest and most respectful gratitude to Professor Theodore Y. Wu, a great scholar and scientist, a true master (who taught me 20 years ago: “always t v to take the path you can learn the most’?.

407

References

1. G. X. Wu, JFluids andStructures, 12, 549-559, (1998). 2. T. Von Karman, National Advisory Committee for Aeronautics (NACA).

Technical Note no. 321, (1929). 3. H. Wagner, National Advisory Committee for Aeronautics (NACA).

Technical Memorandum no. 622, (193 1). 4. 0. M. Faltinsen, Sea Loads on Ships and Offshore Structures. Cambridge

Ocean Technology Series, Cambridge University Press, p. 328 p (1990). 5. A. A. Korobkin and V. V. Pukhnachov, Ann Review Fluid Mech, 20, 159-

185 (1988). 6. T. Miloh, JEng. Mathematics, 15,221-240 (1981). 7. M. Shiffman and D. C. Spencer, Comm on Pure and Appl. Math vol. IV, no

8. M. J. Cooker and D. H. Peregrine, JFluidMech., vol. 297, 193-214 (1995). 9. B. Molin, R. Cointe and E. Fontaine, in 11" Nit Workshop on Water Waves

10. R. Cointe and J. L. Armand, ASME J Offshore Mech. Arctic Eng., 109, 237-

11. R. Cointe, Hydrodynamic Impact Analysis of a Cylinder. MSc Diss,

12. 0. M. Faltinsen and R. Zhao, Workshop on High Speed Body Motion in

13. J. N. Newman, Marine Hydrodynamics. The MIT Press, p. 402 (1978). 14. G. X. Wu, JofFluids andStructures 15,365-370 (2001). 15. W. Peng and D. H. Peregrine, in I f h Int. Workshop on Water Waves and

16. G. X . Wu and R. Eatock Taylor, in. I l l h Int. Workshop on Water Waves and

17. G. K. Batchelor, An Introduction to Fluid Dynamics. Cambridge University

18. C. P. Pesce, J. Appl. Mech., Vol. 70,751-756 (2003). 19. R. L. Seliger and G. B. Whitham, Proc Roy SOC. A 305, 1-25 (1968). 20. J. C. Luke, JFluid Mech., vol. 27, part 2,395-397 (1967). 21. L. Cveticanin, JAppl. Mech., 60,954-958 (1993). 22. H. Lamb, Hydrodynamics. Dover Publications, N.Y., 6th Ed., p. 738

23. D. J. Kim, W. S . Voms, A. W. Troesch, and M. Gollwitzer, A., in 21"

24. A. A. Korobkin, JFluidMech., vol. 318, 165-188 (1996).

4,379-418 (1951).

and Floating Bodies, Hamburg (1996).

243 (1987).

University of California, Santa Barbara, p. 48 p (1985).

Water (Agard Ukraine Inst on Hydromechanics) (1 997).

Floating Bodies, Caesarea (2000).

Floating Bodies, Hamburg (1996).

Press, p. 615, (1967).

(1932).

Naval Hydrodynamics Symposium, Trondheim, Norway, (1 996).

MEASUREMENT OF VELOCITY FIELD AROUND HYDROFOIL OF FINITE SPAN WITH SHALLOW

SUBMERGENCE

S . J. LEE Dept. Naval Arch. & Ocean Eng., Chungnam National Universiy, Daejeon, 305-764,

South Korea

J. M. LEE Chungnam National University, South Korea

D. H. KIM Chungnam National University, South Korea

Lee [ I ] employed the lifting-line theory for hydrofoils of finite span developed by Wu [2] to obtain an approximate formula describing the flow field around the hydrofoil with shallow submergence moving very fast. On top of the basic uniform flow, the induced stream and the downwash were the additional velocity components of the fluid, which appeared in the lifting-line approximation. The behavior of these quantities was investigated for the elliptical distribution of circulation, and it was found out that their variation along the span was not negligibly small, and that the assumption of their constancy is especially poor near the tips.

In order to answer the questions raised by this theoretical work, a set of experiments was planned and carried out for obtaining the velocity field around the hydrofoil of finite span, using a wing of the NACA 0012 section in a circulating water channel. DPN technique was used to measure the velocity field, and the velocity measurements along the span were done for 3 speeds, 3 submerged depths, and 4 angles of attack. The experimental data are compared with the theoretical assumptions, as well as the numerical findings by Lee & Lee [3]. Special care is given to the flow near the tips and in the region close to the leading edge. Though indirect, using the measured data of the velocity, we can also compare the aerodynamic and the hydrodynamic strength of the circulation distribution of a hydrofoil with a given geometry, which was not easy to do previously in the framework of the lifting-line theory.

1. Introduction

In Prandtl’s lifting-line theory, an integro-differential equation for the sectional circulation C, corrected for the downwash w induced by the trailing vortex sheet plays a central role. Following the similar line of thought Wu [2] proposed

408

409

a lihg-line theory for hydrofoils of finite span, and he obtained the following integro-differential equation for the sectional lift coefficient corrected for the induced stream u , which is in the opposite direction of drag, as well as the downwash,

c, = 2z{a(l- 24) - w} (1)

where a is the angle of attack. In survey of the previous experimental findings at NACA, Wu quoted that at

depths larger than four chords, the influence of the free surface is negligibly small, and that in the range of depths between four chords and a half chord lift and drag coefficients are reduced as the submerged depth. And the corresponding value of lift to drag ratio increases to a maximum as the depth decreases until the hydrofoil breaks through the surface, and with the further decrease in depth it decreases very rapidly and eventually to that of the planing surface.

Parkin et al. [4] measured the pressure distribution on two geometrically similar Joukowski hydrofoils and reached a conclusion that even at the shallow submergences the principles of potential theory might be expected to lead to valid and useful results for high speeds. They made a distinction between the two flow regimes, namely the hi h Froude number, Fn , and the low Fn , where the Fn was defined as U / &. Here, U is the speed of the hydrofoil or that of the incoming uniform stream, g the acceleration due to gravity, and c the chord of the hydrofoil. They tested for 2 depths, which are a quarter and a fifth chord, and in the range of depths they tested, the critical value of Fn for dividing the two regimes was observed as 0.61. For Froude numbers higher than this the flow was more like that of deeply submerged hydrofoils, however, on the contrary for Froude numbers lower than this it changed markedly, and the hydraulic jump occurred and even the Kutta condition was not satisfied at the trailing edge. They also showed that the important dimensionless parameters for studying the flow around the hydrofoil running near the water surface were Fn , a , and the depth ratio, ie the ratio of the submerged depth h and c . In the present work, h is the vertical distance measured from the leading edge to the water surface.

The co-ordinate system and a schematic diagram of the problem under consideration are shown in Figure 1, and we note that the half span of the hydrofoil is denoted as b .

Although the spanwise elliptical distribution of circulation is not very often adopted for practical use, it is an important case since for a wing in an unbounded medium it corresponds to a constant downwash along the span, and

410

also to the optimum value of the lift to drag ratio. For hydrofoils near the water surface, Wu [2] also made use of the elliptical distribution of circulation, and thus the result of Lee [l] is based on the same. However, it is not well known that how large the magnitude of the induced stream and the downwash, and their variation along the span of hydrofoils of general plan form.

t'

Fig. 1. The co-ordinate system and a schematic diagram.

Most of previous experimental works on hydrofoils are related to the lift and drag, the wake survey, the location and the structure of tip vortices, the pressure distribution, and surface elevations, and it is hard to find works on the measurement of the velocity distribution near the leading edge. Hence the start of this work, and the current paper is the f is t report on the ongoing efforts of our laboratory and it is hoped that more accurate and usefbl experimental data can be afforded in the near fbture.

2. Experimental Apparatus and Tests

All the experiments reported here were carried out with the CWC at the Chungnam National University. Test section of the CWC is 0.6m wide, 0.8m deep and 2m long. The maximum flow velocity is 1.8m/s, however, for its uniformity over the longitudinal and transverse cross sections it is usual to keep

41 1

the velocity below I d s . More details on the CWC can be found in Lee, et al. [5]. NACA 0012 section was chosen for the hydrofoil, which has the rectangular plan form and whose chord is 8cm, and span 24cm, so that the aspect ratio is 3. Magnitudes of the chord and the span were determined considering the sue of the test section, the upper limit of the flow speed, and the accuracy of the measurements. Accordingly, the aspect ratio was not as large as desired, and it should not be expected that the comparison of the experimental data with theoretical results predicted by the lifting-line approach shows a good agreement but that the comparison sheds a light on the basic understanding of the flow field around the foil of finte span.

Velocity measurements were done using a set of digital PIV system, whch consisted of a laser source, optical device for making a laser sheet, a high speed video camera system, and a PIV S/W and a PIC for handling the whole system and the data. The laser source was made by LEXEL, and its light intensity was 1.2 watts, which was rather weak. High speed video camera system was supplied by PHOTRON, and its model name was FASTCAM-X128OPCI. The PIV S / W was Thinker's EYES 2D, a make of Tientech Co.. For visualizing the flow, poly vinyl chloride seed of specific weight 1.020 made by Yakuri Pure Chemical Co. was used. The camera could take 500 frames per second, and each frame had 1280(horizontal) by 1024(vertical) pixels.

For each case of experiments, namely for a chosen speed of incoming flow, an angle of attack, and a submerged depth, velocities near the leading edge were measured at five longitudinal sections, which were the midspan, both tips and the middle of the midspan and the tips. For each section pictures were taken for two seconds, and since the velocity at the leading edge could not be directly measured due to the strong reflection of the wing body, in order to obtain the value of the induced stream and the downwash there an extrapolation method using the least square fit was employed. Velocity measurements were done for four angles of attack, namely a = 0" , 3", 6" , 9" , for three representative speeds which were 0.3ds, 0.6 d s , 0.85m/s, and for three submerged depths, ie h = 1 . 5 ~ ~ 7 . 5 ~ ~ 13.5cm, and the corresponding depth ratios were 0.19, 0.94, and 1.69, respectively. In order to hold the wing fimzly in its intended inclined positions at different angles of attack, we prepared four adaptors connecting the quarter point of the midspan of the wing and the model-holding system attached to the channel, however, due to the existence of this rod the flow filed around the midspan was inevitably disturbed. For speeds, since it was not practically possible to get the exactly same velocity for the same setting of the rpm of the driving motor, there were some differences between the real speed and the representative value. Altogether tests were done for 36 different cases, and each

412

case needed measurement at 5 sections, therefore totally 180 sectional data were gathered for analysis. Reynolds number corresponding to the typical case was 3.05 x lo4 , and hence the flow could be regarded in general as laminar.

3. Experimental Results and Discussions

As described above, velocities near the leading edge were measured for 36 cases, for which the measured Fn is given in Table 1. Using the representative velocities, we obtain the representative Fn 's as 0.34,0.68,0.96. Lee & Kim [6] proposed a criteria for wave breaking behind a shallowly submerged hydrofoil, and in Figure 2 it is shown with the current experimental cases along with experimental results of Parkin et a1 [6]. According to Lee & Kim's proposal wave breaking is not observable for cases C, D, G, H and I and indeed it was not observed for those cases, though for larger angles of attack it occurred for the case C. It is interesting to note that for the case C, when the angle of attack was smaller namely zero or 3", no wave breaking was observed, however, when it became larger namely 6" or 9" the wave breaking occurred. Furthermore, it should be noted that their proposal was supposed to be valid for the two- dimensional foil, and in the current experiments a wing of the finite aspect ratio was used. We also note that the remark on the critical Fn by Parkm et al. [4] was obtained for a fixed angle of attack 5", and hence the value of their critical Fn should be reduced when we consider the wave breaking for smaller angles of attack.

Although the velocity was measured for 5 symmetrically placed sections, due to the unsymmetry of the incoming flow and other possible experimental inaccuracies, extrapolated data were averaged to yield symmetric results.

In Figures 3 and 4, we show the effects of angle of attack for the case D(the representative F n 0.34, the depth ratio 0.94, and the wave breaking absent), upon the induced stream and the downwash, respectively. It is clearly seen that as the angle of attack increases, the induced stream decreases, while the downwash increases. For the whole range of span the magnitude of the induced stream is larger or comparable to that of the downwash. And the magnitude of the two induced velocities is surprisingly large, and this is probably due to the fact that the aspect ratio of the wing used in the experiments equal to 3 is rather small. The magnitude and the tendency of the change are all in general coincided with the finding of Lee 8z Lee[3] by numerical computations, though the large F n assumption was employed in the numerical coding. As the reduction of the induced stream contributes to the increase of the sectional lift and in turn to the lift of the whole wing, reduction of the induced stream with the angle of attack

413

can be more or less expected. Furthermore, as the angle of attack gets larger, thetrailing vortices become more enhanced, and hence the qualitative behavior ofthe downwash is also anticipated. In Figures 5 and 6 we show the same set forthe case C(the representative Fn 0.96, and the depth ratio 0.19), and we notethat for this case wave breaking occurred for larger angles of attack. Thus aswave breaking sets in with the increase of the angle of attack the induced streamincreases and the downwash decreases abruptly. In this regard, it is worth toemphasize that the effect of wave breaking changes the behavior of the flow fieldso drastically that without giving a due attention to the phenomenon of wavebreaking it is very hard to draw any conclusion from the experimental data.

Table 1. Froude number of tested cases.

h/c

a

Fn

h/c

a

Fn

h/c

a

Fn

0.19

case

A

B

C

0°

0.30

0.66

0.92

3°

0.29

0.66

1.00

6°

0.27

0.68

0.94

9°

0.35

0.59

1.00

0.94

case

D

E

F

0°

0.31

0.66

1.00

3°

0.34

0.66

1.00

6°

0.32

0.65

1.00

9°

0.29

0.58

1.00

1.69

case

G

H

I

0°

0.32

0.66

0.93

3°

0.28

0.65

0.94

6°

0.28

0.64

0.98

9°

0.30

0.66

0.97

In Figures 7 and 8 the induced stream and the downwash, respectively, areshown for a fixed representative Fn being 0.34, for the case A, D and G withthe fixed angle of attack 3°. We first note that in the case A wave breaking

414

occurred and thus if we just compare the case D and G, it is clear that the induced stream decreases and the downwash increases as the submerged depth gets larger. Again, this trend is in accordance with the finding of Lee & Lee [3]. They also found that the behavior of the flow field is reversed with the critical value of the depth ratio 0.5. In Figures 9 and 10 the same set for the case B, E and H is shown. For this set, the representative Fn is 0.68, and the angle of attack is 6". We again note that for cases B and E the wave breaking occurred, while for H did not. Excluding the case H, we are led to the same conclusion as above.

In Figures 11 and 12, we show the case A, B and C for which the depth ratio was 0.19 and the angle of attack was 6'. Although wave breaking was present for all three cases, we see that the induced stream increases while the downwash decreases as the Fn gets larger. We also show the numerical prediction of the induced stream and the downwash, and we observe that the agreement is remarkable even though the wave breaking is present for all three cases. In Figures 13 and 14, we show the same set for the case G, H and I for the angle of attack 9". We note that for these cases there was no wave breaking and that the changing pattern of the induced velocities is not monotonic. However, the numerical prediction is on the right side of the experimental values, namely on the side of the increasing Fn .

Since Lee [ l ] made use of the elliptical distribution of the sectional circulation, which corresponds to the plan form of ellipse, it is hard to make a direct comparison between the theoretical results obtained by him and the present experimental results. However, through the current experiments and the numerical simulations by Lee & Lee [3], it was confi ied that the magnitude of the induced stream itself is not negligibly small and that the spanwise change of the induced stream and the downwash is not small either. Further numerical study is undergoing to make the direct comparison possible, and more physical and numerical experiments with the elliptic plan form is being planned.

4. Conclusions

Although the aspect ratio of the tested wing was rather small due to the limits imposed by the experimental facilities, and the various sources of the inaccuracy in the process of the experiment itself and the measurement were present, we could reach the following conclusions.

First of all, the phenomenon of wave breaking affects the velocity field significantly, and experiments should be well designed taking into account its occurrence in advance.

415

Increasing the angle of attack or the submerged depth while keeping other parameters fixed, the induced stream decreases, while the downwash increases. On the other hand as the Froude number increases while keeping other parameters fured, the induced stream gets larger, while the downwash becomes less. Of course, this interpretation should be received with the due attention to the occurrence of the wave breaking.

Numerical findings and predictions are in general good agreement with the current experimental data.

For more direct comparison with the numerical prediction, a numerical code capable of representing the finite Froude number effects is being developed, and a set of experiments is planned using a wing of an elliptic plan form for more systematically varied submerged depths and the speeds.

Acknowledgments This work was in part sponsored by the Korea Science and Engineering Foundation (KOSEF) under the contract number RO5-2002-000-00695-0, and we would like to express our sincere thanks to the KOSEF.

References 1. S. J. Lee, Proc. 5" Intern. Con$ Hydrodyn., 11 l(2002). 2. T. Y . Wu, J. Math. Phy,, 33-3,207(1954). 3. J. M. Lee and S . J. Lee, Proc. Spring Ann. Meeting SOC. Naval Arch. Korea,

4. B. R. Parkin, B. Perry, and T. Y. Wu, J. Appl. Phy., 27-3, 232( 1956). 5. S. J. Lee, H. T. Kim, and C. K. Kim, Proc. Korea-Japan CWC Workshop,

6. S . J. Lee, and H. T. Kim, J. Hydrospace Tech., 2-1, 1( 1996).

1107(2004).

155(1994).

416

Fig. 2. A criteria for wave breaking.

417

0.45

0.10

0.05

UAC10012 .-3 pn.0.34 hlc-0.91

O . O O - , ' ' ' ' I ' ' ' ' I ' ' ' 1 I ' 1 ' 1 I -0.5 0.5 1

Fig. 3. Effect of angle of attack on induced stream for case D.

0.25

O . O 5 I

Fig. 4. Effect of angle of attack on downwash for case D.

418

Fig. 5. Effect of angle of attack on induced stream for case C.

0.25 - W N

-

- - - _ _ _ - - - - - / -

0.20 - / /

\ \

/... /... -. .. '1

I

\ \

0.10 -

I 0.05 NACM012

F-0.04 WC-0.19

r a

Fig. 6. Effect of angle of attack on downwash for case C.

419

c.

- - ./' -. -. x.

-. -. -. -_ r . ./. 0.05 /./

-0.5 0 0.5 1 O.@J- , ' ' ' ' ' ' I ' I ' ' ' ' I ' I ' I '

Y h

Fig. 7. Effect of submerged depth on induced stream(A, D, G).

420

421

0.15

0.10

0.05

0.00

-0.05

0.35

, , '--

Dl / '.,' ......... -. .

0.20 NACA0012 a-3 .a NrO.19

I- /.".. /.".. ,/' \..

/. '.. \.. '.. '.. '.. ,.."

,/'. '.. - I'

,/" '.. '..' /" '.. - ,/'

I ,/" -

-.\/ '.. - /'

'..

.......... ....... ........... ... ....... .... .......... .... .... I I I I I I I I I I I I I I , . I ..~.

..__ ,.._ - /.. -.._ ..,'

,/ ,I. - ,,,/ - ,..'

Fig. 1 1 . Effect of Froude number on induced stream(A, B, C).

...........................

" ' " " ' " ' ~

Fig. 12. Effect of Froude number on downwash(A, B, C).

422

Fig. 13. Effect of Froude number on induced stream(G, H, I).

0.60 :

------_____ ----..-..-.--..-.=1.--; o,20 =.=r.-.rr.=T.=.=.T.--- -. . - . . -. .

. -. . . -. -. 0.15 :

0.10 :

-. . -. . - - c.1.

NACA 0012 -3 a=6

0.05 y hk0.19

o . O o - ; ' -0.5 0.5 1 " ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I

;h

Fig. 14. Effect of Froude number on downwash(G, H, I).

CHAPTER 4

BIOMECHANICS: MEDICAL


BLOOD FLOW ABNORMALITIES IN SICKLE CELL ANEMIA

ANTHONY TZE-WAI CHEUNG Department of Pathology and Laboratory Medicine, Research-III Building (Suite 3400.) University of California, Davis School of Medicine, UCD Medical Center, Sacramento,

CA 9581 7, USA

In sickle cell anemia (SCA), a single amino acid substitution in the 0-globin gene (Glu to Val) causes the polymerization of S-hemoglobin (HbS) and sickling of the red blood cells in human patients, resulting in vasoocclusion (acute painful crisis). However, the hemorheology of SCA has rarely been studied and the pathogenic mechanism behind this vasoocclusive process has not been clearly defined until recently. Using computer- assisted intravital microscopy to non-invasively identify and quantify real-time blood flow characteristics in the conjunctival microcirculation in SCA patients, we have found numerous steady-state (non-crisis baseline) microvascular abnormalities, including abnormal microvessel morphometry, reduced red-cell velocity, abnormal capillary/arteriole/venule distribution density, and diminished capillary flow. During acute painful crisis, a significant decrease in vascularity coupled with vasoconstriction occurred, giving the conjunctival surface a “blanched” appearance. Rheologic and microvascular characteristics also changed significantly. Enhanced adhesion of sickled red-cells to the activated endothelium resulted in a reduction of blood flow and red-cell velocity. which either slowed significantly or was reduced to a trickle. Vasoocclusion appeared to be a red-cell “log-jamming” phenomenon caused by excessive red-cell adhesion in the constricted microvessels. These rheologic and microvascular changes reverted to steady-state baseline values upon resolution of painful crisis. Studies using a transgenic knockout SCA mouse model showed similar vasoocclusive changes and confirmed the roles of enhanced endothelial adhesivity and pathorheology in vasoocclusion. Excessive red-cell adhesion (via HbS polymerization and red-cell sickling), coupled with enhanced endothelial adhesivity (via endothelial activation and adhesion molecule upregulation), contributes significantly to the changes in rheologic and microvascular characteristics during vasoocclusion; these relationships may have translational significance in the management and treatment of acute painful crisis in SCA patients.

1. Introduction

Sickle cell disease is a compendium of genetic diseases that primarily includes homozygous sickle cell disease (HbSS) -- simply referred to as sickle cell

425

426

anemia (SCA), compound heterozygous disease arising as a combination of HbS and P-thalassemia (HbS-thal), and heterozygous disease (HbS-HbC/HbSC).'92 A single amino acid substitution (Glu to Val) in the P-globin chain of hemoglobin in HbSS patients leads to myriad clinical effects and disease complication^.'^^ Only SCA patients were included in this study. It was hypothesized that vascular pathology (vasculopathy) underlies most of the disease complications and accounts for much of the morbidity and mortality in SCA."' Despite the qualitative description of microvessel sludging and the presence of comma signs in the conjunctival micro~irculation,~~~ real-time quantitative studies characterizing the in vivo microvascular abnormalities of SCA patients have rarely been reported.*-" The paucity of reports on real-time microvascular abnormalities in SCA occurred because of the unavailability of relevant technologies and non-invasive sites to conduct in vivo studies in human patients. Therefore, the development of a technology to study real-time microvascular abnormalities, under steady-state (baseline) and during vasoocclusive (acute painful) crisis conditions, in a relevant non-invasive site was warranted.

To rectify these complications, we have developed a novel real-time technology, computer-assisted intravital microscopy (CAIM), and have also identified the microvascular network of the bulbar conjunctiva (conjunctival microcirculation) as a suitable non-invasive site to conduct these studies in human patients (Figure l)."-I5 The conjunctival microcirculation was selected, not only for its excellent quality of image display, but also for its suitability for longitudinal (follow-up) studies. Because of the unique shape and form of conjunctival vessels, each vessel was identified and re-located for longitudinal studies - each vessel literally served as its own reference control to quantify microvascular changes (Figure 2).

2. Research Design and Methods

2.1. Human studies

A non-invasive study on real-time microvascular characteristics in SCA patients during steady-state (non-crisis baseline) and acute vasoocclusive (painful) crisis conditions has been conducted ( ~ 3 0 ; 1992-2002). The experimental protocols utilized in this study were approved by the institutional review board for human research protection at UCD Medical Center and were in accordance with the Declaration of Helsinki. All patients over 16 years of age or their parents/guardians gave signed informed consent.

427

Morphometry

density and diameter) 4 vA4scA’ + (e.g., Vessel distribution

+ VASVEL + Flow Dynamics (e.g., Red-cell velocity)

( b

Figure I . Figure la shows the schematics of the two computer-assisted intravital microscopy (CAIM) systems used in our laboratory. Figure 1 b shows the non-invasive site (microcirculation of the bulbar conjunctivdconjunctival microcirculation) we have chosen to study the real-time microcirculation in human patients.

428

Figure 2. Two frame-captured images of the same vessels in the conjunctival microcirculation pre- and post-treatment in a clinical trial (unrelated to this study). Note the changes in diameter resulting from extensive vasoconstriction. Figure 2a serves as a reference control for comparison with Figure 2b and can be longitudinally studied.

2.2. Animal studies

After we have identified the microvascular abnormalities and landmark events in SCA, we conducted a series of follow-up studies to further investigate the pathogenic mechanism(s) leading to these abnormalities. Because of ethical concerns in experimentation with human subjects, the human study was limited to focus only on microvascular changes occurring as a natural course of the disease and vasoocclusive events in SCA. Intervention studies designed to investigate the mechanisms underlying the vasoocclusive process were conducted only in animal models. In the past, this was not possible as there was no natural correlate in animals for this unique human disease. However, because of modem advances in molecular biology and biotechnology, transgenic knockout and deficient animal models can now be engineered to have pathological features which are concordant with SCA. The animal protocols utilized in this study were approved by the committee on animal research of the University of California, Davis and University of California, San Francisco.

The following animal models were use: (1) Control mice: C57BL/6 strain (obtained from Charles River

Laboratories, Wilmington, MA) (2) Transgenic knockout SCA mice: Genetically engineered to express

human al-, ‘y-, and PS-globin and lack murine a- and P-globin These mice express only human sickle hemoglobin, and

have hematological, pathological, and rheological features concordant

429

with SCA in human patients (bred in-house in a collaborating laboratory at University of California, San Francisco).

(3) P-selectin knockout mice - Genetically engineered by back-crossing nine generations of P-selectin deficient mice to C57BL/6 mice" (generously supplied by Dr. Diwan, a collaborator at NIH).

2.3. Computer-Assisted Intravital Microscopy (CAIM) for human studies

The microcirculation of the bulbar conjunctiva in SCA patients was videotaped and image analyzed using CAIM (Figure I). The CAIM technology has been utilized successfully in previous studies"-'5 and is briefly described below.

The CAIM system was originally designed and built to study conjunctival microangiopathy in diabetic patients and to evaluate the reversal of microangiopathy in successful simultaneous pancreas-kidney transplanted diabetic patients in vivo." It has been substantially modified recently to study the conjunctival microcirculation in adult as well as juvenile (as young as 2 years old) SCA and diabetic The system was macro-optics based and has an optical magnification of 4 . 5 ~ and an on-screen magnification of 125x. The optical magnification of CAIM was fixed because of its macro design; this non- changeable magnification feature was important in this type of longitudinal/feasibility study as it assured that all measurements on vessel vascularity, morphometry and flow dynamics were quantified on the same basis without a magnification variable.

The CAIM system was located in an assigned intravital microscopy room at UCD Medical Center. The patients were transported to the microscopy room for videotaping of the conjunctival microcirculation for each time-frame in the study. On a few occasions when the patients were too weak or reluctant to be moved, a portable and simplified version of CAIM was used to study the patients by the bed-side. Only SCA patients (ranging in age from 17 to 35 years) and matched controls were selected in this study.

Upon arrival to the microscopy room, the patient was asked to relax for 5-10 minutes and to sit comfortably in front of the CAIM system prior to videotaping, with hidher chin and forehead resting on the chin- and head-rest to minimize vibration. The patient was asked not to touch or rub the eye (normally the left eye) to be studied. If the patient complained about eye irritation, two drops of non-medicated sterile saline was applied. Excessive saline was lightly blotted off by tissue at the corners of the eye whenever appropriate. The patient was again asked to place the chin and forehead on the rests -1 0 minutes after the eye- drop application. The level (height) and angle of CAIM were adjusted to align

430

with the bulbar conjunctiva of the left eye at an angle of approach that would provide the flattest surface for focusing. Anti-red filtered (#58 Wratten green filter) light was focused on the peri-limbal region of the bulbar conjunctiva to enhance vessel visualization. A COHU (Model 2622-1 00, %-inch monochrome CCD format; San Diego, CA) video-camera was used to videotape the conjunctival microcirculation. Based on the on-screen images, constant re- focusing was conducted throughout the procedure to ensure good resolution and sharp image display. It was unavoidable that the images would get in and out of focus because of periodic movement and blinking, especially in children. However, the blurred images and movements did not have much effect on data analysis; only one well-focused frame needed to be captured for the quantitation (computer-assisted measurement via image analysis) of microvessel density/distribution per field and venular diameter, and only 8 successive frames were needed for red-cell velocity. In addition, an adjustment algorithm was incorporated into the imaging software to accommodate limited movement for successive-frame velocity computation. For each patient studied, three to five video sequences were made in different areas in the peri-limbal region of the bulbar conjunctiva. Each area studied consisted of a field of small and large vessels; composed of capillaries (diameter of <1 Opm), arterioles (diameter of -10 to 25pm), smaller venules (diameter of -25 to 65pm) and large venules (diameter of >65pm). Each video sequence normally lasted -1 minute in duration.

Objective data quantification was conducted via computer-assisted image analysis using VASCAN and VASVEL (in-house developed imaging software). ' i . ' 3 - i6 Each video sequence was viewed in its entirety by the principal investigator. Well-resolved video images of interest from each sequence were selected, frame-captured, coded and given to two investigators in the intravital microscopy unit for data analysis; all investigators conducting the analysis were blinded to the source or nature (e.g., medical history) of the video sequences. The results from both investigators were averaged. Video images were frame- captured via VASCAN and each captured image was analyzed and quantified for microvascular characteristics; each field studied using VASCAN measured 8.53mm2 in area. Successive video frames (at least eight) were captured and analyzed via VASVEL for dynamic characteristics (flow and red-cell velocity). Results from each patient in each time-frame were averaged. Because of the unique shape and form of conjunctival vessels, each vessel was identified and repeatedly located in longitudinal studies (e.g., during crisis and post-crisis as a follow-up) and used as its own reference (control) to quantify changes.

431

2.4. Computer-Assisted Intravital Microscopy (CAIM) for animal studies

The CAIM technology for animal studies was developed in this laboratory to videotape and quantifj the intestinal-mucosal microcirculation in rodents (Figure 1). The protocols have been described in detail in a previous pubIicationl6 and are only briefly described below.

C57BL/6 mice ranging in age from four to eight weeks and weight from 18 to 35 gm were used as sources of red blood cells (RBC) and as recipients of injected fluorochrome (XRITC)-tagged RBC to study red-cell velocity and flow dynamics. The SCA and P-selectin knockout (deficient) mice, which were used as RBC donors and/or recipients in these studies, ranged in age from six to ten weeks.

Control and sickle RBC for infusion into recipient mice were from C57BL/6 or SCA mice. Blood obtained by cardiac puncture was collected into sodium citrate. The b u m coat was removed to deplete leukocytes to a level equivalent with that obtained with an accepted cellulose column method, and RBC were washed and labeled with XRlTC for visualization, according to the method of Sarelius and D ~ l i n g . ' ~ Recipient mice were anesthetized with an intraperitoneal injection of sodium pentobarbital (Veterinary Laboratories Inc., Lenexa, KA) at a dose of 0.06 - 0.075 mg/g body weight. Labeled RBC were suspended in phosphate-buffered saline (PBS) to a hematocrit of 25%, and 50 pL were infused by tail vein injection into recipient C57BW6, P-selectin knockout, or SCA mice. These volumes were calculated to obtain a 1:200 in vivo ratio of labeled to unlabeled RBC and to minimize hemodynamic changes from the infused volumes. The intestine of the mice was exposed via a mid-line laparotomy, exteriorized and irrigated via a standardized intravital protocol.'6 Using intravital microscopy, the intestinal-mucosal microcirculation was videotaped for subsequent analysis via VASCAN and VASVEL, described earlier in Human Studies.

In some experiments, the recipient mice were pre-injected with 50 y1 of 600 @ml P-selectin mono-antibody (mAb) RB40.34, 16 U/ml unfractionated heparin (UFH), or the combination of 100 pg/ml L-selectin mAb MEL-14, 400 yg/ml PSGL-1 mAb 2PH1, and 5 mg/ml integrin p2 mAb GAME-46. Each of these reagents or combinations thereof was diluted with PBS to a volume of 50 yl to yield a final effective dosage in 2 ml (approximate total circulating blood volume of an average mouse).

Effects of agonist peptide for murine protease activated receptor 1 (PAR-I) on blood flow dynamics were determined by constant monitoring of RBC

432

movement before and up to five min after topical administration of 50 pL PAR-1 agonist peptide or scrambled PAR-1 agonist peptide (used as control) to the mucosal-intestinal vessels by suffusion. The concentration of PAR- 1 agonist peptide and scrambled PAR-1 agonist peptide varied from 1 to 50 pM. We used the lowest effective concentration as determined by testing each batch of PAR-1 agonist peptide in vitro (by observing the induction of cell contraction on human umbilical vein endothelial cells) and in vivo (by observing the reduction of red- cell velocity of XRITC-labeled sickle RI3C in C57BL/6 mice). The ability of UFH to rescue blood flow from complete stoppage was tested by sufision of 50 pl of 16 U/ml UFH after blood flow had been visibly reduced by PAR-1 agonist peptide suffusion.

2.5. Statistics

All measurements were averaged and reported as mean f SD. Variables and changes were compared using analysis of variance (ANOVA). P values were not presented in the text, but were indicated in the figures whenever appropriate. P values of 10.05 were considered statistically significant.

3. Results

We have successfully identified and quantified numerous real-time microvascular abnormalities which existed during the steady-state (baseline) condition. Rheologic and microvascular characteristics also changed significantly. Vasoocclusion resulted in a reduction of blood flow and decrease in red-cell velocity, which either slowed significantly or was reduced to a trickle. Vasoocclusion appeared to be a red-cell “log-jamming’’ phenomenon caused by excessive red-cell adhesion in the constricted microvessels. We have also identified landmark events which uniquely appeared during a vasoocclusive (painful) crisis -- a significant reduction of venular diameter, a significant decrease in red-cell velocity and a drastic disappearance of small arterioles and capillaries. Using transgenic knockout mice models, we have also succeeded in identifying mechanisms in the pathogenesis of these microvascular abnormalities in human SCA patients.

The results are summarized as follows:

3.1. Human SCA studies

The conjunctival microcirculation in SCA patients during steady-state, vasoocclusion (acute vasoocclusive painful crisis), and post-crisis conditions was recorded on high-resolution videotapes and subsequently analyzed as described

433

earlier. At steady-state (baseline), all SCA patients exhibited most of the following morphometric abnormalities: abnormal vessel diameter (795 15 pm; P<0.05), blood sludging, boxcar blood flow phenomenon, distended vessels, damagedhuptured vessels, hemosiderin deposits, vessel tortuosity, and microaneurysms (Figure 3). There was a decrease in vascularity (diminished presence of conjunctival vessels) in SCA patients compared with non-SCA controls, giving the bulbar conjunctiva an ischemic/avascular appearance in most SCA patients during steady-state. In addition, a significant change in vessel (capillary/arteriole/venule) distribution density also existed. Averaged steady- state red-cell velocity in SCA patients (1.450.8 mm/sec; P<0.05) was slower than in non-SCA controls (2.5h0.6 mm/sec). During vasoocclusiotdcrisis, a further decrease in vascularity (caused by flow stoppage in small conjunctival vessels) and a 36.7?&5.2% decrease in venular diameter resulted, giving the bulbar conjunctival surface a “blanched” appearance. In addition, the conjunctival red-cell velocities either slowed significantly (6.6%*13.1%; P < 0.05) or were reduced to a trickle (<0.01 d s e c or unmeasureable) during crisis. The microvascular changes observed during crisis were transient and reverted to steady-state baseline after resolution of crisis (Figures 4a and 4b). When combined, CAIM represents the availability of a non-invasive tool to quantify microvascular abnormalities in SCA, as well as in other vascular diseases. The ability to identify and relocate the same conjunctival vessels for longitudinal studies uniquely underscores the applicability of this quantitative real-time technology.

3.2. Animal studies

It is an accepted but unproven concept that vasoocclusion in SCA is initiated by abnormal adhesion of sickle RBC to the activated endothelium. Our objective in this study was to investigate whether sickle RBC adhere abnormally to endothelial P-selectin and to identify, if any, the role of P-selectin in vasoocclusion. We found significantly (P<0.05) faster red-cell velocities in P- selectin knockout mice (2.9*0.3 mm/sec) and control mice (3.1*0.2 d s e c ) than in SCA mice (1.6*0.3 mmlsec), which have increased endothelial cell P- selectin expression. Agonist peptide for murine PAR- I , which selectively activates P-selectin in endothelial cells but not platelets, was used to assess the

434

Figure 3. Figure 3a shows a typical view of the conjunctival microcirculation in a healthy non-SCA control subject. Note the even distribution of the conjunctival vessels. Figure 3b shows a view of the conjunctival microcirculation in a SCA patient. Note the uneven distribution of the vessels, the abnormal (wide) venular diameter, sludging of blood [S] in the venules, the boxcar flow pattern [B], and the presence of ischemic areas caused by the significant absence of small arterioles and capillaries [I].

effects of endothelial cell P-selectin on microvascular flow. Suffusion of venules with this agonist stopped flow promptly in normal and SCA mice (-1-2 minutes), but not in P-selectin knockout mice or in control mice pre-treated with anti-P- selectin monoclonal antibody or UFH. Agonist-induced slowing of blood flow in normal and SCA mice was reversed rapidly by suffusion with UFH, provided flow had not already stopped. Leukocyte adhesion was not required for PAR-1 generated vasoocclusion. This study confirmed the pivotal role of P-selectin in abnormal flow dynamics and vasoocclusion in SCA.

4. Conclusion

Our intravital research substantially extends earlier observations by other investigators and offers several additions to the literature. First, our intravital technology represents the first utilization of a conjunctiva-dedicated intravital microscope design that is not based on the slit-lamp (biomicroscope) assembly previously used in other l a b o r a t ~ r i e s . ~ ~ ~ ~ ~ ~ ' ~ The magnification and resolution of the optics are better than biomicroscopes, and the system is extremely easy to operate. Coupling CAIM with videotaping capability offers additional cost, time, convenience, and reliability advantages. In addition, dynamic (red-cell velocity) measurements can be objectively conducted on videotaped sequences, an opportunity that did not exist with still photography in historical studies. When combined, these improvements have enabled us to generate detailed and

435

easily reproducible high-resolution images for real-time morphometric and dynamic studies on the conjunctival microcirculation in SCA patients. The capability of CAIM to frame-capture and objectively quantify (measure) the microvascular characteristics via image analysis further enhances the uniqueness of this technology.

Figure 4. The figures show two frame-captured images revealing the microvascular changes during vasoocclusive crisis (4a) and after crisis resolution (4b). Focusing of the images was aimed at the same location during vasoocclusive crisis and crisis resolution, with the same vessel serving as its own reference control. Figure 4a shows that during crisis, there is significant reduction in vessel diameter and disappearance of small arterioles and capillaries, resulting in extreme avascularity - giving the area a “blanched” appearance. The arrows point at the three vessels targeted for longitudinal comparison after the resolution of crisis. Figure 4b shows an increase in vessel diameter and reappearance of small arterioles and capillaries after crisis resolution. The vessels indicated by the arrows show a significant increase in vessel diameter. In addition, small arterioles and capillaries (indicated by arrowheads) reappear after crisis resolution. (See Figures 4A and 4B in Blood. 2002; 97:3401-3404).

436

Figure 5. The two figures show flurochrome (XRITC) tagged RBC (appearing as bright dots) in the intestinal-mucosal microcirculation in a transgenic knockout SCA mouse. Figure 5a shows that when the red-cell velocity is low or when there is no flow, the RBC

appear as bright round dots. Figure 5b shows that when the red-cell velocity i s high, the flow pathways of the RBC appear as bright streaks instead of dots.

Before this 1 0-year study on real-time microcirculation in SCA, quantitative and non-invasive studies on blood flow in the in vivo human microcirculation have been limited. There are only two easily accessible sites for non-invasive in vivo microcirculation research in human subjects: the nailfold capillary bed and the conjunctival microcirculation. Lipowsky et al have utilized the nailfold capillaries in the fingers of SCA patients and have commented that the micrcovascular organization and vessel size in the nailfold microcirculation were not representative of the microcirculation.8 We used the conjunctival microcirculation because of its easy non-invasive accessibility, excellent quality of image display, and the ability to relocate the same vessels for longitudinal assessment. In addition, the organizational and morphometric characteristics of the conjunctival microcirculation reflect more closely the characteristics of a true microvascular network (e.g., normal presence and distribution of capillaries, arterioles, and venules) and are comparable with those of the microcirculation at the susceptible soft tissue end organ level in SCA.

The results described in the SCA human studies objectively and quantitatively identify, for the first time, microvascular abnormalities which exist in SCA under steady-state baseline conditions. In addition, we have revealed that the microvascular landmark events which occurred during SCA crisis

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(decrease in venular diameter, reduction in red-cell velocity and disappearance of small arterioles and capillaries) are transient and will revert to pre-crisis baseline upon resolution of crisis.

The final piece of the puzzle which can lead to a clear understanding of vasoocclusion is the identification of the cellular mechanism -- the pathogenic process -- which underlies the abnormal adhesion of sickle RBC to the endothelium in the development of the vasoocclusive process. The results described in the animal studies, for the first time, verify the in vivo relevance of P-selectin mediated adhesion of sickle RBC to the endothelium activated by the PAR- 1 agonist peptide. This study reveals the role of endothelial activation and characterizes the abnormal adhesion of sickle RBC to P-selectin in the activated endothelium as the pathogenic process leading to vasoocclusion. Theoretically, if we can block this mechanistic process, we may possibly block or reverse abnormal adhesion and vasoocclusion in SCA crisis. The results from the animal studies have identified a specific mechanism which can be extrapolated to the human situation and therefore, may have translational significance in the development of medications and treatment or management modalities for SCA. Recently, we have completed a Phase-I11 randomized clinical trial (not reported in this manuscript) in which we have successfully utilized an anti-adhesive agent (Poloxamer 188) to reverse sickle RBC adhesion during vasoocclusion (crisis) in nine SCA patients. Other intervention studies based on a design to counteract or ameliorate this abnormal adhesion mechanism are in progress.

This study confirms that computer-assisted intravital microscopy (CAIM) represents the availability of a sensitive, repeatable, and quantitative real-time tool to study microvascular abnormalities in vascular diseases, including SCA. The utilization of the bulbar conjunctiva as a research site provides an ideal, readily accessible, and non-invasive microvascular bed for in vivo research. Moreover, this site offers the additional advantage that the images of the conjunctival vessels are well resolved and easily identifiable for relocation in longitudinal studies. The fact that the experimental approach is objective and quantitative underscores the uniqueness of this technology and its potential as a clinical and/or research tool. Computer-assisted intravital microscopy (to study small vessel vasculopathy) can be used to correlate with transcranial Doppler ultrasonography or magnetic resonance angiography (to study large vessel vasculopathy) in the prediction of premature stroke in SCA. Intravital microscopy can be used to non-invasively study and identify key in vivo landmark microvascular events in different vascular diseases, including SCA, diabetes mellitus, Alzheimer’s disease, and hypertension. The same objective and quantitative technology can also be used to monitor the efficacy of clinical

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trials (e.g., the recently completed Phase-I11 Clinical Trial on Poloxamer 188 as a medication to treat vasoocclusive crisis in SCA) and various therapeutic regimens in allogenic bone marrow andor stem cell transplantation. In each case, because the same vessels can be easily identified and relocated for longitudinal studies, the patient can reliably and appropriately serve as his or her own reference control for longitudinal data interpretation.

Acknowledgement

This manuscript is written as a brief summary of a lecture presented in the Theodore Y.T. Wu Symposium (an ancillary symposium as part of the OMAE 2004 meeting) in honor of Professor Wu’s 80’ birthday. The idea to develop and adapt computer-assisted intravital microscopy for real-time microvascular studies originated in Professor Wu’s laboratory at CalTech when I first tried in vain to utilize the “mega-bug-watcher’’ to quantify cell motility and flow dynamics. Since my training at CalTech, this idea was followed through and further developed at the University of California, Davis. Professor Wu’s influence on my research and professional development is respectfully acknowledged and greatly appreciated.

The subject materials presented in the lecture and the content of this manuscript represent over 10 years of work on instrumentation design and application in vascular biology and hemorheology research. Various portions of the lecture and manuscript have been presented in internationallnational conferences and published in proceedings and journals.

The studies were partially funded by NIH R01 grants HL67432 and HL64396. The contributions of my collaborators (Professor Ted Wun in the human studies and Professor Stephen Embury in the animal studies), fellows, students, and staff are very much appreciated.

References

1.

2.

3.

4.

5 .

Powars D. Chan LD, Schroeder WA. The variable expression of sickle cell disease is genetically determined. Semin Hematol. 1990; 4:360-376. Embury SH, Hebbel RP, Mohaaandas N, Steinberg MH, eds. Sickle Cell Disease: Basic Principles and Clinical Practice. New York, NY: Raven Press, 1994. Bunn HF. Pathogenesis and treatment of sickle cell disease. N Engl J med. 1997;

Platt 0s. Easing the suffering caused by sickle cell disease. N Engl J Med. 1994;

Platt OS, Brambilla DJ, Rosse WF, et al. Mortality in sickle cell disease. Life expectancy and risk factors for early death. N Engl J Med. 1994; 330: 1639-1644.

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330:783-784.

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6. Knisely MH, Eliot TS, Warner L. Sludge blood. Science. 1947; 106:431-435. 7. Paton D. The conjunctival sign of sickle cell disease. Arch Ophthalmol. 1961;

8. Lipowsky HH, Sheikh NU, Katz DM. Intravital microscopy of capillary hemodynamics in sickle cell anemia. J Clin Invest. 1987; 80:117-127.

9. Rodgers GP, Noguchi CT, Schechter AN. Non-invasive techniques to evaluate the vaso-occlusive manifestations of sickle cell disease. Am J Pediat Hematol Oncol.

10. Rodgers GP, Schechter AN, Noguchi CT, et al. Microcirculatory adaptations in sickle cell anemia: reactive hyperemia response. Am J Physiol. 1990; 258:HI 13- H120.

1 1. Cheung ATW. Perez RV, Chen PCY. Improvements in diabetic microangiopathy after successful simultaneous pancreas-kidney transplantation: a computer-assisted intravital microscopy study. Transpl. 1999; 68:927-932.

12. Cheung ATW, Harmatz P, Wun T, et al. Correlation of abnormal intracranial vessel velocity (measured by transcranial Doppler ultrasonography) with abnormal conjunctival vessel velocity (measured by computer-assisted intravital microscopy) in sickle cell disease. Blood. 2002; 97:3401-3404.

13. Cheung ATW, Chen PCY, Larken EC, et al. Microvascular abnormalities in sickle cell disease: a computer-assisted intravital microscopy. Blood. 2002; 99:3999- 4005.

14. Cheung ATW. Ramanujam S, Greer DA, et al. Microvascular abnormalities in the bulbar con.junctiva of patients with type 2 diabetes mellitus. Endocrin Prac. 2001;

15. Cheung ATW, Price AR, Duong PL, et al. Microvascular abnormalities in pediatric diabetic patients. Microvas Res. 2001 ; 63:252-258.

16. Embury SH, Mohaaandas N, Paszty C, et al. In vivo blood flow abnormalities in the transgenic knockout sickle cell mouse. J Clin Invest. 1999; 103:915-920.

17. Paszty C, Brion CM, Manci E, et al. Transgenic knockout mice with exclusively human sickle hemoglobin and sickle cell disease. Science. 1997; 278:876-878.

18. Mayandas TN. Johnson RC, Rayburn H, et al. Leukocyte rolling and extravasation are severely compromised in P-selectin deficient mice. Cell. 1993; 74:54l-554.

19. Sarelius IH, Duling BR. Direct measurement of microvessel hematocrit, red cell flux, velocity, and transit time. Am J Physiol. 1982; 243:H1018-H1026.

66190-94.

1985; 7:245-253.

7:358-363.

DOES INTERFACIAL VISCOSITY EXIST, ITS APPLICATION TO MEDICAL SCIENCE

S . C . LING The Catholic University of America, School of Engineering

Washington, DC 20064, USA

Many major medical problems are now suspected to be triggered by the powerful immune reaction mechanism of the body that has gone awry. If the inflammatory reaction is not suppressed after it has performed its protective function, it can lead to more severe autoimmune reactions. The problem is due to both the excessive release of water from the mucous layer of the microcirculation and the failure for the mind under stressful condition to suppress the inflammatory reaction. Knowing the root cause of the problem, should give one a better handle for managing health.

1. Introduction

It is heartening to observe that Western medical researchers are beginning to note that many major disease processes are triggered by chronic immune, inflammatory, and allergic reactions’. There is also the awareness of Eastern medical concepts, which puts emphasizes on helping to maintain the balance (homeostasis) of the physical state of the body through mental and dietary control2’ ’. It is the objective of this paper to try to combine both Western and Eastern medical philosophies into a unified scientific base. We shall use the new found knowledge about microcirculation as a foundation for addressing the health problem. More specifically, one should utilize all fields of advanced sciences in addressing a complex subject, as it is too easy to miss the root cause when one only depending on a single viewpoint, or specialty. It is hoped that the improved knowledge concerning the microcirculation will contribute to our understanding of how excessive inflammatory reactions in immune and autoimmune problems can be more effectively controlled; thus contributing to the homeostasis of the body and the maintenance of better health.

2. What we know about capillary blood-flow

At present, we still do not have a concise understanding as to how red blood cells can move with such relative ease through the lumen of a capillary that is 20 percent smaller than itself. We do know that all intimal surfaces of the

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circulatory system are lined with a single layer of endothelial cells. The endothelial cells have a thin coat of mucus, which consists mainly of a water solution of long chain mucopolysaccharides. The polymer content is of the order of a few hundred parts per million by weight of water. In addition, the polymer is known to be a type of heparin as shown in Fig. 1. Notice that heparin is a disaccharide of carbon-rings with many sulfate and carboxyl side branches. Possible variations of these side branches are numerous, and they play many major physiological roles. Notice that the electrons associated with the oxygen atoms have large numbers of lone-pair electrons, shown as rabbit-like ears in Fig. 1. These lone-pairs of counter orbiting electrons do not easily commit themselves to chemical bonding, but do actively interact with other molecules. Under normal temperature, they perform nanoseconds of transient polar-bonding with other positively charged ends of polar molecules. An example is when water molecules form a liquid solution known as a mucous-solution. Notice that in Fig. 1 the long chain heparin molecules will have a hgh affinity for water, due to water being a micro dipole-molecule with two positively charged hydrogen on one end of the oxygen, and two lone-pairs of negative electrons on the opposite end. Mucopolysaccharides in solution with water are known to be one of the most negatively charged anions. Large swarms of dipole water-molecules that can feel the inference of the negative-potential field along the backbone chain of the polymer will be struggling to get in touch with it. Under minimum free energy, as when in a fully hydrated state, the polymer will attract tens of layers of water molecules over its backbone chain. Notice that the water molecules that are actively interacting with the polymer will have several of their thermal freedom of motions curtailed. For example, the molecules can no longer easily spin or tumble around. This, in turn, will increase the bulk viscosity of the solution. Conversely, the vibratory thermal-motions for both the lone-pair electrons and hydrogen atoms of water have to be increased to maintain thermal equilibrium condition for the system. This will tend to reduce the bulk viscosity of the mucus. Hence, the experimental result has indicated that the normal bulk viscosity of mucus is only slightly higher than that of pure water. However, at the interface of the mucous layer and the free water molecules, the outer water molecules sense the water molecules of the mucous layer with more intense vibratory thermal-motion. Consequently, the transient polar interaction of the water molecules at the interface is greatly reduced. Measurement of the interfacial viscosity has indicated a value of 40% less than the bulk viscosity of pure water4. This anomalously low interfacial viscosity is tentatively believed to be why red blood cells and blood plasma can move with ease through a small capillary. Since the above experiment was conducted in a tube much larger than the 8 microns capillary, one cannot be sure that the observed effect could still hold in the capillary. However, it is known that red blood cells also have a hghly negatively charged mucous coating. Hence, there will be a strong

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repulsive force between the cells and the capillary wall for fiuther reducing the frictional drag.

Figure 1 . Molecular view of a heparin polymer-chain.

When the body is subjected to stress, injury or infection, the pH of the affected area becomes more acidic. Histamine released into the blood stream under normal pH condition is a monovalent cation. However, under an environment of pH=34 histamine will become a divalent cation that can cross- link the long-chain mucopolysaccharides like the action of a zipper’. This results in the release of the water molecules from the polymer solution, causing the mucus to become very viscous. In severe cases, h s phenomenon can virtually stop the microcirculation. The reddish itchy-bump as a result of an insect bite, or the swollen nasal passage due to inhalant allergens, comes about when water is released from the microcirculation into the extra cellular space. The symptom of the itchy bump and runny nose has led many towards the wrong assumption that the microcirculation has increased rather than became critically slowed and impaired. One also commonly experiences during high mental stress states that

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minor inflammatory reactions can easily blow up into a life threatening state. This is because; under a stresshl state, the mind fails to regulate the immune and inflammatory reactions through the release of cortisol*. Although this is a built- in protective mechanism for the body to prevent excessive bleeding; but in acute states if this continues and becomes a chronic condition, it can lead to many major medical problems. This will be discussed in more detail in the following section. Thus, the maintenance of proper water content for all time in the mucous layer of the circulation system should be of primary importance to medical science. It is known, since ancient times, that the human mind can be trained to achieve self-awareness for various major organs of the body, and thus developed the ability for maintaining a healthy condition for the microcirculation. The popular Eastern “Chi-kung or Yoga” meditation and deep breathing exercise are few of such examples. Mental stress often leads to shallow breathing, resulting in low O2 and high concentration of C 0 2 in the blood. The acidotic effect due to high C 0 2 levels in blood will in turn lead to high blood pressure as well as inflammatory and autoimmune reactions. Therefore the practice of meditation, deep breathing, and dietary control are critical in aiding the body to maintain homeostasis.

3. Medical problems associated with microcirculation

The author first noticed the importance of maintaining proper water content in the blood when he was studying the detailed nature of pulsatory blood-flows in arteries. A large percent of dogs during major surgery tended to urinate a lot, have large increases in blood pressure and heart rate. A corresponding increase in blood viscosity by a factor of four was noted, commonly leading to heart failure. However, it was found that by starting infusions of IV solutions prior to and during surgical procedures, this shock syndrome was essentially eliminated6. Although this procedure has since been adopted as a standard practice for almost three decades, the scientific basis for this is still not well understood. It is, therefore, the impetus of this paper to provide some insight into this problem.

The problem of stomach ulcers was for a long time noted to be associated with high mental stress and an overly active vagus nerve that caused excessive release of histamine and stomach acid. The standard treatment was to cut away the ulcerated portion of the stomach and part of vagus nerve. However, it is now known that a leading root cause for ulcer is due to the toxin released by helicobacter pylori parasite’, whch results in the chronic inflammation of the duodenal wall. The chronic shut down of the microcirculation at the inflamed part causes cell death and the release of excessive acid that ultimately leads to

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heavy bleeding. Similarly, cholera is caused by the toxin released by bacteria, which induces severe inflammation of the intestine. This in turn causes the hstamine to release large amount of body water. The patient can only be saved by timely force-feeding of water rather than by IV infusion.

Arteriosclerosis of the arteries is found to be largely located at the lips of arterial bifurcations and entrances. Clearly, it is related to the extremely high shearing stress that is known to exist there8. This is called the entrance effect in fluid mechanics. Although cholesterol is found at the site, there is no scientific specificity that it is the precursor for the event. Under normal blood pressure, the aorta is highly distensible, and it typically pulsates 40 percent radially. However, under high blood pressure states, the arterial wall is distended to its limit, and becomes rigidg. This causes the heart to pump against a high impedance load; i.e., at systole it has to accelerate a long column of blood all at the same time. The large increase of workload for heart results in a higher demand for more blood flow to the heart; i.e., higher blood pressure. Whle at diastole, the heart valve closes with a high shock, due to the deceleration of the long column of blood. The closing shock causes a hgh-speed squirt of blood into the coronary arteries". The excessively high shearing stress causes the tearing away of endothelial cells lining at the above-mentioned sites as well as at the lips of the heart valve. This causes chronic inflammation with abnormal death and growth of cells over the affected areas, eventually leading to the blockage of arteries and a leaking heart valve. It was also known three decades ago that the widely used hydrogenated cottonseed oil in confectionary foods could cause inflammatory reaction of the arteries. It is only recently that there is a warning to stop using the same oil under a nonspecific label of trans-fat.

The painll symptoms of arthritis, lupus, and gout are caused by the chronic inflammation of the tissue and joints due to infective diseases or deposit of uric crystals. The release of water from the microcirculation into the extra-cellular space causes the slow down of capillary blood flow. Under chronic conditions, it can lead to autoimmune reactions with severe pain and deformation of the affected parts. Thus, by drinking adequate amount of water, practicing deep breathlng, and with combined diet and mind control, for many the development of these deleterious problems can be preventable.

The problems of Alzheimer's disease and autism are commonly believed to be of genetic and biochemical origins. However, it is also known to be associated with inflammatory reactions, as is observed in post-vaccine reactions. When under time extended idammation, the water in the microcirculation is displaced into the cranial cavity. This results not only in the increase of cranial pressure, but also in the quick death of brain cells due to the insufficient supply

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of oxygen. Because brain cells are in most cases irrecoverable, any inflammation of the brain must be treated with almost no time to spare.

4. Conclusions

We have shown that a large number of major medical problems are associated with dammation and the impairment of the microcirculation, caused by the displacement of water from the mucous layer. Thus, the practice for timely replacement and maintenance of water in the circulatory system should be of primary concern to the attending physician; and perhaps no less important, the practice of mind and body control by the patient himself. This training for long- term self-regulation to maintain proper homeostasis is preferable to an extended drug-dependency that can lead to irrecoverable secondary-complications -- the TUO (way or law) of nature".

References

1. Gorman C. and Park A., The fire within, Time, Feb. 23 2004, p.39-46. 2. Stemberg E.M. and Gold P.W., The mind-body interaction in disease, Scientific

America, Special Edition, The hidden mine, Aug. 2002, p. 82-89. 3. Stein J., Just say om, Time, Aug. 4 2003, p.48-56. 4. Ling S.C. and Ling T.Y.J., Anomalous drag-reducing phenomenon at a waterffish-

5 . Pollack G.H., Cells, Gels and the Engines of Life, Ebner & Sons, 2001, p. 137-140. 6. Ling S.C., Atabek H.B., Letzing W.G., and Patel D.J., Nonlinear analysis of aortic

7. Wright K., Gut reaction, Discover, Feb. 2003, p. 24 -25. 8. Ling S.C., Atabek H.B. and Carmody J.J., Pulsatile flows in arteries, Proc. 121h

International Congress of Applied Mechanics, Ed. Hetenyi, Springer-Verlag, 1969,

9. Ling S.C. and Chow C.H., The mechanics of corrugated collagen fibrils in arteries,

10. Atabek H.B., Ling S.C. and Patel D.J., Analysis of Coronary flow field in

11. Dale R.A., Tao Te Ching, Barnes & Noble, 2002.

mucus or polymer interface, J. of Fluid Mech., 1974, 65, p. 499-5 12.

flow in living dogs, Circulation Research, 1973,33, p. 198-212.

p. 278-291.

J. Biomechanics, 1977,10, p. 71-77.

thoracotomized dogs, Circulation Research, 1975,37, p.752-761.

UNSTEADY FLOWS WITH MOVING BOUNDARIES: PULSATING BLOOD FLOWS AND EARTHQUAKE

HYDRODYNAMICS

TIN-KAN HUNG Department of Bioengineering, Department of Civil and Environmental Engineering,

University of Pittsburgh, Pittsburgh, PA. 15668 USA

It seems that there is no commonality between the fluid mechanics characteristics of blood flows and those associated with earthquake hydrodynamics. The former is predominated by nonlinear pulsating flows with strong viscous effects, while the latter is characterized by strong local accelerations with or without a significant viscous effect. In the case of reservoir hydrodynamics during an earthquake, the hydrodynamic pressure can be seriously altered by the vibration of a dam and by the dynamic response of reservoir sediments. The problem can be characterized by unsteady flows with moving boundaries as is the case for pulsating blood flow in an artery or flow with a balloon pumping assist. These problems are very difficult to analyze either analytically or experimentally because the domain of the nonlinear flow processes is varying with time. This paper is to summarize a computational approach to these unsteady flows with moving boundaries, namely mapping the time-dependent domain to a fixed domain.

1. Introduction

The interaction of blood flow with distensible arteries is not only vital to blood pressure but also to blood circulation and flow resistance. The arterial geometry and wall motion have a significant effect on blood flow processes. The effects of arterial stenosis, curvature and wall motion are important to fluid dynamic analysis. Nonlinear pulsating flows associated with an intra aortic balloon pumping or an intra vena cava balloon pumping are also of interest for biofluid mechanics. The inflation and deflation of a balloon alter not only the size of the flow region but also the flow characteristics. Traditionally, engineering mechanics problems were studied when the equations of motion could be simplified while the main characteristics remained unchanged, as evidenced by many milestone analyses such as potential flows and boundary layer analyses.

Paradoxically, an opposite strategy has been used in the author’s research endeavor over the last two decades. The difficulty in solving transient flows with moving boundaries can be handled by transforming the time-dependent

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flow region to a fixed rectangular domain. This approach makes the Navier- Stokes equations in the transformed domain very complicated. However, the geometry of the flow field becomes much simpler and in particular time invariant. The geometric description and the kinematic and dynamic boundary conditions can be expressed as time-dependent coefficients and dynamic parameters in the equations of motion. Furthermore, the boundary layer is stretched so that the finite difference meshes in the computational domain become regular and time invariant while representing time-dependent irregular meshes in the physical domain. The complexity of the equations of motion can be managed in computational fluid mechanics. This approach has led to a series of successful computational analyses for pulsating blood flows in stenotic arteries, curved arteries, intra-aortic balloon pumping, and intra-vena cava balloon pumping.

The same strategy has been employed in calculating nonlinear hydrodynamic pressures on dams during earthquakes. The dam vibration and the surface waves generated during an earthquake alter the flow domain for analysis. The changes in geometry (domain) can be resolved when the flow field is mapped onto a fixed rectangular region while the equations of motion become more complicated. The dynamic interaction between the dam and reservoir water can be well correlated with the vibration of the dam, more specifically the acceleration along the dam face. The surface waves generated by an earthquake are calculated. The viscous effect of reservoir water has been confirmed to be negligibly small (Hung and Wang 1987).

In reservoir hydrodynamic analysis, the effect of sediment in a reservoir was often considered as a damping mechanism in the literature (Fenves and Chopra 1983). It appears to be the case when the hydrodynamic pressure was produced by the dynamic interaction between the dam and reservoir water. However, when the reservoir is subject to earthquake excitation, the dynamic pore pressure in the sediment is coupled with the dynamic pressure of reservoir water (Chen & Hung 1993; Hung & Chang 1995). A new approach for calculating pore pressure and elastic stresses in the sediment is developed in this report.

Another external flow problem discussed in this paper is unsteady viscous flows past a moving and oscillating plate. The complexity of the motion of the plate is characterized by its pitching andor flapping motion which alters the angle of attack when the coordinate system is moving and rotating with the plate.

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2. Pulsatile Flows in Stenotic Arteries

Axisymmetric pulsating blood flows in stenotic vessels were calculated by using an implicit finite difference scheme of the Navier-Stokes equations. For an axisymmetric distensible artery, the radius a(z , t ) can be expressed as a function of the longitudinal axis z and time. In this case, the geometry of the flow field is time dependent. When the radial axis r of the cylindrical coordinate system is normalized by the instantaneous radius at the section, the dimensionless radial axis becomes r/a(z, t) and the wavy lumen is mapped to a fixed rectangular domain for computational flow simulation. In a rapidly accelerated and decelerated flow process, large velocity gradients appear near the wall region. It is imperative to have very fine meshes in the boundary layer and in the zone of flow separation for numerical solutions of the Navier-Stokes equations. This requirement can be accomplished by using an exponential function to stretch the radial axis:

k ( r / a ( z , t) - r* = [ r / a ( z , t)] e

in which k is a stretching factor (Hung et a1 1980). As shown in Fig. 1, the time- dependent physical domain (the r-z plane) is mapped to a fixed rectangle. The non-orthogonal irregular meshes are transformed to regular meshes for computational analysis. In this case, the stretching factor k used is equal to 1.5, and only every other mesh is plotted here. For a rigid stenotic vessel, the radius is no longer varied with time but with the longitudinal axis (i.e.. a = afz) ).

Figure 1. (a) Mesh network in the physical domain. (b) Mesh network in the transformed domain for computation.

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The transformation of the radial coordinate Y to Y* introduces a set of time- dependent geometric parameters in the Navier-Stokes equations. Although the equations become more complicated, the computational domain is now rectangular as the dimensionless radius is simply a unity. The time-dependent irregular mesh (dr) in the physical domain becomes a fixed mesh (dr*) in the transformed domain. All the geometric characteristics of the artery are cast in the equations of motion and the pressure wave equation. This is the main feature and advantage of the computational approach.

Another new feature in the computational analysis is to define the dimensionless velocity components as

where D = the initial inlet diameter = 2a(0, 0)=2ao, p = dynamic viscosity, u,&, t ) = the instantaneous radial velocity of the wall = d (2, t), and Ap, = the maximum pressure drop between points A and B at the inlet and outlet (see Fig. 1). Notice that the dimensionless radial velocity U at a section is normalized by its instantaneous velocity, u,, (z, t), on the wall; that is U=l on the wall throughout the computational analysis. The dimensionless pressure and time are defined by

P = p L / ( D A p ) , T = p t / ( p D 2 ) (3)

where dp(t) ' p a -pR = the instantaneous pressure drop between points A and B in Fig. 1, and p = the fluid density. In this case, the Karman number, K(9, and a dimensionless wall velocity, Wdz, t), are introduced into the Navier- Stokes equations. They are defined respectively by

K = p D 3 A p / (p2 L ) , Uw= p u,&, t)L / (D2Ap, ) (4)

These two dimensionless parameters conceive the dynamic boundary conditions for pulsating flow computation. In our studies (Hung and Tsai 1996 &1997), the velocity on the wall is specified for simplicity as u,,,(z, t) =

u0 (z)Fo ( t ) so that the shape or ratio a(z, t)/a(o, t ) remains the same. The difference of dimensionless pressures between points A and B becomes a constant (i.e., pA -pH = L /D) throughout the computational analysis.

Recapitulating, the computational method maps the time-dependent domain to a fixed rectangular domain. Also, all the geometric, kinematic and dynamic

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boundary conditions become dimensionless parameters in the Navier-Stokes equations. They were referred as forcing functions in the Navier-Stokes equations for calculating unsteady flows with moving boundaries (Hung I98 1). For a rigid vessel, dimensionless wall velocity Uw is no longer needed, and the dimensionless radial velocity can be expressed as

u = p u L / (D’AP,).

Figures 2 and 3 compare the streamlines between the rigid and distensible stenotic arteries. In these figures, the radial axis r is normalized by the instantaneous radius a(0, t ) at the inlet so that the geometry of the figure remains the same as that at T = 0. During the time interval, 0 < T < 0.001 85,

Figure 2. Pulsatile flows in distensible stenotic artery (left) and in rigid stenotic artery (right)

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the pulsating flow is rapidly accelerated by the Karman number shown in Fig. 4b. It reaches the peak flow rate at T = 0.001 85. The transient flow rate can be

the pulsating flow is rapidly accelerated by the Karman number shown in Fig. 4b. It reaches the peak flow rate at T = 0.00 185. The transient flow rate can be expressed by the instantaneous Reynolds number R , and the stream function along the centerline is equal to R/8 (Hung and Tsai 1996 &1997).

Figure 3 . Pulsatile flows in distensible stenotic artery (left) and in rigid stenotic artery (right).

During the wall dilation, the streamlines near the wall are affected by the wall movements, and some of them end on the wall. In comparison with pulsating flow in the rigid stenotic vessel, the outward wall motion delays slightly the flow separation. As the time proceeds, the size and intensity of the vortex continue to grow even though the flow rate (or R ) begins to decrease, reflecting momentum transfer from the main flow to the vortex. As the vortex is being dragged towards the downstream, the second flow separation occurs in the backflow region of the vortex. The intensities of these t w ~ vortices are indicated

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by ry, and ry2 in Fig. 4c, representing fluid circulating around the vortex center. Notice that the vortices remain significant when the main flow is seriously reduced by an adverse pressure drop 4 ( a negative Karman number).

/

I 0 0.004 y t 0 . W -

02

200

R 8 -

100

- 100

I I I I

0 0.004 0.008 fi O2

L . . d ~- 0 0.

Figure 4. (a) Dimensionless wall velocity and radius increment at inlet (top). (b) Flow rate ( Vo , R and A) at inlet of distensible stenotic artery vortex intensities ( W , and W2 ) and flux ( Ww ) induced by wall motion (middle). (c) Flow rate (B) for rigid stenotic vessel, for distensible uniform artery (C), and the Karman number ratio

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As the zone of separation is being elongated, the growth of the second vortex results in a splitting of the main vortex into two vortices ( see T = 0.004 to 0.0055). Because the flow is being decelerated by the negative value of K or Ap, the intensity of the third vortex is insignificant.

The rapid flow deceleration also results in a U-turn flow pattern in the upstream uniform region. This is directly caused by adverse pressure gradients while the pressure gradients in the zone of separation remain to be dominated by the intensity of the main vortex. As the adverse pressure drop vanishes, the main flow is being accelerated when Ap becomes positive again. At T = 0.0065, the small vortex in the upstream is being pushed away from the wall by this mild flow acceleration. It gradually decays during the diastolic phase.

3. Pulsatile Flows in Curved Arteries

Blood flows in curved arteries are unsteady spiral flows. They can be handled in the same approach mentioned above. In addition to the radius (R) of curvature, the flow is controlled by the K6rman number K and a dimensionless radial wall velocity, U = WU, /(U'Ap) . These two dimensionless parameters are cast in the Navier-Stokes equations for computational flow analyses (Hung et a1 1981, 1986, 1987 & 1990). The radial axis in the toroidal coordinate system (r , p, 8) is transformed to Y * by Eq. (1) with the radius a(t) for simplicity. As shown in Fig. 5 , the calculated velocity profile (expressed as p&/ p ) across the diametric plane of a rigid curved tube is in good agreement with the experimental data reported by Alder (1934) for the dimensionless radius of curvature R / a = 20. For the fully developed steady flow, the peak velocity is near the outer bend (the right side). However,

Figure 5 . Comparison of calculated velocity profile (line) on the symmetric plane with experimental results (solid points) of Adler (1934); outer wall on the right end. for blood flow during a systolic acceleration, the interaction of the boundary layer with an irrotational flow in the core region produces higher velocity near

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the inner bend. Figure 6 shows a comparison of the velocity profiles between the rigid and distensible curved arteries with R / q = 3.5. At time

T = /(pa,') = 0.01 , the radius is increased by 8% and the flow reaches its peak. For the same Karman number, the distensible wall results in a significant increase in the longitudinal velocity p h / p (W,,, = the maximum

p h / p ), and secondary velocity pDJu' + v 2 / p (see Fig. 7).

Figure 6. Comparison of pulsating velocity profiles between rigid and distensible curved arteries with R/aO =3.5.

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S...-67 .6

T<»0.000 (Rigid)

S,,,.«169.2

/'- ? y /"

(b) T-0.000 (Distensible)

S..,- 166.3

T~0.a.M (nj.gj.dl (d) T-0.014 (Distensible)

Figure 7. Comparison of secondary velocities between rigid and distensible curved arteries with

R/ao = 3.5.

The entrance effect on the velocity field is demonstrated in Fig. 8. The crosssections indicated by K=2, 3, 4, 5 and 6 are located respectively at a distance of

0.28a0, 0.84tfo, 0.28a0, 0.84a0 1.40fl& 1.96<Zo and 2.52a0 from the inlet.

] 000

Figure 8. Pulsating velocity profiles across the symmetric plane of distensible Curved artery with

R/a<i=3.5 at 7=0.004 (left) at T=0.008 (right), from sections K= 2 , 3, 4, 5 and 6.

The bell-mouth inlet is modeled by a uniform velocity profile without thesecondary flow. The magnitude of the inlet uniform velocity is calculated from

456

the flow rate at the downstream section (K=2) and wall movement. A higher velocity in the inner bend indicates a strong effect of the local acceleration with the curvature effect, indicating a strong interaction between the transient boundary and the irrotational flow in the core region. Due to the radial velocity of the vessel, the velocities ( p m / ,U ) across the symmetric plane reduce from sections K = 2 to 6. Figure 9 shows the development of the secondary flow at a distance of 0.56ao,, 1 . 6 8 ~ and 2 . 8 ~ from the inlet. The intensity of the

T=C. 008

S,,=23.22

T==0.008

s--40 ~ 37

Figure 9. Secondary velocities at a distance of 0.56a0, I .68a0 and 2.8m from the bell mouth inlet; s,= maximumpD(UL +vL)". ' / p .

secondary flows is indicated by S,,, = the maximum pDdu2 + v 2 l p . It

457

increases in the downstream direction (at 1 . 6 8 ~ 0 from the inlet). However, due to the continuous decreases in longitudinal momentum, the secondary flow at a distance of 2 .96~0 from the bell mouth is weaker than that at 1 .68~0. Clearly, the complexity of the three dimensional spiral flows in rigid and distensible vessels can be effectively analyzed by computational analyses.

4. Pulsatile Flows with Balloon Pumping

Intra-aortic balloon pumping has been clinically used as a cardiac assist to many serious patients. Basically, the systolic blood flow is assisted by collapsing an intra aortic balloon, and the flow to coronary arteries can be augmented by inflating the balloon during diastole. Although the flow region around the balloon changes drastically by the balloon motion, the time dependent flow domain can be transformed to a fixed region by the following transformation:

where u(8,z. t ) is the instantaneous radius at a section, and b(t?,z, t ) represents the axis, the radius of the balloon and the catheter radius (refer to bl to b5 in Fig. 9 ). For an axisymmetric flow, both a(z, t ) and b(z, t ) are no longer functions of 0 , and the dimensionless Navier-Stokes equations can be expressed as (Hung 1981):

Figure 10. Definition sketch for blood flow with assisted pumping of balloon: (a) In cylindrical coordinates. (b) In transformed coordinates.

458

au aB a(A-B) au F _- Cl[-+R- I-++- aT aT dT aR F

1 au2 auw auw + U 2 +K[Cl- +--Cl C2 - aR az aR R(A-B)+B

l2 u ap CI a2u a2u 1 = -c, - + C1C2[- + C,C2]- +- +[

aR c2 aR2 az2 R(A-B)+B

aB d ( A - B ) aW F aT aT aT aR F

C,[-+ R ~ 1-+w- aw

+K[C, -+-- c, c2 -

=c, c, ---+C,C2[-+Cc,C2],+-

1 aw a2w + c , c , c , +C,]--2C,C2 -

+ ‘1 [ R ( A - B ) + B aR aRaz

--

1

ap ap Cl a2w a2w +

aR az c2 aR az2

auw aw2 aw2 + uw

(7) aR az dR R ( A - B ) + B

The continuity equation becomes

= O au aw aw U c,--C,C,-+-+ aR art az R ( A - B ) + B

where A = a(’, t)/D, B = b(z ,t) /D, P = p/Ap(t) = p/Apm F(T), F = the time derivative of F(T). The coefficients Cl, C2, C3, C4 are

1 A - B

c, = -

dB d(A-B) C --+R * -82 dZ

a( A - B ) dZ

c, =

( 9 4

459

8 ( A - B ) d 2 B R 8 2 ( A - B ) c, =2c,c, ( 9 4 dZ d Z 2 d Z 2

In addition to these parameters for the balloon motion, the flow is controlled by the Karman number. With these geometric, kinematic and dynamic parameters in the Navier-Stokes equations, the geometry and boundary values for the transformed space become simpler for computational modeling. They were referred as “forcing functions” in the Navier-Stokes equations (Hung 1981). Equation (5) transforms the time-dependent flow domain and computation meshes onto a fixed domain with the same rectangular meshes. This transformation was also effectively employed to calculate cross flow with an oscillating cylinder due to earthquakes in ocean engineering (Chen, Yu and Hung 2005).

Figure 1 1 shows the axisymmetric velocity profiles associated with the pumping of a two-chamber balloon. The flow rate Q is 5 liters/min, corresponding to the Reynolds number R = 4& I nDp = 928. To present the results, the radial dimension of the figure is amplified by a factor of 3. From (a) to (d), the balloon wall inflation and deflation are indicated by the arrows. As shown in (a), the upstream balloon is being deflated while the downstream one is being inflated with a higher radial velocity. The velocity profiles, the radial velocity distribution and shear stresses are altered by the balloon motion. Figure 1 Ob depicts the flow with the upstream balloon being inflated after reaching its hlly collapsed position while the downstream one being deflated. The balloon pumping was employed to enhance blood oxygenation of an intravenous membrane oxygenator (Hung et al 1995).

Three-dimensional blood flow processes associated with intra-aortic balloon counter-pulsation was analyzed by Hung, Natan and Borovetz (1 990). Figure 12 shows a sequential deflation of the balloon during systole. It is based on a constant rate of volume reduction. Figure 13 demonstrates the calculated results of pulsating longitudinal velocity profiles between the aorta and the balloon on the lower half of the vertical diametric plane (indicated by 270’ plane, in the south direction) and that on 300’ plane. When the speed of balloon deflation is much higher, the flow in the core region can be irrotational during early systole.

460

Figure 1 1. Axisyrnrnetric longitudinal velocity profiles associated with intra-vena cava balloon pumping.

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Figure 12. Cross-section of intra-aortic balloon deflation and inflation.

Figure 13. Temporal variation of longitudinal velocity profiles from artery wall to intra-aortic balloon across 270' (upper) and 300' (lower).

5. Hydrodynamic Pressure and Dam Motion

For a dam-reservoir model shown in Fig. 14a, the flow field with surface wave can be mapped to a fmed rectangular domain by transforming the coordinates (x, y, z ) to (x*, y*, z * ) and then to ( X X Z):

462

The function b l (y , z, t ) represents the location in the x-direction of the dam face, b,( y , z ) is that of the upstream end section, and do represents the reservoir water depth. The introduction of bib, z, t ) is to map the vibrating dam face to a vertical plane. Similarly, the functions b3(x, y) and b,(x, y ) indicate the surfaces of reservoir banks. They are used to transform the banks to two vertical planes. Although the water surface, h(x, z, t), is part of the solution to be determined, it is used to map the surface wave to a horizontal plane indicated by y * = 0. The coefficients p and k are used to transform smaller meshes d y near the free surface to regular ones d Y (Hung and Wang 1987; Wang and Hung 1990). Figure 14b shows the hydrodynamic pressure distribution on a dam subjected to a ground acceleration in a diagonal direction (w = 45'9). Figure 14c demonstrates a three-dimensional plot of the free surface produced by diagonal ground acceleration (Wang & Hung 1990).

Figure 15a shows the input data of the horizontal and vertical acceleration components (ug and v g ) of an El Centro Ground motion on October 15, 1979 and the calculated hydrodynamic force coefficient for Pine Flat Dam on Kings River near Fresno, California. Due to the vibration of the concrete dam, the force coefficient CF is much higher than that of a rigid dam (see the dashed line in Fig. 15b). Further understanding of the dynamic interaction between Pine Flat Dam and water in the reservoir can be gained by correlating the dynamic pressure (p*) distribution along with the horizontal acceleration component (u,+ b,) on the dam face of Pine Flat Dam. At t = 5.88 sec, the horizontal ground acceleration uo is much smaller than that of the dam face (see Fig. 15c), and the vertical acceleration V , is downward (equal to -0.168g).

463

Y

Figure 14.

(bottom).

(a) Definition sketch (top). (b) Dynamic pressure on dam (middle). (c) Surface wave

464

1.8

? .2

0 .6

0

-0.6

- 1 . 2

- 1 .a 2.2 2.4 2.6 2.8 3 .O

T i m ( r e c )

-2 .0 ! I 2 .2 2.4 2.6 ' 2:s 3 . 0

Trmc (sec)

1 . 2 , I

Figure 15. Pine Flat Dam; Dashed line: Rigid Dam).

Hydrodynamic pressure and horizontal acceleration component on dam face (solid line:

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6. Flow Past a Plate with Flapping and Pitching Motion

When a coordinate system is set on the center of a moving plate with flapping and pitching motion (refer to Fig. 16a), the Navier-Stokes equations for unsteady two-dimensional incompressible flow can be written in the following dimensionless form (Hung 198 1):

dU 2R, 1 Y " X 2 dU2 auv F, dT RI Rl Rl dT dX dY

(1 + U)- + - - - - - Q 2 X - -- + R, - + C,R, -

dP d2U du 2 d 2 ~ dx ax2 aY aY2

=-Rl -+++C2-+C, -

4 dV 24 1 X d s z dV2 I;; dT R, R, R, dT dX aY

(1 +v> - +- -- LM-- R2 Y *-- - + Rl E v + C l R, -

R12 dP d2V dv d2v =$,-- +-+C2-+C1 -

R, dY dX2 dY dY2 (13)

where the tangential (x) and normal (y) velocity components (U and V) are relative to the moving and rotating plate. They are normalized by the corresponding oncoming instantaneous velocity components u, ( t ) and vo ( t ) ; thatis U =ulu, ( t ) and I/ =vlv , ( t ) .

In this case, the dimensionless velocity in the far field becomes unity. The effects of the transient oncoming flow (in the moving and rotating coordinates) and the angular velocity of the plate are cast in the Navier-Stokes equations by

P P where u) is the rotational velocity of the dimensionless time and pressure are

pL2 u) and R=- (14)

P plate, and L the plate length. The defined by T = p l / p P and

466

Y = p I pUo- . The moving coordinates (x and y) are first normalized by L ( X = x / L and Y* = y 1 L ) and then stretched by a hyperbolic function:

Y = A tanh(BY*) (15) in which A and B are the constants to be selected for finer meshes near the plate. The coefficients C, and C2 in Eqs. (12) and (1 3) are

C, = ABsech2(BY*) and C, = -2BC, tanh(BY*) (16)

In additional to the dynamic parameters RI, R2, and a, the parameters in the dimensionless Navier-Stokes equations include also

aF2 vo @I - v,, F2 (TI F2 =- - V,ll Vll, aT

and F2 =-

In fact, all these time-dependent dimensionless parameters (Rl, R2, Q , F,, F2, F , and F2 ) describe the motion of the plate; they are cast in the Navier- Stokes equations, leaving the dimensionless velocity components U and V to be unity at infinity and zero on the plate. The continuity equation becomes

au R, av -+c --=o ax ' Rl aY

The results shown in this section are associated with a constant flow acceleration from rest with a constant angle of attack a = 15". The prescribed tangential dimensionless velocity R, is linearly increased from zero and reaches 29,000 at T= 2.48x10-' (see Fig. 16). During this period, the moving plate has no angular velocity (i.e.,Q = 0) and the angle of attack is equal to the orientation angle p(= 15") of the plate. Without a pitching motion, the normal dimensionless velocityR2 =R, tan15". The rapid changes in the drag and lift coefficients shown in Fig. 16 can be related to the ratio of the dimensionless local acceleration in the tangential direction:

467

. R1 FI 0 k , = -sec15

R,' +R,' E;

In this flow acceleration period, the fixed angle of orientation ( =15' ) is equal to the angle of attack ( a = 15" ), and the dimensionless local acceleration in the normal direction is proportional to kl. For T larger than 1.48~10~", RI continues to be linearly increased, the plate begins to undergo a sinusoidal flapping motion with R , in an oscillatory pattern (see Fig. 16b). The angle of attack becomes

a(t) = Po + 5" sin5.55(105 T - 2.48) for T > 2.48~10-~ (20)

in which ,YO = 13" . The instantaneous Reynolds number of the flow is

The results of the drag and lift coefficients are also shown in Fig. 16c along with the temporal variations of horizontal and vertical acceleration parameters K I and K2 1

where kl = RI 6 f F, ( R12 + R; ) and k2 = R2 & f F2 ( R; + R; ). The vertical acceleration of the plate indicated by K2 alters the angle of attack and has a stronger effect on the lift and drag coefficients. The presence of a negative K2 which is associated with the upward motion of the plate results in negative values of the l i f t and drag coefficients.

Parallel to the calculation of accelerated flow with the flapping motion, the flow for RI = 29,000 at T = 2.48~10" was used again to simulate flow for the second case. It follows the same rate of increase in RI while the angle of attack indicated by Eq. (20) is solely due to a pitching motion (the last term in the equation). In this case,

a([) = P(t) = 15" + 5" sin 5.55(1O5 T - 2.48) (23 1

468

The third case is characterized by the same temporal variation of RI and a(t) while the angle of attack indicated by Eq. (20) is caused by a combined pitching and flapping motion. That is, the orientation of the plate is

P(t) = 15" + 2.5" sin 5.55(105 T - 2.48) (24)

24 26 28 30 32 34 36 38 40 42 T-106u 1 4 '

r= I 0 6" 1 , L'

Figure 16. Definition sketch of flow past a plate with the coordinate system moving with the plate (top); Time variation of the Reynolds number R and the angle of attack a (middle), Comparison of lift and drag coefficients for flapping motion, pitching motion, and flapping-pitching motion

(bottom).

469

where the second term on the right represents the rotational velocity of the plate. The flapping motion also alters the angle of attack by the same amount of 2.5" sin5.55(10'T - 2.48). Figure 16c shows the comparison of the lift and drag coefficients among Case 1 (flapping), Case 2 (pitching) and Case 3 (flapping and pitching mtion). The oscillations of dynamic forces appear to be stronger for the pitching motion. Figure 17 demonstrates the calculated flow patterns in the fixed frame for Cases 1 and 2 at times when the angle of attack reaches 20' and 10.5'.

0.2

Q.

a i 200 R -32,460

0 = 2 0 0 R -32,460

LI = I0 9 8

R -33.560

Figure 17. Comparison of streamlines between flapping and pitching motions.

7. Hydrodynamic Pressure and Sediment Dynamics

In a previous study of hydrodynamic interaction between the impounded water and the reservoir sediment dynamics due to ground motion (Chen and Hung 1993), the pressure wave equation of the impounded water and that of the pore water are expressed respectively as

470

= O

The pressure wave speed of the impounded water is [ / p]“‘ while that of the pore water is 1 pip] “‘[((I - n ) p + np,,) / np,7 1”’ . For the present study, the momentum and constitutive equations for sediment solids are used to derive the wave equations for the normal elastic stresses 0, and guy :

in which [ ( A + 27) /((I - n)p,, )I”’ represents the wave speed. The shear wave equation can be expressed as:

The last term in this equation indicates the effect of the Stokes drag on shear wave propagation. The dynamic equations for the Stokes drag can be derived as

471

The wave speed is [q /((I - n)p,v]0.5 . For a rigid vertical dam face with a horizontal bottom and sediment surface subject to vertical harmonic motion, one can consider that the shear stress and the normal stress in the x-direction vanish.

Also, Eqs. (25) , (26) and (28) become one-dimensional. Figure 18 shows the calculated results of the hydrodynamic pressure force F on the rigid

I - ds = 50 ft --- ds=100A --- ds=150n

40 -

d s = O n . .. . . .. Cheng's (ds = 0)

Q \

i;

10 -

0 1 2 3 4 5 wh/C

Figure 18. Effects of sediment thickness on the amplitude of dynamic force coefficient CF due to vertical harmonic ground excitation with an intensity of a g=O. Ig.

dam face expressed as a dimensionless force coefficient CF- = Fl(O.5 a g pH ' ) . The results are based on a simple harmonic motion in the vertical direction with an intensity of a g=O.lg. The total depth (H) of water and pore water is equal

472

to 200 ft. The other data include the specific gravity of sediment solid =2.5, the porosity n =0.3, the permeability k=O.OO1jtJ secllb , and the elastic moduli p = 60,000 lb / Jt“ and A = 40,000 lb / Jt“ . The salient results indicate that the hydrodynamic force can be seriously altered by the sediment thickness and the frequency parameter WH f C (the wave speed of water C = 4250 Wsec). The fimdamental natural frequency can be seriously altered by the sediment. The frequency effect appears to spread wider near the natural frequency with the sediment depth.

Acknowledgements

The manuscript summarizes the main ideas behind our previous studies. My efforts and experiences shared with Drs. Tommy Mou-Chang Tsai, Weng-Sheng Cheng, Sheng-Mou Huang, Mou-Hsing Wang, Bang-Fuh Chen, Kuo-Chyang Chang, Harvey S. Borovetz, Thomas E. Natan, Chiuping Chang and Hwaqiang Li remain refreshing and memorable.

Also, I would like to thank Dr. George Bugliarello for recruiting me to Carnegie- Mellon University in 1967 from lowa Institute of Hydraulic Research, and to Dr. Maurice S. Albin for my activities at the University of Pittsburgh from 1975. Their visions and enthusiasm are gracefully accompanied by inspiration and warmth. That same human dimension is evident in Professor Theodore Y.-T. Wu, and his interaction with students and friends. I would like to take this opportunity to salute Professors Bugliarello, Albin and Wu for their inspiration and friendship to students, colleagues and admirers. The support of NSF Grants (GK-31514 and CME76-83420) and NIH Grants (HL-12714, HL-26238, and HL-33934) are acknowledged.

References

1. Chen, B.F. and T.K. Hung, “Dynamic Pressure of Water and Sediment on Rigid Dam,” J. of Engineering Mechanics, ASCE, Vol. 1 19, No. 7, 14 1 1 - 1433, 1990.

2. Chen, B.F., Y.H. Yu and T.K. Hung, “Large-Reynolds-Number flow across a translating circular cylinder with high oscillating frequency,” The Theodore Y. T. Wu Symposium on Engineering Mechanics, 2005.

3 . Fenves, G., and A.K. Chopra. “Effects of reservoir bottom absorption earthquake response of concrete gravity dams,” Earthquake Engineering Structure Dynamics,

4. Hung, T.K. and W.S. Cheng, A Computational of Three-Dimensional Flows in Deformable Curved Tubes, Proceedings of the ASME/ASCE Biomechanics Symposium, 198 I .

5. Hung, T.K. “Forcing Functions in the Navier-Stokes Equations,” Journal of Engineering Mechanics Division, Vol. 107, 643-648, 198 1.

1 1, 809-829, 1983.

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6. Hung, T.K., W.S. Cheng, S.M. Huang, M.C. Tsai and M.H. Wang, “Unsteady Flows with Movable Boundaries,” Proceedings of 20th International Association for Hydraulic Research, Moscow, USSR, September 1983.

7. Hung, T.K. and S.M. Huang, “A Computation of Unsteady Flows Past Airfoil with an Upward and Down Motion,” Proceedings of the 3rd ASCE EMD Special@ Conference, 1146-1 149, 1983.

8. Hung, T.K. and M.H. Wang, “Nonlinear Analysis of Hydrodynamic Pressure on Rigid Dam Motion,” J. of Engineering Mechanics, ASCE, Vol. 113, NO. 4, pp.

9. Hung, T.K., T. Natan, Jr. and H.S. Borovetz, “Fluid Mechanical Analysis of Blood Flows with an Intra-Aortic Balloon Pumping,’’ Proceedings of World Congress on Medical Physics and Biomedical Engineering, 1988.

10. Hung, T.K., Wen-Sheng Cheng, Ding-Fang Li, J.H. Guan and B. F. Chen, “Accelerated Flows in Rigid and Distensible Curved Tubes,” Engineering Science, Fluid Dynamic, A Symposium to honor T. Y. Wu, California Institute of Technology, Aug. 17- 18, 1989, World Scientific, pp. 147-160.

11. Hung, T.K. and B.F. Chen, “Nonlinear Hydrodynamic Pressure on Dams During Earthquake,” J. of Engineering Mechanics, ASCE, Vol. 116, No. 6, 1290-1304, 1990.

12. Hung, T.K. and K.C. Chang, ”Effects of Sediments on Reservoir Hydrodynamic Response,” Proceedings of International Conference on Mathematical M3delling,

13. Hung, T.K., H.Q. Li, H.S. Borovetz, F.R. Walters, P.J. Sawzik and B.G. Hattler, “Intravenous Membrane Oxygenators with Balloon Pumping,” Proceedings of the Fourth China-Japan-USA-Singapore Conference on Biomechanics, 1995, pp. 381-384.

14. Hung, T.K. and Tommy M.-C. Tsai, “Pulsatile Blood Flows in Stenotic Artery,” J. of Engineering Mechanics, ASCE , Vol. 122, No. 9, 890-896, 1996.

15. Hung, T.K. and Tommy M.-C. Tsai, “Kinematic and Dynamic Characteristics of Pulsatile Flows in Stenotic Vessels,” J. of Engineering Mechanics, ASCE, Vol. 123, No. 3,247-259, 1997.

16. Li, D.F. and T.K. Hung, “Pulsatile Flows in Curved Arteries,” Proceedings of the International Conference on Fluid Mechanics, Beijing, China, 34-39, 1987.

17. Wang, M.H. and T.K. Hung, “Three Dimensional Analysis of Pressures on Dams, J. of Engineering Mechanics, ASCE, Vol. 116, No. 6, 1372-1391, 1990

482-499, 1987.

86-100, 1995.

INTERDISCIPLINARY EDUCATION AND RESEARCH EXPERIENCES FOR UNDERGRADUATES IN

MATHEMATICS AND BIOLOGY *

GEORGE T. YATES Department of Mathematics and Statistics

Youngstown State University One University Plaza

Youngstown, OH 44555, USA E-mail: gyatesO ysu. edu

The biological sciences continue to require more analytical methods to advance our understanding of complex biological systems. Collaborations between biologist and mathematicians have and will play an increasing role in future scientific advancements. However, students are not usually exposed to interdisciplinary a p proaches during their undergraduate education. An interdisciplinary project has been initiated at Youngstown State University (YSU) to bring mathematics and biology students into the same classroom and laboratory settings. The goal of this program is to involve undergraduate students from each discipline in active research programs and to develop courses in mathematical biology. Two interdisciplinary courses are described and a summer research program for undergraduates is reviewed. An example project is presented from the summer research program that resulted in a new model for the stationary phase of the microbial growth curve. Through these and other programs, we hope to capture the imaginations of the participating students and bring well-trained and motivated students into the field of Biomathematics.

1. Introduction

The report ‘BIO 2010’ by the National Research Council’ identifies an urgent national need to reform undergraduate education and strengthen the collaborations between different disciplines. Among other suggestions this report concludes, ‘Connections between biology and the other scientific disciplines need to be developed and reinforced so that interdisciplinary thinking and work become second nature,’ and this should occur at an early

‘This work was partially supported by supplementary grant DUE 0337558 of the Na- tional Science Foundation.

474

475

stage in a student’s education. In addition, the majority of faculty members pursuing biological research at leading research or doctoral universities in the United States received their undergraduate degrees in fields other than biology. These and other factors indicate that the traditional undergraduate biology curriculum does not prepare students to undertake modern biological research. Quantitative skills and interdisciplinary training needs to be incorporated into courses in undergraduate education.

The Department of Mathematics & Statistics and the Department of Biological Sciences at Youngstown State University (YSU) are developing a series of biomathematics courses and engaging undergraduate students in interdisciplinary research projects. The first course was taught jointly by Professors Johnston (biology) and Yates (mathematics) and was a project oriented course focusing on biological research topics in field ecology. A second course in mathematical modeling is being developed which emphasizes basic mathematical techniques with applications in biology, and includes statistical and computer applications.

Ample evidence points to effectiveness of research as an educational tool and to the heightened preparedness of students that engage in undergraduate research. We have initiated a summer program where interdisciplinary teams of students were selected and paid for intensive research during the summer of 2004 and 2005. The research projects were jointly advised by Bi- ology and Mathematics faculty and were coordinated with ongoing research projects of participating faculty.

2. Course Development

Interdisciplinary courses are intended to expose mathematics students to biological applications of mathematics and to expose biology students to the power of mathematics in solving biological problems. By including students from both disciplines in the same class, we hope to teach students of different disciplines to communicate. Other goals are to strengthen the quantitative skills of the biology students, to create a mathematical and science community of students, and to develop communication skills by requiring oral presentations and written reports.

To attract students to new courses, the course should count toward the student’s degree requirements. Consideration should be given to the requirements for a major, minor, general education, laboratory course work, honors courses, or other University requirements. If possible, courses should be cross listed in multiple departments so that mathematics students can

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receive credit for a biology course and biology students can receive credit for a math course. Whenever possible the courses should be team taught by a mathematician and a biologist.

Two interdisciplinary courses have been developed at YSU; one in biological/ecological sciences and a second in mathematical modeling. For the first course in field ecology, small student teams composed of a combination of math and biology students collected data and analyzing research problems. The students undertook a week of intensive data collection and analysis at the Gerace Research Station on San Salvador Island, the Bahamas. Teams and projects were selected to intermingle biology and mathematics majors, with the goals of fostering effective interdisciplinary communication and developing complementary skills to achieve their collaborative research objectives.

The second course on mathematical biology is an introduction to quantitative mathematical modeling in biology. Various topics or modules were selected from geometric similarity, population dynamics, ecological modeling, proteomics, bioinformatics, statistics and others. Laboratory sessions are used to teach typical data collection and scientific computing skills utilized in biological modeling.

Throughout the courses, emphasis is on collaborative learning which can be enhanced by making students work in small groups. It is important that the math majors teach the biology students and the biology students teach the math majors. The students should dependent on one another for a truly inter-disciplinary experience. A combination of oral presentations and report writing should be used to foster the collaborative efforts.

3. Research Experiences

The Summer Undergraduate Research Experience (SURE) at YSU was initiated during the 2004 summer. Eight students were engaging in intensive summer research projects at the boundaries between mathematics and biology. Student projects lasted 8 weeks and were jointly advised by faculty members from the Department of Mathematics & Statistics and the De- partment of Biological Sciences. These summer students undertook the following research projects:

1. Mahoning River Project. The goal was to characterize, survey and model the biological, chemical and physical integrity of the Mahon- ing River. The impact of river restoration on the river’s biological, chemical and physical integrity was assessed and modeled. Spe-

477

cial attention was given to the distribution of heavy metals in the riverbank sediment.

2. Zoar Valley Project. The major objective of this field research was to reconstruct and model forest succession in response to multiple interacting disturbance regimes. Zoar Valley provides a unique opportunity to study old-growth forest within the riparian (river and surroundings) zone of a natural and unregulated river.

3. Habitat and Biodiversity. Various diversity indexes were studied to assess their interpretation to environmental issues. Bird, insect and vegetation characteristics were correlated to biodiversity.

4. Proteomics. The role of biochemical and molecular processes associated with morphogenesis in Penicillium marneffei were assessed. The mechanisms that regulate cellular development in this fungus and the interactions between various pathways that determine fun- gal shape and development were also investigated.

The goal of SURE is to capture the imaginations of the participating students and bring well-trained and motivated students into the field of Biomathematics. Students conducted real research, and were required to communicate across the disciplines of biology and mathematics. The interdisciplinary nature of the work and the mixed majors of team members required the students to become conversant in all aspects of the overall project. In this way, the students receive a unique research experience that helps prepare them for a number of potential career choices, including graduate and professional education and industry.

4. Example Project - Growing Curve Analysis

While growing yeast for proteomics experiments, we attempted to model the growth of yeast cells which were grown in a flask mixed with an initial concentration of nutrient solution and left to grow over a period of days. Figure 1 shows a sketch of the four typical stages in the growth cycle1y2 of such batch cultures: (1) a lag phase where little or no net growth is observed, (2) an exponential growth phase, (3) a stationary phase where the population remains nearly constant, and (4) an exponential death or decay phase. The exponential growth and decay phases can be modeled by simple exponential equations, however, the transition from exponential phase to stationary phase and then to the death phase has not been modeled. We developed a new mathematical model to bridge these transitions.

The population model used to model the transition from the growth

478

stationary

t (lime)

Figure 1. tures. Note that the vertical axis is the logarithm of the number of cells.

Sketch of the typical growth curve for microbial cultures during batch cul-

phase to the death phase assumes that the rate of change of the population is proportional to the population,

-- dP(t) dt - k( t )P(t) ,

where P(t ) is the population or number of viable cells in the culture and k ( t ) is the growth rate which is taken as a function of time t . Since new cells are being born at the same time that others are dying, k(t) represents the net growth rate. During the lag phase, k(t) = 0 and the population remains constant. The time spent in the lag phase depends on the time required for the growth enzymes and metabolic pathways for growth to be initiated, and lag phase is not modeled here. During the exponential growth and death phases, k(t) = ko or -kl respectively, where ko and Icl

are positive constants. This gives the classical exponential growth or decay, ~ ( t ) = Piekot or P(t ) = Ple-klt.

We now model the stationary growth phase, and include the end of the exponential growth phase and the beginning of the exponential death phase. During these phases, the concentration of nutrient F(t ) is assumed to limit the exponential growth and eventually, when the nutrients are sufficiently depleted, lead to an exponential decay of the population. During the period

479

just before and after the stationary phase, the growth rate is assumed to be linearly proportional to the nutrient concentration, being positive when F ( t ) is greater than a constant critical level F, and negative when F ( t ) is less than F,,

To complete the model, we assume that the rate of change of the nutrients is proportional to the population,

-- dF( t ) - -aYP(t) . d t (3)

Although the proportionality coefficients Q and p may vary with organism, temperature, nutrient material, and other environmental variables, they are assumed constant and known in this study. The critical nutrient concentration F, is also assumed known for the specific organism, nutrient and growing conditions discussed here. Equations (l), (2) and (3), with appre priate initial conditions, can then be explicitly solved for P ( t ) , k ( t ) and

Combining equations (1) and (2), gives dP(t) /dt = /3 [F( t ) - F,] P( t ) , which can be combined with equation (3) and its derivative to give the second order ordinary differential equation for F ( t ) ,

F ( t ) .

This can be integrated once to give,

-- dF( t ) - E F 2 ( t ) - PF,F(t) + C dt 2 ( 5 )

where C is an arbitrary constant of integration. This first order differential equation is the logistic equation for F ( t ) , and C must be retained to satisfy initial conditions. Although the logistic function is a solution of equation ( 5 ) , we present an equivalent solution using hyperbolic trigonometric functions which attaches more physical meaning to the integration constants. It can be shown that equation ( 5 ) has the solution,

- 2 K F ( t ) = /3 tanh r;(t - t o ) + F, P

where K and t o are arbitrary constants. For convenience we use the constants K and t o , where to locates the symmetry axis and K gives the spread

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or sharpness in the depletion of the nutrient concentration. The constant C is given in terms of K and other specified parameters as

P2F: - 4~~ 2P

C =

Substituting (6) into (2) and (3), we obtain the following explicit expressions for k ( t ) and P(t ) ,

k ( t ) = - 2 ~ tanh K ( t - t o ) (7) 262

f fP P(t ) = - sech2K(t - t o )

which completes the general solution for the population, nutrient concentration and growth rate.

The constants K and t o can be determined once initial data are specified. If the initial population is PO and the initial nutrient concentration is Fo, we impose the initial conditions

P(0) = Po F(0 ) = Fo .

After some algebraic manipulation K and t o can be determined as,

Since the nutrient concentration cannot be negative, we require that F ( t ) 2 0, which gives the condition,

26 F, 2 - tanhrc(t - t o ) P

for all time. This restricts the validity of our solution for time

t 5 t o + - tanh-' K

The right hand side of (13) takes on a maximum value of 2 ~ l P as t approaches positive infinity, and if we take,

F, 2 2 4 P (15)

condition (13) is always true and there is no restriction on t for our solution. Using equation (11) and the square of (15), we find that our solution

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0: -1:

-2:

remains valid for all time provided that

I ' '< 4 3 ' . . 6 'I' B stimc)

----___ ------__kci, --. .-. *.., 3

Figure 3. time t . For PO = 1, FO = 6, F. = 3.3, (Y = 0.5 and 0 = 0.5.

Nutrient concentration F ( t ) and growth rate k ( t ) plotted as a function of

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Figure 2 shows the number of viable cells (or the population) given by (8) as a function of time for specified values of the parameters (PO = 1, FO = 6, F, = 3.3, Q = 0.5 and p = 0.5). The results seem to capture the exponential growth phase and the exponential death phase. The stationary phase is somewhat short in this model. Figure 3 shows the nutrient concentration and growth rate corresponding to Figure 2. Both F ( t ) and k ( t ) are monotonically decreasing functions of time and are given by (6) and (7) respectively.

5. Conclusions

Interdisciplinary courses and research projects have been initiated at Youngstown State University involving students and faculty in the De- partments of Mathematics & Statistics and the Department of Biological Sciences. These efforts involve joint teams of mathematics and biology students working together on course work and on research projects. The goals are to incorporate more analytical components into the Biology curricula, to give more relevant applications for mathematics students and to promote collaborations between the two disciplines. These types of interdisciplinary experiences should be encouraged early in the education process. Further- more, undergraduate research is being integrated into education to better prepare students for advanced degree studies and careers at the frontiers of biology, mathematics and technology.

As an example of an interdisciplinary research project that is accessible to undergraduates, we considered the microbial growth curve. Simple assumptions were made for the growth rate and the concentration of nutrients which led to a system of differential equation which has a closed form solution. Equations ( 6 ) , (7) and (8) represent the results of our new model for the microbial growth curve. The solution is intended to model the transition from the exponential growth phase to the stationary phase and from the stationary phase to the exponential decay phase of the growth curve. Although not developed to model the exponential growth and decay phases of the growth curve, the model exhibits all the features of the growth curve with the exception of the lag phase and it predicts a somewhat abbreviated stationary phase.

Acknowledgments

This paper is dedicated to Professor Theodore Yao-tsu Wu whose pioneering efforts in mathematical biology inspired this work. Thanks are given to Carl

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Johnston, Chet Cooper, Tom Smotzer, Angela Spalsbury, Nate Ftitchey and Doug Faires. The field ecology (Bahamas) and SURE 2004 students are gratefully acknowledged. This work was made possible by supplemental funding from the NSF grant DUE 0337558.

References 1. BIO 2010. Transforming Undergraduate Education for Future Research Bi-

ologists. National Research Council of the National Academies. National Academies Press, 2003.

2. Prescott, L.M., J. P. Harley & D. A. Klein. Microbiology, fifth edition, Mc- Graw Hill, 2002.

3. Marr, A. G. Growth Kinetics, Bacterial. In Encylopedia of Microbiology, 2nd ed., vol. 2, J. Lederberg, editor-in-chief, 394-403. San Diego, Academic Press, 2000.

IN VITRO STUDY ON THE INTERNAL DESIGN OF PROVOX 2TM VOICE PROSTHESIS

HORACE H . LAM U Department of Mechanical Engineering, The Universit), of Hong Kong,

Hong Kong, CHINA

Natural-voice quality and convenience have boosted the popularity of prosthetic voice rehabilitation method in the last decade. However, Candida colonization always causes leakage problem and limits valve life. In our study, convergent cores of different outlet radii have been used to substitute the original core of Provox 2TM voice prosthesis. It is found that prosthetic performance has been improved for core outlet radius down to 1.5 mm, at which valve opening angle has increased by 31.27 %, force constant increased by 125.48 % and airflow resistance by 25.72 % only. Exact dimensions for the prosthetic geometry require further in vivo investigation. Guidelines

1. Introduction

Total laryngectomy is a procedure in which larynx of a patient, with malignant tumor, is surgically removed. Patients who have undergone total laryngectomy will have their respiratory tracts separated from their digestive tracts. Breathing is made possible with an opening (tracheostoma) on their necks; however, they will lose their natural voice-producing ability. Fortunately, in the past few decades, professionals have introduced various methods, e.g., electro-larynx, esophageal-speech, etc., to restore voice for larygectomized patients.

Blom and Singer [ 13 pioneered a post-laryngectomy voice rehabilitation technique in whch a tracheoesophageal fistula (TEF) was punctured surgically on the wall between the trachea and the esophagus. A valved prosthesis, designed to maintain the TEF patency and permit uni-directional pulmonary airflow, was then placed at the fistula [2]. After this simple surgery, with the technique learnt from speech therapists, patients could then produce voice easily by obstructing the tracheostoma on their necks and diverting pulmonary air into the buccal cavity.

In the past two decades, the design of voice prosthesis has undergone significant evolution, i.e., from duckbill slit-valve to hmged flap-valve, and from esophageal flange to bi-flange. These modifications were aimed at improving the performance of the voice prosthesis [2-41. Compared with other voice rehabilitation methods such as the “robotic” electro-larynx and the time-limited

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esophageal speech, tracheoesophageal prosthesis (TEP) offers laryngectomized patients a better quality and greater quantity of speech.

Different brands of voice prosthesis with slight variation in design, either indwelling ( e g , ProvoxTM) or non-indwelling (e.g., Blom-SingerTM), are commercially available and much comparison has been carried out among them. In terms of airflow resistance, one might perform better in particular studies [4- 71. However, under continuous mechanical and biochemical influences at the esophageal side, the one-way valve in all voice prosthesis malhctions gradually. Leakage of esophageal contents into the trachea occurs, either around or through the voice prosthesis. Causes of leakage include gradual dilation of the TEF, granulation formation and incomplete valve closure due to Candida colonization on the valve surface [ 8-1 01.

In our study, convergent cores with different outlet radii have been used to substitute the original core of the Provox 2TM voice prosthesis. It is proposed that by increasing a d o w velocity, less time will be used for phonation air transmission and the chance of leakage across the prosthesis can be lowered. Moreover, the valve opening force increases with airflow velocity. A corresponding increase in valve closing force can give better closure of the prosthetic valve. In vitro experiments have been carried out to obtain the relationships among valve opening angle, force constant, airflow resistance and airflow rate at different core outlet radii.

2. In Vitro Experimental Study

Samples of new Provox 2TM (Atos Medical, Sweden) voice prosthesis are donated by Queen Mary Hospital for our in vitro experimental study. All of the prostheses are equal in length (6mm). Provox 2TM voice prosthesis is made up of two components: the external shell which is made of medical-grade silicon rubber; and the internal core which is made of fluoroplastic [l 13, as shown in Fig. 1 and 2. The inlet and outlet radius of the core are 2.85 mm and 2.00 mm respectively. In our experiment, the original core is replaced with custom-made convergent cores, of same inlet radii (2.85 mm) but different outlet radii (1.00, 1.25, 1.50, 1.75 and 2.00 mm), as shown in Fig. 3. The modified prosthesis is then inserted into our in vitro artificial throat for measurements.

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The in vitro artificial throat is made up with the following components: a pump is used to supply air at an airflow rate from 0.0001 to 0.00023 m3/s, w i h which the normal phonation airflow rate of 0.00015 m3/s [3] in laryngectomized patients is included; a flow-meter is used to measure airflow rate flowing into the acrylic replica at steady air supply; a low-pressure differential transducer (Model LP; DJ Instruments, Billerica, MA) is used to measure the pressure variation in the acrylic replica. The signal generated is then monitored with a multi-meter. The acrylic replica is basically a 250 ?-long cylindncal hollow tube, of 25 mm inner diameter and 3 1 mm outer diameter, as shown in Fig. 4. A digital camera (Model 885; Nikon) is used to take high resolution photos on the valve opening mechanism.

Figure 4. Acrylic replica. Figure 5. Illustration of the experimental setup.

The modified prosthesis is inserted into a puncture site at the acrylic replica and secured with silicon-gel. The whole experimental setup is shown in Fig. 5. Regulated airflow is then applied to the replica, pressure variation across the voice prosthesis is measured with the differential pressure transducer, and photos are taken with the high resolution digital camera. The above procedures are repeated three times for an average value for each convergent core.

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3. Theoretical Study

The relationship between prosthesis valve opening mechanism and other physical parameters, such as tracheal pressure, prosthesis elasticity, etc., can be obtained by slmplifjmg the real situation. Consider a control volume in the prosthesis, enclosing the valve, as shown in Fig. 6.

i 2 Figure 6. Illustration of laterial-sectional view of voice prosthesis with convergent core. Region 1 and 2 represents tracheal inlet and esophageal outlet of the voice prosthesis respectively.

We assume airflow across the voice prosthesis is horizontal, steady, one- dimensional, invisid and incompressible. The continuity equation and the momentum equation then become

(44 - % 4 3 ) + R = P Q(us (2)

where Q (m3/s) is the volumetric airflow rate, ui ( d s ) , Ai (m2) and Pi (Pa) are the airflow velocity, cross-sectional area and pressure at region i respectively, p (kg/m3) is the air density at 37OC, R (N) is the force acting on the control volume by the valve.

Assume the cross-sectional area at tracheal inlet equals to that at esophageal outlet, and neglect the airflow velocity change due to core convergence,

A, = A 3 . (3)

Substituting Eq. (3) into Eq. (2), we can express R as,

R = -(s - P3)A1,

R, = k e , ( 5 )

(4)

It is assumed that the valve closing force can be described by Hooke's Law,

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where k (N/m) is the force constant and e (m) is the horizontal extension. During leakage, the horizontal extension is small and can be expressed in terms of 8,

e = r 2 8 , (6)

where 8 (rad) is the valve opening angle, as shown in Fig. 6. The valve closing force Rc is equal in magnitude but opposite in direction as

force R thus the relationship between force constant k, valve opening angle 6 core outlet radius r2, core inlet area A l and trans-prosthesis pressure difference (P&) becomes,

4. Results and Discussions

Figure 7 shows the experimental results on the relationship between valve opening angle 8 and airflow rate Q for different core outlet radius r2. Generally, valve opening angle increases with increasing airflow rate. However, the increasing manner is not uniform but S-shaped. The valve opens approximately 0.1 rad from 0.00010 to 0.00013 m3/s; 0.2 rad from 0.00013 to 0.00020 m3/s and 0.1 rad from 0.00020 to 0.00023 m3/s. The S-shaped tendency is more significant for small core outlet radius.

On the other hand, there seems to be a maximum valve opening angle of about 0.65 f 0.05 rad at high airflow rate. The experimental maximum valve opening anlge, at which upper part of the S curve converges, can be explained by the posterior convex shape of the valve, which inhibits the valve root from further bending. At particular airflow rate, small core outlet radius always results in greater valve opening angle, which means more air can be transmitted within the same period; or less time is required to transmit the same quantity of phonation air.

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0.0 ~

0.09 0.12 0.15 0.18 0.21 0.24 Airflow Rate (10.' rn3/s/s]

Figure 7. Experimental results showing relationship between valve opening angle and airflow rate. Keys on the right indicate the core outlet radius r2.

There is no significant difference between Provox 2TM and MOO prototype, since the dimensions of their valves are exactly the same though there is slight difference between their core geometries. In our prototypes, valve radius is set to be 0.3 mm longer than core outlet radius. So valve sue decreases as the core converges. Pressure decreases gradually from the projected core center to the valve edge. The reduction tends to be gentler as core outlet radius decreases from 2.00 mm to 1.50 mm, and becomes steeper again as core outlet radius further decreases to 1 .OO mm.

The first phenomenon can be explained by the valve position. Valve opening angle of small core outlet radius prototype is small comparing with that of large core outlet radius prototype at same airflow rate. In R200 prototype, valve inclines 5" from vertical only. When pulmonary air strikes on the valve, airflow direction changes abruptly. So there is a steep pressure reduction across the valve surface. As core outlet radius decreases, the prototype has a larger valve opening angle at the same airflow rate. In R150 prototype, valve makes 15" to the vertical. Airflow direction changes gently when striking on the valve. So there is gradual change of pressure distribution across the valve surface.

Under the same principle, pressure reductions across the valve surface should be milder for small core outlet radius prototypes. However, in RlOO prototype, even the valve makes 25" to the vertical, the resolved components of pulmonary airflow striking on the valve still possess high kinetic energy. So it causes pressure to change abruptly across the valve surface again.

Under certain circumstances, such as asymmetric adhesion between valve and valve sit due to mucus accumulation; the valve may not open as perfectly

490

balanced as it is designed. If the valve is suffering from abrupt pressure variation, such unbalanced opening condition may cause irreversible damage to the elasticity of valve root. Leakage will occur soon before Candida colonizes the prosthesis. Moreover, Candida may grow more rapidly in the less mobile region under unbalanced valve opening condition. And it will deteriorate the voice prosthesis more rapidly. In contrast, gentle pressure variation across valve surface is definitely beneficial to prosthesis performance. Under the same unbalanced valve opening condition, pressure distributes relatively evenly on the valve surface, causing less hazardous effect on the valve root. As a result, the service life of voice prosthesis can be extended.

Figure 8 shows the experimental results on the relationship between force constant of the valve k and prosthetic core outlet radius r2. Force constant increases exponentially when core outlet radius is reduced. A greater force constant means a higher rigidity of the valve and consequently a greater pressure is required to push the valve open. It also implies that the modified prosthesis is less susceptible to leakage.

Valve opening force increases with airflow velocity. A corresponding increase in valve closing force is demanded and which can be satisfied by raising the rigidity of the valve. As a result, the valve-hardening (decrease in prosthesis elasticity) effect caused by Candida colonization can be reduced.

35

30

3 25 2 5 20 2 s 15 ti e 10

5

+

0 1 .oo 1.25 1.50 1.75 2.00

Core Outlet Radius [mml Figure 8. Experimental results showing relationship between force constant and core outlet radius.

In patients using TEF to generate speech, the durability and efficiency of the voice prosthesis is of great importance. Patients generally prefer prosthesis of lower airflow resistance [12]. According to the above studies, slightly convergent core is favorable in prosthesis design. Core of too small outlet radius

49 1

Core outlet radius rz (mm)

Percentage increase in valve opening angle (%)

is undesirable due to its abrupt pressure distribution across valve surface and high trans-prosthesis airflow resistance. Table 1 shows comparison among cores of different outlet radii.

2.00 1.75 1.50 1.25 1.00

0 1 1.64 3 1.27 47.64 74.18

Percentage increase in force constant (%)

Percentage increase in airflow resistance (%)

0 41.68 125.48 310.15 602.38

0 6.54 25.72 66.36 97.24

According to our experimental results, converging the core outlet radius to 1.50 mm will increase the valve opening angle, force constant and airflow resistance by 3 1.27, 125.48 and 25.72 % respectively. At this core outlet radius, the prosthesis design is benefited by the convergence. Further converging the core outlet radius to 1.25 mm will increase the valve opening angle, force constant and airflow resistance by 47.64, 310.15 and 66.36 % respectively. At this core outlet radius, the disadvantage of convergence becomes significant.

It is recommended that a convergent core is absolutely beneficial to the design of voice prosthesis. The increase of valve opening angle can allow phonation air to flow across the voice prosthesis within a shorter time, lowering the chance of leakage. Moreover, raising the rigidity of the valve reduces the material deterioration effect caused by Candida. However, there should be a limitation because airflow resistance becomes significant at too small core outlet radius. Based on our study, an optimum core outlet radius occurs between 1.5 to 1.75 mm. For the exact dimensions, further in vivo investigations have to be carried out.

Acknowledgements

The author is grateful to Prof. Allen T. Chwang for his excellent supervision and extraordinary guidance. Special thanks should also be given to Dr. Paul Lam of the Queen Mary Hospital for introducing the problem and supplying the free samples of Provox zTM voice prosthesis.

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References

1. Blom ED, Singer MI and Hamaker RC, 1981, “Further Experience with Voice Restoration after Total Laryngectomy”, Ann Otol Rhino1 Laryngol, 90, pp. 498-502.

2. Blom ED, 2000, “Tracheoesophageal Voice Restoration: Origin - Evolution - State of the Art,” Folia Phoniatrica et Logopaedica, 52, pp. 14-23.

3. Eerenstein SEJ, Grolman W and Schouwenburg PF, 2002, “Downsizing of Voice Prosthesis Diameter in Patients with Laryngectomy,” Archives of Otolaryngology Head Neck Surgery, 128, pp. 838-841.

4. Karschay P, Schon F, Windrich J, Fricke J and Henmann IF, 1986, “Experimens in Surgical Voice Restoration Using Valve Prostheses,” Acta Otolaryngologica, 101,

5 . Verkerke GJ, Geertsema AA and Schutte HK, 2001, “Airflow Resistance of Airflow Regulating Devices Described by Independent Coefficients,” Annals of Otology, Rhinology and Laryngology, 110, pp. 639-645.

6 . Heaton JM and Parker AJ, 1994, “In Vitro Comparison of the Groningen High Resistance, Groningen Low Resistance and Provox Speaking Valves,” Journal of

7. Jebria AE3, Gioux M, Henry C, Devars F and Traissac L, 1989, “New prosthesis with low airflow resistance for voice restoration following total laryngectomy,” Medical and Biological Engineering and Computing, 27, pp. 204-206.

8. Leunisse C, Weissenbruch R, Busscher HJ, Van der Mei HC, Dijk F and Albers FWJ, 2001, “Biofilm Formation and Design Features of Indwelling Silicone Rubber Tracheoesophageal Voice Prostheses - An Electron Microscopical Study,” Journal of Biomedical Materials Research, 58, pp. 556-563.

9. Busscher HJ, Geertsema-Doornbusch GI and Van der Mei HC, 1997, “Adhesion to silicone rubber of yeast and bacteria isolated from voice prostheses: Influence of salivary conditioning films,” Journal of Biomedical Materials Research, 34, pp. 201 - 210.

10. Ackerstaff AH et al., 1999, “Mulit-institutional Assessment of the Provox 2 Voice Prosthesis,” Archives of Otolaryngology Head Neck Surgery, 125, pp. 167-173.

1 1 . Hilgers FJM and Schouwenburg PF, 1990, “A New Low-Resistance, Self-Retaining Prosthesis (ProvoxTM) for Voice Rehabilitation after Total Laryngectomy,” Laryngoscope, 100, pp. 1202-1207.

pp. 341-347.

Laryngology and Otology, 108, pp. 321-324.

A POTENTIAL ROLE FOR MUSCLE PUMP-GENERATED INTRAVASCULAR SOLITONS IN MAINTENANCE OF TISSUE-

ENGINEERED BIOREACTORS IMPLANTED IN BONE

H. WINET Department of Orthopaedic Surgery, UCLA/Orthopaedic Hospital, LA, CA 90007, USA

C. CAULKINS-PENNELL Department of Biomedical Engineering, UCLA, Los Angeles, CA 90007, USA

J. Y . BAO Departmen t of Orthopaedic Surgery, UCLA/Orthopaedic Hospital, LA, CA 90007, USA

Cell-seeded scaffolds are prime candidates for engineering regeneration of injured tissue. When engineered to keep seeded cells alive so they can communicate with host tissues, the porous implant is termed a “bioreactor”. Bioreactors must be quickly vascularized for seeded cells to survive while establishing physiological connections with host cells. Surgery kills tissue adjacent to implants, creating a nutrient transport problem because of resulting delay in bioreactor vascularization. Bone is a poroelastic matrix which moves its fluid contents when bent by attached skeletal muscle. Such percolation is termed %one interstitial fluid flow” (BIFF). We have proposed that during exercise muscles attached to bone compress vessels within and adjacent to their compartments in such a manner as to create soliton pressure waves that propagate upstream in veins, which have resisting valves, and downstream in arteries. This phenomenon is well known in exercise physiology as the “muscle pump mechanism” and is responsible for preventing pooling of blood in the lower extremities of the body under normal gravity conditions. Some of the affected vessels branch to bone vessels that form the microcirculatory beds in bone. Microvessels are significant contributors to BIFF in that the solitons they convey increase transmural transport, i.e. capillary filtration. We here propose that enhanced BIFF resulting from soliton-driven capillary filtration penetrates bioreactor implants enhancing both nutrient exchange and fluid shear stress on seeded cells. Data supporting the hypothesis were gathered from rabbit tibia1 cortex microcirculation using a bone window implant and intravital microscopy by stimulating the gastrocnemius muscle with transcutaneous electrical nerve electrodes. They include evidence that 1) capillary filtration in bone microvessels can be convective in the absence of muscle pump activity, 2) capillary filtration is enhanced by muscle pump activity and 3) when the muscle has been released (detached from the bone), so as to eliminate poroelastic load, muscle pump soliton effects include capillary flow interruption.

493

494

1. Introduction

Tissue engineered scaffolds provide a culture environment for inducing surrounding host tissue to overcome a pathological resistance to healing. Once implanted these matrices must establish communication with host cells to initiate

the induction process. Such contact is not

trivial in bone because implantation surgery creates a ring of necrotic (dead) tissue around the prepared site. Thus, live host cells lie beyond a rigid--although porous--wall. If the scaffold is loaded with inducing agents (growth factors or cytokines) controlled release must begin when scheduled. If seeded with donor inducing cells (e.g. mesenchymal stem cells) the scaffold is termed a "bioreactor" and communication with host cells will have to occur quickly. It has become

evident that diffusion of communication molecules is too slow to cover the

siphysis (EPI), Figure 1. Typical long bone showing e distances between scaffold and host metaphysis (MET) and diaphysis (DIA) sections. in time to assure implant device Typical forces on the bone are indicated by arrows

(which are general indicators, not vectors) M (skeletal effectiveness. Moreover, seeded muscle, MU), G (gravity) and R (reaction at joint). suffer hypoxia and other effects of Cortex (C) is bounded externally by periosteum (P)

and internally by endosteum (E). Veins are indicated insufficient nutrition when transport is by EV, MV, cvs and limited to diffusion. Scaffolds must be vascularized to overcome this limitation, usually via angiogenesis of host endothelial cells'". Once they have penetrated the scaffold, capillaries deliver nutrients and pick up waste by a process termed "nutrient exchange", which consists of filtration (delivery) and reabsorption (pickup) through the

vessel wall.

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Fluid which has extravasated during capillary filtration is interstitial fluid. It flows through the bone matrix and around bone cells before returning to the bloodstream directly by blood capillary reabsorption and indirectly by lymphatic capillary absorption. This bone interstitial fluid flow (BIFF) has another important role in the effectiveness of tissue engineered scaffolds. It is the source of the mechanical signal which modulates bone cell behavior. Accordingly, engineering of any scaffold for bone tissue must take into account the influence of mechanical forces experienced by the bone organ. Specifically, it must address BIFF resulting from these mechanical forces. The hypothesis proposed herein, takes the thought process one step further. It considers how

Figure 2. Cross section through cortical bone showing BIFF pathways as indicated by extravasation and percolation of blood-borne ferritin molecules. *-shaped structures are osteocytes in their lacunar compartments, which are connected by canaliculi. Halos mark percolation fronts advancing away from the center of each osteon where Haversian canals containing capillaries are located. If not reabsorbed by blood vessels downstream ferritin follows front progression toward periosteum and lymphatic capillaries. PPL indicates the perivascular and MPL the matrix prelymphatic channels (there are no lymphatic capillaries in cortical bone). The large source artery at the cartoon bottom is a nutrient artery branch. (from Kelly and BronV3 with permission). BIFF flow is always from the endosteal toward the periosteal boundary. However up to 1/3 of capillary bloodflow is in the opposite direction from periosteal arteries.

BIFF is generated and how it may be enhanced to prime host bone cells and scaffold- seeded cells for tissue engineered signaling.

The general environment for any model of bone cell modulation by bone mechanics is illustrated in Figure 1. A typical long bone has skeletal muscle attached to its periosteum (P) by a tendon (T). During muscle contraction a force is generated (M) which pulls on cortical bone (C). The cortex above the tendon insertion will undergo compression and that below it will undergo tension if it is closer to the ground. Gravity (G) generates this tension, which will become compression during weight-bearing. R indicates the reactive force at the joint. Arteries supplying diaphyseal (DIA) bone come

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from three general pathways: 1) nutrient arteries (NA) which pass through foramens (holes) in the cortex, 2) periosteal (P) arteries which are supplied directly by branches from extra-osseous arteries or through tendons (T) and 3) metaphyseal arteries (MA) which originally entered metaphyseal bone. All extraosseous vessels share compartments with skeletal muscle (MU). Bloodflow paths in veins effluxing bone are numerous and

WRES

Figure 3. Section from midshaft of typical cortical bone showing general poroelastic geometry. There are essentially two material regimes, cortical bone and marrow. Wet cortical human bone has a modulus of elasticity of the order 2 x 10’’ N/mz2. It is by no means brittle. Pores in cortical bone are occupied by blood vessels and carry interstitial fluid similar to that in other organs. Muscles inserted at the pexiosteal surface by Sharpey’s fibers pull on cortical bone during contraction, creating compression in the part of the cortex they cover and tension in the more distal part. Either stress can be enhanced or reduced by gravity loads. Marrow, the material that occupies the medullary canal, in contrast, is a mixture of fat globules, saline and air. It is somewhat compartmentalized such that measurements of intramedullary pressure (IMP) vary from location to location within the same bone. In general, however, IMP pressure is higher than pressure outside bone.

difficult to The current model explaining how mechanical forces on bone control BIFF is a

replacement for the traditional model known as “Wolff s Law”. Wolff s model is strictly solid mechanical and was thought to be implemented by a piezoelectric current generated in bone matrix under mechanical stress 1 3 ’ . It was disproven by demonstration that almost all of any electrical current generated by mechanical stress in normal wet bone is due to streaming potentials ’.

This electrokinetic model takes advantage of the observation that volume fraction of water in cortical bone is 13%69 and molecules at least as large as femtin percolate through the matrix via solvent drag from osteon to osteon at rates well in excess of diffusion18,1 9,43,46,51,68 . A cartoon of such percolation is presented in Figure 2 which shows a cross-section of cortical bone. This bone consists of an array of osteons

497

penetrated by blood vessels. which run down their Haversian canals. Small channels called “canaliculi” branch out from the canals toward a network of compartments called “lacunae”. Each lacuna contains an osteocyte.

The electrokinetic model proposed that the streaming potentials developed from zeta potentials between flowing ions in bone fluid and fixed bone cell membrane charges. In vitro experiments have confirmed bone cell response to streaming potentials 12955*56376-

. However, for about 20 years investigations into the process by which BIFF- caused bone streaming potentials were inconclusive because it could not be demonstrated, primarily because direct measurements could not be obtained, that threshold cell receptor activation potential values measured in vitro were achieved in vivo under normal conditions 12’.

In 1990 Frangos and colleagues proposed that bone cells respond to fluid shear stress in a manner similar to endothelial cellsg’. Control of endothelial cell activity by blood flow-generated fluid shear stress is well e~tablished~~. Given the high incidence of redundancy in evolved tissues and the common ancestry of the two cell types from mesenchymal stem cells, such a prediction is logical and has been confirmed for bone cells in vitro 34936*44390. There are important differences between endothelial and bone cells. For example, osteocytes and their processes (see Figure 2) are surrounded by relatively thin fluid (not necessarily Newtonian) annuli in the lacunar and canalicular compartments, rather than relatively large volumes of flowing blood. Although no direct measures of fluid flow in vivo over bone cell surfaces have been obtained, in vitro experiments indicate that a shear stress stimulation threshold producing 1 % strain” and a streaming potential threshold of 1 O p are effective.

79,86,122,123

2. What drives BIFF?

#at is unclear at this point is the source of hydraulic pressure gradients required to drive BIFF. It has been proposed that BIFF is generated by incompressible fluid shifts resulting from pressure and tension cycles on interstitial fluid during bending of bone by muscle contraction and weight-bearing whether during locomotion’5~45~47~83~84~86or posture m a i n t e n a n ~ e ~ ~ ’ ~ ~ . The basis for the current model is Biot’s poroelasticity theory4 which predicts fluid percolation rates based on pore geometry and distribution coupled with matrix elasticity14315. The simplified environment for this model is summarized in Figure 3. The cartoon shows a section of diaphyseal bone represented as interstitial fluid-saturated porous walls surrounding a non-Newtonian marrow core. Vessels are omitted for simplicity. Any mechanical stress on the bone tube distorts its wall and forces shifts in the interstitial fluid seen as changes in the BIFF illustrated in Figure 2.

There have been tissue-level observations supporting the general concept of the mode147,48,68,1 18 , and initial bone bending-based model calculations predicting fluid shear stress on in vivo osteocytes indicated that it reaches magnitudes sufficient to cause the required 1% strain. However, more recent model refinements with real measures indicate that flow rates generated by poroelastic bending alone (i.e. disregarding weight loading) are not sufficient to reach receptor activation threshold16340383.

2.1. What is the mechanical contribution of cardiovascular dynamics to BIFF?

Since fluid enters bone interstitium primarily by leaking out of blood capillaries one may

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reasonably ask what the pressures driving capillary filtration contribute to BIFF? Published reports modeling flow generated by bone bending either I ) state that flow in capillaries driven by heartbeat alone generates capillary filtration rates too small to exceed activation threshold shear stress of osteocytes, or 2 ) simply ignore trans-capillary pressure gradients~5,45,47,83,84,86,87,~09,110,124 . McCarthy and Yang have criticized this limitation of a pressure generating mechanism to p~roelasticity~~. Hillsley and Frangos 33 suggested that capillary filtration is an important source of the BIFF which stimulates bone cells by fluid shear stress. In addition, they proposed that endothelial cell secretion of cytokines such as NO (nitric oxide), which affect osteoblasts, is enhanced by capillary filtration. Otter found that cortical bone streaming potential signals correlated highly with circulation generated IMP pressure oscillation^^^ that developed in the absence of

Figure 4. Cortical bone showing hypothetical pathways of bloodflow. Skeletal muscle is located above lymphatic capillaries in the figure. Blood flows into cortex via nutrient artery branches which branch into arterioles and capillaries of osteons and Haversian canals. Arrows in vessels indicate bloodflow. Valves (<) are shown in veins. Capillaries are shown leading to and from an osteon where they exchange nutrients for waste from cortical bone cells. Interstitial fluid also exits cortical bone via pre-lymphatic channels which lead to lymphatic capillaries and lymph vessels (shown). Arrows in muscle indicate shortening and widening during contraction. Vessels adjacent to muscle are compressed and soliton waves propagate in blood. Solitons from the skeletal muscle pump are indicated with S. Compression of poroelastic matrix slows bloodflow in cortical capillaries such that soliton pressure increases. Musclecaused bending and weight bearing compression/tension of the poroelastic cortical matrix probably normally dominate vascular soliton effects. During exercise, however, muscle pump and heartbeat-generated pressures will further increase intravascular pressure.

weight bearing and muscle contraction; but he drew no conclusions about the ability of these potentials to stimulate bone cells. Pressures have not been measured in cortical bone, but normal oscillations in vessels of canine tibia medullary canals have amplitudes of about 15 mm Hg, rising from a baseline of 25 mm Hg78. The notion that capillary filtration can help drive BIFF is consistent with the evolutionary doctrine that redundancy increases survival odds, and allows us to consider factors controlling blood pressure as contributors to mechanical modulation of bone cells. The idea of enhancing BIFF by increasing capillary filtration is not new. Saphenous vein compression causes increased capillary filtration and periosteal bone formation in canine

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2.2. How does the muscle pump affect bone microcirculation?

Gravity complicates body fluid distribution. In the horizontal body blood is pumped through bone microvascular beds by baseline heartbeat. Distribution is modulated by local switching of vascular pathways in response to tissue oxygen demand. Extravascular fluid tends to settle downward under gravity, but the vertical distances are short and oxygen is sufficiently distributed to make local hypoxia insignificant. Prolongation of this condition during bed rest, however, has significant physiological effects including atrophy of muscle and o s t e ~ p e n i a ~ ~ ~ ~ ’ ~ ’ ~ ~ .

When the body orients vertically, extravascular interstitial fluids shift downward and the heart must increase output to insure initial venous return to the right atrium and normal flow to cephalic arteries. If a body depends on the heart alone for upward bloodflow, gravity eventually decreases venous return and cephalic delivery, allowing blood to pool in veins-which have 6-8 times the volume of arteries4’--of the lower extremities, and the individual faints. Muscle-pump activity in limbs increases bloodflow and “milks” the vessels to return blood distribution to safe levels. In the lower extremities pumping alone would be ineffective if veins and lymphatic vessels were not valved so as to prevent backilow. In the upper extremities the consequences are easily demonstrated by increase in brachial artery pressure 25-30 mmHg coupled with a decrease in blood flow-even during hand-grip exercises-as the forearm is moved from below to above the heart l e ~ e 1 ’ ~ ~ ” ~ ~ . Under non-exercise conditions postural activity is required to maintain body balance. It has been suggested that maintenance of postural balance by muscles produces “persistent low-magnitude” strains, which because they are spread out over long periods of time may produce sufficient mechanical stimulus to modulate bone cells94. Such a stimulus would probably be purely poroelastic, however, since associated vascular pressures would be small.

We reason that skeletal muscle, acting through a muscle pump mechanism increases the rate of capillary filtration by increasing capillary hydraulic pressure via contraction of skeletal muscle in compartments adjacent to bone. Exercise magnifies the affect by increasing baseline blood pressure through increased heartrate and muscle pump activity. Two anatomical circumstances suggest how the mechanism operates: 1) bone influx and efflux vessels outside bone are contained within fascia-bounded compartments which include skeletal muscle and 2) eMux vessels (veins) are valved.

When the muscle pump contracts all vessels within its compartment are occluded as indicated in the cartoon of Figure 4. At the same time, two intravascular solitons (solitary pressure waves), one traveling upstream and one downstream propagate away from the occlusion site in the vessels’ incompressible fluid. Solitons moving toward the bone are upstream in veins and downstream in arteries. Venous valves close and block capillary blood eMux(with the exception of collaterals that do not empty into the involved muscle compartment). Pressure in arteries rises and is transmitted to the nearest downstream capillary bed. If such a bed is in cortical bone, intravascular capillary hydraulic pressure will increase in varying degrees depending on a variety of local conditions. As a result the transluminal pressure gradient increases speeding up capillary filtration and BIFF as indicated in Figure 6, which is a magnification of a region of Figure 4. Note that a scaffold implant has been added to this cartoon and will be discussed below.

Muscle pumping also increases IMP. Contraction of the quadriceps muscle causes a 30mm Hg or more rise in femoral IMP9,49,58,’0’ . That this effect is not exclusively

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poroelastic is suggested by evidence that 1) BIFF in cortical bone is from the endosteal to the periosteal bounds$, 2) there is a simultaneous rise in blood pressure upstream” and certainly downstream of the muscle and 3) marrow is not a Newtonian incompressible fluid”. There must surely be a secondary poroelastic effect through the vasculature. Specifically, bloodflow tends to increase on the tension side and decrease on the compression side of cortical bone during physiologically relevant bending65. As a result blood flowing in a nutrient artery branch will be diverted from compressed cortical bone toward marrow microcirculation. One consequence of diversion may be an increase in leakage from medullary sinusoids, which could increase IMP enough to collapses both sinusoidal and continuous capillaries in the medullary canal-a classical Starling resister effect-resulting in both medullary and cortical ischemia”. However, the long exposure periods of high IMP required for such edema seem to occur only under pathological conditions such as ischemic osteonecrosis22 and bone marrow edema syndrome37. In support of a poroelastic connection it has been observed that a physiologically relevant peak bending load of 6 0 0 ~ ~ on a turkey ulna generates an IMP of 64 mmHgg8.

One must take care in applying measured values to theoretical models. IMP in human tibia has been measured to vary from 35 mmHg in the supine to 46 mmHg during walking64. Interestingly, the corresponding standing pressures were about 85 mmHg during ~eight-bearing~~. Tibia1 IMP standing pressure in the goat is 15.5 mmHgIz6, and serves as a caveat against: extrapolating from quadripeds to bipeds. One must also be careful to avoid extrapolation from IMP measures to BIFF in adjacent cortical bone. Notwithstanding a probable cortex poroelasticity link to pressure buildup in marrow nutrient artery branches, attempts, to establish a causal relationship between IMP and cortical blood flow have not been consistently successfu17~”’.

2.3. Affect of exercise on bone microcirculation

Pressures generated in muscle during exercise can be considerable. Interstitial values as high as 570 mmHg96 have been recorded. At the low end would be pressures generated during normal “postural” contractions25x94. One might speculate that the muscle pump soliton threshold for generating capillary filtration sufficient to add a significant fluid shear stress vector to BIFF is reached during limb movements sufficient to preventing blood pooling under gravity.

Exercise appears to increase bloodflow to bone, but the results are uneven. “Exhaustive” exercise in rats increases metaphyseal bloodflow, decreases diaphyseal bloodflow and does not effect marrow bloodflowlog. Short term treadmill exercise of adult dogs increased bloodflow in soft tissue around joints, but had no effect on juxtaarticular boneIo6. Longer term treadmill exercise caused a steady increase in cortical bone bloodflow from 1.6 to 2.5 ml/mir1/100g”~. It has also been observed that during exercise vascular resistance in bone increases two to fourfold while vasodilation in adjacent muscle increases29.

The effect on muscle and bone microvasculature of increased bloodflow during exercise is complicated by the fact that 1) local hematocrits and shear stress on RBCs will influence both the rheology and rate of oxygen delivery, 2) shear stress on endothelial cells upregulates the release of a number of agents which alter both permeabil- ity72,105,’1’3’2g and caliberg5 of vessels and 3) hydrostatic pressure will rise or fall as a function of ‘2’, increasing the potential for convective flows. It should come as no surprise that results from muscle cannot be extrapolated to bone.

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Values for cortical vessel luminal pressures have not been reported as it is difficult to measure in this hard tissue. Calculation from trans-cortical pressure gradients is problematic. Even though blood is an incompressible fluid, it is non-Newtonian in small vessels and permeability of capillary walls is not uniformly distributed"; consequently, neither is the transmural pressure gradient.

Another confounding factor is "crosstalk" between endothelial cells and bone cells via cytokines/chemokines. Fluid shear stress on endothelial receptors upregulates their expression of peptides which stimulate bone ce11s1~13~23~30~39~57~73~82~112~133 , and bone circulation is correlated with BMU work rate in~rease '~ . '~~ . In addition sinusoidal

marrow capillaries may contribute to BIFF by making fluid available to the cortical matrix. Indeed, their contribution has been proposed to directly account for the rapid transport of large molecules through cortical bone63. All of these effects must eventually be coordinated in a total physiological model.

2.4. Capillary filtration in cortical bone

How convective are bone capillary filtration flows? How do flow rates reflect neighboring skeletal muscle activity? This is not known. McCarthy and Yang63 have attempted to set up a total transport model by superposing solutions to transport equations through 1) the capillary wall using the Renkin-Crone capillary permeability (P)-surface area (S) product equation, PS = -Qln(l-E), where Q is blood volumetric flow rate and E extraction, 2) the extravascular space between capillary wall and osteon wall using Fick's second law, where the source concentration is determined by PS and 3) the bone matrix using a compartment diffusion model which attempted to include uptake of solutes by metabolizing bone cells. Transport in this model was considered to occur by diffusion and it allowed for high rates by arbitrarily increasing concentration gradients. When McCarthy62 attempted to account for transport of albumin by assuming diffusion in an in vivo experiment he found rates of transport so high that he had to conclude that the flow was convective. He doubted that cortical vessels were capable of such flow and proposed that the more permeable medullary sinusoids provided the logical pathway62. However, it has been well established that similar continuous capillaries in other organs are capable of convective transmural transp01-t~~.

Nakamura and Wayland established a model to determine if macromolecular extravasation in mesentery was by diffusion or convection. Analyzing single vessels they reasoned that if the concentration gradient between a source (vessel center) and a sink (some point distance x from the vessel center) decreased its slope faster than the rate predicted for the molecule in question by Fick's second law, then the parsimonious conclusion was that the difference was convective transport. The gradient was represented by the ratio of concentrations at time t, C,(x,t)/C,(O,t), with Co tending to be constant because the capillary was continuously perfused. In a given experiment with fluorescent macromolecules extravasating, concentration was directly proportional to fluorescence intensity, so the ratios were the same. Temporal change in

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concentration/fluorescence at x is predicted by equation { 1 } their error function form of Fick's law. Here D is the reference difision coefficient for the tested solute. A relative diffusion D* is computed by replacing Cx/Co with experimentally measured fluorescence intensity values. Where D* > D, transport cannot be due to difision alone, and it is presumed that the amount of increase is a measure of convection. Certainly, this measure will not be sufficient as a predictor of bone cell activating bone matrix percolation, but it will be necessary.

3. Demonstration of effects of exercise on microcirculation

In order to demonstrate that microcirculation contributes directly to convective transport in any tissue, it must first be demonstrated that capillary filtration is convective. Two measures of capillary filtration commonly used are capillary filtration coefficient Pfi and hydraulic conductivity Lp These quantities are related by the equation Pf= S Lp 27 where S is the permeable capillary surface area. In most cases the value of S is not known, so it is difficult to determine if Pf is due to an Lp value large enough to indicate substantial convection or to substantial vessel recruitment66. Cases where convection was thought to have been demonstrated in a subject at rest93 have been questioned by some who propose the result may have been an artifact of methodolog?. The consensus appears to be that in a horizontal body at rest Lp is dif€usive66.

It is, however, generally accepted that Lp is convective when capillary pressure is sufficiently highg7 andor flow is sufficiently fast to create shear stress which will stimulate endothelial cells to increase capillary permeabilid', as occurs during exercise. In the latter case endothelial nitric oxide (eNO) is proposed to be a major local effector'32. During exercise cardiac output share received by muscles rises from 15-20% to 80-90%, at least five fold, controlled by local metabolic factors"'. The abundance of studies on the physiology of skeletal muscle microcirculation during exercise is matched only by the paucity of similar studies in bone. The pressure increase in skeletal muscle arteries is not as great as the increase in muscle perfusion. For example, gastrocnemius blood flow increased from 4.7 to 10.3 ml/lOOml/min during human rhythmic foot loading against a wall, yet corresponding blood pressure only increased from 82 to 85 mmHg6'. In contrast pressure changes may be substantial in capillaries (of the order 18 Initial rise of capillary filtration driving pressure (= Jo /P' where .Jo is flux through capillary wall) in cat sympathectomized skeletal muscle may rise to 58 mmHg during graded exercise'. In an "exercised" cat muscle organ preparation with the sympathetic input intact, capillary pressure rose from 16.7 to 32 mmHg in one study6' and by 12.2 mmHg in another, leading to "marked" capillary filtration6'. With exercise and vessel dilation, increased vessel dilation and the resultant decrease in resistance and pressure observed by Masped' indicates a complex relationship in muscle blood flow. In a number of rat-on-treadmill studies skeletal muscle showed an increase in Pf of 27%98-'00. Our inability to accurately predict microvascular response to increased perfusion during exercise may be linked to 1) hypervolemia which is delayed and may continue for at least 24 hours post exercise32, 2) albumin extravasation which is linked to hypervolemia and changes the transluminal oncotic pressure gradient32, 3) the formation of temporary shunt ve~sels''~, 4) decrease in venous resistance (increase in vascular cond~ctance)"~, which in one study presented as a pressure drop of 45 mmHg'" and 5) varies from vein to vein7', and 6) neuroendocrine, autocrine, paracrine and other secreted regulatory agents which effect both permeability66 and perfu~ion'~~'' add to the complexity of BIFF.

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4. Experiments which will test the model

Verification of the model requires demonstration that increase in blood flow (which we have measured) is not compensated for by bone arteriolar constriction. Further, it must be shown that convective capillary filtration can occur in response to a muscle pump released from its tibia1 insert. Finally, it must be shown that extravascular flow generated by the muscle pump in the absence of poroelastic load is a significant component of 1) flow percolating through the lacunar-canalicular system and stimulating osteocytes and/or 2) perivascular flow stimulating osteoblasts and/or 3) perivascular flow stimulating osteoclasts.

It would appear that the initial step in testing the model would be to determine if capillary filtration in cortical microvasculature 1) is convective or 2) can be made convective by the muscle pump. Preliminary data in support of this step have been obtained.

4.1. Preliminary data in support of the model

Vpicd Lhe Scan lor Cardrd Group I 100

180

180

140

t i 120

f 100

00

80

-

40

20

Figure 5 . Effect of TENS on capillary filtration pre-and post 30 and 60 minutes stimulation. Peaks are dye in vessels.

Capillary filtration of large

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fluorescent molecules was used to indicate level of transport through large pores (assuming the two-pore modelg3). Fluorescein isothiocyanate Dextran 70kDa (FITC- D70) was injected into a rabbit and its entrance into cortical vessels in the compartment of an optical bone chamber implant (BCI) observed via intravital microscopy. Analysis of the recordings with an image processing program produced fluorescence intensity profiles across one vessel at three times. These results show an effect of muscle stimulation on capillary filtration. In previous work we have reported evidence that convective capillary filtration occurs in bone microvasculature without requiring muscle contra~tion'~~. In this observation the ratio of change in intensity at a given distance x from the vessel was equivalent to the ratio of FITC-D7O concentration change, C(x,t)/C(O,t), at x over the same time period. Equation { I } was applied and compared with the same ratio for free diffusion of FITC-D70. For the profiles measured D* > D indicating that convection was indeed taking place.

Enhancement of capillary filtration in cortical bone by skeletal muscle contraction has also been tested. Figure 5 shows a comparison of two histograms of the fluorescence intensity incidence as a function of emission level. These data represent three vessels in a BCI compartment which is 2mm in diameter and lOOpm thick. FITC-D70 was injected before and RITC-D70 after 30 minutes of muscle stimulation. Stimulation came from a Tonatronic@ model TMS DT 8000 transcutaneous electrostimulator applied externally over the gastrocnemius muscle. The TENS pattern was 30 volts at 80 mA and 4Hz. Absence of extravasation is normal for these large fluorescent molecules. Under TENS, however, capillary filtration is indicated by a rise in extravascular fluorescence. The D* data and plots in Figure 5 support the notion that convective capillary filtration is present in bone capillaries.

4.2 Implications of BIFF for tissue engineering

BIFF enhances transport to and from any bone implant, as can be appreciated by examining Figure 6. Tissue engineered porous implants benefit from faster 1) perfusion to and 2) percolation within their matrix129. For cells seeded in the device BIFF may be the difference between life and death. The difference derives not only from nutrient exchange, but from removal of eluted products of degradation of an erodible implant. For example, it has been proposed from in vitro studies that lactic and glycolic acid monomers from degrading poly(1actide-co-glycolide) (PLGA) scaffolds are the cause of late-term reactions to erodible bone fixation devices1I3. Although it has been shown that in vivo pH changes in host tissue surrounding an eroding PLGA implant are too small to be physiologically significant" acidic monomer concentrations at the interface between polymer shell and attached seeded cells may be sufficient to inhibit cell function. BIFF- driven percolation would dilute such monomer buildup. The most important long term effect of BIFF on scaffold biocompatibility is its influence on vascularization. By enhancing communication between host cells and scaffold, increased fluid transport will not only jump-start host response time, but will give vectorial direction to the response. If host cell stimulating agents, such as the angiogenic Vascular Endothelial Growth Factor (VEGF) were scaffold-bound they would reach their targets earlier in the healing process than would be the case under diffusion-limited conditions. Addition of enhanced transport to the engineering of a scaffold would complicate controlled release calculations. Erosion of the scaffold would increase the engineering challenge. But the

505

rewards of enhanced effectiveness and flexibility of application would be worth the extra effort.

Figure 6. Cortical bone with an implanted cell-seeded bioreactor (SCAFFOLD) showing hypothetical pathways of fluid transport to and from impant and capillaries in cortical bone. Scaffold cells are gray discs with 'nuclei'. Black arrows indicate bloodflow and white arrows (other than identifiers) indicate extravascular transport, including BIFF. Flow out of (filtration) and into (absorption) capillaries as well as that into and out of the scaffold are summarized as "nutrient exchange". Nutrient exchange delivers nutrients to the implant and picks up waste which is retum'ed to the venular capillaries. BIFF also transports fluid to pre-lymphatic channels in the cortical bone, which lead to lymphatic capillaries Enhancement of BIFF by muscle pump action enhances exchange with the bioreactor, providing nutrient support to seeded cells, enhancing transport of cytokines loaded into the scaffold and increasing the rate of scaffold erosion. Muscle-caused bending and weight bearing compression/tension of the poroelastic cortical matrix normally dominate vascular soliton effects. During exercise, however, muscle pump and heartbeat-generated pressures should become significant contributors to BIFF.

As it percolates through a bioreactor scaffold BIFF fluid will pass over seeded cells. Given a sufficient trans-device pressure gradient, these cells will experience fluid shear stress andor streaming potentials. Stimulation of membrane mechano/electrokinetic receptors will generate a variety of responses including, cytokine/growth factor secretion, motility, proliferation, growth and transformation resulting from induction. The type, maturity and differentiation level of each cell will determine its response. For mesenchymal stem cells in a bone implant development of bone or vessels would be a logical objective.

The frequency of redundancy in organic evolution leads one to expect more than one pathway to generating threshold BIFF for stimulating bone cells. Accordingly, there may be circumstances under which required flow could be generated without poroelastic deformation. Two circumstances where this approach would be beneficial are in microgravity and disuse osteoporosis. The former occurs in astronauts and the latter in bedridden patients. Both result in osteopenia which may severely reduce bone strength. An externally applied device which could emulate muscle pump effects may provide the needed intravascular propagating pressure solitons. Such devices have been applied c~inica~~y~6.20,21.26,53,54,Sl,l 16,l17.121

506

Acknowledgements:

Particular thanks are owed to David(Kambiz) Rabie who assisted in the performance of much of the work in the Bone Chamber Laboratory reported in this report. This work was supported by the h s Angeles Orthopaedic Foundation, The Cora Kaiser Fund of Orthopaedic Hospital and the UCLA Department of Biomedical Engineering.

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CHAPTER 5

BIOMECHANICS: ZOOLOGICAL


CREEPING FLOW AROUND A FINITE ROW OF SLENDER BODIES

EFRATH BARTA AND DANIEL WEIHS Faculty of Aerospace Engineering

Technion, Ha ifa 32000, Israel

The flow through and around a finite row of parallel slender bodies moving at constant low speed in a viscous incompressible fluid is studied. The motion occurs under creeping flow (R<<I) conditions. This row is an abstraction of a comb-wing configuration suggested for use for flying vehicles of mm size, that operate in a creeping flow regime, utilizing the viscous effects to cany along enough fluid so as to approximate continuous surfaces. The bodies are taken as rod-like ellipsoids of slenderness ratio smaller than 0.01 and described by a distribution of singularities. Results for the drag on the individual bodies, as well as for the full row are presented. The row is shown to act very much like a continuous surface, with over 95% of the flow moving around, and not through the comb, in most cases, with a potential saving of tens of percents in wing weight. Parametric results for number of rods, rod density (ratio of inter-rod distance to rod length), and slenderness ratio are presented, indicating that most of the losses occur near the row ends and rod tips, as expected.

1. Introduction

The Stokes approximation for low Reynolds number flows (creeping flow) has been very successful and useful in describing the flow phenomena occurring under these conditions, which in air or water usually mean very small bodies moving at low speeds (Happel & Brenner, 1973, for example). However, the slow decay of boundary effects has resulted in serious difficulties in analyzing problems in which multiple bodies are involved.

Recent developments in microelectronics and power supply technology enable designing extremely small flying vehicles, with wingspan of O(lO”m). Such vehicles move at speeds of up to O( lO-’m/s), thus approaching the Stokes flow regime. One of the cardinal problems in designing such minuscule vehicles is weight (mass). This constraint caused us to look for weight reducing options, leading to the idea of utilizing the slow decay of boundary effects in Stokes flow by building non-continuous comb-like structures as aerodynamic surfaces. Such comb-like surfaces will act as full wings as the fluid in the spaces between the solid parts will presumably be dragged along, and oncoming flow will be deflected around the structure.

Recently, the senior author and colleagues (Zussman, Yarin & Weihs, 2002, Naveh et al. 2003) showed that for vehicles moving in the low Reynolds number range, the aerodynamic surfaces do not have to be continuous to produce the aerodynamic lift required. This was done by manufacturing what is essentially an artificial dandelion seed, and showing that the deceleration obtained by a porous web is equivalent to that of a continuous surface.

Ths leads to the possibility of using a comb-like array of rods that will serve as wings. Such a configuration can save up to 80% of the wing weight, while producing lifi roughly equal to a continuous wing of the same dimensions.

515

516

Sunada et al. (2002) showed that even for R=10, comb-wings can produce 80- 90% of the forces produced by a solid surface of the same dimensions. Comb- wings of this form are found in the insect order of Thysanoptera (Thrips), insects with wingspan of lO”m or less.

An existing model of flow around an infinite row of cylinders (Tamada & Fujikawa, 1957, see also Ayaz & Pedley, 1999) was used in the studies above to estimate force on the row as a function of the rod dimensions and distances between members of the row. Results for sinking speeds showed that the predictions of this model are within 5% of the experimental results when the Reynolds number is low enough. However, several assumptions relating to neglecting the effects of the row being finite were required to enable applying the Tamada & Fujlkawa -Ayaz & Pedley model. Most significantly, an infinite row forces the flow to move between rods for any given spacing value. In a finite row, when the distance between rods is small enough for viscous “closure” the fluid approaching the comb sees a continuous surface and escapes the low velocity and high drag area between row members by flowing around the structure as a whole, thus justifying the continuous wing model. The explanation for the good agreement of the infinite row model with experimental results lies in a reasonable, but theoretically unjustified step made in our previous studies (Zussman, Yarin & Weihs, 2002, Naveh et al. 2003). This entailed assuming that the increase in drag on each rod in a dense row can be translated into a smaller percentage of fluid passing between the rods in a finite row with the limit of almost no fluid moving between the rods. Thus a 40-fold increase in drag predicted for a certain density by the Tamada & Fujikawa model was assumed to mean that only 2.5 % (1/40) of the flow passes between rods. This assumption proved a posteriori to be good, but a reliable calculation of flow both around and within a finite comb is still needed.

In this paper we present a consistent model for the creeping flow around and withn a finite row of extremely elongate bodies, thus avoiding the paradoxical results of two-dimensional flow on the one hand, and the limitations of having infinite rows on the other.

Our approach is based on the Johnson (1980) and Barta & Liron (1988) axial singularity distribution model that yields integral equations of the 2nd kind for the intensities of the singularities. In contrast to other methods that entail singular solutions (Kim & Karrila 1991) the advantages of this approach is its rigorous use of the various types of singularities, the uniform validity of the solution over the whole surfaces of the bodies and its computational simplicity. In Sec. 2 we specify the assumptions that underlie the model and write the resultant equations. The limitations and the strength of the model are demonstrated by analyzing the errors involved with it in Sec. 3. Results giving a theoretical basis for the experimental experience that showed the row of rods to behave like a continuous wing are presented in Sec. 4. The possible applications of this work are discussed in Sec. 5 and Sec. 6 concludes the paper.

517

2. Formulation of the Model

We describe the flow field induced by a finite row of parallel slender ellipsoids immersed in an unbounded incompressible fluid, moving at R<<1 (Stokes flow). The motion can be in any direction relative to the ellipsoids. Our method may be applied for a row of slender bodies of different shape (including those which have axes with nonzero curvature) with minor modifications.

We normalize length parameters with respect to half body length, and denote by E the slenderness ratio defined as the maximal radius of the ellipsoid and the distance between the centers of two adjacent bodies by d (d is obviously larger than 2~ and may vary from several E’S to quite a large number in a sparse array). The number of ellipsoids m, can vary from 2 upwards, limited only by computation time. Motion of the row induces flow around and between the ellipsoids. We compute this flow field in order to elucidate the role and importance of the interactions between the bodies and in order to determine the range of parameters ( E , d and m values) for which the ellipsoids will “drag” the fluid between them and the whole row of slender bodies will approach the action of a continuous surface. Thus, the Stokes equations:

v p = pv2u (1)

are solved for the velocity u and pressure p within a medium with viscosity p where either the velocities of the bodies or the total forces exerted on them are given.

The most intuitive approach to tackling this problem is to distribute Stokeslets (Kim & Karrila 1991) on the faces of all the ellipsoids. Sellier (1999) did so and formulated a Fredholm equation of the 1’‘ kind that was asymptotically solved for a single slender particle. A much simpler solution was suggested by Liron & Barta (1992, 1996): Assuming that given forces and moments are exerted on the ellipsoids, the intensities of the Stokeslets (the stresses) are determined by solving integral equations which are Fredholm equations of the 2”d lund. However, numerical solutions based on this method are lengthy, as each slender body has to be represented by many surface sub- elements. Thus, a solution for an array of many such bodies involves a huge system of equations.

We present an attractive alternative to the above approach, where we distribute the singularities on the axes of the ellipsoids thus restricting the number of the unknowns considerably. As the errors resulting from h s simplification are not clear a-priori, a detailed analysis of the errors involved is performed in Sec. 3.

518

1.

2.

3.

4.

5 .

2.1. Basic Assumptions

All ellipsoids are identical and have the same given velocity i.e., we deal with a resistance problem (as opposed to the above mentioned mobility problem where the forces are specified). It is sufficient to solve for 3 motions along 3 orthogonal principal directions due to the linearity of the Stokes equations. Motion in any direction is a linear combination of those. The rods have no angular velocity. Each body is represented by a distribution of singularities along its axis between its foci. This actually means that a circular ring (at any altitude) that encircles any ellipsoid and is close enough to it has a uniform velocity. To a leading order only Stokeslets and Doublets may be considered. The contribution of other types of singularities is secondary. The intensities of these two kinds of singularities satisfy the same relation determined for simpler cases by Johnson (1980) and by Barta & Liron (1988).

2.2. The Integral Equations

We use a coordinate system tangential, normal and bi-normal to the bodies' major axis e,, en, eb (Happel & Brenner, 1973). While in general these unit vectors change along the centerline, for ellipsoids they coincide with the familiar Cartesian e,, ey , ex axes, see Fig. 1.

. . . . , d

......

Figure 1. Schematic description of the configuration.

A

e

......-.-.*..

Henceforth we write integral equations for the intensities of the Stokeslets by equating the given velocities v, u, w along the e,, en, eb directions at an

519

arbitrary point ( S, 4, w ) on the surface of an ellipsoid to the expressions of the velocities induced by the singularities. Using a uniformly valid asymptotic expansion, Johnson (1980) wrote integral equations of the second kind for the intensities of the Stokeslets required to describe the motion of an isolated slender spheroid. Barta & Liron (1988) have extended his solution to account for motion of two parallel bodies where the distance between their axes is comparable to the body's length, O(1). In the present solution, dij is of O(E) so we extend the integral equations to include the effect of Doublets of neighbor bodies (bearing in mind that a-posteriori justification for utilizing their method for the flow field around a row of m densely packed bodies is needed).

Denote by a i ( s ) , p i ( S ) the intensities of the Stokeslets and Doublets respectively that are situated at point s on the axis of the i-th ellipsoid (i=l ,.. .m) and act in the j-th direction (j=s, n or b) and define:

(3) 2 e2= 1- E

Then, we assumed that the relation that resulted in a uniform velocity on the surface of each of the bodies for motion of two distant ellipsoids (Barta & Liron, 1992) is valid here:

E L

2 pi (s) = -a! (s)(e2 - s2) i=l, ... m j=n, s, b. (4)

The velocities at point ( s,rl ,w ) i.e., at the cross sectional plane situated at height s that has the polar coordinates (rl, w ) are:

e m e

ui (s, 4, w) z (2L + 1)a; (s) - j -e

e

i=l, ... m ( 5 )

520

(s - s‘)2 3E2(e2 -d2) a; ( S ‘ ) [’ + ( ( ~ 4 ) ~ + di;} [’ - ~ { ( S - S I ) ~

E2 m (e2 -sr2)aY(s’) +-C J ds’ i=l,. . .m 2 j=1 j#i -e Jm3

a; (s) - a; (s’) Wi(S,r, ,W) z(2L+l)a;(s)- J ds’ +

-e 1s - s’(

3c2 (e2 - s’~) * (1 - )+ 2 { ( ~ - s ‘ ) ~ + doz}

j+i

d,; 3c2 (e2 - sr2) a)’ (S’) [’ + {(s-s’)~ + d,;} [’ - ~{(s-s’)~ + di;} I) d+d,i+doids’ E2 m (e2 - S ’ ~ ) ~ ~ ( S ‘ ) +-c J ds’ i=l,. . .m 2 j t i j=1 -e ,/m3 (7)

where:

L = h(2/E) (8) and d, is the distance between the centers of ellipsoids i and j.

motion along e, or % involves singularities in both directions. For motion along en the only nonzero singularities are in this direction while

3. Evaluation of the Model

The integral equations Barta & Liron (1988) developed for two bodies that are separated by a distance of O( 1) were shown to be accurate within a relative error of O(~ln&). We now prove that the relation between the intensities of the singularities (4) is still optimal for distances of order E. The error involved with applying these integral equations is estimated afterwards.

521

3.1 Asymptotic justification of the equations.

The ellipsoids are rigid bodies, so that in order to satisfy the no-slip boundary conditions, the induced velocity at point ( S,rl,y ) must be independent of W . In order to get a uniformly valid asymptotic expansion we follow Johnson’s (1980) method and write u far from a given singularity (outer expansion (‘I), near the singularity (inner expansion (i) with a stretched coordinate) and then “tailor” both expansions by an outer-inner expansion (Oil . (The reader is referred to Johnson’s work for a more comprehensive discussion of the method.)

For the sake of brevity we deal here just with three bodies which are d=ks apart and are moving along the en direction. When dealing with a larger row we’ll have to account for the presence of the more remote bodies in the same manner as we do here for the adjacent bodies. Due to symmetry we expect just two different values of intensities: al(s) and az ( s ) for the edge and middle bodies respectively (the same holds for the intensities of the Doublets) and we explicitly write the velocity induced on the surface of the central ellipsoid as follows:

Outer expansion: The total velocity that Stokeslets + Doublets situated on the three axes at height s’ induce at point ( S, rl , W ) on the middle ellipsoid is:

Note that in the outer expansion the contribution of a singularity located at the body under consideration is identical to the contribution of the parallel point on the adjacent body.

Using a stretched coordinate (7:

Inner expansion:

(3 = (s - s‘) / & (10)

the distance vector that connects the singularity point s’ with ( s, r, , is written as:

)

r = E [ c e , + q - c o s ~ en + ( k - q . s i n y ~ ) e b ] (11)

where:

522

In Eq. (1 1) k is replaced by 0 or -k when the singularity is on the middle or opposite side body respectively. The velocity induced by the singularities located at s’ is:

1 1 3 + 1

3 P: (s)

T[ do2 +q2 +k2 -2kq.siny do2 +q2 +k2 +2kqesiny

+q2 00s’

+ o(a,p/E2) (13)

0 Outer-inner expansion: Write the outer expansion by utilizing the stretched coordinate to get the following expression for the velocity:

523

The induced velocity at (s,r,,y) is a result of the contributions of all the singularities distributed between the foci of the bodies therefore we get that to the leading order:

0 2 e

u(s,q,y) E ] u"'(s,q,y;o)Edo+ ](u(~)(s,~,~;s~)-u(~~)(s,~,- 0 1 -e

where (3, = (e+s)/ E, o2 = (e-s)/ E.

In order to satisfy the rigid- body no -slip condition we seek a relation between a (s) and P (s) that will render u to be y~ -independent. We specify here just the y -dependent terms that we get after substitution of (9), (13) and (14) in (15):

1 do2 +q2 +k2 +2kq,s iny

+ 1 -0, do2 +q2 +k2 -2kq-siny

1

do2 +q2 +k2 +2kq.siny 3 + 1

du2 +q2 +k2 -2kq.s iny +q2 - cos2 y[ 3

1 1 3 + 3

do2 +q2 +k2 -2kq-siny do2 +q2 + k 2 +2kq.siny

1

do2 +q2 +k2 +2kq+siny 5

+ 1

do2 +q2 +k2 -2kq.siny -3q2 * cos2 y[ 5

IW(3 (16) In case of a sparse row- out of the above three integrals we retain just the second one to leading order. This is found to be equal

(1 - s2) cos2 y or: null when (4) is satisfied. In

1 - e2s2 - a; (SN 2P; (4 to 2(

E2(e2 -s2) any other case- the first integral involves logarithmic expressions while the third one- does not. Therefore there is no relation, may it be (4) or any other that yields a \cI -independent velocity. The optimal choice would be to use (4) as we

524

did above but this means that the induced velocity depends on \v to a leading order through the following term:

(e - s) + J(1- es)’ + k2 - 2 k J a sin y~

-(e + s) + J(1 + es)2 + k2 - 2 k 4 z sin w

(e -s) +d(1- es)2 + k2 + 2 k d z s i n y

-(e+s)+J(l+es)2 +k2 + 2 k J z s i n y

The w-dependency is very weak near the foci of the bodies andor wherever the row is sparse (k relatively large) but is not negligible in any other case. For an ellipsoid on the edge of the row (i=l or m) only one of the logarithms above appears in the equation therefore the W-dependency is stronger.

3.2 Method of Numerical Solution and Error Analysis The integral equations (5-7) have to be solved numerically in order to compute the values of the intensities of the singularities. As Johnson suggested the two “intuitive” approaches are: 1. Iterative solution (used by Barta & Liron, 1988): under the integral sign use the current a value and simply integrate to yield the next iterative value outside the integral sign. 2. Convert the integral equations to a system of linear equations where each integral is replaced by an appropriate summation (e.g., by the rectangle or trapezoid rule). Using the iterative approach is simple and appealing however it proves to be problematic for dense or large rows (where d is very small or m is large). In such cases the solution oscillates and the convergence of the oscillations to the solution depends on the initial guess. Variations of h s simple method that consider combinations of couple of previously computed iterations improved the results for just some of the cases within the parameters range.

The “linear system approach” proved to be effective for every parameters set and was therefore chosen for the solution of the integral system. The interval of integration, 2e is divided to sub-intervals each of ds length. We validate in any case that a change in ds won’t significantly change the results i.e., we are converging to the solution of the integral equations. However, as was discussed above, the integral equations involve inaccuracies (which might be non- negligible when the ellipsoids are densely packed) and we wish to estimate the effect of those inaccuracies on the solution. We apply the following well-known

525

theorem to get an upper bound to the error of the solution, viz., the error involved with solving the linear system (A+GA)(x+ 6x)=b satisfies:

K(A) is the matrix condition number and is quite a small number (a typical situation for stable systems like the Fredholm equations of the 2"d kind). The analysis above determines the relative error of the matrix coefficients for the case of just three bodies moving along the en direction. Comparing the integrands in (15) with those that appear in ( 5 ) yields, after using Taylor expansions, an error of O(E /d2) for the coefficients on the ellipsoids at the edges of the row and O(E 'Id3) otherwise. This error estimation is being used to compute llGAll in (18). For the examples presented in the next section we found that the upper bound for the relative error in a(s) is less than 2%, the actual value depending on the specific chosen vector norm, on ds- the length of the integration subintervals and m a d y on the values of the parameters m, E and d.

A priori we have no better way to estimate the accuracy of the solution process but posteriori we may validate it by: 1. Comparing with known solutions - We verified that the solutions here

indeed coincide with previously known solutions for the case of two relatively distant ellipsoids (Barta & Liron, 1988). For motions along the tangential or bi-normal directions we verify that the secondary intensities computed at the centers of the ellipsoids are extremely small (should be zero by symmetry considerations). Computing the induced velocity on the surfaces of the bodies to check whether it coincides with the dictated no slip value. Specifically, we compute u( s, rl , y~ ) for various points on the surface of any of the ellipsoids:

2.

3.

where we sum k over the three orthogonal directions j=n,s,b. r=l r I and r is the radius vector that connects point (s ,r1,y) and s'- the location of the singularities. Faultless solution would yield a velocity of unity at all points and the deviation from this is an indication of the errors we made. Note that Eq. (19) involves multiplication of the solution (a and /3) by the inverse of the distance from the axis. It thus necessarily enhances the inaccuracies at the ends of the bodies where this distance becomes extremely small. We used Eq. (19) to estimate the accuracy of all the solutions given below and found that on the

526

average the error for u( S, r, , y~ ) equals about 1 % at most near the center of the row and might be as high as 4% at the “ends” of the row. It is expected that the errors would increase towards the ends of the row since our assumption of circumferential symmetry is illegitimate there. We have no way to decide whether the errors found in the solutions at the tips of the bodies are due to the “checking process” (i.e., validation through velocity computations) or due to the method. This does not have practical implications, as actual combs will usually have supports at the ends anyway.

Based on the above error analysis and on many numerical examinations we can justify the claim that the velocities that we compute for any point within the central part of the fluid are usually accurate to within 1% error and the velocities at the peripheral part of the row are accurate w i h n 4% error at most. Increasing m, the number of the bodies in the array andor increasing the gaps between the bodies andor using more slender bodies reduces the errors involved.

4. Results Equations (5-7) were converted to a system of linear equations and the distribution of the intensities of the Stokeslets (actually: the stresses along the axis of each ellipsoid) computed. Henceforth we concentrate on the physically relevant properties such as the total drag acting on each body and the resultant flow field in the vicinity of the bodies. The range of parameters is based on comparison with “real life situations”: E varies between 0.01 and 0.001, d equals (5-50)* E and the no. of rods, m is 10-80.

4.1 Drag

Assuming that the bodies move along each of the 3 orthogonal directions e,, en, eb with unit velocity, the drag exerted on the i-th ellipsoid is given by:

e

D/ = 8 . n ~ Jai(s)ds j=n, s, b i=l, ..... m (20) -e

Note that the drag always has a component in the direction of motion only. Although motion along the tangential or bi-normal directions involves stresses along the other direction as well, these stresses are asymmetric with respect to the center of the body and don’t contribute to the total drag on each body. Those secondary stresses are relatively small. For example, in a row of 10 ellipsoids with d=10& and &=1/300 moving along the eb direction, the maximum local stress in the e, direction (attained at the foci of the first or last body) is one seventh of the maximum local stress along the eb direction. Along the major part of the axis the secondary intensities are much smaller in comparison to the “main” intensities. The secondary intensities evoke a moment on each body. For

527

e, direction

e. direction

a direction

motion along e, direction this actually means that bodies on opposite sides within the row tend to tilt in opposite directions as to increase the spacing between their “heads”. The sum of all the moments is null due to the a- symmetry with respect to the center of the row.

Figure 2 displays the distribution of the drag force exerted on the various ellipsoids normalized by the drag for an isolated ellipsoid as obtained by Chwang & Wu (1975) for motions along the 3 orthogonal directions. Obviously, the presence of adjacent bodies moving in the same direction lowers the drag exerted on the body since they induce a flow within the surrounding fluid so that each body is actually immersed in a fluid that flows in the direction of its motion. The closer the ellipsoid is to the center of the row, the lowest drag exerted on it due to the effect of the near-by ellipsoids which is hghest there. As was found for motion of two bodies (Barta & Liron, 1988) here, as well, the most prominent interaction occurs when the motion is along the bi-normal direction while the least significant one occurs in motion along the tangential direction. Hence, the smallest drag forces are found for motion along the bi- normal direction, see Fig. 2. The sum of all the forces exerted on the bodies is presented in Table 1. The higher the interactions between the bodies are, the less force is needed in order to make the row move therefore- a denser row involves reduced total drag compared to sparser one and motion along the eb direction involves reduced total drag compared to motion along the other directions.

5 .O 8.3 12.5

4.4 6.9 10.1

3.1 4.6 6.5

Table 1 . The over-all drag forces exerted on a row of 50 ellipsoids with E = 1/300 moving along the 3 orthogonal directions for various d gaps, normalized by the drag on a single ellipsoid.

d=lO& d=25~ d=50&

Next we look at the effect of varying the slenderness ratio. Note that by fixing m and the ratio d / ~ but changing E we actually change the “aspect ratio” of the whole row i.e., we change the ratio between its total length and width. We would expect that reducing E or the total span of the row will result with reduced over-all drag. Comparison of Tables 1 and 2 demonstrates this. Moreover, it shows that wherever the interactions are more significant, the sensitivity with respect to E is lower (because then even E as high as 11300 is small enough as to be very effective) e.g., when d=lOE the over-all drag for ~=1/1000 is 77% and 68% of the over-all drag for ~=1/300 for motions along the eb and e, directions respectively. This implies that for “slender rows” the effect of dealing with “multi-bodies configuration” is quite similar for motions along any direction where for more “squared” rows there is a significant difference between motions along different directions.

528

e,direction 3.4 5.1

e. direction 3.2 4.5

6 direction 2.4 3.2

7.3

6.1

4.2

drag

0 . 4 -

0 . 3 5 -

0.3:

0.15

0.1

' i 10 20 30 40 50

Figure 2a. Motion in the normal direction.

drag

0.15

0.1

' i 10 20 30 40 50

Figure 2b. Motion in the tangential direction.

529

drag

Figure

0 . 3

0 . 2 5

0 . 2

0.15

0.1

0 .05

* i 10 20 30 40 5 0

Figure 2c. Motion in the bi-normal direction

Total drag (normalized by the drag for an isolated ellipsoid) of the i- ... ellipsoid in a row of

50 ellipsoids, where the spacing d is equal to 5 0 ~ (top curve), 2 5 ~ (middle) and 1 0 ~ . E = 11300.

Figure 3 shows the dependence of the drag on m. The more ellipsoids we add to the row, the lower the drag becomes. For E = 11300 h s monotonic dependence reaches an asymptotic value when mz.80, indicating that extending a “wing” composed of slender rods beyond a certain limiting number of rods results in diminishing returns. Reduction of E involves more intense dependence of the drag on m thus slightly increasing the asymptotic m value (compare Figs. 3a. and 3b). This form of dependence of the drag on m is found for motion in the other dnections with slightly different numerical values. While Fig. 3 presents the effect of extending a given row, Fig. 4 presents the effect of increasing m but keeping the same span of the row, i.e. increasing density of the row. The more ellipsoids contained in a given space, the lower is the drag on each of them. Moreover, the densely packed situation is characterized by almost uniform forces throughout the row (except for the few bodies on the edges) while the variations between the bodies in the “sparsely packed situation” are considerable. Figure 5 shows the minor effect of changing the slenderness ratio. Replacing the ellipsoids in a given row by ellipsoids that are three times more slender results in slightly less pronounced interactions.

530

" O I : 0 . 2 5 -

0 . 2 -

drag

0 . 3 :

0 . 2 5 :

0 . 2 :

0 . 1 5 .

0 . 1 :

0 . 0 5 :

' r n 2 0 40 60 80 1 0 0

Fig 3a E = 1/300

drag

' r n 2 0 40 60 80 1 0 0

Figure 3b E = 1/1000

Figure 3. The total drag exerted on ellipsoids on the edge (top curve) and in the center of a row moving along the normal direction as a function of m, 10sms100. d = 1 0 ~ .

531

drag

' LengthTotal 0.2 0 . 4 0 . 6 0 .8 1

Figure 4. The total drag exerted on ellipsoids within a row of length 1 moving in the normal direction for E = 11500 and m=l1 & d=50& (top curve) or m=51 & d=lO&.

drag

i 10 2 0 30 40 50

Figure 5. The total drag on each of the ellipsoids within a row of 51 bodies moving in the normal direction for E = 1/500, d=lOi (top curve) and E = 1/1500, d=30&.

4.2 Velocities Another important outcome of the model is the velocity of the fluid that surrounds the ellipsoids. Using ( 1 9) we can compute the velocity anywhere, but in order to get an indication whether the row actually acts like a continuous

532

0 . 9 .

0 . 8 6

surface it is sufficient to compute velocities in the middle of the gaps between the bodies, the midpoints, where we expect the local minimum velocity.

Figure 6 presents those velocities for a row of 10 slender (E =1/300) ellipsoids where d=lOE. In a way, th~s figure demonstrates the same trends demonstrated by Fig. 2, namely: The closer we are to the center of the row the hgher are the interactions (therefore- the low drag and the high velocity of the fluid). The highest interactions occur in motion along the bi-normal direction while the lowest interactions occur in motion along the tangential direction. However, the drag computation provides global information only, while the fluid velocity is a local trait that sheds more light on the situation. Comparing the velocities at three altitudes (positions along the length of the rods) it is clear that a uniform velocity is obtained for the principal part of the comb and the only significant change of velocity occurs near the end of the bodies (at approximately 10% of the length). This result is explained when one considers the solutions computed for the Stokeslets’ intensities- it turns out that for each ellipsoid which is not on the edge a(s) is almost constant along the major part of the axis and then steeply increases towards the foci of the body, see Fig. 7. Moreover, the variation of a(s) from one body to another is minor for most of the row (significant changes just near the boundaries- for the 1-2 extreme ellipsoids). Since the effect of the Stokeslet dissipates rapidly the velocities at any given location are affected m a d y by the singularities at this location (and its near surrounding) and therefore will obey the same rules as the local intensities.

U

Figure 6a. Motion in the normal direction.

533

0.92.

0 . 9 .

U

i 2 4 6 8

Figure 6b. Motion in the tangential direction.

0 . 9 6 0.971

i 2 4 6 8

Figure 6c. Motion in the bi-normal direction.

Figure 6. The velocities at the midpoints between the ellipsoids at heights: s=O, s=0.5e, s=0.95e where m=10, E =1/300 and d=lOE.

534

intensity

0 . 0 3 5 :

0 . 0 3 :

0 . 0 2 5 -

0 . 0 2 :

0 .015 :

' SQ

Fig7a. E = U300

intensity

0 .03

0 .025

0 . 0 2

0 .015

0.01

' SQ 0.2 0 . 4 0 . 6 0 . 8 1

Fig 7b E = 1/1000 Figure 7. The distribution of intensities of Stokeslets along the half-axis of the first (top curve), second (middle) and middle ellipsoid (bottom curve) for a motion along the normal direction where m=10, and d=IOE.

Figure 8 demonstrates the effect of extending the row on the fluid velocity. Note the difference between the situation presented by Fig. 3 where we

535

concluded that the asymptotic value of m should be about 80, and the present situation where for the same d and E values the fluid flows with the ellipsoids for the lion’s share of the row when m250. Further extension would increase the velocity of the fluid a little, however the deviation from a coherent motion for the central portion of the row is already within the limits of accuracy for m=50. This difference is probably due to the effect of increasing m, which is much more pronounced at the ends of the ellipsoids than near their centers.

Figure 9 shows that a row of 50 ellipsoids acts as a continuous wing for all practical purposes when d r 2 5 ~ . Further increasing of the gaps between the bodies induces a non-uniform velocity; the fluid lags a little after the ellipsoids in the central area of the row and lags even more towards the ends. In case that d=50~ further extension of the row (higher m values) will result in a more homogenous fluid velocity across the row, but the fact that the fluid lags after the ellipsoids won’t change e.g., increasing m from 50 to 80 will raise the maximum resultant velocity just slightly (from 0.937 to 0.944). In other words, the most effective way to approach a continuous surface behavior is to lower the distance between rods d. Increasing the number of rods m will have a secondary effect and decreasing their slenderness ratio E will be even less effective.

U

0.965

0 . 9 6

0 .955

0 . 9 5

0.945

i 2 4 6 a

u

. i 10 20 30 40 50

Figure 8. The velocity along the normal direction at the midpoints between the ellipsoids at s=O (the equatorial plane) for m=10 (top figure) and m50. In both cases: E = 1/300, d=lO&.

536

1

0.98

0 . 9 6

0 . 9 4

0 .92

0 . 9

0 . 8 8

. i 1 0 20 30 4 0 5 0

Figure 9. Velocity along the normal direction at the midpoints between the ellipsoids at s=O (the equatorial plane) for m=50, E = 1/300 where d = 5 ~ (top curve), d=lO&, d=25~ and d=50& (bottom curve).

5. Discussion

We have applied the singularity method in order to compute the creeping flow field in the vicinity of a row of slender ellipsoids moving in any given arbitrary direction within a stationary medium. Utilizing an axial distribution of singularities proved to be surprisingly accurate and enabled us to solve for the drag forces exerted on the bodies and for the velocity induced in their surroundings with a relatively low computation effort. Naturally, one wishes to enhance the accuracy of the solution in a rigorous way. We found that the axial distribution of Stokeslets and Doublets presented here produces the best results. We checked the possibility of distributing Quadrupoles and Stresslets as well. Johnson (1980) proved that the accuracy of the solution for an isolated ellipsoid would improve after distributing Stresslets and Quadrupoles with intensities of O ( ~ E * ) and O(ac4) respectively. For the present configuration using such intensities might improve the circumferential a-symmetry by O(E) at most (while the needed correction is of order O( 1)). The option of using higher intensities is excluded because it would interfere with the circumferential symmetry that the singularities induce on “their own ellipsoid”. The limitations on the accuracy of the method of axial distribution are expected due to the assumption of symmetrical flow around each ellipsoid. This assumption is inherent to the method but is physically inaccurate. We analyzed the implication of this assumption and showed that it yields satisfactory results.

We could not compare our quantitative results with other solutions such as Tamada & Fujdcawa (1957) and Ayaz & Pedley (1999) who solved for an infinite row of infinitely long cylinders. They found that reducing d would result in higher drag- in contrast to our computations. This discrepancy is a result of the fact that small d implies very high shear gradient in their configuration (as

537

the fluid cannot “escape” around the row, but has to “find its way” withn densely packed bodies). In the present, more realistic case it implies very low shear gradient (as the fluid is being dragged with the bodies). Therefore, the infinite row case is not the limit of the finite configuration for m-m but it is entirely a different situation. Cheer & Koehl(l987) solved the case of motion of two cylinders and qualitatively reached similar conclusions.

6. Conclusion

The ability to predict for which values of the parameters a row of slender ellipsoids behaves like a continuous surface for all practical purposes may serve as an important tool in designing of wings for minuscule flying or swimming vehicles.

1.The presence of neighboring bodies has a tremendous effect on the hydrodynamic forces, and it is significant even if the bodies are far away (e.g., in Fig. 2 the forces are smaller than half of their isolated body value even when the distance between bodies equals 50 radii. See also Tables 1 and 2).

2. The forces depend on both the total span of the row and the density of the row (Figs. 3 and 4).

3. The interactions between the bodies are less pronounced for bodies of higher slenderness (Fig. 5).

Some qualitative conclusions:

7. Acknowledgements

The authors wish to thank the Deborah Fund at Technion, and the Fund for Promotion of Research at Technion for their generous support.

References

Ayaz, F. and Pedley, T.J. 1999 Flow through and particle interception by an infinite array of closely-spaced circular cylinders. Eur. J. Mech.B/Fluids 18(2),

Barta, E. and Liron, N. 1988 Slender body interactions for low Reynolds numbers - Part 11: Body-body interactions. SIAM J. Appl. Maths 48, 1262-1280. Chwang, A.T. and Wu, T.Y. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes Flow. J. Fluid Mech. 67,787-815. Cheer, A.Y.L. and Koehl, M.A.R. 1987 Paddles and rakes: fluid flow through bristled appendages of small organisms. J. Theor. Biol. 129, 17-39. Happel, J and Brenner, H., Low Reynolds number hydrodynamics, Noordhoff, 1973,553 p. Johnson, R.E. 1980 An improved slender-body theory for Stokes flow. J. Fluid Mech. 99 4 1 1-43 1. Kim, S. and Kamla, S.J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.

173-196.

538

Liron, N. and Barta, E. 1992 Motion of a rigid particle in Stokes flow: a new second-kind boundary-integral equation formulation, J. Fluid Mech. 238, 579- 598. Naveh, R., Yechieli, R., Sachyani, M. and Weihs, D. 2003 MEMS Based Structure for Miniature Aerial System ISRAMEMS 2003, Tel Aviv, Nov. 2003. Sellier, A. 1999 Stokes flow past a slender particle. Proc. R. SOC. (Lond.) A 455,

Sunada, S., Takashima, H., Hattori, T., Yasuda, K. and Kawachi, K. 2002 Fluid- dynamic characteristics of a bristled wing J. Exp. Biol. 205,2737-2744. Tamada, K. and Fujikawa, H. 1957 Quart. J Mech. & Appl. Math, 10,425-431. Zussman, E., Yarin, A.L. and Weihs, D. 2002 A micro-aerodynamic decelerator based on permeable surfaces of nanofiber mats. Experiments in Fluids 33,315- 320.

2975-3002.

THEORY AND NUMERICAL CALCULATION OF HOVERING FLIGHT OF A DRAGONFLY

HIROSHI ISSHIKI IMA (Institute of Mathematical Analysis) Osaka-Sayama, Osaka 589-0022, JAPAN

Several kinds of birds and insects can make hovering flight. Those birds and insects other than a dragonfly generate the upward force by orienting their bodies upward. However, the dragonfly can generate the upward force with the body remaining horizontal. Furthermore, the dragonfly can realize perfect hovering without moving the body horizontally and vertically.

The secret may exists in the fact that the dragonfly has so-called tandem foil system consisting of a pair of fore and aft foils and it can oscillate them independently and freely. The foil rotations have the three degrees of freedom. A single foil can generate vertical force, of course, but if the fore and aft foils of the tandem foils oscillate with opposite phases, the horizontal force cancels completely. Furthermore, the induced velocity of one foil interferes with the motion of the other foil, and there is a possibility of producing a bigger vertical force with smaller energy consumption.

In the present theory, vortex theory of inviscid fluid is applied. Specifically, a two dimensional nonlinear theory of oscillating foils is formulated by using a nonlinear model of shed vortices. Numerical calculations based on the theory are conducted.

1. Introduction

Dragonfly may be one of the best flyers among flying creatures on earth. It can hover with the body keeping horizontal. It can change its direction 180 degrees instantly. It can change the altitude several meters suddenly. Some of them can fly at the speed of 50 kmm.

The secret may exists in the fact that the dragonfly has a tandem foil system consisting of a pair of fore and aft foils and it can oscillate them independently and freely. The foil rotations have the three degrees of freedom. A single foil can generate vertical force, of course, but it generates rather big lateral force. However, if the fore and aft foils of the tandem foil system oscillate with opposite phases, the horizontal force cancels completely. Furthermore, the induced velocity of one foil interferes the motion of the other foil, and there is a possibility of producing a bigger vertical force with smaller energy consumption. The efficient foil motion is a key to determine the energy consumption. Obtaining the efficient foil motion in case of low advance speed may not be an easy task. For the purpose, an appropriate theoretical model may be required. A

539

540

model based on vortex theory (Kochin, Kibel & Roze (1964) and Basu & Hancock (1978)) may give a tool to satisfy the request.

In the present theory, vortex theory of inviscid fluid is applied. Specifically, a two dimensional nonlinear theory of oscillating foils is formulated by using a nonlinear model of shed vortices. Numerical calculations based on the theory are conducted. Ths theory can treat a big foil motion at low advance speed such as at hovering flight. According to this theory, a tandem foil system generates a pair of shed vortices of mutually opposite phases. Since the slipstream is nearly equal to the addition of the slipstream made by the motion of each foil, the thrust force may not decrease much even when the foil motions overlap.

The effect of the leading edge vortex (Ellington & Ushenvood (2001)) is very important in very low Reynolds number flow such as discussed in the present paper. However, the effect is not included in the present study and left in future.

2. Vortex Theory of a Tandem Foil System

The thrust and the efficiency generated by an oscillating foil are discussed on the basis of vortex theory. For simplicity, a two-dimensional theory of a pair of tandem foils is treated.

In Figure 1, motion of an oscillating tandem foil system with advance speed, trajectories of shed vortices and slipstream made by the shed vortices are illustrated. The fore and aft foils make motions with opposite phases. The x-axis is directed downward, and the foil motions correspond to hovering flight of a dragonfly. The tandem foil generates the upward thrust.

The Reynolds number based on wing tip speed of flapping motion at hovering flight of a dragonfly is estimated as 2.1 x lo4, where the chord and span of each of four foils are 0.01m and 0.04~1, the amplitude of the flapping 60deg , and the frequency of the flapping 30Hz . The Reynolds number is rather small, but it may still allow us using vortex theory to estimate the fluid dynamic performance of the flapping.

The coordinates are shown in Figure 1. Let t refer to time. The fore foil makes a sinusoidal surge or flapping motion h(t) of the amplitude b/2 and the circular frequency w :

(1) b 2

h(t) =-cos(wt)

The pitch or feathering motion is denoted by a(t) (see Figure 3). Let the foil advance speed be U and the shed vortex flow downstream relative to the foil at the constant speed of U + AU, . The trajectory of the shed vortex may be given as

541

1 The relation between the frequency w and the wavelength a is given as

2n a

(3)

The shed vortex y(x , t ) may be obtained by the continuity theorem of vortex as

w = - (U + AU,,)

where r(t) is the vortex bound to the foil. The distance between the fore and aft foils is denoted by w . The aft foil makes surge and pitch motions with phases opposite to those of the fore foil. The bound vortex of the aft foil r,,(t) then satisfies

r,, ( t i = -r(t)

Figure. 1 . A pair of tandem oscillating foils with advance speed (The solid line refers to the foil motion and the broken line to the trajectory of the shed vortex).

542

The trajectory of the shed vortex of the fore and aft foils q(x, t) and q, ( x , t) is given as

(6) 1 V ( X , t ) = -qu ( X , t) = - cos - ( X - (U + AUy ) t ) !2 (:

And the x and y components of the induced velocity u(x, y, t) and v(x, y, t) can now be written as

where w is the distance between the fore and aft foils as shown in Figure 1, the fust and second terms on the right side are the velocity components induced by the shed vortex of the fore and aft wings.

A relation between the shed vortex velocity AU,. and the velocity of the slipstream:

is assumed. This equation is used to determine the shed vortex velocity AUy . The momentum conservation law in the time average flow holds, i.e.:

(9)

approximately, when the time variation is small. The time averaged velocity distribution of the induced velocity component u is assumed as (+ Figure 2)

The appropriateness may be given by Figure 6b. Substitution of Eq. (10) into Eq. (9) gives

2K 2% fnimpr(t)(h, (o,t) - v(o,h(t),t))dt

543

=32pU’b[-!&+&(%)’], when w 2 b

2 2(w-b/2)’ + b ’ ( ~ - b / 2 ) ~ 3b4 3b’ - -(w - b/2)- - b’ 5 6 16 160

+-[- (w- b/2)3 w + - ’ - yr}, when w < b b4

8 3 + - W + ( w - b/2)4 W - 2

(1 1b) These equations may be used to obtain the initial value of AUl in iterative calculation, that is AUT = U, (0) .

’t w + b/2

W F

0 AU

F A Y ) - b/2

Figure 2. Time averaged velocity distribution in slipstream in case of a tandem foil system.

Figure 3 shows the inflow to the foil. The relationship between pitch motion or the geometrical attack angle a( t ) of the foil and the bound vortex r(t) may be given as

where C,(a) is the lift coefficient and c is the chord length. If the lift coefficient C, (a) is approximated by

C,(a) = 2 a a (13)

544

Then, a is obtained as

+ r(t) n c J(u + u(0, h(t), t))' + (- h, ( t ) + v(0, h(t), t))'

The effective angle of attack ag ( t ) is given as

Figure 3. Thrust generated by a fore foil.

The thrust T ( t ) , the lateral force F ( t ) and the input power P(t) of the fore foil are calculated by

T ( t ) = p W ( h , (0,t) - v(0, h(t) , 0 ) W ) = - p W @ + U ( O , h ( t ) , 0 )

(16a) (16b) ( 16c) P(t) = -F(t)h, (0, t )

and the mean thrust T , the mean lateral force F of the fore foil and the efficiency T,I are written as

- T = 6"" T( t ) dt

F = f"" F(t ) dt

27r

2n -

- P = 27r P""P(t)dt

~

The reduced frequency LT (inverse of the advance ratio), the thrust coefficient C, , the power coefficient C, and the efficiency T,I are defined as

545

WC c=- U

T c, =- -u2c P 2 -

T c, =- - U’C P 2

TU q = - P

3. Numerical Calculation

Numerical calculations are conducted on the basis of the theory discussed in the previous section. A tandem foil system is treated. First, cases with the given bound vortices are discussed. Cases with the given foil motions are then discussed. The unknowns of the former problem are foil motions, and those for the latter problem are the bound vortices. Since the problems are nonlinear, the solutions must be obtained by iterative procedures. The former problem is rather simple, but some ideas are required for the convergence of the calculation of the latter problem. In the former problem, the direct unknown is the slipstream velocity AU,. , and the foil motions are determined by using AU,. . The unknowns for the latter problem, namely the slipstream velocity AUy and the bound vortex r , must be determined simultaneouly. In the iterative procedures, the unknowns are not renewed 100 %, but the 50 % of the old values are preserved in the iterations. The convergence was judged by monitoring AU,. .

3.1. Motion of a Tandem Foil System With Bound Vortices Given

The bound vortex r(t) is given as r(t) = -r, in(^ t + cr1)

The calculation of the velocity components induced by the shed vortexes (+ Eq. (7)) is conducted by using the following more specific equations:

546

2 8 (U + AU,)’ +- b2 m2 sin2( ?!! (4 - (U + AU,)t))

4 a

1+ - sin2 - ( ( r - (U+AU,) t ) ,I (: ( ; (’a ( x - t y + y - - c o s - ( ( r - (U+AU, ) t )

2z , (U+AUy)2 + - m 2 s ~ ’ ( ~ ( ( r - ( U + A U r ) t ) ) bZ 4 a

547

2n

£0\t--(C/ + AC/J

J(C7 + AC/,)2 + -j«2 sin2f^(^ - (17

sina 1 V a

(20b)

Let b ,U , co , £ri and F, be given. First, the initial value of At/j, must

be determined. For the purpose, Eq. (11) is solved for A£7. Let this be theinitial value of At/r . The wavelength a of the shed vortex is obtained by Eq.

(3). Substitution of these into Eq. (20a) gives an approximation of M(OO,O,?) .

Substituting this into Eq. (8), the renewed value of At/^, may be obtained. Then,

a is renewed by Eq. (3). The iteration of these calculations converges to thesolution of the problem.

In the following calculations, the calculation condition specified in Table 1is used. The convergence of the calculation is carefully checked.

Table 1. Calculation condition with the bound vortex given

ItemAmplitude of Foil Surge

MotionAdvance Velocity

Frequency

Amplitude of Bound Vortex

Phase of Bound Vortex

Chord Length

Notation

6/2

U

f

r,*nc

Value

0.03m

0.001 m/s

30 Hz

0.025 m 2 / s

-ISdeg

0.01m

The pitch motion a of the fore wing is obtained by Eq. (12) or (14).The effects of the distance W between the two foils on the geometrical and

effective attack angles a(t) and a^(t) of the fore foil are shown in Figures

4a~b. Those on the thrust T(t), the lateral force F(t) and the input power P(t)are shown in Figures 5a~c.

548

.-

0 0.01 0.02 0.03 t ... (332)

Figure 4a. Effects of the distance w between the two foils on the variation of the geometrical attack angle a .

15 - g 10 s - 5

: o a, -5

-10

5

Q.

- 15 - 20

0 0.01 0.02 0.03 time ... (sec)

Figure 4b. Effects of the distance w between the two foils on the variation of the effective attack angle Q;/I .

-g- 0.15 . 5 0.1 h

0.05

0 0 0.01 0.02 0.03

time ... (sec)

w=0.015

w=0.06

Figure 5a. Effects of the distance w between the two foils on the variation of the thrust T(r) of the fore wing.

549

0.04 0.03 0.02

z 0.01 : o

- 0.01 LL -0.02

- 0.03 - 0.04

-w=0.045

\

v

Y

0 0.01 0.02 0.03 time ... (sec)

Figure 5b. Effects of the distance w between the two foils on the variation of the lateral force F ( t ) of the fore wing.

02

0.15 h

E

3 0.1 Y

: 0.05 - Y

Q O

- 0.05 0 0.01 0.02 0.03

time ... (sec)

Figure 5c. Effects of the distance w between the two foils on the variation of the input power P(r) of the fore wing.

In Figure 6a-b, the distribution in x and y direction of the time averaged induced velocity u are shown. Figure 6b supports the appropriateness of an assumption that the distribution in y direction of the time averaged induced velocity u is approximated by Eq. (10).

550

1 4 Q 1 2

w=0.03 - - - -W=0.045 - - - - . - .*0.06

Z ’ : 0.8 0.6 0.4

0 = 0 2

- 3 - 1 1 3 5 XI a ... (- )

Figure 6a. Effects of the distance w between the two foils on the distribution in x direction of the induced velocity u where unVg is averaged in -b/2 < y < w+ b/2 .

2 r I 1

I I I -0.5 I I

-0.1 -0.05 0 0.05 0.1

I I

Figure 6b(l). Distribution in y direction of the time averaged induced velocity u where uaVg is

averaged in one cycle ( w = 0.0 15m , b/2 = 0.03m ).

2

1.5

1 0.5

0

I -0.5 I I

-0.1 -0.05 0 0.05 0.1 Y ’.. (m)

~ -w=0.03

Figure 6b(2). Distribution in y direction of the time averaged induced velocity u where uovg is

averaged in one cycle ( w = 0.03m. b/2 = 0.03m ).

551

2

1.5

1

0.5

0

0.5 -0.1 -0.05 0 0.05 0.1 0.15

I Y ...

Figure 6b(3). Distribution in y direction of the time averaged induced velocity u where urn,# is

averaged in one cycle ( w = 0.045m , b/2 = 0.03m ).

0.5

= o m

I

-0.5 ' I

'0.1 -0.05 0 0.05 0.1 0.15

Y ... (m)

Figure 6b(4). Distribution in y direction of the time averaged induced velocity P where uaYb is

averaged in one cycle ( w = 0.06m , b/2 = 0.03m ).

In Tables 2, the effects of the distance w between the two foils on the thrust T per foil, the power P per foil and the thrust to power ratio TIP are given. The advance speed is U = O.OOlm/s, virtually zero. The thrust to power ratio T / P = 1.715 of a single foil (calculated in advance) for the same thrust is bigger

than the thrust to power ratio for a tandem foil system (+ Table 2). However, if the load of the single foil is doubled, the thrust to power ratio T / P = 1.127 (calculated in advance) is smaller than that for a tandem foil system.

552

Table 2. Relationship between the thrust T per foil, the power P per foil and the thrust to power ratio TIP and the distance w between the two foils

w = 0.01 5m w = 0.03 w = 0.045 w = 0.06 Thrust T 0.08358 0.08258 Power P 0.06185 N/s 0.06651 0.05505 0.04826 Thrust to

Power Ratio 1.370 s/m 1.276 1.518 1.711

3.2. Bound Vortices of a Tandem Foil System With Foil Motion Given

In a problem where the bound vortex T(t) is given and the pitch motion a(t) is solved, the basic unknown of the problem is the shed vortex velocity AUr . The pitch motion is obtained as the result. The convergence of the iteration procedure is straightforward. On the other hand, in a problem where the pitch motion a(t) is given and the bound vortex T(t) is solved, the basic unknowns of the problem are the bound vortex T ( t ) and the shed vortex velocity AU, . The convergence of the iteration procedure needs more care.

Besides what was discussed in Section 3.1, some additional details are required. The attack angle is increased from the zero lift state. The Fourier analysis of bound vortex T( t ) is conducted, and only the fimdamental frequency component is flowed downstream. This may be problematic from the theoretical viewpoint and may be a kind of an expedient. The numerical results seem to be reasonable notwithstanding this limitation.

Let b , U , w and a be given. First, r(t) is obtained from Eq. (12) or (14), where u(O,h(t),t) and v(O,h(t),t) are neglected. The wavelength of the shed vortex a is obtained by Eq. (3), where the shed vortex velocity AU? is neglected. AUr is then determined from Eq. (8) where u(co,O, t ) is calculated by Eq. (20a). Substitution of these into Eq. (20) gives an approximation of u(0, h(t), t) and v(0, h(t), t) . Substituting these into Eq. (14), the renewed value of r(t) may be obtained. Then, a is renewed by Eq. (3). The iteration of these calculations may converge to the solution of the problem.

First, the same problems as discussed in Section 3.1 is considered. Namely, the pitch motion a(t) that is obtained as the solution in Section 3.1 is given as the input condition of the problem in this section, and the bound vortex T( t ) is solved. T( t ) obtained in this way is compared with the one assumed in Section 3.1. Almost the same value is obtained.

553

Item Amplitude of Foil Surge Motion

Advance Velocity

This problem is not directly related to the hovering flight, but the purpose is to show the ability of the algorithm

Table 3. Calculation condition with the foil motion given

Notation Value b/2 0.015 m U 3.0mIs

Frequency Amplitude of Foil Pitch Motion

Phase of Foil Pitch Motion Chord Length

f 30 Hz a 25 deg E , -1Odeg C 0.01 m

30 20

i o

-20

g 10

%-I0

Figures 7a-b shows the geometrical attack angle a(t) (i.e. input condition) and the effective attack angle acfl (t) . The variations of the thrust T ( t ) per foil,

the lateral force F ( t ) per foil and the input power P ( t ) per foil are shown in Figures 8a-c. The effects of the distance w between the two foils may be read.

-30 0 0.01 0.02 0.03

t ... (sec)

-~=0.045 ----~=0.06 I

Figure 7a. Variation of the geometrical attack angle a(?)

h 20 2 15 I? 10

i 5 - - 0

I -5 -10 I - ~ = 0 . 0 6 - 15 - 20

~=0.015 - w=0.03 '

c ----~=0.045 a,

0 0.01 0.02 0.03 time ... (sec)

I I

Figure 7b Effects of the distance w between the two foils on the vanation of the effective

angle aqf

attack

554

-------

0.14 - 0.12 E

Y 0.08 2 0.1

2 .4- 0.06 i= 0.04 0.02 0

0 0.01 0.02 0.03 time ... (sec)

Figure 8a. Effects of the distance W between the two foils on the variation of the thrust T ( t ) of the

0.15 E 0.1 5 0.05

L O - 0.05 - 0.1

h

-0.15 ' 1

0 0.01 0.02 0.03 time ... (sec)

Figure 8b. Effects of the distance w between the two foils on the variation of the lateral force F(r) of the fore wing.

0.5 1 - 4 0.4 2 E 0.3 z 02

0.1

0 Y a

-0.1 ' f

0 0.01 0.02 0.03 time ... (sec)

w = 0 . 0 1 5

/-w=O.O6 1

Figure 8c. Effects of the distance w between the two foils on the variation of the input power P(t) of the fore wing.

555

Figure 8c. Effects of the distance w between the two foils on the variation of the input power P( t ) of the fore wing.

The effects of the distance w between the two foils on the thrust coefficient C, per foil, the power coefficient C, per foil and the efficiency q are shown in Table 4.

Table 4. Relationship between the thrust coefficient C, per foil, the power coefficient

POWerCoefft. c, 1.0884 1.078 1 0.9744 0.9573

4. Conclusions

A theory based on vortex theory was developed for analyzing performance of an oscillating foil advancing at low speed, and numerical calculations were conducted to verify the theory. Although a comparison with experiments has not yet been conducted, the numerical results seem appropriate from theoretical viewpoint. Introduction of viscous effects will make it possible to compare the theory with experiments (Scherer (1968), Jones and Platzer (1999)).

As a theoretical task, the convergence when the foil motion is specified should be reexamined to improve its convergence property. An extension of the theory to three-dimensional problems should also be conducted in future.

From a viewpoint of practical applications, the determination of the efficient foil motion for thrusting and hovering of a flying robot is very interesting. This topic may be extremely important to realize the long flight, since the efficient foil motion saves the energy consumption and elongate the battery life. In the present paper, it is confi ied that the tandem foil system has no disadvantage from the point of the energy consumption. What kind and amount of plus it may have is an interesting problem left in future.

The effect of the leading edge vortex (Ellington & Usherwood (2001)) is very important in very low Reynolds number flow such as discussed in the present paper. However, the effect is not included in the present study and left in future.

Acknowledgements

The author would like to express his sincere thanks to Dr. H. Narita, Senior Advisor for Science & Technology, Office of Naval Research Global-Asia and

556

Prof. K. Nakatake of Department of Marine Systems Engineering, Kyushu University for their valuable advices and warm encouragements.

References

1. Basu, B. C . and Hancock, G. J., 1978, The Unsteady Motion of a Two- Dimensional Aerofoil in Incompressible Inviscid Flow, J. Fluid Mech., part

2. Ellington, C. P. and Usherwood, J. R., 2001, Lift and Drag Characteristics of Rotary and Flapping Wings, , Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications edited by T. J. Mueller, Progress in Astronautics and Aeronautics, Vol. 195, AlAA, pp.23 1-248.

3. Jones, K. D. and Platzer, M. F., 1999, An Experimental and Numerical Investigation of Flapping-Wing Propulsion, AIAA Paper No. 99-0995.

4. Kochin, N. E., Kibel, L. A. and Roze, N. V., 1964, Theoretical Hydrodynamics, International Publications, P.2 14-2 18.

5. Scherer, J. O., 1968, Experimental and Theoretical Investigation of Large Amplitude Oscillating Foil Propulsion Systems, U. S. Army Engineering Research and Development Laboratories Contact Number DA-44-009-

1, pp. 159-178.

AMC-1759 (T).

A NUMERICAL STUDY ON FLUID DYNAMICS OF BACKWARD AND FORWARD SWIMMING IN THE EEL

ANGUILLA ANGUILLA*

WENRONG HU, BINGANG TONG Department of Physics, Graduate School of the Chinese Academy of Sciences,

Beijing, 100039, China

HA0 LIU Department of Electronic and Mechanical Engineering, Chiba University, Japan

In addition to forward undulatory swimming, eels have the ability to swim backward. A computational study on fluid dynamics of backward and forward swimming performance in the European eel Anguilla anguilla is presented with a special focus on the analysis of its propulsion mechanism. A two-dimensional geometric model of the European eel is approximated by a NACA0005 airfoil, which represents the middle horizontal section of the eel body. Kinematic data of the backward and forward swimming eel are based on the experimental results of the European eel [S ] .

Results are presented for the backward and forward swimming hydrodynamics, indicating that the backward swimming mechanism is somehow like a ‘reversed forward’ swimming one, i.e., thrust is generated when a jet-stream is formed in the wake with a reversed Khrman vortex street. However, the backward swimming utilizes a constant- amplitude mode, which can perform larger force generation for maneuverability (maneuverability-based swimming) but achieve lower propulsive efficiency compared with the linearly-increasing amplitude mode in forward swimming (efficiency-based swimming).

An extensive study of the effects of frequency, amplitude and wavelength of the traveling wave on the backward swimming hydrodynamics shows obvious similarity with those observed in forward swimming and an extensive comparison is conducted between the forward and backward swimming to elucidate why the eels select different locomotion modes and how each matches their forward or backward swimming.

1. INTRODUCTION

It is well known that most aquatic vertebrates swim by a traveling wave propagating from head to tail. However, a few fish species can swim backward

This work is supported by the NSFC Project No. 10332040 and the CAS Project No. KJCX-SW- L04.

557

558

by reversing the direction of the traveling wave along the body in case they need to escape from their enemies or maneuver in dense vegetation or maze-type environment. All these species such as eels, moray eels and congers [ 181 have elongated and flexible bodies and swim forward in the anguilliform mode. In this mode, more than one-half of a sine wave occurs along the length of body during swimming. Most previous studies of the undulatory swimming focus on the forward swimming. Much experimental studies have been developed to explain forward swimming performance of fish [15], Miiller et a1 Ell], Wolfgang et a1 [19]. Several excellent analyses based on the inviscid potential flow theories by Lighthill [6], Wu [21] and Cheng, et al. [3] have revealed many key points of undulatory locomotion. More recently, Carling, et al. [2] simulated the self- propelled anguilliform swimming by modeling the fluid with the Navier-Stokes equations, and the motion of the body with Newton’s laws of motion. However, there are some problems of the streamlines pattern in this paper, which conflicts with the experimental results [l I] . Carling, et al. [2] , Liu [8][9] and Wolfgang et a1.[19] simulated numerically the forward undulatory swimming, which enhanced our understanding of the importance of viscous fluid phenomena in estimating the propulsive efficiency.

The hydrodynamics and propulsive mechanism associated with the backward swimming, however, have hardly been taken as a subject. D’aoClt and Aerts [5], for the first time, provided a detailed description of the backward swimming kinematics in the European eel, Anguilla anguilla. They observed two major kinematic differences between the forward and backward swimming.

In this paper, we use a two-dimensional computational fluid dynamic modeling method to study the unsteady hydrodynamics of the backward undulatory swimming in the European eel, Anguilla anguilla, with a specific focus on the analysis of the propulsion mechanism associated with the vortex structure and the dynamic features. Furthermore, an extensive comparison is conducted on the unsteady hydrodynamics between the forward and backward swimming to elucidate why the eels select different locomotion modes and how each matches their forward or backward swimming.

2. METHODOLOGY

2.1. Geometric model

According to the three-dimensional waving plate theory [3], we know that the angulliform mode in undulatory locomotion has the properties of two- dimensional flow when the traveling wave amplitude keeps constant or increases

559

slightly along the body length, and approximately a complete wavelength can be observed at any instant. In fact, the forward and backward swimming mode of eels is just the case, and hence a two-dimensional model may be an appropriate approximation in studying the feature of its undulatory swimming. Based on the body shape of the European eels [ 5 ] , we employ a NACAOOO5 airfoil to approximate the middle horizontal section of the eel body as illustrated in Fig. 1.

(a)

(b)

'I 0 4

Fig. 1 . Definition of the eel model (a) Body shape of the European eel [ 5 ] (b) NACA0005 airfoil

2.1. Navier-Stokes equations solution

A robust, in-house CFD solver of two-dimensional Navier-Stokes equations is utilized in this study, which has been applied to many realistic biological problems successfully [8][9].

The governing equations are the two-dimensional, incompressible, unsteady Navier-Stokes equations written in strong conservation form for mass and momentum. Introducing the artificial compressibility method and transforming these equations into a generalized curvilinear coordinate system with a moving boundary, the nondimensionalized governing equations can be rewritten as:

where

In the preceding equations, p is the pseudo compressibility coefficient; p is pressure; u and v are velocity components in Cartesian coordinate system (x , y); t denotes physical time and z is pseudo time. Re is the Reynolds number. V(t) are the area of the cell ( i , j ) constructed by four grid points, and l(t) denotes its

560

four edges with unit outwards normal vectors of n=(nx, nJ. Subscripts x and y express derivatives with respect to x and y. To analyze the moving object of an undulating swimmer that continuously deforms, a body-fitted mesh system being regenerated at each time step is introduced. This leads to a contribution from grid velocity Ug=(ug, vg) to the governing equations.

Discretization of the governing equations is implemented in a way of finite volume method. A third-order upwind differencing scheme is used for the convective term in a manner of the MUSCL scheme (ultimate conservative differencing scheme[l7]), and the viscous term is evaluated by a Gauss integration method resulting in a 2nd-~rder central differencing. An implicit factorization approximate method, based on the Euler implicit scheme, is employed for the time integration. In the time-accurate formulation, the time derivatives of the velocity components in the momentum equations are differenced using a frrst-order, two-point, backward-difference formula and an inner-iteration is carried out to satisfy the equation of continuity at each physical time step.

Because the computational model performs time-varying undulating swimming, a moving grid system is hereby introduced, which guarantees the grids to fit the deformed boundaries at each physical time step. To incorporate the dynamic effect due to the rapid deforming boundary, pressure divergence at the surface stencils is used for the pressure condition, which is derived from the local momentum equations.

2.2. Kinematics of swimming

In the present analysis, the lunematic data of the backward and forward swimming are based on the study of the European eel[5].

The total length of the European eel is 0 . 2 2 ~ which is chosen as the reference length in the non-dimensional analysis. The initial geometry of the computational model is a NACA0005 airfoil, which well approximates the two- dimensional configuration of the eel [5].

The lateral motion of the centerline of the eel can be described by a sinusoidal function:

where a(x) represents amplitude, A is wavelength, T is period, h(x,t) is the transverse displacement of the body center line, t is time, and x is the coordinate in the x-direction corresponding to the body length. Introducing reduced

561

frequency of k = 2@/(2U), where f is frequency, L is body length and U is swimming speed, Eq. (2) can be rewritten as:

h(x,t) = a(x) sin[2z(x/;l) - 2kt]. (3) Based on the observed data [ 5 ] , the swimming speed in both forward and

backward swimming is taken as 0.1742ms-'. Hence, the Reynolds number is calculated to be Re = UL/v = 2.5 x lo4 , where V is kinematic viscosity, with a value of 1.533 x 10-6m2s-'. The wavelength of the traveling wave in forward and backward swimming is approximately 0.92 [18]. Note that there are two major kinematic differences in forward and backward swimming. Firstly, the slope of wave frequency against swimming speed is significantly higher in backward swimming than that in forward swimming. Correspondingly, the reduced frequency is calculated to be 4.7124 in forward swimming and 7.854 in backward swimming. Secondly, the amplitude of the traveling wave shows apparent discrepancy in the two cases: the amplitude in forward swimming can be fitted in a form of a(x)=0.07x+0.01 whereas it keeps constant with a(x)=0.05 in backward swimming.

2.3. Hydrodynamic performance

Navier-Stokes equation, which is for an incompressible fluid motion is: 1

Re dvldt = -vp + - v 2 v , (4)

in which the stress tensor ( aij ) in an incompressible fluid takes the form such as:

oii = +l/R.(hi/8xj + h j / a x i ) ( 5 )

On a basis of the solutions of the Eq.(l), that the velocity and pressure are computed at each discretized cell center, the total force acting upon the body surface can be obtained by integrating the pressures and shear stresses over the body surface, such that:

Note that, for the high Reynolds number flow about a slender body like an eel, it is reasonable to consider that the averaged thrust comes from the pressures acting on the body surface whereas the shear stresses contribute merely to the

562

mean drag force. Hence, the thrust (FT) , drag (FD) and the resultant force on x- axis (F,) in the backward swimming can be defined as:

F, = FT - FD . (9) The power required for the undulating swimming, i.e., the work done in

a unit time, can be considered as the work to undulate the body such that,

where p is water density and Ai(x) is the width (area of unit cross section) of the section i on the center line. All these force-related parameters can then be nondimensionalized such that we get:

c F , = C T - c D . (14)

?j-=c,.uJc,. (15)

For an undulating swimmer, we define the mechanical propulsive efficiency, i.e., the rate of effective work done, in a time-averaged manner, such that:

3. RESULTS AND DISCUSSION

3.1. Validation test of the computational method

A variety of validation tests have been undertaken [7][8][9] to assess the reliability of the present method. Here we illustrate the friction drag on the NACA0005 airfoil which is in excellent agreement with the analytical results of 2D flat plate of finite length [ 141 (Fig. 2 ).

In order to validate the present modeling method with moving grid system, an extensive study of the flow past an oscillating NACA0012 airfoil was also conducted with the grid cells rotating with the airfoil simultaneously in the

563

inertial reference frame at each iterate time step. A comparison with experiments is illustrated in Fig.3.

. Present result -Analytic result ’I

40 100 1000 Re

Potential llowtheory

St

0.7 o,6 [ - Present numerical resun

Experimental result - Potential llowtheory

/ 0.5

0.3 0.41 0.1 0.2 0.3 0.4

St

Fig.2 Friction drag coefficient of a NACA0005 airfoil Fig.3 Mean thrust coefficicient of an oscillating NACA0012 airfoil (Re=l.2x 104)

3.2. Present grid system

A grid system with 140x70 grids is generated around the eel body, with grids clustered to the eel body, in particular around the nose and tail. The minimum grid spacing adjacent to the body surface is determined by the empirical formula *A = o . l l a = o . o o 6 . To avoid unstable reflection of the solution at the open boundary, a distance far from body is taken the eight times of the body length. Fig.4 shows the local grids near the eel body.

Fig.4 h a 1 grids near the body

4

- Fig. 5. Average thus (‘r ) and average resultant force

on x-axis direction ( ‘Fx )against reduced frequency in backward swimming (a(x)=0.05, Re=2.5x1O4, n=0.92) .

3.3. Propulsive mechanism of backward swimming

It is known that during forward swimming, thrust generation may be achieved when the formation of a reverse K h t i n vortex street is observed in the wake [8]. In backward swimming, our results indicate that the mechanism of

564

the thrust generation is somehow like a reversed replica of that in forward swimming, i.e., the thrust is generated when the time-averaged flow in the wake forms a Kirmin reversed street, which generates a jet-stream opposite to the swimming direction.

However, an undulatory eel can produce thrust only under proper parametric conditions. Fig. 5 illustrates the average thrust ( cr) and the average resultant force in x-axis direction ( c~= ) against reduced frequency (k) in backward swimming. Both forces increase with increasing the reduced frequency. At the point where the reduced frequency is 7.5, the average resultant force in x-axis direction is nearly zero, which implies that the average thrust just balance the average drag at the point and hence the eel can swim backward at a uniform speed. Though the eel is observed to swim backward at a reduced frequency of about 7.854 [5] , the present result is a very good approximation with consideration of the fact that three-dimensional effects were neglected among other approximations (as described above).

(a) k=3.141 (b) k=7.854 (c) k= 1 2.566 Fig. 6. Vorticity contours at three different reduced frequencies. Solid lines denote positive vorticity, broken lines negative vorticity. The left end of the object is the snout of the backward swimming eel. (Re = 2 . 5 ~ lo4, a(x)=0,05, 1 = 0.92 ),

(a) k=3.141 (b) k=7.854 (c) k=l2.566

broken lines negative values. The left end of the object is the snout of the backward swimming eel (Re=2.5x1O4, a(x)=0.05, n=0.92).

Fig. 7. Velocity contours at three different reduced frequencies. Solid lines denote positive values,

F l ~ ~ ~ ~ 96)-

(a) k=3.141 (b) k=7.854 (c) k=12.566 Fig.8. The momentum flux on a far-field transverse plane in the wake. The left end of the object

is the snout of the backward swimming eel. (Re=2.5X1o4, a(x)=0.05, 1=0.92).

Fig. 6 shows the vorticity contours at three different reduced frequencies where the wavelength is 0.92, the amplitude is a(x)=0.05, and the reduced frequencies are taken as 3.141, 7.854 and 12.566, respectively. In the case (a) where no thrust is generated (Fig. 5), the K h B n vortex street is left in the wake, i.e. two zigzag vortices appear at a specific space apart of such opposite senses as the vortices behind a blunt body. In the case (c) where thrust (Fig. 5) is generated that leads to acceleration in the backward locomotion, the wake pattern is more like a KBrmBn vortex street but with the vortices rotating in a reversed direction. In the case (b), the thrust (Fig. 5) is mostly balanced by the drag, and the eel swims backward approximately at a uniform speed. Here, in the wake, the vortex centers of the reversed K h B n vortex street are observed to be aligned approximately, that is consistent with the new statement by Wu [ 2 2 ] .

r=OT k0.2T l=0.4T

k0.6T

r=OT

l=0.8T (a) k=3.141

t=T

r=0.2T t=0.4T

r=0.6T t=0.8T ( b ) k=7.854

t=T

566

t=OT t=0.2T h / I ' , '.. I . . . .

t=0.4T

t=0.6T t=0.8T ( c ) k=12.566

t=T

Fig. 9 Pressure contours at three different reduced frequencies. Solid lines denote positive pressure, broken lines negative pressure. The left end of the object is the snout of the backward swimming

eel. (Re = 2.5x104, a(x)=0.05, a = 0.92).

The velocity contours at three different reduced frequencies are shown in Fig. 7. Note that here the velocities are calculated by subtracting the swimming speed from the velocities in the flow field. Furthermore, the sequent images as shown in Fig. 8 illustrate the momentum flux through a far-field transverse plane in the wake. From both Fig.7a and Fig. Sa, it can be seen that when the frequency is low (k=3.141), the Kirmin vortex street forms a jet-stream resulting in a momentum loss in the wake, which prevents the eel from backward swimming. When the average thrust just balances the average drag at the reduced frequency of 7.854, the wake is of a non-momentum one (with the momemtum flux over the cross plane integrated to be zero) (Fig. 7b, Sb). Therefore, the vortex centers of the reversed K6rmin vortex street is aligned approximately (Fig. 6b). When the reduced frequency is 12.566, a strong jet-stream is observed obviously that is induced by the two rows of stagger vortices at a specific space apart, generating a net flux of the wake momentum, i.e., thrust on the eel body surface (Fig. 6c, 7c, Sc).

To further elucidate the thrust generation mechanism, Fig. 9 is depicted to show the pressure contours at the three different reduced frequencies. The absolute value of pressure increases as the frequency increases. The pressure distribution varies with the body wave traveling. It is seen that the pressure distributions are quite different in the drag- generation case (Fig. 9a) from those in the thrust-generation cases (Fig. 9b,c). In Fig. 9a, the pressure adjacent to the

567

body surface keeps negative throughout the stroking period and hence drag but no thrust is generated. In Fig. 9b-c, there exist both positive and negative regions simultaneously over the stroking period. The alternating positive and negative pressure zones traveling from the snout down to the body tail create thrust. We can also see that the pressure-based forces act mainly on the anterior part and the middle part of the body in the drag-generation case, but in the net thrust- generation case the pressure-based force is observed acting almost on the posterior part and the middle part of the body.

3.4. How the traveling wave affects backward swimming hydrodynamics?

Here we discuss the effects of the undulatory frequency, wavelength and the amplitude profile on the backward swimming hydrodynamics of the eels.

3.4.1. Propulsive eficiency versus undulatory JLequency

As mentioned above, both the average thrust and average resultant force in x direction increase when the undulatory frequency increases. As shown in Fig. 10, the power required for undulating swimming also increases with increasing frequency. According to the elongated-body theory of Lighthill [6] on the basis of the linear inviscid assumption, the propulsive efficiency may be calculated such as,

where v = fAdenotes the wave speed and h ' (L) is the local slope at the tail tip, which obviously influences the propulsive efficiency.

Accordingly, the propulsive efficiency should monotonously decrease with increasing frequency. However, in the present study it is not the case. There are two main differences. Firstly, the propulsive efficiency reaches the maximum when the reduced frequency is 6.282. Corresponding to this frequency, the Strouhal number (St=fA/u) is 0.2, which is defined as a production of the oscillating frequency ( f ) multiplied by the width (A) of the wake, divided by the swimming speed (U). Based on the analysis of airfoil characteristics in pitching and heaving motion, Triantafyllou et al. [16] pointed out that the propulsive efficiency shows an optimal variation against the Strouhal number, with the maximum between 0.25 and 0.35. Secondly, the propulsive efficiency in backward swimming is much lower than that predicted by the elongated-body theory of Lighthill [6] (Fig. lo), which is thought due to the viscous effect.

568

Fig.10 Average power coefficient and propulsive efficiency versus Strouhal number St in backward swimming, (a(x)=o.05, Re=2Sx1O4, A=0.92),

Fig. 1 1 Propulsive efficiency versus Strouhal number St in the forward swimming, (a(x)=O.Mx+O.Ol, Re=2.5x1O4,A =0.92).

On the other hand, the present results show that the forward swimming achieves much better propulsive efficiency (Fig. 11) than the backward swimming (Fig.lO), which agrees with the experiment results [5 ] . Again, due to the viscous effect, the computed propulsive efficiency of the forward swimming is also lower than that based on the elongated-body theory (Fig. 11).

3.4.2. Amplitude effect

Fig. 12 shows the variation of the dynamic features with increasing amplitude in backward swimming. All the average quantities of the thrust coefficient ( F ~ ) , the resultant force coefficient in x direction ( ' F X ), the power coefficient ( c~ ) and the propulsive efficiency ( ) increase as the amplitude increases, which is quite similar with those in forward swimming.

-

3.4.3. Wavelength effect

The variation of dynamic features with the wavelength in backward swimming is illustrated in Fig. 13. The computed results show that the average quantities such as the thrust coefficient ( c~ ), the resultant force coefficient x direction ( c~= ) and the power coefficient ( F~ ) increase with increasing wavelength. In contrast, the propulsive efficiency (7) decreases with increasing wavelength as predicted by the elongated-body theory in forward swimming.

569

Fig. 12 Average values of thrust coefficient (c*),

resultant force coefficient in x direction (CF' ),

power coefficient ( C f ) and propulsiveefficiency ('Jversus amplitude in backwardswimming. (<* = 0.92 ;jt=7.854; Re = 2.5xio4

;

a(x) = 0.03, 0.05 and 0.08).

Fig. 13 Thrust coefficient (cr), resultant force

coefficient in x direction (Cf<), power coefficient

(cc) and propulsive efficiency (') versuswavelength in backward swimming. (a(x)=0.05;

*=7.854; Re = 2.5x:o4;/l = Q.80, 0.92,1.00

and 1.20).

3.5. Discussion of locomotion modes in forward and backwardswimming

Note that the locomotion modes in the forward and backward swimming ofthe eels are obviously different and here we beg a question: why does an eelutilize the swimming mode with constant waving amplitude only in the backwardswimming but not in the forward swimming, in which only the mode of linearly-increasing waving amplitude has been found?

Firstly, we compare the dynamic performance in two different wavingamplitude modes of (1) the constant-amplitude mode, a(x)=O.OS, and (2) thelinearly-increasing amplitude mode, a(x)=0.07x+0.0l, but with the samemaximum (0.08), at the same frequency in the forward swimming (Table 1). Inmode (1), the friction-based drag coefficient is lower than that in mode (2) butthe thrust coefficient and the propulsive efficiency also show lower values; themaximal side force coefficient shows much higher value in mode (1) but thepower coefficients are almost the same in two cases. It is reasonable that thehigher thrust, the higher propulsive efficiency and the lower side force mean thatthe dynamic performance in mode (2) (the linearly-increasing amplitude mode)is much better than that in mode (1) (the constant-amplitude mode) in forwardswimming.

Table 1 Comparison of two amplitude modes in forward swimming (I) (k=4.7124,/i

cr,CD

cr

t'Linax

c.i

Mode (1)a(x)=0.08

0.039

0.0210.060

0.460.0880.68

Mode (2)a(x)=0.07x+0.01

0.045

0.0250.070

0.200.0900.78

570

t=OT

t=O. 6T

M . 2 T

f=0.8T (1) a(x)=0.08

t=OT F0.2T

t=0.6T t=O.BT (2) a(x)=0.07x+0.01

t=0.4T

t=T

I ' I

t=0.4T

t=T

Fig. 14 Pressure contours of two modes in forward swimming (Re = 2.5 x LO', k=4.7124, = 0.92).

constant-amplitude mode, a(xF0.08; (2) linearly-increasing amplitude mode, a(x)=0.07x+0.0 1. Fig.14 shows the pressure contours in the two cases. The forces mainly act

upon the anterior and middle parts of the body in mode (1) but on the posterior and middle parts in mode (2). Note that in mode (1) the undulatory amplitude of 0.08 along the whole body is very large, which is identical to the maximum of the undulatory amplitude at the tail tip in mode (2). Accordingly, much larger force is generated and such a large force is associated with larger muscle strain. Our results agree with the conclusion reached in the experimental study [5 ] that muscle strain is pronounced in the anterior part during backward swimming, and in the posterior part during forward swimmingh fact, there are more organs and

571

skeleton in eel's head, pronounced lateral head movements is not favorable for movement and perception [ 5 ] . Thus, in actual swimming, the amplitude in backward swimming is not as high as that in forward swimming. In order to generate sufficient thrust, eels must use the higher frequency with the lower amplitude in the backward swimming.

Table 2 Comparison of two amplitude modes in forward swimming (n) (Re = 2.h 10',1=0.92) and backward swimming

Table 3 Comparison of modes in forward

Furthermore, let us examine what happens if an eel uses the same locomotion mode as in backward swimming during forward swimming. The computed force-related quantities are illustrated in Table 2. In this case, although greater thrust generation is achieved than in the linearly-increasing amplitude mode in the actual forward swimming, the side force and the consumed power are also much higher, which results in much lower propulsive efficiency in the constant-amplitude mode (Table 2). In the continued long-time forward swimming, a swimming mode capable of achieving higher efficiency with lower power consumption with lower muscle strain (eficiency-based swimming) is desirable. Therefore, an eel has no reason to use the constant-amplitude mode in the forward swimming. Note that larger side force is created in the constant- amplitude undulatory mode in both forward and backward swimming, which implies that this may be the mode in case the eels do maneuvering (maneuverability-based swimming). Actually, the backward swimming mode is observed when the eels perform their maneuverability such as in dark and complex environments, and as a 'burst' escape response for brief periods only.

Note that, by means of the present analysis (Table 2 and Table 3), we see that the backward swimming hydrodynamics is not an exactly reversed replica of those as in the forward swimming. The thrust in the backward swimming is much lower than thatin the forward swimming even though the locomotion modes are

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absolutely the same (see the second column in Table 2 and Table 3). In order to compare the backward and forward swimming, we choose the corresponding frequencies to guarantee their thrust generation on the same order: k7.854 in the backward swimming and k3.700 in forward swimming (Table 3). Although the thrusts in both cases are almost the same: a margin (0.0048 and 0.0061), much larger side forces and the consumed powers are generated in the backward swimming but very lower propulsive efficiency is achieved. In fact, in backward swimming, the movements are not as pronounced in the caudalmost part of the animal, but are larger along the whole body length and associated with much higher estimated muscle strains than in forward swimming [ 5 ] . The higher muscle strains correspond to the higher side force in backward swimming. The higher consumed power is associated with that white muscle powers the backward swimming. We know that white muscle, which possesses primarily anaerobic, glycolytic metabolism, is the fastest fiber type in fish, capable of both high shortening velocities and relatively fast rates of activation and relaxation [4]. In contrast, the red muscle becomes most functional in the forward steadily swimming. Red aerobic muscle has low rates of activation, relaxation and shortening velocities. And its gear ratio is a quarter of that of the white muscle [12]. Therefore, the different locomotion modes used in an eel’s forward and backward swimming may be the best choice associated with their propulsive mechanism, physiology and ecology.

4.

1.

2.

3.

Conclusion

The backward swimming mechanism is somehow like a ‘reversed’ forward swimming one, in which a thrust is generated when a jet-stream is formed in the wake with a reversed K h n i n vortex street. When the eel swims backward approximately at a uniform speed, the vortices of alternating sense are observed to be aligned approximately that is consistent with the new statement by Wu [22]. Difference in swimming kinematics between the backward and forward swimming c o n f i i the idea that the constant-amplitude-based backward swimming achieves larger force generation for maneuverability (maneuverability-based swimming) but with lower propulsive efficiency compared with the linearly increasing amplitude-based forward swimming (efficiency-based swimming). Effects of frequency, amplitude and wavelength of the traveling wave on the backward swimming hydrodynamics have similar influences on the propulsion mechanisms in both the anguilliform forward swimming and the

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backward swimming. 4. There exists a best matching between the swimming mode and the

propulsion mechanism in the anguilliform forward and backward swimming.

References

1.

2.

3.

4.

5 .

6. 7.

8.

9. 10. 11.

12. 13. 14. 15. 16.

17. 18. 19.

20. 21. 22.

Carling, J., Williams, T., and Bowtell, G. , J. Exp. Biol. 201, 3143- 3 166( 1998). Carling, J. C., Bowtell, G. and Williams, T. L., Mechanics and Physiology ofAnimal Swimming (ed. L. Maddock, Q. Bone and J.M.V. Raper), pp. 1 19-1 32. Cambridge University Press( 1994). Cheng, J. Y., Zhuang, L. X. and Tong, B. G., J.Fluid Mech. 232,341 - 355(1991). CurtinN. A. and Woledge R. C., J. Exp. Biol. 140, 187-197(1988). D’aoiit K. and Aerts P., J. Exp. Biol. 202,1511-1521(1999) Lighthill M. J., J. Fluid mech. 44,265-301. (1970) Liu, H., Proceedings of the 1995 ASMEIJSME Fluids Engineering Annual, FED 215, 105-121 (1995a). Liu, H., Wassersug, R. J. andKawach, K., J. Exp. Biol. 199, 1245-1260 (1996). Liu, H. and Kawachi, K., J. Comput. Phys. 155,223-247 (1999). Lu, X. Y., Int. J. Comput. Fluid Dynamics. 16(1), 65-82(2002). Miiller, U. K., van den Heuvel, B. L. E., Stamhuis, E. J., Videler, J. J., J. Exp. Biol. 200, 2893-2906( 1997). Rome, L. C., Sosnicki, A. A., Am. J. Physiol. 260, C289-C296 (1991). Roshko, A., NACA Report 1191(1954). Schlichting H., Boundary layer theory. McGraw-Hill, Inc., 174-190( 1979). Stamhuis, E., J. Exp. Biol. 198,283-294( 1995). Triantafyllou, G. S., Triantafyllou, M. S. and Grosenbaugh, M. A., J. Fluid Struct. 7, 205-224 (1993). Van Leer, B., J. Comp. Physics 23,276-299(1977). Videler, J. J., Fish Swimming. London: Chapman & Hall (1993). Wolfgang, M., Anderson, J.M., Grosenbaugh, M.A., Yue, D.K.P. , Triantafyllou, M.S., J. Exp. Biol. 202,2303-2327(1999a). Williamson, C. H. K., J. Fluid Mech., 206, 579-627(1989). Wu, T. Y., Adv. Appl. Mech. 11,l-63(1971). Wu, T. Y., Adv. Appl. Mech. 38,291-352 (2001).

IMPULSE EXTREMIZATION IN VORTEX FORMATION FOR APPLICATION IN LOW SPEED MANEUVERING OF

UNDERWATER VEHICLES

KAMRAN MOHSENI Department of Aerospace Engineering,

University of Colorado, Boulder, CO 80309-0@9, USA E-mail: mohseniQcolorado. edu

Compact zero-mass pulsatile jet actuators are proposed for low speed maneuvering and station keeping of small underwater vehicles. The flow field of such jets are initially dominated by vortex ring formation. To this end, impulse extremization in vortex ring formation is considered. It is shown that a pinched-off vortex ring characterizes the extremum impulse accumulated by the leading vortex ring in a vortex ring formation process. An appropriate scaling for vortex ring impulse is found and the limiting values of the non-dimensionalized impulses are established. An estimate for the non-dimensional impulses is provided by equating their values from the slug model with their values from a vortex in the Norbury family of vortices. For a vortex ring generator with constant kinetic energy and circulation generation rate, the pinched-off vortex ring has a maximum impulse If’ = 11 normalized by the circulation and energy. On the other hand, for a vortex ring generator with constant rate of circulation generation at a constant translational velocity, a pinched-off vortex ring produces a minimum impulse I:d = 0.12 normalized by the circulation and translational velocity. Direct numerical simulations of vortex ring formation and vortex ring pinch-off process are performed and the estimated values of the non-dimensionalized impulses are confirmed. A formula for calculating the translational velocity of a vortex ring, generated through an impulsive ejection of fluid from an orifice, is presented. Variation of the transla, tional velocity as a function of the stroke ratio and the non-dimensional mean core radius is calculated. Application of such vortex generators as zero-mass pulsatile jets for low speed maneuvering of underwater vehicles is also demonstrated. The actuators could be used in two ways: (i) to improve the low speed maneuvering and hovering capabilities of traditional propeller driven underwater vehicles, (ii) and as a synthetic jet for flow control at higher cruising speeds. A model for calculating the rotation rate of the underwater vehicle is also proposed and verified.

1. Introduction

A starting jet is usually characterized by the roll up of the ejected shear layer from a nozzle or an orifice and the formation of vortex rings. Vortex rings has captured the attention of many researchers over the last century.

574

575

Vortex rings have relatively simple and persistent three dimensional structure and at high Reynolds numbers they decay slowly. The generation, formation, evolution, and interactions of vortex rings have been the subject of numerous investigations (see, e.g., Shariff and Leonard [l] and the references in there). In this study, we focus our attention on a specific characteristic of vortex ring formation; namely the impulse extremization in vortex ring formation and its connection with the vortex ring pinch-off phenomenon.

The amount of impulse imparted to the flow in the form of a vortex ring depends strongly on tbe vortex formation process. In particular, it depends on the rate of providing the invariants of motion during the formation process. In a laboratory, vortex rings can be generated by the motion of a piston pushing a column of fluid through an orifice or nozzle. The boundary layer at the edge of the orifice or nozzle will separate and roll up into a vortex ring. The experiments of Gharib et al. [2] have shown that for large piston stroke versus diameter ratio ( L I D ) , the generated flow field consists of a leading vortex ring followed by a trailing jet (see Figure 4). The vorticity field of the formed leading vortex ring is disconnected from (pinched-off) that of the trailing jet at a critical value of LID (dubbed the “formation number”), at which time the vortex ring attains a maximum circulation. The formation number was in the range of 3.6 to 4.5 for a variety of exit diameters, exit plane geometries, and non-impulsive piston velocities.

In this paper our goal is to predict and explain the impulse accumulation in the leading vortex ring without resorting to dynamics. The vortex ring pinch-off process occur at a limiting stroke ratio LID where the generating apparatus is no longer able to deliver energy, circulation and impulse at a rate comparable with the requirement that a steadily translating vortex ring has maximum energy with respect to kinematically allowable perturbations. The physical explanation is that for short stroke length the system relaxes to a steadily translating vortex ring. Increasing the stroke length results in a larger vortex ring. Note that any vorticity generation mechanism has its own specific rate of energy, circulation and impulse generation. In particular, the relative rate of providing these invariants of motion to the system is important in determining the final properties of the resulting vortex ring. For a cylinder piston mechanism these rates are approximated by the slug model (see e.g., [3]). For high stroke lengths the conventional cylinder piston mechanism is not able to provide energy compatible with a single steadily translating vortex ring at that circulation and impulse

576

[2]. Consequently, for large stroke length the system relaxes to a periodic array of vortex rings. To generate vortex rings with higher non-dimensional circulation, one needs to alter the rate of delivery of invariants of motion. For a cylinder piston mechanism, Mohseni & Gharib [3] proposed a time varying exit diameter, that was later verified numerically [4].

Mixing entropy maximization offers an alternative explanation of the vortex ring pinch-off process besides the energy minimization approach in Kelvin’s variational principle. From this point of view, any vortex ring generator can be viewed as a tool for initializing an axisymmetric flow with a particular rate of generation of invariants of motion. Each vortex ring generator has a specific rate for providing the flow with the kinetic energy, impulse, circulation, etc. In this picture, at small strokes (small formation number L I D ) one will find that all of the initial vorticity density will coalesce into a steadily translating vortex ring. As the stroke length increases the size and strength of the resulting vortex ring increase. This process persists until a critical formation number is reached, when the vortex generator is no longer able to provide invariants of motion compatible with a single translating vortex ring. Equivalently, beyond the critical formation number a single vortex ring at equilibrium (steadily translating) that maximizes the mixing entropy for a given energy, impulse and circulation is not possible. In this case the leading vortex ring is disconnected from the trailing jet and will relax to a steadily translating vortex ring with the translational velocity Ut, dictated in the maximum entropy principle. For very large strokes (greater than twice the critical formation number) successive vortex rings will pinch-off from the the trailing jet. This scenario was already verified in the numerical simulations of the vortex ring pinch-off process [4]. The general observation in these simulations was that the main invariants of motion in the pinch-off process are the kinetic energy, circulation and impulse. The higher enstrophy densities did not play a significant role as long as the Reynolds number was relatively high [5] .

We have also developed a simple and low cost pulsatile j e t propulsion scheme for small underwater vehicles; see Mohseni [6] and the cover story of the New Scientist [7]. Prototypes of a pulsatile jet generator using the Helmholtz cavity actuators are designed and built for underwater maneuvering and propulsion. As described below, our scheme is likely to be used in two ways. First, it can be combined in a hybrid design fashion with propellers to improve the low speed maneuvering and hovering capabilities of traditional propeller driven underwater vehicles. Second, the same actuators can be sued for flow control and drag reduction at higher cruising

577

speeds. Our objectives in this study are multi-folded. In the next section we

show that the limiting stroke ratio for-a pinched-off vortex also characterizes an extremum state for accumulation of impulse in the leading vortex ring. Appropriate scaling in such a process are identified and an estimate of non- dimensional impulses for the leading vortex ring are offered by a theoretical model. We will also derive a relation between the translational velocity of the leading vortex ring and the piston velocity. Direct numerical simulations of the pinch-off process are presented in section 3 and the limiting impulses are calculated directly. Potential of pulsatile jets for propulsion are considered in section 4. Application of pulsatile jets for low speed maneuvering of underwater vehicles is studied in section 5. This includes the development of Colorado underwater vehicle test beds, synthetic jet actuator prototypes, a model for calculation of drag momentum and the required pulsatile jet moment to overcome that. Finally our concluding remarks are summarized in section 6 .

2. Impulse Extremization in Vortex Ring Formation

Consider an arbitrary generator of axisymmetric vortex sheets. A popular example is a cylinder-piston mechanism, where a cylindrical shear layer is ejected from the exit of the cylinder at a particular speed, approximately the piston velocity Up. It is known that cylindrical vortex sheets are unstable and will role up to a vortex ring. The vortex ring has an induction velocity causing it to accelerate an the vortex ring grows. This combination of vortex ring enlargement and acceleration continues until the vortex sheet is unable to inject any more vorticity to the leading vortex ring. In that case the leading vortex ring is detached from the vortex sheet and the remaining vortex sheet goes through an instability process again to form a new vortex ring. Continuation of this process in axisymmetric flows will eventually result in the formation of a periodic array of vortex rings with toroidal diameter h and spacing 1. Each vortex will move with a translational velocity Ut,. The period of vortex ring shedding, then, will be T = l /Utr . In such a process the parameters involved are the geometrical scales 1 and h, the translational velocity Ut,, and the main invariants of motion in Euler equations: Energy E , impulse I , and total circulation I'. One can expect a functionality of the form f(1, h, Ut,, E , I , I?). A straightforward dimensional anaIysis results in

578

where

1 rut, tnd = - = - , non-dimensional formation time,

h

(3) non-dimensionalized impulse based on E and P,

(4) non-dimensionalized impulse based on Ut,. and l’.

Both parameters on the right hand side of equation (1) can be interpreted as non-dimensional impulses. Ifd is non-dimensionalized by the circulation and energy and I:d is non-dimensionalized by the circulation and translational velocity. I f d and I:d are closely related to the non-dimensional energy E n d = E/r3 /211 /2 and non-dimensional circulation r n d = r/~ut2T/3, respectively, defined in Mohseni and Ghaxib [3]. In this study we are interested in impulse extremization. Consequently, non-dimensional impulses I:d and I:d are more appropriate parameters than the alternative variables E n d and r n d .

Following Mohseni and Gharib [3] we consider a relaxational model for the vortex ring pinch-off process. We argue that the formation of vortex rings at relatively high Reynolds numbers is mainly an inviscid process. Therefore, the invariants of motion, namely the energy, impulse, and circulation must be the same initially and after the formation of vortex rings. Note that apart from the energy, impulse, and circulation all the other invariants of motion (higher order enstrophy density moments) are lost during the mixing process and will not significantly affect the formation process IS].

There are various experimental, numerical and engineering methods to generate vortex rings. Each of them has a particular rate of generation of invariants of motion. In order to predict the limiting formation number for each vortex generator one needs to estimate the rate of injection of invariants of motion for that particular vortex ring generator. The non- dimensional impulses (3) and (4) are then formed and equated to an estimate of the same quantities for the resulting pinched-off vortex.

Following Mohseni and Gharib [5] we approximate the initial state by a column of Auid with diameter D = 2R and length L moving at a constant velocity Up (piston velocity) and the final state is approximated by a vortex in the Norbury family of vortices. The slug model is characterized by the the following relations for the energy, circulation and impulse of the ejected

579

fluid:

(7) 1 1 I = -rD2LU - - r D 2 r

The slug model provides an estimate of the invariants of motion initially injected in the medium. On the other hand, we assume that the invariants of motion after the formation can be estimated by the invariants of motion for a vortex in the Norbury family of vortices. Norbury vortices are steady solutions of Euler equations with one parameter a, non-dimensional mean core radius. The vorticity density distribution WIT is constant in each vortex. Norbury vortices cover vortex rings of small cross section as a approaches zero to Hill's spherical vortex for a = 4. Using the non- dimensionalization employed by Norbury one can write

4 p - 2

E = (Ra212)2 EN (8) r = (Ra2Z2) l!rN (9) I = (Ra2Z2) Z3IN (10)

where the subscript N indicates the corresponding non-dimensional quantity for a Norbury vortex. Here, R = w / r is the vorticity density, and 1 is the vortex radius [8].

By equating the non-dimensional impulses I t d and I:d from the slug model with the same quantities calculated from the Norbury family of vortices one obtains

We assumed that D M 1. Note that the right hand sides of equations (11) and (12) are functions of a only. It is not surprising that these relations are the same equations obtained from the energy and circulation extremization obtained in [3]. In order to proceed we need a relation between the piston velocity Up and the translational velocity Ut,. An approximation Ut, = Up/2 was offered in [3]. Here we derive an explicit formula for the translational velocity.

580

2.1. Translational Velocity of a Vortex Ring

Velocity of a steadily translating axisymmetric vortex is a constant of motion in inviscid flows. Theoretical prediction of translational velocity of thick vortex rings is a challenging task. Translational velocity depends on the rate which the invariants of motion are provided to the system. In the slug model we can estimate the translational velocity by the Roberts’ formula [9]

”’ = s E / constantrand vortex volume

Considering zero variation in the vortex ring circulation, equation (5) yields 6E I’ E S L Up E 6 L 61 L L SI 2 L 61

To simplify the variational derivative on the right hand side of equation (14), one can employ equation (6) for the impulse to calculate

(14) - =-- - -

2 1 R 61 = -6R

By substituting SI into equations (14) one obtains

6E - Up E R b L 61 2 2LISR _ _ - - - -

6L In order to proceed we employ Norbury’s result to calculate -. From 6R equation (11) one obtains

6L L Sa - = - + 2g’(a)R- SR R SR The constraint on the volume of the vortex ring is employed to derive an equation for Scr/SR . Norbury [8] provided results for the vortex ring volume as a function of the mean core radius, namely V, = f (a)R3. f (a) , reproduced in Figure 1, is a function of the mean core radius a only. Hence

Now equation (17) can be recasted as

Finally, the translational velocity in equation (14) can be written as

581

Figure 1. Volume of core in Norbury vortices [S].

This function is depicted in Figure 2 using Norbury's data [8]. There are slight fluctuations in this graph, which is a consequence of calculating derivatives from relatively coarse data in Norbury [8]. For most values of the mean core radius, particularly for 2 5 L/D 5 5 the slug model predicts a translational velocity around Up/2 . This is consistent with a previous assumption in Mohseni and Gharib [3]. At higher values of a which corresponds to larger stroke ratio L / D in the slug model there is a significant drop in the translational velocity. This is a limitation of the slug model which assumes that the relative rate of injecting invariants of motion is constant and not dependent on the stroke ratio. However, in reality, the formation and growth of boundary layers at the orifice walls will change the effective velocity and diameter of the ejecting fluid. It is expected that inclusion of such effects results in better prediction of the translational velocity for larger stroke ratio.

0.2 0.2

I 1.5 4 6 8 a UD

a5

(4 (b) Figure 2. vortices) with (a) the mean core radius, and (b) the stroke ratio LID.

Variation of the translation velocity of a vortex ring (in slug model - Norbury

582

2.2. Impulse Extremization

By substituting the translational velocity from equation (20) into equation (12) one obtains

The right hand side of equations (11) and (21) are only a function of the mean core radius a. These equations are depicted in Figure 3 using Norbury data. The two curves intersect somewhere in the range 3 5 LID 5 4, which characterizes the properties of a pinched-off vortex ring.

Knowing the mean core radius for a pinched-off vortex (see Figure 3) one can estimate the values I f d and I i d from the Norbury data. For a vortex ring generator with fixed kinetic energy and circulation as in equations (5) and (6), the pinched-off vortex ring has a maximum impulse I$. This number is estimated to be around I f d M 11. On the other hand for a vortex ring generator that delivers the same amount of circulation at a constant translational velocity, a pinched-off vortex has a minimum impulse l i d .

This number can be estimated to be around r i d "N 0.12. An explanation for the behavior of I f d and I L d is offered in section 3.3.

It is important to note that the above value for the formation number and the estimated values for I F d and I L d are only achieved as long as the rates of generation of the integrals of motion are constant during the formation process. One can argue that in order to change the values of I f d

I I 0 0.5 1 1.5

a.

Figure 3. mean core radius of the Norbury vortices.

Formation number LID of the slug model as a function of the non-dimensional

583

and the rates of injection of the invariants of motion into the flow need to be modified during the formation process. Two possible techniques axe feasible: i) varying the nozzle diameter or iz) the speed of the shear layer (speed of the ejected fluid slug) during the formation process. Accelerating shear layer results in a larger vortex ring while decelerating shear layers result in a smaller vortex ring and consequently smaller formation number.

3. Direct Numerical Simulation

In this section direct numerical simulations of vortex ring pinch-off process is performed and the non-dimensional impulses I:d and Ifd are directly calculated. Several numerical techniques for generating vortex rings have been implemented in the past. Stanaway and Cantwell [lo] used a given vorticity distribution and its corresponding velocity field as the initial condition. Nitsche [ll] implemented a vortex sheet model to reproduce vortex generation experiments by Didden [12] for small stroke ratios. Numerical solutions of the low Mach number Navier-Stokes equations for small stroke ratios was used by Madnia [13]. Another method is to prescribe an axial velocity profile Vz(r) at an inlet, in an attempt to model the injection of fluid through the nozzle of the experimental apparatus (141.

We believe that the extremum values of ILd and I f d axe independent of the vortex ring generator as long as the rates of injection of the invariants of motion are constant during the formation process. To this end, we use an alternative method for numerical generation of vortex rings, namely by application of a non-conservative force directly to the momentum equation. This method has previously been used, for impulsive forces, in order to generate vortex rings with certain desired properties [4,15,16]. Similar technique is used in this study and is reported in the next section.

3.1.

We use a body force aligned with the axial coordinate of an axisymmetric cylindrical coordinate system (5, r ) which has the form:

Vortex Generation by a Non-Conservative Forre

fz(r,x,t) = c F ( t ) G(5) H ( r ) (22) Here, C is a constant with units of circulation representing the circulation amplitude, and F , G axe functions with units of inverse time and inverse length, respectively, and H is a non-dimensional function. We use a regularized step function for F , namely F ( t ) = - (tanh (at(to - t ) ) + tanh (at(t - t o - 2'))) / (22') where at is a time scale

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which controls the smoothing and T is the duration of the force. We have experimented with different values of at, and the results presented below use at = 2.0R2/C, where R is the radial extent of the forcing region. This is short enough such that the vortex formation process was essentially independent of its value. The functions G and H control the spatial variation of the flow generated by the force. For the radial distribution, we use H ( r ) = erfc(& % ) I 2 where a is a non-dimensional constant. This is a smoothed “top-hat” function and a,. controls the smoothing of the profile. a, is reminiscent of the boundary layer thickness at the exit of the nozzle in experiments. It controls the shear layer thickness of the resulting flow. For the axial distribution, we again use a regularized delta function G(z) = &e-a(2)z/ (a,&?) where a, controls the axial extent over which the force is smeared. The ellipticity of the vorticity distribution in this case is controlled by ax.

The above non-conservative forcing introduces four independent non- dimensional parameters that can be written as: a,./R, a z /R , TC/R2, and Re = C/u . By analogy with the cylinder-piston vortex generators, T controls the stroke length. Larger T corresponds to larger LID. In all the calculations presented in this study we make sure that the Reynolds number defined based on the (measured) circulation of the leading vortex ring and viscosity is above 1000 to avoid significant deviation in some of the invariants of motion due to viscous dissipation.

3.2. Numerical Technique

An existing code for the axisymmetric compressible Navier-Stokes equations is employed in this study. In the cases presented here the maximum Mach number in the flow was 0.2. Consequently, for our purposes the results are essentially incompressible. While the numerical method is inefficient for computing low Mach number flows, the CPU requirements for the present problem are minimal. An explicit fourth-order Runge-Kutta method is used for time advancement and a sixth-order compact finite difference scheme [17] is used in both the axial and radial directions. The polar coordinate singularity is treated by the technique proposed in [MI. A buffer zone is used at the boundary to absorb the waves. From the location of the applied force, the computational domain extends to -3R and 21R in the axial directions and about 4R in the radial direction. The grid spacing is &R, and the time step is 6 R/a where a the sound speed. The grid independence of the solution was established by using higher time and spatial resolutions.

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In order to calculate the non-dimensional impulses we need to calculate the following integrals of the motion for the leading vortex ring:

I? = wdxdr . (23) J E = n w$dxdr, J I = 1 WT2dxdT,

Initially, the velocity field is differentiated to obtain the vorticity. In order to distinguish between the leading vortex ring and the rest of the vorticity, we set the vorticity to zero outside of a closed iso-contour of vorticity around the leading vortex core, at which point the vorticity is 2 % of the maximum vorticity in the core. To obtain the stream function, $ from the vorticity distribution, an integral equation [19] e.g., is employed, where the equations (23) are integrated numerically with a 4th-order accurate quadrature scheme. The translational velocity, Ut, is estimated by tracking the position of the maximum vorticity in the leading vortex core.

3.3. Numerical Results

In this section, we vary the four parameters associated with the forcing and establish the conditions under which the pinch-off time and the normalized impulses Ifd and &, of the leading vortex rings are independent of the values of the forcing parameters.

Vorticity contours at several different stages of the formation process are presented in Figure 4 for a typical case with a,/R = 0.2, a,/R = 0.2, TC/R2 = 25.3, and Re = 3600. The Reynolds number based on the circulation of the leading vortex ring is 2800. In figure 4(a), a vortex ring with small core size is generated shortly after the onset of the forcing. As the vortex ring grows in size (figure 4(b)) it translates downstream due to its induction velocity. In figure 4(c) the spiral structure of the core is well-established, and at the lowest vorticity level depicted in the figure the tail is disconnected from the leading ring.

The non-dimensional impulse Ifd is directly calculated in a series of direct numerical simulations. Figure 5(a) shows the computed values of Ifd for these cases. For constant energy and circulation the impulse of the leading vortex ring is maximized at the pinch-off time. An explanation can be offered by employing the relationships for energy and circulation from the slug model in equations (5) and (6). For constant circulation and energy one can derive a relation between L and D, namely LID2 = c1, where c1 is a constant. Therefore, for constant energy and circulation an increase in the stroke ratio LID is equivalent to an increase in the diameter (LID = c1D). Since for constant circulation, impulse is proportional to D2, increasing the

586

b

h

(b)

h

(c>

4 8 10

X

0 ' " ' " " ' " " ' " " " ' 2 4 6 8 10

X

0 " " " " " " " ' ~ ' 2 4 6 8 10

X

Figure 4. Time evolution of vorticity contours in a direct numerical simulation of the vortex ring pinch-off process. Vortex ring formation at different times; (a) tC/R2 = 26.83, non-dimensional vorticity (wR2/C) contour levels (min=O.O8, max=0.78, increment=0.07) /3.16 (b) tC/R2 = 44.72, contour levels (min=0.05, max=0.65, increment=0.06) /3.16 and (c) (min=O.O6, max=0.66, increment=0.06) /3.16. The detachment of the trailing jet from the leading vortex ring is evident in (c).

tC/R2 = 53.66, contour levels

stroke ratio LID results in an increase in the impulse of the leading vortex ring. Consequently, for constant circulation and energy, the accumulated impulse in the leading vortex ring is maximum for the largest achievable stroke ratio before a pinch-off occur; that is for the limiting formation number. Values of maximum Ifd for a pinched-off vortex in figure 5(a) are consistent with the estimated value of Ifd M 11 in section 2.2.

Computed values of I id for several pinched-off cases are presented in Figure 5(b). The value of the non-dimensional impulse I E d for a pinched-off vortex ring in Figure 5(b) matches accurately with the theoretical prediction of M 0.12 in section 2.2. In contrast to I f d , for constant circulation

587

a a

Figure 5. Vortex rings with different forcing duration. (a) IFddr (b)lzd . The line corresponds to the total impulse of the leading vortex ring during the formation and after pinch-off, while the symbols correspond to impulse of the leading vortex ring. TC/R2 = 9.11, Case 3 (---); TC/R2 = 14.2, (0); TC/R2 = 25.3, (+); TC/R2 = 39.5, (0).

and translational velocity, the impulse of the leading vortex ring will decrease with the stroke length LID. An explanation can be offered by using the slug model. To simplify this analysis we assume that the translational velocity is approximately half of the piston velocity (see Figure 2). For a vortex generator with constant circulation and translational velocity one can show that the stroke length L must be a constant. On the other hand, for constant circulation, the impulse is proportional to 0'. Consequently, for constant circulation and translational velocity, an increase in the stroke ratio LID or equivalently a decrease in diameter D results in a decrease in the accumulated impulse in the leading vortex ring. Therefore, the minimum value of Izd is achieved at the limiting stroke ratio for a pinched-off vortex.

4. Pulsatile Jet Propulsion

The propulsion scheme suggested here is loosely inspired by the propulsion of cephalopods (e.g. squid and octopi), salp, and jellyfish [20-251. Squid (see figure 6) use a combination of fin undulations and a jet which can direct thrust at any angle through a hemisphere below the body plane. Their complete range of locomotory behavior rivals that of reef fish. Jet propulsion swimming of the squid is accomplished by drawing water into the mantle cavity, and then contracting the mantle muscles to force water

588

out through the funnel. The funnel, which is directly behind and slightly below the head, can be maneuvered so as to direct jets in a wide range of directions. Another example of pulsatile jet locomotion is jellyfish swimming [26], which relies upon repeated contractions of an umbrella-shaped structure, or bell. During contraction, circular subumbrellar muscles pull the sides of the bell inward, reducing the volume of the subumbrellar cavity, and forcing water out through the velar aperture. Water is drawn back into the subumbrellar cavity during the relaxation phase. The jellyfish can optimize its propulsion by controlling the diameter, velocity, and profile at the exit of the velar aperture.

Figure 6. Squid locomotion by pulsed jet.

Weihs [25], Seikman [23], and recently Krueger and Gharib [27] have shown that a pulsed jet can give rise to a greater average thrust force than a steady jet of equivalent mass flow rate. In a pulsed jet, an ejected mass of fluid rolls into a toroidal vortex ring which moves away from its source. A continuously pulsatile jet, therefore, produces a row of vortex rings (see Figure 6). At high pulsing frequency, the jet structure can become increasingly turbulent.

Vortex ring jets can be generated using a variety of mechanical devices.

Ls

Figure 7. Cylinder piston mechanism.

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While a squid generates vortex rings by muscle contraction around the mantle, one of the simplest ways to generate vortex rings and pulsatile jets in the laboratory is the motion of a piston pushing a column of fluid through an orifice; the so-called cylinder-piston mechanism (see Figure 7). This system provides a simplified approximation to natural pulsatile jet generation, and it is amenable to experimental, computational, and analytic study. When the piston pushes fluid through the cylinder, the boundary layer of the fluid expelled from the cylinder will separate and roll up into a vortex ring at the orifice edge. Experiments [2] have shown that for large enough ratios of piston stroke versus diameter ( L I D ) , the generated flow consists of a leading vortex ring followed by a trailing jet. See Figure 8(a) for experimental results, and Figure 8(b) for corresponding numerical simulations.

L - a

t

LID= 2.0 "

2

I

UD = 3.8

4 UD r l ( r )

Figure 8. Left: Experimentally obtained fluid vorticity profiles during the vortex ring pinch-off process (for various L/D formation numbers) [2]. Right: Numerical simulation of vortex ring formation at various formation numbers [18]. Only one half of the symmetric jet cross section is presented.

It was both experimentally [2] and analytically [3] observed that the limiting stroke L I D occurs when the generating apparatus is no longer able to deliver energy, circulation and impulse at a rate comparable with the requirement that a steadily translating vortex ring has maximum energy with respect to kinematically allowable perturbations. Mohseni and Gharib [3] suggest that the properties of the leading vortex ring are the final outcome of a relaxation process, dependent only on the first few integrals of the motion (the energy, E , impulse, I, and circulation, I?). Mohseni [5] argued that the energy extremization in Kelvin's variational principle has

590

a close connection with the entropy maximization in statistical equilibrium theories. Numerical evidence for a relaxation process in axisymmetric flows to an equilibrium state has been provided by Mohseni et al. [18] in a direct numerical simulation of the vortex ring pinch-off process. Similar phenomena are observed in the alternating vortex shedding behind bluff bodies (see [ZS]).

In squid and jellyfish, the exit diameter of the cylinder (mantle or bell) varies during the expulsion of fluid. This technique optimizes propulsive output. We have recently shown that a time varying shear layer velocity mechanism can also manipulate pulsatile jet behavior (see [3,18]). While the mechanisms here are even more complicated than the piston-cylinder model, this model does provide useful guidance on the overall physical phenomena at work.

5. Application of Pulsatile Jets for Low Speed Maneuvering of UUVs

Unmanned Underwater Vehicles (UUVs) will play a major role in the future environmental control and monitoring, underwater archeology, search and rescue missions, and underwater battlespace. Two main categories of unmanned underwater vehicles are autonomous underwater vehicles ( AUVs) and remotely operated vehicles (ROVs). AUVs operate for relatively long periods underwater without direct human guidance while ROVs are pow- ered and teleoperated via a tether connected to a surface command ship.

Most AUV designs (e.g. WHOI’s REMUS and MIT’s Odyssey) have traditionally been based on a propeller thruster combined with control fins (or shrouded thrusters) to propel and steer the vehicle. Such designs are often streamlined (torpedo-like body shape) and optimized for low drag during forward motion. Maneuvering control forces are generated by lift or deflection forces created by fluid flow over the control surfaces. At cruising speeds, and for relatively uncluttered spaces, this paradigm is extremely efficient and effective. However, at low speeds and in tight spaces the magnitude of the available control forces drops significantly. Consequently, such vehicles are difficult to dock. As a result much current effort is devoted to the development of docking mechanisms.

ROVs, which are not designed for cruising, typically follow the so-called “Box Design” or a multi-pontoon design. Better low speed maneuvering and control are achieved by sacrificing the low drag body-of-revolution design and adding multiple thrusters at different locations and directions.

591

MBAFU’s Tiburon and WHOI’s JASON [29] are among successful ROV designs in this category. Successful AUV designs in this category include WHOI’s ABE [30] and SeaBED and Stanford’s OTTER [31]. While Pro- pellers are excellent when they work at constant speed they will be less efficient for small motion corrections when the motions of propellers involve less than a full shaft rotation. This results in degraded control precision and possibly periodic oscillations of the vehicle’s position.

In summary, underwater maneuvering (especially at low speeds) and docking procedures represent a major challenge in the design of AUVs and ROVs. To this end, experimental platforms for testing, evaluating, and developing a low speed maneuvering (LSM) capability for UUVs are recently developed at the University of Colorado [32,33]. A novel pulsatile jet technology is proposed that could overcome many of the shortcomings described above for low speed maneuvering of AUVs and ROVs, and enable new types of lower cost micro-AUVs. As described below, this propulsion scheme has no protruding components that increase drag, has very few moving parts, and takes up relatively little volume. Such hybrid designs which incorporate both a main propeller and a distributed set of pulsatile jets will improve low speed AUV performance. While propellers clearly perform best at cruising speeds, pulsatile jets can significantly augment low speed maneuverability, and enable occasional loitering/hovering actions.

5.1. Colorado Undemuater Vehicle Test Beds

Special design of UUVs are required in order to implement, demonstrate, and evaluate fully maneuverable self-contained hybrid underwater vehicles that combines pulsatile jet actuators with propeller-based propulsion schemes. To this end, the first phase of designing, building, and testing a Remote Controlled (RC) underwater vehicle, HydroBuff (see figure 9), was completed in early 2003 [32]. The first version of the HydroBuff is 1.4 m long, uses a conventional propeller and control surfaces, and is remotely controlled up to 5 ft depth. Below this depth, communication with the vehicle is not reliable. The vehicle is designed with 1% positive buoyancy, so in case of communication loss, the vehicle comes up to the surface.

A new lighter and shorter (around 1 m) underwater vehicle was recently designed and built at the University of Colorado at Boulder [33]. The new vehicle, Remote Aquatic Vehicle (RAV), can house up to four SJAs within the vehicle body, and will have an active buoyancy system (see Figure 9). RAV will be used as a platform for testing the performance of the SJAs for

592

Figure 9. UUV test beds at the University of Colorado. (Top) HydroBuff: A remotely controlled unmanned underwater vehicle [32]. (Bottom) RAV: An RC UUV capable of housing 4 pulsatile jet actuators [33].

low speed turning capabilities and high speed drag reduction. RAV is also designed with an expandable payload section capable of carrying various sensors for telemetry. This vehicle will serve as a model test-bed for hybrid vehicle designs that combine pulsatile jets with conventional propellers and torpedo-like bodies.

5.2.

While the piston-cylinder model is attractive for theoretical studies and ease of experimental set-up, there are more practical means to generate pulsatile jets. To this end, prototypes of pulsatile jet generators using the Helmholtz cavity concept are designed and built [33,34]. Various actuation techniques can be employed for actuating the diaphragm. These includes, but not limited to, using solenoid plungers, acoustic speakers, electrostatic and piezoelectric actuation. In this design the inward movement of a diaphragm draws fluid into a chamber (Figure 10). The subsequent outward diaphragm movement expels the fluid, forming a vortex ring or a jet depending on the formation number. Repetition of this cycle results in a pulsatile jet. Because of the asymmetry of the flow during the inflow and outflow phases, a net fluid impulse is generated in each cycle, even though there is no net mass flow through the chamber over one cycle.

Figure 11 shows the structure and appearance of a pulsatile jet actuator prototype [33,34]. The driving diaphragms consist of a rigid disk

Synthetic Jet Actuator (SJA) Prototypes

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Figure 10. Synthetic jet actuator concept: (Left) Fluid entrainment; (Right) vortex ring formation.

with a flexible surround. Currently a solenoid actuator is used to generate the diaphragm motion. The fluid pushed by the moving diaphragm exits through an orifice. The experimental prototypes also allows easy substitution of different sized orifices and different sized chambers. In this way, physical parameters can be easily varied so that theoretical models (see below) can be compared against actual experimental results in different parameter regimes. This design has many advantages including its simplicity, very few moving parts, compactness, and no high tolerance (and therefore costly) components.

5.3.

The input design parameter for low speed maneuvering of UUVs is the revolution per minute (RPM) turn rate requirement or equivalently the angular velocity w. From the required RPM one can calculate the drag moment experienced by the vehicle. We estimate the drag forces experienced

Analysis of Synthetic Jet Actuators

( 4 (b) (4 Figure 11. CU Boulder Synthetic jet prototype [33,34]: (a) CAD model of the actuator design. (b) Plunger and solenoid assembly. (c) Actual fabrication of the synthetic jet actuator.

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Figure 12. Test of the synthetic jet actuator [34]. (a) Using SJAs to rotate a 4 inch diameter tube. (b) installation of SJAs on an 8 inch diameter tube, illustrating minimal impact of actuator on hull design.

by a submergcd tube (see Figure 12) in rotation around an axis normal to its symmetry line. Since a differential element of the tube at a radial distance of r away from the rotation axis has a local velocity of V = rw, which increases with distance from the rotation axis, the differential elements experience different drag forces. These forces can be estimated from drag data for flow behind a cylinder with diameter d at the local Reynolds number Re = ( T w ) d / u . Note that three dimensional, cross-flow, and flow- vehicle interaction effects are ignored in this simplified analysis. The total drag moment of a tube of length L rotating with an angular velocity of w around its middle can be approximated by (ignoring external flow effects)

or by changing the integration variable to the local Reynolds number R e

where CD is the drag coefficient behind a cylinder at the local Reynolds number. The SJAs are expected to provide at least equal moment on the vehicle to overcome the drag moment.

In order to estimate the moment produced by the SJAs, we use the slug model (see e.g., [3,35]) to approximate the thrust or impulse during the jet expulsion from the Helmholtz cavity. We assume the optimal formation number of L,/D M 4 [2,3,5,18] for the ejecting slug of fluid with length L, and the jet exit diameter D. Since water is incompressible, the volume of the ejected jet (see figure 13)

TD2 v, = -L, 4

595

is equal to the volume displacement of the Helmholtz cavity due to the displacement of the diaphragm

where h is the plunger stroke and D,, and D,, are the diameter of the cavity and the plunger, respectively. Consequently, the exit diameter (assuming L,/D M 4) is related to the stroke length of the plunger (or diaphragm) through

_ - Ls D:a+D:, - h 2d2

Therefore, for optimal vortex formation, assuming L,/d M 4

or

By knowing the stroke length of the plunger and its frequency one can easily estimate the generated impulse from the slug model to be pD2L,Uj /4 , where p is the fluid density and Uj = 2Ls f is the exiting jet velocity (proportional to the plunger velocity) during the e-xpulsion period. An estimate of the moment produced by the SJAs can now be easily obtained by multiplying the SJA force with its moment arm. For a pair of actuators with a separation distance of 1 the net moment M ~ J A can be estimated to

D L u l I - ,

ejected fluid

I _' -. I I

diaphragm

Figure 13. Actuation of a synthetic jet actuator.

596

be

M ~ J A = 16npD4 f21. (25)

Results of these calculation for the 4.5 inch test tube shown in figure lZ(a) are reported in figure 14. In order to accommodate for the reverse momentum during the ingestion part of the actuation a momentum adjustment factor of two is used. This is justified based on the calculations reported by Mittal et al. [36]. Calculated momentum drag in equation (24) is also shown in the Figure. The part of the SJA moment curves above the drag moment value represents enough actuation moment to overcome the drag. Figure 14 shows that the required drag moment can be overcome with various actuator exit diameters consistent with the optimal formation number of 4. Therefore, for a given solenoid stroke, one can estimate the optimal length of the ejected fluid, exit diameter, and attainable rotation rate of the submerged tube. Similarly the velocity of the solenoid actuation (or its frequency) can be related to the jet velocity at the exit of the cavity. Consequently, for a given cavity geometry, exit diameter, solenoid actuation frequency, and solenoid stroke, one can calculate the SJA moment. This is also depicted in figure 14 as a function of the actuation frequency for various exit diameters.

0 10 20 30 40 50

f Wz)

Figure 14. rpm.

Thrusting moment vs. actuation frequency for various exit diameter at one

597

Larger exit diameters require less actuation frequency, and higher solenoid force for the specified duty cycle. The ability of the SJAs to rotate a submerged tube was demonstrated as depicted in Figure 12 in order to validate the models presented in this section. Our test results closely matched the hydrodynamic thrust model of Figure 14. More detailed account of the effect of various actuation parameters (D, D,,, Dcy, f , velocity profile of the diaphragm, exit hole length, cavity and exit hole geometry) is the subject of a future publication.

6. Conclusions

Impulse extremization of vortex ring formation during an impulsive ejection of fluid through an orifice or nozzle was studied. It was shown that the vortex ring pinch-off state also characterizes the impulse extremized state during the vortex ring formation. This is a complementary result to the energy maximization in Gharib et al. [2,3] and the entropy maximization in Mohseni [5]. Consequently, the pinched-off state not only characterizes the energy and entropy extremization but also the extremum impulse state during the vortex ring formation.

An appropriate impulse scaling for vortex ring formation were found. It was shown that the pinch-off process can be characterized by the non- dimensional parameters I$ = I r 3 / E 2 and ILd = Iu,",/r3. we derived an equation for the translational velocity of a vortex ring as a function the mean core radius a or the stroke length LID. This relation improves the accuracy of our technique based on a combination of the slug model and Norbury vortices. An estimate for Ifd and ILd was provided by equating their values from the slug model with their values from a vortex in the Nor- bury family of vortices. For a vortex ring generator with constant kinetic energy and circulation generation rate, the pinched-off vortex ring has a maximum impulse Ifd x 11. On the other hand, for a vortex ring generator with constant rate of circulation generation at a constant translational velocity, a pinched-off vortex produces a minimum impulse ILd x 0.12. Direct numerical simulations of the vortex ring pinch-off process are also performed and the non-dimensional parameters Ifd and Iid are directly calculated. Direct numerical simulations confirmed the estimated values

A novel mechanism for low speed maneuvering of underwater vehicles is proposed where zero-net mass pulsatile jet actuators are employed to achieve maneuvering capability at low speeds without sacrificing the low

for Ifd and Izd.

598

drag body-of-revolution designs. The same actuation mechanism could be used for flow control and drag reduction at higher cruising speeds. The actuation mechanism is simple, has very few moving parts, has no protruding components that increase drag, and takes up relatively little volume. It is expected that such a hybrid design which incorporate both a main propeller and a distributed set of pulsatile jets to improve low speed UUV performance.

The Helmholtz cavity actuators are designed based on the required moment to overcome the drag moment of the underwater vehicle at a given rotation rate. Simple equatidns are derived in order to calculate the vehicle’s drag moment and the generated moment of the actuators. Consequently, by knowing the geometrical characteristics of the underwater vehicle and its expected rotation rate one can design a zero-mass pulsatile jet actuators to accommodate the rotation requirement. The most relevant parameters in the design of the actuators are the plunger stroke and diameter, cavity diameter, jet exit diameter, and actuation frequency.

7. Acknowledgments

The research in this paper was partially supported by the National Science Foundation contract 11s-0413300. The underwater vehicles in Figure 9 were designed as part of senior projects supervised by K. Mohseni and S. Palo in academic years 2002-2003 and 2003-2004. The author would like to acknowledge the help from the involved students: M. Baumann, W. Dalbec, E. DeKruif, I. Morikawa, J. Novick, M. Rhodes and M. Schade for the HydroBuff [32] and M. Allgeier, K. DiFalco, D. Hunt, D. Maestas, S. Nauman, J. Poon, and A. Shileikis for the RAV [33]. The actuator in Figure 12(c) was designed as part of a summer independent study under the supervision of K. Mohseni by L. Copperberg, L. Georgieva, C. Madsen, L. McCrann, and E. Thomas [34]. The author would like to thank S. Palo for his contribution to the development of Colorado UUVs. The author also benefited from discussions with J.E. Marsden and J. Burdick on control aspects of UUVs.

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28. K. Mohseni. Studies of two-dimensional vortex streets. AIAA paper 2001- 2842, June 2001. 31st AIAA Fluid Dynamics Conference and Exhibit, Ana- heim, CA.

29. D. Yoerger, J. Newman, and J.-J. Slotine. Supervisory control system for the jason rov. IEEE Journal of Oceanic Engineering, 11(3):392-400, 1986.

30. D.R. Yoerger, A.M. Bradley, and B.B. Walden et al. Surveying a subsea lava flow using the autonomous benthic explorer (abe). Int J. Syst. Sci.,

31. H.H. Wang, R.L. Marks, T.W. McLain, S.D. Fleischer, D.W. Miles, G.A. Sapilewski, and S.M. Rock. Otter: A testbed submersible for robotics research. In ANS 1995, 1995.

32. M. Baumann, W. Dalbec, E. DeKruif, I. Morikawa, J. Novick, M. Rhodes, supervised by K. Mohseni M. Schade, and S. Palo. Senior design project: Hydrobuff R5L unmanned underwater vehicle. April 2003.

33. M. Allgeier, K. DiFalco, D. Hunt, D. Maestas, S. Nauman, J. Poon, supervised by K. Mohseni A. Shileikis, and S. Palo. Senior design project: Remote Aquatic Vehicle. April 2004.

34. L. Copperberg, L. Georgieva, C. Madsen, L. McCrann, and supervised by K. Mohseni E. Thomas. Summer project: Zero-mass change synthetic submarine maneuvering jets. August 2003.

35. J.O. Dabiri and M. Gharib. A revised slug model boundary layer correction for starting jet vorticity flux. Theoretical and Computational Fluid Dynamics,

36. R. Mittal, P. Rampunggoon, and H. S. Udaykumar. Interaction of a synthetic jet with a flat plate boundary layer. AIAA paper 2001-2773, 2001.

29(10):1031-1044, 1998.

17(4) :293-295, 2004.

CHAPTER 6

HYDRODYNAMICS: MHD, VISCOUS AND GEOPHYSICAL FLOWS


MHD SELF-PROPULSION OF DEFORMABLE SHAPES

T. MILOH Faculty of Engineering, Tel-Awiv University, Tel-Aviv, 69978, Israel

E-mail: milohOeng.tau.ac.il

The paper is dedicated to Prof. T.Y. Wu: A great scholar and teacher who has inspired so many of us, on the occasion of his 80th birthday with best wishes for many happy returns!.

It is demonstrated analytically that an insulating deformable body can self-propel itself in a quiescent conducting and incompressible fluid by means of an externally applied uniform magnetic field. In particular, it is proven that such a body can accelerate persistently from rest by executing periodic controlled surface deformation. The newly found MHD self-propulsion mechanism is more pronounced compared to that existing in non-conducting fluids.

1. Introduction

It is well known that a deformable body can self-propel itself through a fluid by changing its shape in a periodic manner, proving a physical mechanism is available for exchanging momentum between the body and the surrounding fluid (see recent comprehensive review on the subject by 18). Vortex shedding from the tips of the body into the wake and the conse- quent induced lift, is one example for such a mechanism which exists in real fluids. In such a case, the presence of a non-vanishing fluid viscosity is essential even though viscous effects may be ignored with respect to inertia, as demonstrated in the classical contributions of 15,5 and 16. It is also in order to mention here the other extreme case, namely the effect of self-propulsion of oscillating profiles in creeping flows (Stokes regime) where inertia effects are ignored altogether with respect to viscous ones (i.e., 1494911t13 and others). The latter case is easier to handle analytically since the velocity field and the induced pressure can be found directly from solving a linear bi-harmonic equation.

One can then pose the following question: can a body self-propel itself through an inviscid and incompressible medium, or in other words, can a

603

604

fish swim in a perfect fluid? This interesting problem has been analysed in great lengths by 1 2 , 1 7 9 6 9 7 9 1 . Saffman l2 should be probably credited as the first to demonstrate that the answer to this query is yes!. He was able to show it by considering a deforming nearly spherical shape and using a four- term expansion for the surface deformation in terms of tesseral harmonics. The analysis in fact rendered a stronger result, namely that even a periodic surface deformation with zero-mean over a single period, which preserves both the volume and centroid location of the deforming spherical shape, can result in a persistent self-propulsion velocity starting from rest. In order to evaluate this velocity it is essential to carry the analysis to O(c2), where E

is a typical deformation amplitude. have reconsidered Saffman’s case of a deforming

sphere, by using a full axi-symmetric tesseral expansion (i.e., employing infinite series instead of only four terms) and releasing the assumption of isochoric deformation by also allowing the sphere (bubble) to pulsate. They were able to correct few algebraic flows in S a h a n ’ s l2 analysis and provided the following general statements (valid of course only for an axisymmetric deforming sphere) : a) The induced persistent self-propulsion velocity is quadratic in E even for a non-isochoric (pulsating) surface mode. b) Self-propulsion is created by non-linear interaction between the symmetric and skew-symmetric terms in the surface deformation expansion. c) Maximum effect (i.e., largest locomotion velocity) is attained when the time-dependent symmetric and skew-symmetric terms are all out of phase. Further extensions for general (non-spherical) deforming shapes are also possible by considering the induced forces and moments which exert on such shapes and the resulting trajectories. For a general account on the subject the interested reader is referred to lo.

The fact that a deformable body can indeed propel itself in an otherwise quiescent infinite expanse of perfect fluid, is definitely not apparent since it seems at first glance to violate the D’Alambert paradox. Nevertheless, such a self-propulsion effect is rather restricted in practice mainly for two reasons; a) the persistent velocity is small being quadratic in the deformation amplitude e; b) for a body starting from rest, the resulting impulsive mean propulsive velocity implies infinite acceleration at t = O+ which may be physically unrealistic. Motivated by the current interest on flow-control around insulating bodies in conducting fluids (such as sea water or plasma) by imposing a uniform magnetic field 9, it is suggested to study the effect of MHD self-propulsion of deformable shapes. In order to do so, we consider a quiescent conducting fluid in the presence of a uniform magnetic field. The

Benjamin & Ellis

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fluid magnetic Reynolds number is assumed small. A smooth (streamlined) body starts to deform periodically from rest. It is shown that in this case the body acquires a persistent acceleration which is compatible with the condition that V = 0 at t = O+. Furthermore, the self-propulsion effect is more pronounced in the present case since the acceleration is O ( E ) and the resulting propulsive velocity increases with time after the deformation has commenced.

2. General formulation

We consider an electrically conducting and incompressible Newtonian fluid in an unbounded 3D domain. The governing equations which include the Lorentz term are

(2.1) Dv Dt ~ - = - V ~ + J X B + ~ V ~ ~ ; V . ~ = O

where v is the ambient velocity, p denotes the pressure, B is the applied magnetic induction and J is the induced current in the flow field. The fluid density p and the kinematic viscosity u are taken to be constants. Using Ohm's law (neglecting induced magnetic fields) the electric density current can be written as

J = o ( E + v x B) (2.2)

where E denotes the ambient electric field and 0 represents the electrical conductivity of the fluid. Maxwell's equations and the requirement that V J = 0, admit for an irrotational flow the solution B = BO = const and imply that V x E = '17. E = 0. Thus, the induced electric field can be generally expressed as,

E = Eo-lBolVe , V2e=0 (2-3)

where EO is the uniformly applied electric field and e is the disturbance electric potential due to the presence of the magnetic field, subject to a proper decay condition at infinity. Representing the ambient magnetic field as BO = IBolb, where b denotes a unit vector in the direction of Bo, allow us to express the boundary conditions applied on the surface of any non- conducting and uncharged surface S, as

de d n - I s = -(v x b . n) Is

where n is a unit normal vector to S directed into the fluid. Equation (2.4) assures that the normal component of Jevaluated on S vanishes. It enables

606

us together with Eq. (2.3) to determine uniquely the induced electric field in the fluid.

Let us next consider the case of an insulating deformable body embedded in an otherwise conducting incompressible fluid. Expressed in a Cartesian coordinate system attached to the body centroid, let the time- dependent surface of the body be denoted by S(r, t) and the surface deformation velocity by Vd. Since S is a stream surface one gets

For simplicity we assume that the body starts to deform at t = O+ and at the same instance it is also subject to the influence of externally applied uniform magnetic and electric fields (Bo, &) respectively. If the surface of the body is assumed clean 3, the initial motion induced in the fluid is primarily irrotational since at the outset inertia effects govern the viscous ones. Thus, it is plausible to express the deformation velocity Vd in terms of a deformation potential 4, i.e., 'ud = V4, where V2$ = 0 and 4 -t 0 away from the deforming surface. Under these circumstances, (i.e., pure deformation) the velocity term appearing in both Eq. (2.2) and Eq. (2.4) can be replaced by V$ and the Lorentz term in Eq. (2.1) representing the initial MHD pressure gradient in the fluid is given by,

V ~ L = a(E0- 1 BO I Ve + 0 4 x Bo) x Bo. (2.6)

If the body is rigid 0 4 = 0, the boundary condition Eq. (2.4) renders e = 0 and thus the 'hydrostatic' Lorentz pressure gradient is constant aEo x Bo. This will give rise to the well-known buoyancy-like MHD force - aVE0 x BO exerted on a body of volume V.

In order to compute the 'hydrodynamic' MHD propulsive force resulting from the last two terms in the r.h.s. of Eq. (2.6) we refer to the methodology of * and multiply Eq. (2.6) by Vpi. where pi(i = 1,2,3) denote the linear unit Kirchhoff potentials such that V2pi = 0 and I s= ni. The pressure force F(F1, F2, F3) is then obtained by integrating the product VPL Vpi over the semi-infinite fluid volume V+ surrounding the body. Thus,

S,, VPL * V p i d = V ( p ~ V p i ) d = - pLnidS (2.7) Fi = S,+ s,

or when substituting the non-buoyancy terms in Eq. (2.6)

607

Applying Stokes and Green’s theorems to Eq. (2.8) enables us to express the latter also as

The fact that the MHD pressure force acting on the deformable body cannot be expressed only in terms of surface integrals is connected with the physical stipulation that these loads are connected with the Joule energy dissipation throughout the entire fluid volume.

3. Example: Axisymmetric deformable sphere

In order to demonstrate the effect of the MHD self-propulsion, let us consider the simple case of a deforming spherical shape with a time-dependent radius given in polar coordinates as,

00

r (e , t ) = a(t)[i + ~ ( e , t ) ] = a(t ) [ l+ CE,(~)P,(P)] ; P = case (3.1) n=O

where Pn(p) denotes the Legendre polynomials and E~ = O(E) . This deformation pattern preserves azimuthal rotational symmetry about the axis 0 = 0. Associated with Eq. (3.1) there is a deformation potential

Applying the impermeable boundary condition to second-order on the

avt + u(1+ r ] ) + a- l ( l - p2)r],$, = $r + ar]&, (3.3)

deformable surface Eq. (3.1) yields,

;on T = a(t).

where u = % is unrestricted. Eq. (3.3) implies that to first order 1 3

= $(ZnV).

X o = - k ; A , = d , + k e , , n > l (3.4)

where k = 3: =

k = 0 and If the deformation is isochoric, i.e., the volume is preserved at all time,

Substituting Eq. (3.1) in Eq. (3.5) yields, 0 0 2

En - O ( E 2 ) .

n

608

In order to isolate the dynamic effect of self-propulsion which is enhanced by momentum exchange between the deformable surface and the surrounding fluid from the purely kinematic one, we impose the restriction that the surface deformation does not change the location of the body's volume centroid, i.e.,

which after some tedious algebra and using the orthogonality properties of the Legendre polynomials, gives,

Thus, for isochoric deformation both EO and ~1 are O ( 2 ) and E, = O ( E ) for n > 1. If the deformation contains a volume-mode (pulsation or source flow), then co = O ( E ) and k can be in principal O(1). Since el = O(E') , one needs to find also A1 to second-order. In order to do so, let us multiply Eq. (3.3) by p and integrate both sides over [-1,1]. Following a procedure similar to one gets,

n + 2 A1 = d l + k & l + 3 x (2n + 1)(2n + 3) (in + k ~ n ) ~ n + l (3.9)

n

n + l (dn+l+ kEn+l)En. +9x n ( 2 n + 1 ) ( 2 n + 3 )

Finally, differentiating Eq. (3.8) with respect to time and substituting back into Eq. (3.8) leads

A1 = - 3 x ( 2 n + 3)-l(dn + I C E , ) E ~ + ~ = O(E 2 ). (3.10)

So far we have established the relationship between the coefficients of the deformation potential Eq. (3.2) and the prescribed deformation pattern Eq. (3.1). The next and final step is to evaluate the self-propulsive MHD force given in Eq. (2.9) which arises due to the interaction between the deformable body and the applied magnetic field. In order to preserve the axial symmetry of the problem at hand, we assume that Bo is directed along the same axis and without loosing generality we can also choose EO = 0. The first integral on the r.h.s. of Eq. (2.9) depends on the electric potential e and must be determined by solving the boundary value problem Eq. (2.4) with v = V$ and b = (l,O,O). Because of symmetry it can be easily

n

609

verified that Is= 0 and thus e E 0 (a well-known result in symmetric MHD problems).

The second integral term in the r.h.s. of Eq. (2.9) is identified as the deformation Kelvin impulse. The only surviving term is that which is directed along x1 (axis of symmetry). Since it is essential here to work to second order, we substitute

nl= P - C E n ( 1 - P 2 ) p n ( P ) + o ( E ~ > (3.11) n

for the x1 component of the normal n and

which eventually leads with d S = -27ra2[1 + 2 C E ~ P ~ ( P ) ] (see ' ) and Eq.

(3.10) to m

& I d s = 27ra4X1 = -67ra4 c ( 2 n + 3>-'(in + I C E ~ ) E , + ~ . (3.13) I . n

What remains now is to compute the last integral in Eq. (2.9) which can be also written as &+ g$db', where cp1 = -$v denotes the Kirchhoff potential. In order to evaluate the partial deriviates of the various potentials with respect to 2 1 , it is useful to use the following relationship (see 2, p. 135);

(3.14)

Thus, by virtue of Eq. (3.2) one gets

n m

where

(3.16)

610

which yields finite values only if m = n f 1 and m = n - 3.

in Finally, combining Eq. (3.4), Eq. (3.13), Eq. (3.16) and Eq. (2.9) result

Thus, for an isochoric deformation (V = 0 ) , the force experienced by the deforming body and the resulting acceleration is generally 0(c2). However, if the sphere is allowed to pulsate (i.e., V # 0 ) , the self-propulsion effect is more pronounced and is O(E) . Furthermore, a single-mode expansion Eq. (3.1) with only ~3 # 0 is sufficient to obtain the propulsive force which according to Eq. (3.17) is of O ( E ) and given by F1 = & a uBiV ~ 3 .

4. Discussion

Let us consider for example the case of a massless deforming spherical bubble with a time-dependent radius given by Eq. (3.1), preserving axial symmetry. If the ambient fluid is non-conducting (i.e., u = 0 ) or BO = 0 , then following Benjamin and Ellis the deformable bubble will acquire a self-propulsive velocity given by

Thus, the velocity of the bubble centroid due to surface deformation is always quadratic O(c2) in the deformation amplitude. It arises from nonlinear interactions between symmetric (n-even) and skew-symmetric (n- odd) surface modes. For periodic deformation these two modes must be out of phase in order to obtain maximum mean (over a period) self-propulsion velocity. The minimum number of non-vanishing terms in Eq. (3.1) according to l2 is four, i.e., n = 0 , 1 , 2 , 3 ( E n = 0 for n > 3). For an isochoric deformation Eq. (3.6) and Eq. (3.8) yield EO = -$ E; - $ c i and c1 = -% € 2 ~ 3 . The self-propulsion velocity according to Eq. (4.1) in this case is simply V, = 9 i z ~ 3 . For non-isochoric (changing volume) deformation there is no restriction imposed on EO = O ( E ) and Eq. (4.1) renders

On the other hand, the same deforming bubble when embedded in a conducting fluid (u # 0 ) in the presence of an external uniform ambient

27 d & = w z ( v E 2 ) & 3 .

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magnetic field Bo, attains a self-propulsion acceleration given by Eq. (3.17) divided by its added-mass. The bubble can in principle accelerate from rest with a vanishing initial velocity, such an effect is not possible for a non-conducting fluid medium. Moreover, the minimum required number of terms in the expansion Eq. (3.1) in this case is only one (n = 3) with cn = 0 for n # 3. Following Eq. (3.17) the self-propulsion acceleration A, is simply given by A, = auB,2~3. It is thus shown that unlike the self-propulsion velocity V, = O(e2) the MHD self-acceleration is more evident being of a higher order, i.e., A, = O(E) . It is incited by the interaction between the simple volume (source) mode u and the third (skew-symmetric) mode ~ 3 . For a periodic a( t ) and E g ( t ) the two must be out-of-phase in order to achieve a maximum mean acceleration over a single period.

References 1. T.B. Benjamin and A. Ellis, Self-propulsion of asymmetrically vibrating

bubbles, J . Fluid Mech. 212, 65-80 (1990). 2. E.W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea

Pub. Co. N.Y. (1965). 3. D. Legendre and J. Magnaudet, The lift force on spherical bubble in a

viscous linear shear flow, J. Fluid Mech. 368, 81-126 (1988). 4. J.M. Lighthill, On the squirming motion of nearly spherical deformable

bodies through liquids at very small Reynolds number, Commun. Pure Appl.

5. J.M. Lighthill, Note on swimming of slender fish. J . Fluid Mech. 10, 321-344 (1960).

6. T. Miloh, Optimal self-propulsion of a deformable prolate spheroid, J. Ship Res. 27, 121-130 (1983).

7. T. Miloh, Hydrodynamic self-propulsion of deformable and oscillating bubbles, Mathematical Approaches in Hydrodynamics, S.I.A.M. (ed. T. Miloh) 21-37 (1993).

8. T . Miloh, The motion of solids in inviscid uniform vortical flows, J.Fluid Mech. 368, 81-126 (2003).

9. T. Miloh, Magnetohydrodynamics in non-uniform flow fields, (to be published) (2005).

10. T. Miloh and A. Galper, Self-propulsion of general deformable shapes in a perfect fluid, Pmc. Roy. SOC. Lond. A. 442, 273-299 (1993).

11. R. Purcell, Life at low Reynolds, Ann. Phys. 45, p. 3 (1977). 12. P.G. Saffman, The self-propulsion of a deformable body in a perfect fluid,

J. Fluid Mech. 28, 385-389 (1967). 13. A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds

number, J. Fluid Mech. 198, 557-586 (1989). 14. G.I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R.

SOC. Lond. A. 209, 447-461 (1951).

Math. 5, 109-1 18 (1952).

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15. G.I. Taylor, Analysis of the swimming of long and narrow animals, Proc. R. SOC. Lond. A . 214, 158-183 (1952).

16. T.Y. Wu, Swimming of a waving plate, J. Fluid Mech. 10, 321-344 (1961). 17. T.Y. Wu, The momentum theorem for deformable body in perfect fluid,

Schiffstechnik. 23, 229-232 (1976). 18. T.Y. Wu, On theoretical modeling of aquatic and aerial locomotion, Adv.

A p p . Mech. 38, 291-353 (2001).

NONLINEAR ANALYSIS ON TRANSITION TO FORM COHERENT STRUCTURES*

JUNW Department of Mathematics and Statistics, University of Vermont, I6 Colchester Ave

Burlington, VT 05401, USA

M YANG Department of Mathematics and Statistics, University of Vermont, I6 Colchester Ave


Through examples in fluid mechanics and geophysical fluid dynamics, this study concerns nonlinear behavior of small disturbances as they evolve in time to form coherent structures. The mathematical problem is interesting as solutions to the linearized equations are unstable when a parameter in the model passes through a critical value. For an example in fluid mechanics, we use a set of shallow water equations to model the free- surface flow down an inclined open channel, including the effect of internal dissipation. We use a combination of perturbation and Fourier series methods to derive evolution equations for the periodic asymptotic solution. We present asymptotic and numerical results to show that our theory predicts the solution accurately for both transient and quasi-steady phases. As an example in geophysical fluid dynamics, we use Eady’s model to study the behavior of small disturbances around a basic zonal flow in the case of baroclinic instability. We show that for a set of special parameters our method with some dominant modes captures the formation of coherent structures. Weakly nonlinear analysis for a general case is difficult due to the complex dynamics of the problem which leads to chaos. One of such example is also given.

1. Introduction

Transition phenomena are common in many physical situations, especially fluid flow. One group of transitions occurs when some basic state of the nonlinear system become linearly unstable. In this case, small disturbances superimposed on the basic state will grow in time and this provided the mechanism for the transition process. One good example is the study of the roll waves in an inclined open channel; see Whitham (1974), Dressler (1949), Needham & Merkin (1984), Yu & Kevorkian (1992) and Yu & Yang (2003). The governing

This work is supported by the Vermont-NASA EPSCoR Program and the University of Vermont.

613

614

equations of the mathematical model are the shallow water equations for mass and momentum conservation (see for example, Kevorkian (1 995)),

h, +uh, + hu, = 0,

u2 hum F 2 R

h(u, +h,+uu,)=h--+-,

where h is the free surface height normal to the sloping bottom and u is the flow speed parallel to the bottom averaged over h . The undisturbed flow speed is F (Froude number), and R is the Reynolds number. The hu, 1 R term in the second equation of (1.1) accounts for internal energy dissipation.

Equations (1.1) without the internal energy dissipation term ( R = 00 ) were studied by Whitham (1974), Dressler (1949) and Yu & Kevorkian (1992). The basic state h = 1, u = F is linearly unstable if F > 2 (Whitham (1974)), i.e., for any small disturbance imposed on the basic flow, the response predicted by the linear theory of the original governing equations has amplitude that grows exponentially in time. Dressler (1949) showed that it is necessary to have F > 2 in order to find discontinuous quasi-steady periodic solutions that he called roll wuves. Weakly nonlinear evolution of a marginally unstable problem was solved using multiple-scale expansion by Yu and Kevorkian (1992). Kevorkian et al. (1995) studied a general class of hyperbolic conservation laws for this problem. A review paper on these studies including a linearly unstable example in solid combustion was given in Yu (2004).

The model with R = O(1) was studied by Needham & Merkin (1984), Yu et al. (2000) and Yu & Yang (2003). Needham & Merkin (1984) derived the linear stability condition,

2R F I - R - k 2 ’

where k is the wave number of the disturbance. They also showed that periodic quasi-steady continuous waves exist when the basic flow is locally unstable. Yu et al. (2000) and Yu & Yang (2003) considered the evolution of arbitrary initial disturbances and their relation to the calculated quasi-steady waves. Marginal instability where the wave number k is close to the critical wave number k, , i.e.,

0 < (k, - k ) l k , = E << 1, (1.3)

with

615

was assumed. Yu et al. (2000) focused on weakly nonlinear long waves (k CC 1 ) and multiple scale expansions were used to derive a generalized Kuramoto-Sivashinsky equation that governs the dominant asymptotic solution for large time. Yu & Yang (2003) used a combination of perturbation and Fourier series methods to derive evolution equations for the asymptotic solution corresponds to a single mode initially, i.e., they analyzed the asymptotic behavior of the solution of (1.1) subject to the initial conditions

h(x, 0) = 1 + &ao exp(ih) + &a: exp(-ih),

u(x, 0) = F[1+ &bo exp(ih) + (1.5) exp(-ih)],

where E measures the amplitude of initial disturbances and

la01 = 0(1), p o l = O(1). (1.6)

By using up to order O ( E ~ ) of the governing equation (l.l), asymptotic periodic solutions correct to O(&*) were obtained. The asymptotic solution was shown to remain valid in both transient and quasi-steady phases.

In Section 2 we summarize the result of the combination of perturbation and Fourier series methods. In Section 3, we extend the method to an example in geophysical fluid dynamics. There, we use Eady’s (1949) model to study the behavior of small disturbances around a basic zonal flow in the case of baroclinic instability. Model formulation for the Eady’s model can be found in Pedlosky (1987), with the governing equation

( a -+z- a ) 4+ {a@q ----- a@@} = 0, at ax hey a x *

where 4 is the stream function perturbation,

is the potential vorticity perturbation and S is the stratification parameter. Variables x, y and z are longitude, latitude and vertical coordinates,

616

respectively, and t is the time. We assume periodic condition along x direction with L,, being the period. The lateral boundary conditions are

=0, aty=Oandy=L,. (1.9) ax Finally, conditions for bottom ( 2 = 0) and top ( z = 1 ) boundaries can be evaluated fiom

=o. (1.10)

Linear instability analysis by a normal mode method as well as bounds on the phase speed and growth rate can be found in Friedlander (1980) and Pedlosky (1 987). Nonlinear dynamics of a slightly unstable baroclinic wave for a two-layerf-plane system was studied by Pedlosdy and Klein (1991). Mu and Shepherd (1994) analyzed nonlinear stability of the Eady's model. In Section 3 we summarize results of using the combination of perturbation and Fourier methods for a nonlinear analysis of the linearly unstable problem.

2. Channel Flow Example

We start with an asymptotic solution to (1.1) subject to the initial condition (1 S). We focus on marginally linear unstable solution where (1.3) holds for k . Proceeding as in Yu & Yang (2003), we make a change of dependent variables in (1.1):

1 F

g(x , t ) = h(x,t)-1, v(x, t ) = - [ u ( x , t ) - F ] . (2.1)

We also make a change of independent variables

y = x - Ft - F 6 vo(t ')at', T = t ,

where,

617

Due to the periodic boundary conditions, we assume the asymptotic solution in the form

From (1 .3), linear theory implies that the 1" mode in (2.4) is marginally unstable and all the other higher modes are stable. We consider special initial condition ( lS) , where we only have nonzero 1'' mode initially and higher modes all have zero initial value. For a weakly nonlinear analysis, we assume

go = O(&), vo = O(&), g, = O(dm'), w, = O(&'"'), (2.6)

and show that these lead to a consistent perturbation expansion in (2.4). Substituting (2.1) -- (2.6) into (1.1) and collecting coefficients up to the 2"d mode in m and the 31d order of & , we obtained equations for go, v0, g,, w,, g, and w,. It can be shown that go, W, and w, can be solved from these equations, leaving only differential equations for v0, g, and g, as the evolution equations. The exact form and the derivation of the evolution equations are rather involved. They can be found in Yu & Yang (2003) and are omitted here for brevity.

Once the values of vo, g, and g, have been calculated, the solution for h and u to order is then available through (2.4) and (2.1) as

Xx, t> = 1 + g, (TI exp(iky) +

u(x, t ) = F[I + vo ( T ) + w, ( T ) exp(iky) + w,* ( T ) exp(-iky)

exp(-iky)

(2.7) +g, (T) exp(2iky) + g; (TI exp(-2iky) + W3),

+w, (T) exp(2iky) + w; ( T ) exp(-2iky)l+ O ( E ~ ),

where y and T are defined in (2.2). To illustrate the above idea we compare, in Figure 1, surface height h from (2.7) (dotted line) with the numerical integration of (1.1) (solid line) for a simple example with a, = i / 2 and

618

b,, = 0 in (1.5). More examples, as well as analysis on formation of coherent quasi-steady waves can be found in Yu & Yang (2003). Here, we show in, Figure 2, the comparison of the asymptotic and numerical quasi-steady waves for the above example.

X

Figure 1 , Numerical (solid) and asymptotic (dashed) values of h for t = 50.

Figure 2. Numerical (solid) and asymptotic (dashed) values of h for quasi-steady traveling waves.

619

3. Eady's Model

We now extend our method of combining perturbation and Fourier series methods to study the behavior of small disturbances around a basic zonal flow in the case of baroclinic instability using the Eady's model. Similar to (2.4), we assume the asymptotic solution have the form:

m=-M n=l m#O

B,, (t) exp(-a,z)] exp(irnkx) sin(nly)} + N"

n=l

where A-,, = A:,,, B-,,, = B:,, and

The form of (3.1) with (3.2) was based on the solution of the linearized equation of (1.7), see for example, Mo (1995). The linear stability condition for the mode associated with index of m and n is

amn 2 a, = 2.3994. (3.3)

We choose parameters S, k and 1 such that only the mode associated with a,, is unstable and all the other modes are stable. Also, we choose the initial conditions such that only the mode associated with a,, is nonzero. Further, we consider only the interaction between the modes of a,, and a,,, and assume that

A,, - B,, = O(E), A,,, - Bo, = O(E'). (3.4)

One interesting feature of (3.1) is that it has already satisfied the nonlinear equation (1.7), periodic condition along X direction and the lateral boundary conditions (1.9). Therefore, one must determine A,, and B,, fkom the nonlinear boundary conditions (1.10) by solving a system of ordinary differential equations. The technique can be extended to include even more general form of (3.1), for which the nonlinear governing equation is satisfied.

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The procedure ultimately leads to the technique of projecting differential equations onto a boundary under the boundary conditions. (See Wu (2001), for example.)

Now, proceeding as in Yu and Yang (2005), we substitute (3.1) into boundary conditions (1.10) to obtain complex evolution equations for A,, , B,, , 4,, , and Bon . It can be shown that there exists a quantity A such that 4,, and Bo, depend linearly on A , and A, , and B, , can be represented as a quadratic polynomial of A . The evolution equation for the quantity A is given

by

1 d 2 A ---= V, +V,A+V,A2 + & A 3 , k2 dt2 (3.5)

where coefficients V,, V, , V, and & depend only on parameters S , k and I, and initial values A, , (0) and B, , (0). (The form of these coefficients can be found in Yu and Yang (2005).)

To demonstrate our asymptotic solution, we study a special case where S = 0 . 2 8 6 , k = n , Z = Z , a,, =0.99ac, A , , ( 0 ) = 3 ~ 1 0 - ~ and B,,(O) = - 3 . 2 3 ~ 1 0 ~ . We show that in this case our asymptotic solution captures the coherent structure of the Eady’s model. In Figure 3, we compare amplitude lAll(t)l calculated by the asymptotic method with that from the numerical computation, which included terms in (3.1) with M = 6 and N = No = 13 . (It can be shown that in this case contribution from these terms in (3.1) well approximates the nonlinear solution.) Figure 4 shows the phase diagram. As can be seen in Figures 3 and 4, the coherent structure of the nonlinear solution was well captured by our asymptotic solution. However, the dynamics of the nonlinear system for a general case is quite complex and, in general, chaotic. One can not expect our asymptotic solution, which has the form (3.1) and includes only interaction between the modes of a, , and a,, , to capture the complex behavior of the nonlinear solution. In Figures 5 and 6 , we show only the numerical computation (which included terms in (3.1) with M = 6 and N = No = 13 ) for a typical case of S = 0.0625, k = 6.255, l=n , a,, = 0 . 7 3 a c , A , , ( 0 ) = 3 ~ l O - ~ and B , , ( o ) = - 3 , 2 3 ~ l O - ~ . The scale of the wave number in this case comes from the linear stability analysis

621

and, as demonstrated in Pedlosky (1987), it is in good agreement with the observed scale of synoptic atmospheric disturbances.

k = h , S4.286. tx,,=0.9Srxe - simulation approximation All(0)=3x1 0". - _ I _ _

0 005 O - 1 Bl,(0)=-3.23~10'

0.004

7

6 0 003

0.002

0,001

0.OOO 0 100 150 250

time

Figure 3. Numerical (solid) and asymptotic (dashed) amplitudes for a special marginally unstable case.

0.0003 -

0.0002 -

0.0001- n 9 P

o.oo00 -

-0.ooD1 -

-0.0002 -

k4.n. S-0.286, rt,,=0.99trc - simulation

A,, (O)=3x1OJ * approximation

0.000 0.001 0.002 0.003 0.004 0.W

14 t I

Figure 4. Phase plan for the case in Figure 3.

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0030,

0.025 -

0.020 - - 9

0 015 -

0.010 -

0005-

k=6.255. I==, S=0.0625, t~,,=0.73cz~

A, ,(0)=3~10’~, 8, ,(0)=-3.23~10”

Figure 5. Numerical amplitudes for a typical case.

0.018 1 k=6.255, I=x. S=0.0625, ~ill=0.73tLC A,,(0)=3xfO”, B,,(0)=-3.23x104

0.015

0 012 4

0.000 0.005 0.010 0.015 0,020 0.025 0.030

&,I

Figure 6. Phase plan for the case in Figure 5.

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4. Conclusions and Outlook

We have done an asymptotic and numerical study of two weakly nonlinear, marginally linear unstable problems in fluid mechanics and geophysical fluid dynamics. As the solutions to the linearized equations are unstable when a parameter in the model passes through a critical value, it is crucial that we account for the cumulative effect of small nonlinearities to obtain a correct description of the evolution over long times. A combination of perturbation and Fourier series methods has been proven useful for such nonlinear analysis.

For the second example of the Eady’s model, we obtained mathematically the coherent structure (see Figures 3 and 4) for the nonlinear solution corresponding to a set of special parameters. These parameters are largely different from the scales of the synoptic atmospheric disturbances. However, they might correspond to some other atmospheric andor oceanic processes or they may be realized in experiments (see for example, Mo (1995)). Further researches through observational studies of atmosphere and ocean, especially using remote sensing data, are needed in order to identify more patterns of atmospheric and oceanic disturbances. Finally, numerical simulation and analysis of the chaotic solutions can help us gain understanding of the complex dynamical system in the general cases.

References

1. R. F. Dressler, Comm. on Pure andAppl. Math., 2, 149 (1949). 2. E.T. Eady, Tellus, 1,33 (1949). 3. S. Friedlander, An Introduction to the Mathematical Theory of Geophysical

Fluid Dynamics, North-Holland Publishing Company, New York (1980). 4. J. Kevorkian, Partial Differential Equations: Analytical Solution

Techniques, Chapmen and Hall, New York (1993). 5. J. Kevorkian, J. Yu and L. Wang, SIAM J. Appl. Math., 55, No. 2, 446

(1995). 6. J. D. Mo, Y. M. Zheng and B. N. Antar, FluidDyn. Res., 16, 251 (1995). 7. M. Mu and T. Shepherd, J. Atmos. Sci., 51, No. 23, 3,427 (1994). 8. D.J. Needham and J. H. Merkin, Proc. R. SOC. Land., A394,259 (1984). 9. J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, New York

(1987). 10. J. Pedlosky and P. Klein, J. Atmos. Sci., 48, No. 10, 1,276 (1991). 11. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New

12. T. Y. Wu, A h . Appl. Mech., 37, 1 (2001). 13. J. Yu and J. Kevorkian, J. Fluid Mech., 243,575 (1992).

York (1974).

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14. J. Yu, J. Kevorkian and R. Haberman, Studies in Appl. Math., 105, 143

15. J. Yu and Y. Yang, Studies in Appl. Math., 111, 1 (2003). 16. J. Yu, Observation, Theory, and Modeling of Atmospheric Variability,

(Chief Editor, Xun Zhu), World Scientific Publishing Company, 589 (2004).

17. J. Yu and Y. Yang, Nonlinear Evolution of Small Perturbations in the E a 4 ’ s model, In Preparation (2005).

(20UU).

LARGE-REYNOLDS-NUMBER FLOW ACROSS A TRANSLATING CIRCULAR CYLINDER

BANG-FUH CHEN Department of Marine Environment and Engineering, National Sun Yat-sen University,

Kaohsiung, Taiwan 804. E-mail: chenbf@,mail.nsysu.edu.tw

YI-HSIANG W Department of Marine Environment and Engineering, National Sun Yat-sen University,

Kaohsiung, Taiwan 804

TIN-KAN HUNG Departments of Bioengineering and Civil and Environmental Engineering

University of Pittsburgh

Offshore structures often encounter waves, currents and earthquake excitations. Assessment of hydrodynamic forces on structures can be learned from viscous flows past an oscillating cylinder. During an earthquake excitation, the relative velocity Vo between cross flow (current) and an oscillating cylinder (induced by ground motion) could be very large. In this computational flow simulation, the flow is developed from an impulsive acceleration with a constant cross flow characterized by Ro = 3000 along with a large oscillatory velocity VC of the cylinder.

1. INTRODUCTION

Viscous flows across a circular cylinder have drawn a significant attention in the last two decades. The interaction between a constant cross flow and a moving cylinder was reported by Dutch et al. (1998) for a low Reynolds number (100 to 200) of the cross flow with an exciting frequency about 0.2 redlsec which can be characterized by a small Keulegan-Carpenter number Vo /(2ao) between 5 to 10. For flow across a cylinder in a marine environment during earthquakes, the frequency can be 10 to 20 times higher than that in Dutch’s study and the Reynolds number can be one or two-order of magnitude larger.

In this study, the cylinder is oscillating with lugh frequency under a cross flow of & = 2paVdp =3000, and Vo /(2ao) = 1. The motion of the cylinder can be represented by its Reynolds number & = (2paV~/p)cos(oTi-z). The

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complexity of the transient flow processes associated with the oscillating cylinder is resolved by mapping the time-dependent flow region to a fvred domain for the computational analysis. The computational procedure has been well tested and validated for transient flow past a stationary cylinder. The numerical solutions are in good agreement with those reported in the literature.

2. METHOD OF APPROACH

The two-dimensional unsteady flow can be solved from the vorticity transport equation expressed in a polar coordinate (7,4) moving with the oscillating cylinder (indicated by point C in Fig. 1) as

where y is the stream function, w" the vorticity, and vthe kinematic viscosity. The stream function can determined from

As shown in Fig. 1, bz(4, t) represents the radial distance of the outer boundary of the flow field in the moving frame indicated by point C. Due to the motion of the cylinder, b2 and the flow domain are varied with time. The complexity and difficulty associated with such a time-varied domain can be resolved by transferring Eqs. (1) and (2) from the moving coordinate (7,4) to a fixed polar coordinate system (7,6) through

FI case = s+FCOS# (3)

(4) 1

qe) = {[b, (4, t ) + s(t> cos 41' + [ ~ ( t ) sin #1'>7 in which S is the horizontal displacement of the cylinder. In this case, the origin (point 0' in Fig. 1) of the fixed coordinate system is located at the cylinder center at the onset of flow acceleration. Also, the computational domain or B(8) becomes time invariant. The coordinate system is further transformed to

I F - a @ = - 4 r = B(e) - a 2r

( 5 )

627

By this transformation, the cylinder face is mapped onto r* = 0 and the outer boundary onto r* = 1, and the region onto @ = 0 and 2 (Hung 1981; Chen 1997). Thus, the moving cylinder surface is transformed to a fixed value (r * = 0) and the whole computational domain is mapped onto a rectangular region. Finally, the coordinates r* is stretched to produce fmer grid sizes near the cylinder face. This transformation results in irregular and time-dependent meshes in the fluid domain while representing regular meshes (AR x A@) in the computational domain (Hung and Wang 1987; Hung and Chen 1990).

The numerical solutions were obtained from the fixed coordinate system with a constant oncoming flow. The distribution of the stream function on the moving cylinder can be integrated directly from the velocity of the cylinder and prescribed along r* = 0 in the R-@ plane. On the outer boundary of the computational domain, the irrotational flow is not altered by the moving cylinder when the boundary is far away from the cylinder.

3. RESULTS AND DISCUSSIONS

In order to prevent numerical instabilities, the Courant-Friedrichs-Lewy condition and the necessary condition for stability are followed along with the grid Fourier numbers. In a test of three different time steps 0.05,O.Ol and 0.002, the surface vorticity distribution is almost identical for AT = 0.01 and 0.002. In order to assure numerical stability and accuracy, a smaller time step AT = 0.001 is used in the calculations of the present study.

The numerical accuracy was warranted after a parametric study of the mesh sue. The mesh size is primarily controlled by the stretching coefficient kl. In this study, kl = 3, AR = 1/250, AQ = 1/200, AT = 0.001. Due to Eq. (6), the radial mesh sizes adjacent to the cylinder are extremely small. The computational code is written in a parallel computing mode on an IBM SP2 supercomputer with eight CPU in all the simulation cases. The computational procedure has been successfully tested for an impulsively accelerated flow for R = 3000. As shown in Fig. 2, the present numerical results are in good agreement with that reported by Ta Phuoc LOC and Bouare (1985).

The rest of the results is for a flow produced from rest by an impulsively accelerated cross flow characterized by & = 3000 together with an oscillatory motion of the cylinder. The dimensionless cylinder velocity is indicated by & =

3000 cos(oT + n). The flow patterns shown in Fig. 3 begin with an onset potential flow at T=O. The stream function is obtained from Eq. (2) with vanishing vorticity. They are plotted in the fvred frame. When the coordinate

628

system is moving with the cylinder, the unsteady flow processes can be characterized by an instantaneous Reynolds number of the flow past the cylinder:

R = & - & = 3000 - 3000 COS(CDT+X)

which is equal to 6000 at T= 0. From T = 0 to 0.5, the cylinder is moving in the upstream direction with a dimensionless velocity expressed as a Reynolds number &, whch is vaned from zero to -3000 during this time interval. Thus, the instantaneous Reynolds number R of the flow is reduced from 6000 to 3000. At T = 0.5, the dimensionless velocity of the cylinder is momentarily equal to zero. The flow pattern in the fixed frame is the same as that observed in a coordinate system moving with the cylinder. Recapitulating, the transient flow is produced by an impulsive acceleration from rest with R = 6000, followed by flow deceleration associated with the cylinder motion. The vortices produced in this short duration are rather weak in comparison with those generated by Ro without the cylinder motion.

From T = 0.5 to 1.5, the cylinder is moving in the downstream direction. The instantaneous Reynolds number reduces to zero (R = 0) at T = 1. From T =

1 to 1.5, the instantaneous Reynolds number of the flow increases from zero to 3000 as & reduces from 3000 to zero. Again, the flow pattern in the fixed coordinate system is the same as in the moving frame at T=1.5. As the cylinder moves back towards the upstream and completes the first cycle of oscillation at T = 2, the instantaneous Reynolds R reaches 6000 again. The flow field is no longer irrotational as was the case for the onset flow acceleration. The viscous effect can be noticed from the asymmetry flow pattern along the north-south axis of the cylinder center. At T = 2.5, the cylinder velocity momentarily vanishes again, the zone of separation is much larger than that T = 0.5, and each of the symmetric vortices has been elongated and split into two vortices. In the fixed frame, the effect of the cylinder motion in the cross flow direction can be further seen from 2.5 < T < 3.5. From T=3 to 3.5, the instantaneous Reynolds number increases from zero to 3000 while & decreases from 3000 to zero. Further flow acceleration is associated with the cylinder moving back upstream and completing the second cycle of oscillation at T = 4.

Figures 5 and 6 show the flow patterns for the 3rd cycle of oscillations. At T = 4, the oscillating cylinder is moving upstream, passing the center of its oscillation, and the instantaneous Reynolds R = 6000. Although the line of separation cannot be seeing in the fixed frame, the growth of vortices with time is evident. Comparing the flow patterns at T = 2.5 and 4.5, one can see the flow separation and the growth of vortices when the cylinder reaches the upstream

629

end of the oscillation with R = 3000 and & vanishes momentarily. At T= 3 and 5, the cylinder is moving with the same velocity of the cross flow and the Reynolds number of the flow vanishes R= 0. The curvilinear flow patterns are characterized by streamlines moving with the cylinder. The line of separation and vortices can be seen again when the cylinder reaches the downstream end of its oscillation at T = 3.5 and 5.5. Due to the strong effect of the cylinder motion or & on the cross flow, the vortices are practically symmetric instead of the phenomena of the von K a d n vortex street.

4. CONCLUSION

As the motion of the oscillating cylinder continues, the zone of separation becomes bigger with more vortices being developed. The intensity of a vortex (amount of flow rate circulating around it) can be increased through energy transfer from the main flow or decreased through energy dissipation. The pressure and shear along the cylinder are calculated and the drag force is correlated with the oscillating cylinder and the time variation of the Reynolds number. The study was supported by the National Research Council of the Republic of China.

REFERENCES

1. Chang, C. C. & Chern, R. L. (1991). ”A numerical study of flow around an impulsively started circular cylinder by a deterministic vortex method.” Journal of Fluid Mechanics, 233,243-263.

2. Chen, B.F. (1997), “3D nonlinear hydrodynamic analysis of vertical cylinder during earthquakes. I: rigidmotion”, J. Engrg. Mech., 123 ( S ) , 458-465.

3. Hung, T.K. 1981, “Forcing function in Navier-Stokes equations”, J. Engrg. Mech.,

4. Hung, T. K. & Chen, B.F. (1990), “Nonlinear hydrodynamic pressures on dams”, J. Engrg. Mech., ASCE, 116(6), 1372-1391.

5. Hung, T.K. & Wang, M.H. (1987). “Nonlinear Hydrodynamic Pressure on Rigid Dam Motion.” J. Engrg. Mech., ASCE, 113(4), 482-499.

6 . Dutsch, H., Durst, F., Becker, S. & Lienhart, H. (1998). “Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers.” J. of Fluid Mechanics, 360,249-271.

7. Ta Phuoc LOC, (1980). “Numerical analysis of unsteady secondary vortices generated by an impulsively started c circular”, Journal of Fluid Mechanics 100,

8. Ta Phuoc LOC & Bouard, R. (1985) “Numerical solution of the early stage of the unsteady viscous flow around a circular cylinder: a comparison with experimental visualization and measurement”, Journal of Fluid Mechanics 160, 93-1 17.

ASCE, 107, EM3,643-648.

11 1-128.

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Fig. 1 The definition sketch.

(iii) Streamline at T = 4 2.0

1.5

1.0

0.5

0.0 .1.50 .1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00

Fig. 2 Comparison of streamline patterns for Re=3000, at T=4;(i)Phuoc LOC & Bouard (1985) flow visualization, (ii)Phuoc LOC & Bouard (1985) numerical result and (iii) present analysis.

631

Streamline at T = 0 Streamline at T = 0.25

Streamline at T = 0.5 Streamline at T = 0.75

Fig. 3 Flow patterns observed in the fixed frame; T=O to 0.75.

Streamline at T = 1

2

1 I m om 100 nm 3w 4w sm 6 W 7w 800

Streamline at T = 1.75

Streamline at T = 1.5

Streamline at T = 2 3

2

1

0

1

2

1 .2.m -1m om I M 200 3w 4m sw 6 W 7w BW

Fig. 4 Flow patterns observed in the fixed frame; T=l to 2.

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Streamline at T = 4 Streamline at T = 4.25


Fig. 5 Flow patterns observed in the fixed frame; T = 4 to 4.75.

Streamline at T = 4 Streamline at T = 4.25 3.

2

1

0.

1.

2

3 -2.w - l W ow 1.w ZW 3.03 I W 5.w 6.00 7w I W LW l W ow rw ZW 303 I W 5w 600 7w I W


Fig. 6 Flow patterns observed in the fixed frame; T = 5 to 5.75.

A SPECTRAL METHOD FOR THE MASS TRANSPORT IN A LAYER OF POWER-LAW FLUID UNDER PERIODIC FORCING

LINGYAN HUANG Department of Mechanical Engineering, The University of Hong Kong, Pokjiulam Road,

Hong Kong

CHIU-ON NG Department of Mechanical Engineering, The University of Hong Kong, Pokjiulam Road,

Hong Kong

ALLEN T. CHWANG Department of Mechanical Engineering, The University of Hong Kong, Pokjiulam Road,

Hong Kong

A Fourier-Chebyshev collocation spectral method is employed in this work to compute the Lagrangian drift or mass transport due to periodic forcing in a thin layer of non- Newtonian mud, which is modeled as a power-law fluid. Because of the non-Newtonian rheology, these problems are nonlinear and must be solved numerically. On assuming that the solutions are of the same permanent waveform as the pressure loading, the governing equations are made time-independent by referring to a horizontal axis that moves at the same speed as the wave. The solutions are periodic in the horizontal direction, but are non-periodic in the vertical direction, and the computational domain is therefore discretized according to the Fourier-Chebyshev spectral collocation scheme. In this study, the spatial derivatives are computed with a differentiation matrix. In order to incorporate the boundary conditions, the matrix diagonalization technique is used to solve the matrix equation, and all the definite integrals in the vertical direction based on the collocation points are performed by the modified Clenshaw-Curtis quadrature rule. The developed method is applied to compute the motion of the mud. The comparison between the numerical results and the analytical solution in the Newtonian limit shows the good accuracy of the spectral method.

1. Introduction

Mass transport of bed mud (i.e. cohesive sediment deposits) under surface waves is an important consideration when predicting the transport of nutrients or contaminants and the long-term change of bed level in a coastal sea. Most mud- type suspensions will exhibit shear-thinning or pseudo-plastic behaviors when the concentration of very fine particles becomes sufficiently high (Whitehouse et al. [7]). Essentially the effective viscosity of the fluid mud will decrease with an

633

634

increasing shear rate. Such a non-Newtonian behavior of the fluid will render the mass transport problem much more difficult to solve than that for a Newtonian fluid. The problem is nonlinear, and therefore, unlike the Newtonian case, the mass transport velocity cannot be found simply by first time-averaging the second-order differential equation, which instead must be solved in full before the steady component can be separated from the complete solution. This work aims to show the application of a spectral method to solving numerically for the mass transport of a power-law fluid under periodic forcing.

In the following sections, we shall first briefly explain how the nonlinear equations of mud motion under wave pressure are obtained, and then present a spectral numerical scheme by which the equations are solved for the mud mass transport. The present problem consists of a shallow layer of power-law fluid mud, which is driven directly by an applied periodic pressure load on the free surface. The focus here is on the development of a fully discrete spectral collocation method used to find solutions of the first- and second-order nonlinear equations of motion of the mud. Since the mass transport, by definition, is a Lagrangian quantity, we shall base our study on the Lagrangian description.

To solve numerically a partial differential equation, the finite difference or finite element methods are often used to compute the spatial derivatives; but these methods will typically require a large number of nodal points in order to yield satisfactory results. As a promising alternative, spectral and pseudo- spectral methods have been greatly advanced in recent years. The spectral method distinguishes itself from the finite difference and frnite element methods by the fact that the global information is incorporated in computing a spatial derivative. The spectral method can yield greater accuracy for a smooth solution with far fewer nodes and therefore less computational time than the finite difference and finite-element schemes. In this paper, the nonlinear equations of motion are solved by a Fourier-Chebyshev collocation method, in which the solutions are numerically approximated by global interpolation polynomials rather than truncated series expansions as used by Ozkan-Haller and Kirby [5 ] .

2. Theoretical formulation

We assume that there exists a single uniform layer of power-law fluid mud lying on a horizontal rigid bed and subject to a periodic external pressure load on the free surface. See Fig. 1 for a definition sketch of the problem.

The constitutive law for a power-law fluid is given by . "-1 .

zy = py yy ( i , j =x,z), (n < 1),

635

where zii is the shear stress, yii = aui / a x j +auj /axi is the shear rate,

y = ,/m is the shear rate amplitude, p > 0 is the consistency parameter,

and n is the flow index, which is taken to be 0 < n < 1 in view of the fact that marine mud is typically shear thinning (or pseudo-plastic).

We present here a dimensionless formulation in which the horizontal and vertical lengths are normalized with respect to the wavelength of the forcing and the undisturbed depth of the layer, respectively, and the time is normalized with respect to the period of the forcing (see Fig. 1). Denoted by (a, 6) are the undisturbed horizontaYvertica1 coordinates of any fluid particle, and (x , z ) the instantaneous coordinates of the particle at time t L 0. The corresponding components of the particle velocity are then given by (u, w) = (i, i), where the

overdot denotes the time derivative. The Lagrangian description is to express all the variables as functions of a, 6 and t. There is no motion at the rigid bottom of mud, which is fixed at S = z = -1. On the other hand, shear stress is assumed to be zero on the free surface, where the instantaneous elevation of the free surface at S = 0 is given by 7 = z (a , 0, t). The dominance of pressure over shear stress as a driven force to the mass transport of bed mud has been shown experimentally by Isobe et al. [3].

surface pressure forcing p = -P,cos(ka-ot)

surface elevation

power-law fluid particle displacements x = x(s6,t) z = z(c1,6, r)

6 = -h .........................................................

Figure 1. Definition sketch of the problem

636

The governing equations below have been derived from the Lagrangian form of the Navier-Stokes equations and boundary conditions by means of a perturbation analysis based on the shallowness of the problem E = kh << 1. Further details can be found in Ng [4]. The present study is primarily concerned with finding the numerical solutions by a spectral method.

In the first-order dimensionless problem, the shear stress is

and the governing equation of the power-law fluid motion is

where Fr = o2 I gk2h is the Froude number, i12 = ptsn-2 I ph2 is the square of the normalized Stokes boundary layer thickness, and is a prescribed amplitude of the pressure load. By the change of variable 5 = a - t, the above equation can be converted into a steady form as

where u(l) (5,s) = i ( l ) = -ax(') /a{ is the first-order horizontal velocity, and the

overbar denotes depth averaging. Equation (2.4) is subject to the boundary conditions that u(') = 0 at 6 = -1, &(I) 186 = 0 at 6 = 0, and u(') is 2z-

periodic in {, i.e. u(') (5) = u(l) (5 + 2 ~ ) . It suffices to find solutions in the domain O < { I 2 z and - 1 < S I O .

After finding d ) , the other variables can be computed as follows:

6

J1) (4,s) = ju'"ds , $) (0 = 8) (s = 0) . -1

At the second-order, the shear stress is found to be

637

and the governing equation of the mud motion is

where C (given in Ng [4]) is a lengthy forcing term composed of sums of

products of the first-order solutions, and u(') (5,s) = X(') = -ax(') / 86. The

boundary conditions for (2.8) are: u(') = 0 at 6 = -1, &(') / d S = 0 at S = 0,

and u(') is 2s~periodic in 5. Once u(') is found, the mass transport velocity can be evaluated from its 5-average

3. Spectral collocation method

Since the physical problem is 2z-periodic in the horizontal direction, but non- periodic in the vertical direction, the Fourier-Chebyshev collocation techmque is adopted to solve numerically the two-dimensional differential equations given above.

For the sake of simplicity, let us explain the essence of the adopted spectral method through a one-dimensional case. Instead of working with a truncated series expansion in the traditional spectral methods, we construct the numerical approximation uN (x) to a fimction u(x) through a global interpolation polynomial based on the grid values u(xj) (Gottlieb and Hesthaven [2])

according to

where u N ( x ) is an interpolation of the function u(x), i.e. uN(xj)=u(x.), J

andZj(x) represents the Lagrange interpolation polynomial based on the grid

points xi, which takes the form

638

The derivative of u(x) at the grid point xi is numerically evaluated by

d l . (3.3)

where Dv is the differentiation matrix.

3.1. Periodic direction: Fourier method

In the horizontal direction, Fourier collocation method is applied to the periodic variables, and the collocation points in {-direction are chosen to be equally spaced so that Cj = 2 n j / Nx, j E [1, Nx] . The numerical approximation

uN ( 5 ) can be expressed by a global periodic interpolation polynomial as

with uN (ti) = u ( t j ) and the corresponding differentiation matrix Ds in the

case of even collocation being

3.2. Non-periodic direction: Chebyshev method

In the vertical direction, the non-periodic physical variables are written as Chebyshev approximations, with which the Chebyshev collocation method is used for evaluating the vertical derivatives and the Clenshaw-Curtis quadrature for evaluating the vertical integrations.

639

3.2.1. Definition and general properties

The Chebyshev polynomial of the first kind T, ( x ) is a polynomial of degree k

defined for the canonical interval x E [-l,l] by

Tk ( X ) = cos(k c0S-l X ) , k = 0,1,2, ..., (3.6)

and therefore, -1 2 Tk I 1 (Peyret [6]). From trigonometric relationships, we get the recurrence relation on the Chebyshev polynomials

Tk+l ( X ) - 2XTk ( X ) + Tk-l ( X ) = 0, k 2 1 , (3-7)

which allows us to deduce the expression of the polynomials Tk , k 2 2 , from To = 1 and = x , and the recurrence relation on the derivatives

3.2.2. Chebyshev collocation method

At the collocation points (Gauss-Lobatto points)

x j = c o s - , j=O, 1 ,..., N , ( Y ) the Chebyshev approximation uN ( x ) for x E [-l,l] is defined by

j = O

with uN (x i ) = u (xi ) , and hi ( x ) is a polynomial of degree N defined by

( - 1 r ' (1 -.')Ti ( x ) hi ( x ) =

C j N 2 ( x - X j ) ,

(3.9)

(3.10)

(3.1 1)

where co = c, = 2 and cj = 1 otherwise (Peyret [6]).

As the non-periodic hc t ions in our problem are bounded in the range [-l,O] , we define Sj in the vertical direction as the images of the Nz+l

Gauss-Lobatto poing xi by the transformation

640

1 Sj = -( 2 x j - 1) = +[ cos ( j z / Nz) - 1 1 , j E [O, Nz] , (3.12)

with the ( N z + 1) x ( N z + 1) Chebyshev spectral differentiation matrix D, being

, i = j = O , 2Nz2 + 1

1 3 ci (-1)’”

, i * J , I-- c j si-sj - cos ( j z / N z )

1 - cos’ ( jz / N z )

9

, O < i = j < N z

2Nz2 +1 , i = j = N z -- 1 3

(3.13)

2, i = O or Nz 1, otherwise

where ci =

In order to minimize the round off errors for the calculation of the first- order derivative, we use the correction technique of Bayliss et al. [l] to compute the diagonal entries D,i,i by

NZ

D ~ ~ , ~ = -C D ~ ~ , j , i = 0, ..., NZ . (3.14) j=O j+i

To calculate the second-order differentiation matrix Df) , we fist calculate

a provisional differentiation matrix if) as the square product of the matrix D6, then repeat the application of the correction technique of Bayliss et al. [l].

The choice of the images of Gauss-Lobatto points ensures that the grid points in the physical domain are concentrated close to the bottom and surface boundaries.

3.2.3. Numerical integration

The definite integral such as Z = uN (S)dS based on the collocation points

Sj = - [ cos ( j z / N z ) - 11 , j E [0, Nz ] is accomplished by the Clenshaw-Curtis

P, 1 2

641

quadrature rule. Modifying the Clenshaw-Curtis quadrature from [-l,l] to the

interval [-1,0], we get

(3.15)

where the weights mi are 1/2 times of the analogues given by Peyret [6] .

4. Solution method

4.1. Boundary conditions

In order to explain how to handle different types of boundary conditions for second order equations, let us use the Newtonian limit of Eq. (2.4) as a simple example:

The domain $2 in which the equations (4.1H4.3) are solved is confined to

0 < 6 I 2n and -1 I 6 50. The solution u(') (4,s) is approximated by the

polynomial u t ) (C,S) of degree at most equal to Nx in the &direction and Nz

in the Gdirection. The spectral collocation approximation to problem (4.1)- (4.3) is based on the Fourier-Chebyshev mesh defined by

2n 1 Nx 2

ti =-i, i ~ [ l , N x ] , Sj =-[cos(jn/Nz)-11, j ~ [ O , N z l . (4.4)

We denote the inner discretized domain by RN = ( ~ i y 6 j ) y i = 1, ..., Nx,

j = 1, ..., Nz-1 and the discretized boundary by r = (&,Sj ) , i = 1, ..., Nx,

j = 0, Nz. Equation (4.1) is forced to be satisfied by the polynomial u,$) (g., S,) at every inner collocation points (ti, S j ) E R,, and the boundary conditions

(4.2) and (4.3) are applied at the boundary points (ti, d j ) E r.

642

Now the vertical boundary conditions can be taken care of by following the matrix diagonalization method as used by Pey-ret [6] for the one-dimensional Helmholtz equation, which has general Robin boundary conditions as follows:

The application of the collocation method leads to an algebraic system for the unknowns u$) (ti, Sj ) , For the diagonalization

procedure, it is convenient to eliminate the vertical boundary values by boundary conditions (4.2) and (4.3) so that the unknown row vector is

i E [ 1, Nx] , j E [ 0, Nz].

U(‘) (i) = (u$’ ( t i , d1), u$’ (ti, S2) ,..., u$’ (ti, 6Nz-l)) , i E [1, Nx] . (4.7)

The boundary values u i ) ( ti, 6,) and u i ) (ti, &) are expressible from the

boundary conditions by

where

= cO,+cN,- -‘O,-‘N,+ (4.10)

(4.1 1)

(4.12)

b0,j = -co ,+P+D~o,~ - C ~ , J - D ~ ~ ~ , ~ , j = l , . . . , Nz-1, (4.13)

co,- =-P+Dso,N~, co,+ =a- + P - D G N ~ , N ~ ,

cN,+ =-P-DGNz,O, cN,- =a+ +P+DSO,O,

bN,j =-CN,-P-DGN~,~-~N,+P+DS~,~, j = 1 , . . . , Nz-1, (4.14)

and DSm,j, m = 0 ,..., Nz, j = 0, ..., Nz are obtained from Eqs. (3.13) and (3.14).

Due to the effect of boundary conditions, the differentiation matrices

643

corresponding to the &derivative along each ti line are changed into the

(Nz - 1) x (Nz - 1) matrices D t ) and D f ) with

Dbm,j (’1 - - D6m,j (’) +,( D6m,0 ( 2 )b , .+Dg!NzbN, j ) , 0 I m , j = l , ..., Nz-1. (4.16)

For an efficient application of the matrix diagonalization technique in the two-dimensional case, the discrete system approximating (4.1H4.3) can be written into the matrix form as

where dl) is the matrix of dimensions Nxx(Nz-1) consisting of the inner unknowns in R, , that is,

(4.18) i = l , ..., Nx, j= l , ..., Nz-1.

In Eq. (4.17), Dr and D f ) are matrices of dimensions N x x N x and (Nz-1)

x(Nz - I), respectively. Finally, H is an Nxx(Nz - 1) matrix containing Nx

row vectors H = [ H ~ 3‘ , i = 1, ..., NX , with

4.2. Solution by iteration

The first- and second-order governing equations (2.4) and (2.9) discretized by the Fourier-Chebyshev collocation method are nonlinear systems. In order to avoid the impact of the initial guess to the convergence of the nonlinear systems, a pseudo-transient continuation has been used to solve the governing equations. The steady state is practically achieved when the relative errors between two successive pseudo-time steps reach a given level of tolerance (say, lo”), and the results are then the solutions to the original governing equations. With a simple

644

explicit pseudo-time-discretization scheme, the pseudo-transient transformation of Eq. (2.4) can be written as

(4.20)

where u f ) and uk+l (1) are the k th and (k +1) th iteration results of the solutions,

respectively, and At is the pseudo-time step that can be adjusted to control the stability of the iteration.

The space discretization scheme is the same as that for the simple example in the last section. Now we may express the discrete system for the recurrence Eq. (4.20) into a matrix form as

where U(’) is the matrix of the inner unknown in ZZ, as in the last section. The treatments of boundary conditions uN (1) (c i ,Sj ) for (c i ,s j ) E r are the same as

Db -(*) Db (1) =Db (2) intheNewtonianlimit.

(4.7) and (4.8). However, the differentiation matrix corresponding to

3/38 in the governing equation will be derived by use of the relationship

To start the iteration, we use the tensor product (also called “Kronecker product” or “direct products”) to get the initial guess, but use the form of (4.15) to save the computing time and memory of the computer in the iteration procedure.

The solution procedure for the second-order Eq. (2.8) is basically the same as that for the first-order equation described above.

645


5.1. Accuracy test

In this section, the accuracy of the numerical treatment is verified using the comparison between the analytical and numerical solutions to the first-order mud motion for the Newtonian limit.

For the case of Ps = 1.0, d = 1 .O and Fr = 1.0 , the analytical first-order vertical and horizontal displacements given by Ng [4] at the free surface 6 = 0 can be expressed by

$) = z ( ~ ) (6 = 0 ) = % [ Z(6 = O)eic] = 0.022 cos 8 + 0.33 12sin c , (5.1)

x(l) (6 = 0) = %[.?(6 = O)ei5] = 0.4959~0s 5 -0.0412 sin 5 , (5.2)

which are in the ranges of q(’) ~[-0.3319,0.3319] and x(’)(S=O)E

[-0.4976,0.4976].

At the collocation points Nx x (Nz + 1) = 50 x 46, the numerical solutions of

the first-order vertical and horizontal displacements at the free surface are given

by $ ) E [-0.3320,0.3320] and x(l) (6 = 0) E [-0.4965,0.4965] , where #)

and x(’) can be obtained from u(l) by the Clenshaw-Curtis quadrature rule in the vertical direction and the trapezoidal rule in b e horizontal direction, respectively. The maximum errors of #) and x(’) are 0.02% and 0.22%, respectively, and therefore we adopt the nodal discretization Nx x (Nz + 1) =

50 x 46 in our all-numerical calculations. The computation in the work has been performed with MATLAB programs known as “M-files.”

5.2. First-order motion

The present problem depends on four input parameters, i.e. n , A’, Fr and P, . A smaller value of n corresponds to a stronger non-Newtonian character, or more specifically, a higher rate of decrease of the effective viscosity with the shear rate.

We first recall the first-order motion of the fluid mud for the Newtonian limit. Figure 2 shows the horizontal velocities u(’) at n = 1 with Ps = Fr =

A’ = 1.0, which is a simple harmonic function of the phase variable 5 like the pressure load, that is, the positive maximum value in each 6 appearing near 6 = irc 12 and negative maximum value near 6 = 3~ 12, and zero velocity near

646

4 = 0, K and 2 ~ . Figure 3 shows the counterparts at n = 0.2 for a non- Newtonian shear hnning mud, which are no longer in simple harmonic motion. The fluid mud seems to be nearly motionless over a frnite fraction of the phase near 4 = 0, z and 2z, where the magnitude of the pressure load is weak. This phenomenon can be explained with the shear-thinning rheology of the mud, which shows a tremendously large effective viscosity at a low shear stress. The fluid will not be in motion until the applied load is high enough to overcome the resistance, which happens near the peak of the periodic pressure load at 6 = z / 2 and 3~ / 2. Even after the mud starts to move, the magnitudes of the velocity for the non-Newtonian fluid n = 0.2 are smaller than those for a Newtonian fluid at the pressure load P, = 1 .

0.5

U ('1

- -1- - - - - - - - r------ ,i - - - - - -

I I I \

0

Figure 2. The Fourier-Chebyshev numerical solution u(') of the first-order governing equation of

motionfor n = l (pS = F ~ = A * =1.0)

647

0 5

Figure 3. The Fourier-Chebyshev numerical solution U ( l ) of the first-order governing equation of

motionfor n = 0 . 2 (pS =Fr=a2 =1.o)

5.3. Second-order motion

Since the mass transport is the (-average in a period of the second-order velocity u ( ~ ) and the Fourier collocation points are equally spaced in the period, the numerical solutions of the mass transport across the depth can be obtained

by the &-average of the Nx horizontal velocity u ( ~ ) . Figure 4 gives a comparison of vertical profiles of the mass transport velocity uL as a function of n with P, = 1.0 and P, = 2.0, respectively. It is noticeable that the vertical velocity profile tends to be more uniform near the free surface as the value of n becomes smaller. Figure 4 shows the mass transport velocity increases sharply near the bottom where the shear stress is higher. Therefore, the use of images of Gauss-Lobatto points in the vertical direction can indeed enhance the performance of the numerical method.

Figure 4 shows that the mass transport velocity near the free surface decreases with a decreasing value of n when P, = 1.0. The results are however

648

dramatically different when the pressure load is increased to P, = 2.0 : the mass transport velocity then increases as the flow index is lowered. Note that the mass transport velocity for the highly pseudo-plastic fluid is very sensitive to the pressure load. As P, is doubled from 1.0 to 2.0, the maximum value of mass transport velocity for n = 0.6 increases from nearly 0.2 to 2.0. This illustrates that the increase of the applied load may lead to considerable diminishing of the effective viscosity for a pseudo-plastic fluid after overcoming the resistance to motion. The Fourier-Chebyshev numerical solutions are found in close agreement to the results of Ng [4] obtained by a second-order implicit finite difference scheme.


A Fourier-Chebyshev collocation spectral method has been employed in this work to compute the Lagrangian drift or mass transport velocity due to periodic forcing in a thin layer of non-Newtonian mud, which is modeled as a power-law fluid. The results are found to be in very good agreement with the analyhcal solutions in the Newtonian limit and the numerical solutions by finite differences of Ng [4]. Since far fewer nodes are needed, the computational time is much reduced as compared with that of a finite difference method. Typically the convergence to a steady state can be achieved in a few minutes with a 1.70GHz CPU personal computer. From the calculations and comparisons, the Fourier-Chebyshev collocation spectral method is demonstrated to be accurate and stable when applied to solving the present system of nonlinear equations.

The first- and second-order numerical results reveal the sensitivity of the fluid motion to the pressure load with the shear-thinning character. It is clear from the profiles shown in Fig. 4 that the mass transport velocity increases sharply near the bottom where the shear stress is higher.

Acknowledgments

The work reported in this paper constitutes part of the Ph.D. thesis of the fist author, under the supervision of the second and third authors, at the University of Hong Kong. Financial support by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Projects HKU 7081/02E and HKU 7 199/03E is gratefully acknowledged.

649

0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

s

-

-

- - -

-

-

s

0 I I L -0.1

-0.2 -

-0.3 -

-

-0.4 -

-0.5 1 //a

UL

Figure 4. Vertical profiles of mass hansport velocity UL as a function of n 2 with 1 = F, = 1.0 and (a) 8 = 1.0 or (b) 8 = 2.0.

650

References

1 . A. Bayliss, A. Class and B.J. Matkowsky, Roundoff error in computing derivatives using the Chebyshev differentiation matrix. J. Comput. Phys. 116,380-383 (1994).

2. D. Gottlieb, and J.S. Hesthaven, Spectral methods for hyperbolic problem. J. Comput. and AppliedMath. 128,83-131 (2001).

3. M. Isobe, T.N. Huynh and A. Watanabe, A study on mud mass transport under waves based on an empirical rheology model, Proc. 23rd Int. Con$ Coastal Eng. 3093-3106 (1992).

4. C.O. Ng, Mass transport in a layer of power-law fluid forced by periodic surface pressure. Wave Motion, 39,241-259 (2004).

5 . H.T. Ozkan-Haller and J.T. Kirby, A Fourier-Chebyshev collocation method for the shallow water equations including shoreline runup. Applied Ocean Research, 19,21-34 (1997).

6. R. Peyret, Spectral Methods for Incompressible Viscous Flow. Springer- Verlag, New York (2002).

7. R. Whitehouse, R. Soulsby, W. Roberts and H. Mitchener, Dynamics of Estuarine Muds. Thomas Telford, London (2000).

THE COHESION AND RE-SEPARATION OF PARTICLES IN SLOW VISCOUS FLOWS

REN SUN Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai,

200240, China

ALLEN T. CHWANG Department of Mechanical Engineering, The University of Hong Kong, Pokjiilam Road,

Hong Kong

The coagulation and re-separation process of two spherical particles in a slow viscous flow is investigated analytically. Both particles may differ in size, and the configuration of the particles is arbitrary relative to the direction of the driven flow. By using the multi-harmonic transformations between two coordinates and the extended successive reflection method, the complete solution to the exterior velocity field around the two submerged particles is obtained. It is used to determine the hydrodynamic interaction between the two particles and then to predict the coagulation and re-separation process of the particles.

1. Introduction

Interacting particles in an unbounded slow viscous flow are relevant to areas as disparate as chemical engineering, biological engineering, soil mechanics, geophysics, materials science and the microelectronics industry. It pertains to the motion of microorganisms, the transport of blood cells in arteries, and the flocculation of granular matter such as slurries, colloids and composites. Several theoretical approaches for solving the equations of low-Reynolds-number flow around two spheres have been developed for many years. These include those using reflections [I], bispherical coordinates [2-41, tangent-sphere coordinates 1151, collocation methods [6] and twin multipole expansions [7] etc.

Our purpose here is to investigate how a spherical particle captures another one in a uniform creeping flow, and under certain conditions, these two touching particles driven by the flow would separate from each other again. The two particles are released in a low-Reynolds-number flow, so the general solution for the flow field outside the two particles may be expressed in harmonics and biharmonics based on Lamb’s representation of the solution for Stokes’ flow. In order to proceed with the problem analytically, sets of transformations of

651

652

harmonics and biharmonics between two coordinates are given. By using these transformations and the extended successive reflection approach, the complete solution to the velocity field of the two spherical particles submerged in a slow viscous flow is obtained analpcally. The present method is accurate and produces results in iterative form that are suitable for computation. Moreover, every corrective series is established just to satisfy the impenetrable boundary condition just like images in an ideal flow [8], and only the final series satisfies the no-slip boundary condition. In this manner, the items in the final series expressions decay by the order O(l/s2) faster than those obtained by the pure reflections, so that all the final series converge even when the two spherical particles contact each other. Consequently, analytic expressions in closed form for the hydrodynamic interaction forces and moments exerted on the two particles are determined. Based on these accurate expressions in iterative form, several dynamical cases from capture to re-separation are obtained numerically from the dynamical equations of motion. These numerical results are compared with available experimental data, and the comparison shows good agreement on the whole coagulation and re-separation process.

2. The Solution of Two-Sphere Problems

2.1. Governing Equation and Boundary Conditions

Two spherical particles of radii R , and R, with densities p, and p, respectively are released in an Smite fluid of density p and viscosity p, whose velocity at infinity is U , = 1.0 . To simplify the analysis, the two particles are considered to make planar motion in the x-y symmetric plane, as shown in Fig. 1. Under the assumption of incompressible creeping flow, the fluid velocity u and pressure p are governed by the familiar Stokes equations together with the continuity equation

v p - p v = u = 0, V ' U = 0, (1)

and the boundary conditions are

u=ui+uie ,xr ; . on q = R i ( i=1,2) ,

(2) u +U,e, as r 03,

where ui is the translational velocity of particle i, q its angular velocity. Here eo denotes the unit vector along the positive direction of the corresponding axis.

653

FIG. 1. Sketch of two spherical particles and the corresponding coordinates.

2.2. Analytical Solution

Two relative coordinates, (x' ,y ' ,z) and (X' ,Y' , z ) , fixed on the individual particle centers are introduced for convenience, which can be turned into spherical ones by

X ' = r2 sin 0, cos p ,

z = r2 sin 0, sin p. y' = r, case, , and (3) x' = 5 sine, cosp,

= 5 sinel s inp,

From the general solution given by Lamb [9] , the velocity field outside an isolated particle translating and rotating in a Stokes flow with velocity U, at infinity can be written as

r2

6P u = Uoex +B,V(x'/r:) + B2V(y'/r:) -L{AIV(x ' / r : ) + A,V(y'/r:)}

(4) 2 C +-(Ale, + A2ey.) ++(x'eyf - y'ex,) ,

3PF-I r1

where A's, B's and C, are constants to be determined by the boundary conditions. Following Sun & Hu [S], we construct the velocity field u in the

surrounding fluid around a two-particle system as

where boldfaces h's and H's are velocity disturbance vectors in relation to the two-particle system. One may refer to Sun & Hu [8] for fiuther details. Here

654

A,,,, B,,, and C, are constants just like ones mentioned above but for the two particles. Substituting (5) into

(2) leads, with some manipulation, to ten algebraic equations of Ai,, , B,,, and C, (i, j = 1,2),

--+A+ 2B.. A , . V, sin(-+ jr a) = u:,, for i, j = 1,2, Ri' Mi 2

1 3O0 A,., 1 1 a, 3B3-ij O3 B,,, {- + - C a + - {- - - C p,!z;.,l} + - C a g,;+ Ri' k=O P 2Ri k=O k=O

for i, j = 1,2 , (6)

4 A C R 2 for i= 1 and 2, +(-I), 3-iJ 112 i = 0,

7P

where u:,, ( i, j = 1,2 ) denotes the jth velocity component of particle i in the (x',y',z) coordinates, and all the notations are the same as ones in Sun & Hu [8]. Here values of A,,, , B,,, and Ci are simultaneous solutions of the above ten algebraic equations.

2.3. Hydrodynamic Forces and Torques on Individual Particles

In order to investigate the coagulation and re-separation process of the particles, it is important to predict the hydrodynamic forces and torques on individual particles. The forces q and torques exerted by the fluid on particle i due to its motion and interaction with the other are derived by

where Ci is the surface of particle i, n the outward normal to the surface. Here Il .n is the radial component of the fluid stress tensor II . Evaluation of the surface integrals in (7) gives, after some mathematical manipulation, the following components in the (x ' , y', z) coordinates

655

for i = l and 2. Here the corresponding notations are defined in a manner similar to the foregoing ones. One easily find that letting s + co , for these sixteen algebraic equations in (6) and (8), all coupling terms related to s vanish, and thus

Fh. = 6zpRi (U0 c o s a - ~ ~ ! ~ ) , cjr = 6zpRi(-U0 sina -uI2),

qz = -8~puw,R,', (9)

for i=1,2. Expressions in (9) are the well-known results of an isolated sphere moving in a Stokes flow. In (8), in addition, force expressions include such flow parameters related to the rotational effect and those with the translational effect appear in torque expressions as well. This implies that translation would be coupled with rotation in the two particle system. The phenomenon can not be seen in a potential-flow situation if the solid particles are spherical.

656

3. Results and Discussion

Thls research presents analytical expressions for the exterior velocity and the hydrodynamic interaction between two spherical particles. This is a main contribution of the present work. Both Eqs. (6) and (8) are employed to determine the hydrodynamic interaction between two spherical particles and predict their dynamical behaviors in a creeping flow. To this end, the numerical calculation is the vital resort of solving these algebraic equations. As each of coupling terms in the equations decays by the rate of 1/s2 , the truncated series at k = 50 would have errors smaller than the error tolerance of five significant figures even for the touching case. The motion of the particles is governed by

du . dt

dw . dt

M i - - = q + f ; : ,

J i L =7;.+zi,

where Mi and Ji are the mass and moment of inertia of particle i, f , and zi are the restraining force and moment exerted by the other particle, which would not be equal to zero only for the case of two particles in contact. By means of these equations, we may explore how a spherical particle captures another one in a uniform creeping flow for the quasi-steady situation.

3.1. A sphericalparticle drifting around another fuced one

For a particle dnfling around another fixed one, there are two situations in their coagulation and re-separation process. One gives that when a smaller moving particle is captured by the larger fixed one, it keeps rolling without slip on the surface of the large one until it finally runs away, as shown in Fig. 2. Another seen from Fig. 3 demonstrates a pure gliding situation. Note from the figure that the captured small particle would slide on the large one and then goes away in some situation, but there is a situation, in which the smaller particle can not slip away although it keeps rotating. Moreover, it is easily observed from these figures that trajectories of the moving particle are asymmetric with respect to the x axis.

657

~ ......... --.-- ....... y l R l ' . 5 y $ q 10 I. ............ (. ......................... .....

0 5

0 e" -Iw -Iw

05--. 10-

Alp - 26

.m

FIG. 2. (a) Trajectories of a drifting spherical particle around another fixed one for the rolling case. (b) Variation of the orientation angle of a drifting spherical particle around another fixed one for the rolling case.

.....

1.5

y /R , ................. ................. . . . . .. ,..'

0.5

2 1 0 1

(a) xlR,

FIG. 3a. (a) Trajectories of a drifting spherical particle around another fixed one for the sliding case. (b) Variation of the orientation angle of a drifting spherical particle around another fixed one for the sliding case.

3.2. Two driping spherical particles

The above phenomena are also observed €or the two simultaneously drifting spherical particles. Figure 4 shows for given sue and density ratios, the hf t ing particle captures a slightly smaller one, and then the smaller particle rolls on the surface of the larger one until it ultimately leaves. The leaving position is at a point on the larger particle which is almost farthest in the transverse direction. And the two particles rotate in opposite directions. For the case of two identical particles in size and density, one can note fiom Figure 5 that the two particles translate in the same way, keeping an unchangeable configuration in the creeping

658

flow, except for deviating slightly from the stream toward the trailing particle side, independent of the separation distance. And they rotate with the equivalent angular velocity in opposite directions, respectively.

FIG. 4. (a) Coagulation and re-separation process of two drifting particles. (b) Variation of the orientation angles in the coagulation and re-separation process versus xlR,. (c) Orientation angle a of the line joining the two particle centers versus xlR, .

4. Conclusions

There are two situations in the coagulation and re-separation process of two drifting spherical particles. One gives that a particle captured by the other in the flow keeps rolling on the second particle mtil it finally rolls off. Another demonstrates the captured particle would slip on the catcher before it leaves. For the case of two identical particles in size and density, two particles move at the same translational velocity and rotate at an equal angular velocity but in opposite directions.

659

Acknowledgments

This research was sponsored by the National Natural Science Foundation under Grant No. 10372060 and by the Hong Kong Research Grants Council under Grant Number HKU 7 19 1/03E.

References

1 .

2. 3. 4. 5. 6. 7. 8. 9.

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice- Hall, 1965). M. Stimson and G. B. Jeffery, Proc. Roy. SOC. London A l l l , 110 (1926). A. Goldman, R. G. Cox and H. Brenner, Chem. Eng. Sci. 21,1151 (1966). M. H. Davis, Chem. Eng. Sci. 24, 1769 (1969). A. Nir and A. Acrivos, J. Fluid Mech. 59(2), 209 (1973). P. Ganatos, R. Pfeffer and S. Weinbaum, J. Fluid Mech. 84( l), 79 (1978). D. J. JefieyandY. Onishi, J. FluidMech. 139,261 (1984). R. Sun and W. R. Hu, Phys. Fluids 15(10), 3015 (2003). H. Lamb, Hydrodynamics (Cambridge University Press, 1932).

3

FIG. 5. (a) Trajectories of two drifting identical spherical particles. @) Variation of the orientation angles of two drifting identical spherical particles versus x l R , . (c) Orientation angle a of the line joining the two particle centers versus xlR, .

ON COHERENT VORTICES IN TURBULENT PLANE JETS GENERATED BY SURFACE WATER WAVES

CHIN-TSAU HSU Dept. of Mechanical Engineering, Hong Kong University of Science and Technology

Hong Kong, China

JUN KUANG Dept. of Aerospace and Mechanical Engineering, University of Southern Calgornia

Los Angeles, California, USA

Coherent vortices in turbulent plane jets as generated by surface water gravity waves were investigated experimentally. Experiments were carried out in a water tank of 6m long, 0.4m wide and 0.4m high with a water depth of 34cm. Plane jets were generated by ejecting water vertically from a slot orifice of a jet generator mounted on the bottom of the tank. The ejected water has the same temperature as the ambient water to avoid buoyancy effect. Regular surface gravity waves were generated by a flap-type wave- maker on one end of the tank and eliminated by a wave-absorber on the other end. Turbulent plane jets were visualized with a laser-induced fluorescence (LIF) technique and measured with a laser Doppler Velocimetry (LDV) system. It is found that strong alternative coherent vortices like the vortex-street structure were generated and developed in the self-similar region of the plane jets when the surface waves were imposed. Two vortices of opposite sign are generated for each period of the surface gravity waves. The mean properties of the plane jets in a wave environment remain self- similar as occurs in turbulent plane jets in a stagnant ambient. However, the jet spreading in wave environment is much larger than those without waves. Hence, the generation of the coherent vortices enhances the jet mixing. The magnitude of the wave-induced velocities, wave-associated mean stresses and wave-induced turbulent Reynolds stresses were obtained using a phase average scheme. The results show that the wave-induced flow field is dominated by the fundamental mode and the contribution from high order harmonics is negligible.

1. Introduction

Turbulent plane jet is one of the simplest turbulent shear flows and has been studied for decades. The works of Heskestad El], Gutmark & Wygnanski [2], Andreopoulos et al. [3], and Kuang et al. [4], among many others, provided the statistical features of the plane jet experimentally. Another interesting phenomenon, the coherent structure inherent in the turbulent jet, had also been studied by many investigators. De Gortari & Goldschmidt [5] and Cervantes &

660

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Goldschmidt [6] measured a turbulent plane jet using hot-wire anemometers. They detected the vortex structures in the turbulent plane jet by correlating velocities measured at two points on opposite sides of the jet centerline. More detailed spatial correlation measurements by Oler & Goldschmidt [7] and visualizations by Goldschmidt et al. [8] suggested the existence of a two- dimensional vortex-street-ldce structure. In a follow-on investigation, Browne et al. [9] reported a similar coherent structure occurred in the initial region of a plane jet. Their measurement suggested that the decay of centerline velocity in the interaction region be correlated to the onset of the asymmetric structures. Dracos et al. [lo] investigated the coherent structure of turbulent plane jets in bounded fluid layer; their flow visualization results showed clearly the vortex- street-like structures in the flow field of the plane jet.

In the present study, an experiment on the turbulent plane jet in a wave environment was carried out in a water tank. The temperature of the jet water was the same as the ambient water in the tank so that buoyancy effect was eliminated. A laser-induced fluorescence (LIF) technique was used to visualize the flow field. When the surface water waves were imposed on the vertical plane jet, large eddies of alternative sign similar to the Karman vortex-street were visualized in the plane jet. This represents a high order oscillatory flow in the jet flow field as perturbed by small amplitude surface waves. The velocity of the plane jet was measured with a laser Doppler Velocimetry (LDV) system. A phase average scheme was used to process the LDV signal to decompose the mean, the fluctuation and the wave-induced velocity and stresses. It is found the mean properties of the plane jets in a wave environment remain self-similar as in the case of a turbulent plane jet in a stagnant ambient. However, the jet spreading in a wave environment is larger than that in a stagnant environment.

2. Experimental Setup

Experiments were carried out in a 6m long and 0.4mx0.4m (wide x high) water tank, with a water depth of 34cm. The schematic of the experimental setup is showed in Figure 1, which consists of a water tank, a wave-maker, a wave absorber, a jet generator and a water re-circulation circuit. A plane jet was generated by injecting water vertically through a 2mm slot orifice of the jet generator of 39.5cmxllcmx8cm in dimension mounted on the bottom of the tank. Two platforms of lOOcm long, 39.5cm wide and 8cm high were installed on each side of the jet generator to generate a uniform water depth with an effective water depth of 26cm so that the ratio of the water depth H to the jet orifice width d (=2mm) is H/d=130. Two outlets located at the far ends of the

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tank bottom were connected to a water pump to create a water re-circulation circuit, which supplied water to the jet generator and maintained a constant water depth during the experiment. Progressive surface waves were generated using a paddle-type wave-maker installed at one end of the tank, and eliminated using an absorber at the opposite. Wave frequency in the present experiment was 1.6Hz, corresponding to a wave slope ka,=0.046 of small amplitude waves, where k is the wave number and a, the wave amplitude. The jet exit velocity Wo was adjustable by controlling the flow rate to the jet generator. The details of the jet generator design can be found in h a n g et al. [4].

I , A

wavemaker

c) water tank

H

V ,

8- valve 0 - ~owraterneter b - check valve

Figure 1 . Schematic sketch of the experimental setup

The evolution of the turbulent plane jet was visualized using a laser induced-fluorescence (LIF) method. The LIF system consists of a 300mW Argon-Ion laser, a fiber-optic cable, a collimator and a Powell lens. After being collimated, the laser beam is directly through the Powell lens to produce a laser sheet of about 1.5mm thick with uniform intensity. A water solution of Rhodamine 6G (C28H31C1203) was used to seed the jet flow through a 0.6mm inner diameter tube merged with the jet orifice. A green beam with a wavelength of 5 14nm excites the Rhodamine 6G water solution to emit high intensity yellow light with wavelength 570nm. Images were recorded using a 3-CCD digital video camera and captured using a DV raptor card in a PC computer.The flow field of the plane jet was measured with a LDV in a back-scattering mode. Velocity components u and w in the x and z directions were measured. The fiber-optic probe of the LDV system was aligned carefilly to maintain the measurement volume in the x-z plane. The probe was mounted on a computer-controlled 3-D traverse system. The accuracy of the traverse system is f0.01mm in each

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direction. Water was filtered before filled into the tank and degassed to eliminate the unwanted air bubbles. To improve the signal quality the jet flow field was seeded with fine titanium oxide (TiOz) particles of 1-5pm in diameter.

3. Flow Visualization

Figure 2(a) shows the image of the plane jet when the wave frequency is 1.6Hz and the jet exit velocity is W0=1.33mls, i.e., Re=W&-2340. For comparison, the visualization of the plane jet injected into a still water, i.e.,pO, at the same water depth and jet exit velocity is presented in Figure 2(b). The images in Figure 2 show an obvious difference for plane jets in stagnant and wave environments. For the case with the surface waves in Figure 2(a), Karman vortex-street-like coherent structure was observed. However, for a plane jet in a stagnant environment, no vortex-street-like coherent structure was observed. Previous work by Kuang et al. [4] shows that the turbulent plane jet impinging onto a free surface in a water of H/z=130 is divided into three regimes: the zone of flow establishment (ZFE) in O<Z/d<30, the zone of established flow (ZEF) in 30<z/d<100 and the zone of surface impingement (ZSI) in lOO<Z/d<130. In the initial zone of ZFE, Figure 2(a) shows no vortex near the jet orifice and the plane jet was observed to deflect oscillatory in a wave environment.

(a)f=1.6Hz (b) pure jet

Figure 2. Coherent structures of a plane jet in wave environments for the jet exit velocity W0=1.33mlS.

Similar phenomena were observed by Chyan & Hwung [l I] for a round jet in a wave environment. At about z/d=30, the generation of the first eddy was observed. Note that the location of z/d=30 represents the entrance of ZEF for the

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turbulent plane jet. So the coherent vortex structure is expected to initialize and develop in ZEF. When the jet deflects in opposite directions, the vortices with alternative signs are formed and move to the downstream. The process is very much llke a vortex-street shedding behind a cylinder. Wavelet analysis by Kuang & Hsu [12] showed that the vortex shedding frequency was at the surface wave frequency. The Strouhal number defined with vortex shed frequency, the jet exiting velocity and jet width as St=fd/Wo is 0.0024. Moving downstream of the jet, the eddy increases its scale but decreases in intensity. The decay of vortex intensity was observed by the intensity decrease of fluorescence dye. There are about four vortex pairs developed in ZEF of the jet as shown in Figure 2(a). The visualization of the plane jet presented in Figure 2 shows also that the jet spreading is wider in a wave environment as a result of stronger mixing of ambient fluid due to the existence of alternative vortices.

4. LDV Measurements of Jet Flows

4.1. Mean Flow Field and Jet Spreading

The flow visualization showed the generation and development of the vortex-street-like coherent structure in the self-preserving region. However, the long time averaged results of the plane jet remain axisymmetic with the maximum velocity occurred along the jet centerline at x=O. Figure 3 shows the results of the mean velocity distributions at z/d=30,40, 50, 60 and 75 in the ZEF for Wo=1.33mls with surface waves off=1.6Hz. The self-similarity of the mean velocity profiles was confi ied by the figure. The Gaussian profile with fitted constant of ~ = 2 2 as obtained by fitting to the velocity data at f=1.6Hz was also plotted in Figure 3. This value is smaller than ~ = 4 8 for plane jets in a stagnant environment. Note that a small K represents a wider spreading of the jet. Hence, the plane jet spreads wider in a wave environment than in a stagnant environment. This is consistent with the flow visualization shown in Figure 2 under the same ex erimental conditions. The spreading constant a calculated using a = 1 /1.2 P K is 0.179, which is larger than 0.12 for a pure plane jet obtained in Kuang et a1 [4]. The jet spreading can also be quantified by the increase of the half-velocity points xIIz along z. For each longitudinal mean velocity profile in the ZEF, the value of xl/z was obtained by curve fitting the profile to a Gaussian. The result was plotted as a dashed line in Figure 4.

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t

t 0.5

0 5 0.1 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.1

Figure 3. Mean velocity profiles in ZEF of the plane jet at z-stations of z/d=30, 40, 50, 60 and 75 with frequencyf=1.6Hz.

Figure 4. Jet spreading for the plane jet of Re=2340, H/d=130 in the cases with (dashed line) and without (solid line) surface waves.

For comparison, the data of jet spreading in a stagnant ambient was also plotted as a solid line in the figure. Figure 4 shows that the jet spread linearly even in a wave environment. The regression of the data gave the spreading constant PO. 179, consistent with the value obtained above.

4.2. Wave-Induced Flow Field

A periodically oscillatory quantity in wave perturbation flow fields is decomposed as

= 2 sin(wt - B E ) + harmonics (1)

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where is the amplitude and Qz the phase angle of the fundamental mode. Here, g can be any quantity interested, such as the wave-induced velocity or stresses. A detail description of the phase average scheme to obtain g", and Qz was presented in Kuang and Hsu [13]. Figure 5 shows the distributions of (a) I% f w, and (b) 0, at z/d=30,40,50,60 and 75 along the lateral x direction at the jet exit velocity of Wpl .33ds and Re=2340 when the wave frequency is f=1.6Hz. As shown in Figure 5(a), the fundamental mode of the wave-induced velocity had almost zero amplitude along the centerline of the plane jet. Figure 5(a) shows that the maximum amplitudes occur at a position of q=+O.16. The phase 6, in Figute 5(b) shows to jump at the positions of q=O and f0.16.

N

ow

025

0.20

0.45

0.40

005

om

-4.0 4.1 0.8 0.4 0 .2 0.0 0.2 0.1 011 0.8

q=X/(z--.J

0

Figure 5(a). Amplitude distributions of wave-induced velocity in ZEF of the plane jet at Wp1.33mls, Re=2340 in waterH/d=130 with surface waves off=l.6Hz.

4.5 4.4 43 0 . 2 4.4 0 0 0 4 0 2 0.3 0 4 0 5

*-xl(z.zJ

Figure 5@). Phase distributions of wave-induced velocity in wdirection in ZEF of a plane jet at W p l . 3 3 d s , Re=2340 in water H/d=130 with surface waves off=1.6Hz.

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Figure 6 shows the wave-associated mean stresses distributions along the x location at z-stations of z/d=30, 40, 50, 60 and 75 in the ZEF, for W0=1.33ds, Re=2340 with surface waves atf=1.6Hz. Even though the curves in Figure 6 is quite similar to that shown in Figure 5(a), a major difference between them is that Figure 5(a) contains only the fundamental mode of wave-induced velocities

and u" , whle Figure 6 consists of the all harmonic contributions. The amplitude of fimdamental mode is zero at q=O in Figure 5(a) while E2 =O.O04W: at q=O in Figure 6 since harmonics were included. The comparison of Figure 6 to Figure 5(a) shows that the contribution due to the harmonic components is relatively small, i.e., the non-linear effect is negligible.

"'- t

Figure 6. The wave-associated mean stresses distributions in ZEF at z-stations z/d=30, 40, 50 and 60 of plane jet at W ~ l . 3 3 d s ,Re=2340 in water H/d=130 with surface wavesf=1.6Hz.

5. Conclusion

A turbulent plane jet emanating from the bottom in a surface wave environment was investigated experimentally in a water tank. Coherent structure of the plane jet was visualized using LIF and the velocity field measured using LDV. The visualization of the plane jets shows the existence of a Kannan vortex-street like coherent structure in the ZEF when surface waves were imposed. The coherent vortex structures in the plane jets enhance the mixing of the ambient fluid and cause wider jet spreading.

The mean flow properties in the ZEF, such as the linear spreading of the plane jet and the Gaussian distribution of the mean velocity, remain similar to the plane jet in a stagnant ambient. The measurement results confirm a wider jet spreading in a wave environment. The wave-induced components were decomposed by a phase average technique. The results show twin-peak feature for the fundamental modes in the wave-induced velocity, with a maximum value

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occurring at a point of about ~=f0.16. The wave-induced flow field is dominated by the fundamental mode and the contribution from hgh order harmonics is negligible.

Acknowledgments

Thls work was supported by the RGC of the Government of HKSAR under the Grant No. HKUST625W02E.

References

1. G. Heskestad, “Hot wire measurements in a plane turbulent jet” J. Appl. Mech., 32,721 (1965).

2. E. Gutmark and I. Wygnanski, “The planar turbulent jet” J. Fluid Mech., 73, 465 (1976).

3. J. Andreopoulos, A. Praturi and W. Rodi, “Experiments on vertical plane buoyant jets in shallow water” J. FluidMech., 168, 305 (1986).

4. J. Kuang, C. T. Hsu and H. H. Qiu, “Experiments on vertical turbulent plane jets in water of finite depth”J. Engg. Mech. 27(1), 18 (2001).

5. J. C. de Gortari and V. W. Goldschmidt, “The Apparent Flapping Motion of a Turbulent Plant Jet - Further Experimental Results” ASME J. FZuids Eng. 103, 119 (1981).

6. J. G. Cervantes and V. W. Goldschmidt, Trans. ASME I: J. Fluids Engng., 103,119 (1983).

7. J. W. Oler and V. W. Goldschmidt, “A vortex-street model of the flow in the similarity region of a two-dimensional free turbulent jet” J. Fluid Mech., 123,523 (1982).

8. V. W. Goldschmidt, M. K. Moallemi, and J. W. Oler, “Structures and Flow Reversal in Turbulent Plane Jets” Phys. Fluids, 26(2), 428 (1983).

9. L. W. Browne, R. A. Antonia and A. J. Chambers, “The interaction region of a turbulent plane jet” J. Fluid Mech., 149,355 (1984).

10. T. Dracos, M. Giger and G. H. Jirak, “Plane turbulent jets in a bounded fluid layer” J. Fluid Mech., 241,587 (1992).

11. J. M. Chyan and H. H. Hwung, “On the interaction of a turbulent jet with waves” J. Hyd. Res., 31,791 (1993).

12. J. Kuang and C. T. Hsu, “Wavelet analysis on coherent structure in a turbulent plane Jet under surface wave action” (submitted) (2005).

13. J. Kuang and C. T. Hsu, “An experimental investigation on submerged vertical turbulent plane jets with surface waves” (submitted) (2005).

A DYNAMIC MODEL FOR STRONG VORTICES OVER TOPOGRAPHY ON A p PLANE

HUNG-CHENG CHEN, CHIN-CHOU CHU AND CHIEN-CHENG CHANG Institute of Applied Mechanics, College of Engineering,

National Taiwan University, Taipei 106, Taiwan, Republic of China E-mail: hcchen, chucc, changccQiam.ntu. edu.tw

In this study, we propose a simple dynamic model for predicting tracks of strong cyclonic vortices over topography on a p-plane. The dynamic model bears a kinematic relationship that reveals a sophisticated dependence of the velocity component on the local gradient of the topography, effected by the vortex Rossby number, planetary vorticity gradient (the p-effect) and topographic vorticity gradient as well as other effects. Calculations of strong vortices over a bell-shaped mountain with the dynamic model reveal interesting track patterns of cyclonic motion. The results of the dynamic model are also confirmed by solving a more detailed shallow water model; the vortex tracks obtained from these two approaches show excellent agreement.

1. Introduction

The complexity of strong, geophysical vortices interacting with topography has attracted significant attention during the past serveral decades. Exam- ples such as oceanic eddies meandering over the continental shelf or tropical cyclones (TC) drifting over complex terrains have revealed many interesting and complicated flow features. Among many important aspects, the prediction of vortex trajectory over topography is of particular interest for operational forecasting. In previous studies, numerical experiments using shallow water equations l l 2 , quasi-geostrophic vorticity equations 31495 or primitive equations 6,7z8 and laboratory experiments 4,9910 carried out in a rotating tank have successfully captured the dynamical features of the vortex/topography interaction such like the vortex drifting, the Rossby wave wakes and the meandering of vortex tracks.

In spite of these successes, the mechanism with the path of a cyclonic vortex is still not fully understood for the path may be influenced by many physical and geometric factors. It would be useful to have an analytical theory to reveal how these factors are interwoven to influence the motion

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of cyclonic vortices. The purpose of this study is to propose a dynamic model to help elucidate the mechanism of the tracks of the strong cyclonic vortices over topography. The key assumptions made are:

(i) The strength of the cyclonic vortex remains constant when approaching the topography.

(ii) During its evolution, the variation of the mean height of the vortex is proportional to the variation of the mountain height.

(iii) Each topography is associated with a geometric modification factor that accounts for the effect of induced circulations produced by interaction between the primary votex and the topography.

The dynamic model is applied to predict the trajectory of a cyclonic vortex encountering a bell-shaped topography of elliptic platform at different maximum heights. The initial condition is a Rankine vortex, while its initial velocity is determined by solving a much more involved shallow water equation.

2. The Dynamic Model of Vortices over Topography

2.1. Derivation

Figure 1 is schematic of the physical problem that sketches all relevant length scales. First, the flow behavior is investigated following the law of conservation of potential vorticity (PV) ''

Dll - =o, Dt

where II = (fo + Poy + 6) /Ho is the potential vorticity and fo is the Coriol- lis parameter, PO is the beta parameter and C is the relative vorticity of the fluid. The fluid layer depth H o ( q y , t ) can be expressed as HO = Do+q-hs, where DO is the unperturbed depth, q(z, y, t ) is the free-surface depression (negative) due to the motion and hB(z, y) represents the bottom topogra-

In order to apply (1) to the entire cyclone, we assume that the motion of the cyclone has the mean height profile hE as shown in Fig. l(a), and define the associated mean unperturbed depth D = D0/2. The effective mean layer depth H which can be expressed as H = D + q - hE. Assume small variations of the surface depression q and a gentle mean height profile hE with respect to the mean unperturbed depth D, i.e., q / D << 1, hE/D << 1.

phy.

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These assumptions enable us to recast Eq. (1) in the following form

for a strong cyclonic vortex (Ro M 0(1 - 10)).

(a) SIDE VIEW

I

1

(b) LOCAL VIE!,^: - - - - v; cou2y

(c) TOP VIEW Y

higher f

lower f

Figure 1. plane. Here h M is the maximum height of bottom topography and vv is the maximum vortex depression. Note that the figure is not plotted on scale, e.g., Do = 5000m, h M = 250Om and q,, M 408m for vortex conditions: V, M 40m/s, R, w 150km; also hE is much smaller than hB .

Schematic of a strong cyclonic vortex approaching a topography on a

Let r = &/V, be the reference time, where V, is the maximum

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azimuthal velocity and & is the corresponding radius. We have a set of non-dimensional variables defined by

Since (i) f o /D is constant and makes no contribution and (ii) poy << fo for p plane approximation, we can multiply (2) by T ~ D and neglect the higher order terms to obtain

- = o , DII* Dt*

0 where the non-dimensional potential vorticiy

(4)

II* = pop* + Pkhk - p:q*

+ c* (1 + Ro, (/3ih& - p,q*)) = const. (5)

The above expression contains the four non-dimensional parameters involved. They are (i) the vortex Rossby number Ro, = Vm/fo&, (ii) the planetary p number & = ,BoR$/Vm, (iii) the topographic /3 number & = foRmhM/VmD, and (vi) the vortex p number p: = foRmqv/VmD.

The proposed dynamic model predicts that at any instant the non- dimensional velocity e* of a cyclonic vortex consists of two contributions:

where pG is the initial velocity in the absence of topography and c; is the meridionally adjusting velocity.

Next, we derive an expression of the velocity component of cy. For a strong cyclonic vortex approaching the topography, it is reasonable to assume that the mean surface depression qc and the mean relative vorticity cc at the central vortex region (of radius &) are approximately constant. Equivalently, we make the assumptions

bq* x0, and bc* x0. (7)

From (5) and (7), the small time variation of the non-dimensional PV can be approximated as

bII* M poby* + (1 + Rove,*) ,8$bhk = 0,

by* =-a bh; along the direction o f f*,

( 8 )

or, alternatively

(9)

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where the dynamic model constant a! is defined as

A key proposition is made here that the small uariation of the mean height profile bh; is proportional to small variation of the topographic height bh;, that is,

bh> M K 6h;, (11)

where the mean profile proportional constant K is approximately of the order of 0(10-3) N O(10-2). It is noted that under the present assumptions the path of the cyclonic vortex is modified only along the meridional direction y . From (9) and (ll), a meridionally adjusting velocity (MAV) f; can be defined to be

+ by* + dh; + V* E lim -ev = -a!K-ey E -a~Se' , , 21 6t*+0 bt* dt*

where Zv denotes the unit vector directed toward north. The instantaneous rate of change of topographic height S along the direction of q* can be derived from the relationship

dh; h;(q + (@cos2y + I?;)&*) - h&(q) 7 (13) -- - lim

dt* 6t*+0 bt* where denotes the current position of the vortex center. Refer to Fig. l(b); y represents the angle between the unit normal of topography n' and the vertical axis direction i. It is noted that, in (13), the geometrical factor ms2y = 1/(1 + ( d h L / d ~ * ) ~ + (dh ; /ay* )2 ) corrects the magnitude of the velocity @ by first projecting it along the topography and then back in the horizontal direction. Substituting (12) into (13), we solve for S to obtain

S = wsz y (q: -V*h;) . 1 + cYK%

Combining (12) and (14), we have the final expression of the meridionally adjusting velocity

Equation (6) with (15) constitute the proposed dynamic model (DM) which predicts the drift velocity and the position of strong vortices over topography on a P plane.

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2.2. Discussion

The dynamic model reveals several interesting features.

The factor /I&/& indicates that the planetary vorticity gradient and the topography gradient act in the opposite sense to accelerate the cyclonic vortex. When approaching the topography, (qt - V*h&) is generally positive. The q: is negative, and therefore the cyclonic vortex adjusts its proceeding direction toward south. When leaving the topography, (?$ - V* h&) is generally negative. The fg is positive, and therefore the cyclonic vortex re-adjusts its proceeding direction toward north. If there is no topography ( P i = 0 ) , it is evident from the model that the cyclonic vortex will proceed with its original direction.

Nevertheless, one important factor missing from the dynamic model is the effect of induced circulation produced by the interaction between the cylconic vortex and the topography. The factor is less important in approaching but becomes significant when the cyclonic vortex re-adjusts its direction toward north. For this purpose, we introduce a geometric modification factor 8 into the dynamic model constant

Since the net induced circulations are typically in the sense of the primary cyclonic vortex, the factor 8 is expected to be greater than 1.

3. The Shallow Water Model

The full shallow water model (SWM) l1 used in the present study can be expressed as

%+ v * (fiH0) = 0,

g + g v (hB + Ho) = - (fo + POY) 2 x fi, (17)

(18)

where fi = (u, v) denotes the horizontal velocity, g is the reduced gravitational acceleration. In the present study, we choose a value of g = 1.96ms-' which properly represents the buoyancy force in the real atmosphere as suggested in Ref. 12.

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I" *I. D "I m m I"

I " ' I '

-.@> .., - @ ' > ..,

" ... . I

Figure 2. The path of a cyclonic vortex over a bell-shaped mountain approaching from different locations. The dotted lines represent the paths in the absence of topography. The circles represent the vortex center positions obtained by the SWM. The dashed lines represent the dynamic model results with 0 = 1 while the solid lines represent the results predicted by the dynamic model with 0 > 1. The value of 0 varies for different topographic heights (1.5 for h M = 1500rn, and 2.0 for h~ = 2500rn). The mean profile proportional constant K. is a fixed 0.015 for all cases presented here.

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3 .I. Initial Conditions

Similar to Ref. 1, the initial velocity and height field for a Rankine vortex are expressed as

and

where ve(r) denotes the azimuthal velocity of the vortex and T the radial distance of each fluid particle to the vortex center. The SWM was numerically integrated using a public domain, finite-volume code CLAWPACK l3 and was performed with a grid spacing 5km in a 640 x 640 rectangular grid. The vortex conditions are: DO = 5km, cC = 5.33 x 10-4s-1 and R, = 150km.

3.2. Topography Condition

The shape for the island topography is given by

hB = h M / (1 + a i 2 ( x - x ~ ) ~ + bi2(y - ya)2)1’5, where x, and ya the coordinate of the mountain center, ah and bh are the mountain half-widths in the x and y directions, respectively. The topography conditions are: ah = 40km and bh = 120km with h M = 1500m or 2500m. In this study, we have adopted values that roughly represent the location and geometric shape of the Island of Taiwan similar to Ref. 7. All the DM calculations are integrated using a small time step 6t = 40s in view of the reference time T NN 3750s. The topographic gradient V*h> is calculated analytically deriving by (21).

4. Results and Discussion

Figure 2 shows the predicted paths by the dynamic model with comparison to the calculated results by solving the shallow water model as well as to the paths in the absence of topography. The initial location of the vortex for each case from north to south is spaced by 50km on the right side 300km of the topography. Both the predicted tracks calculating from these two approaches consistently show a turning-to-north tendency as they get

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across the topography. Notably, the track deviations by the existence ofthe topography comparing to the free vortex runs are getting larger whenthe impinging location of the vortex shifts to the north.

16 h _^ 32 h130 121 122 123 124 125 12G 116 117 116 ' 122 123 12* 125 126

22 h'?18 117 118 119 120 121 122 123 134 125 126 'l

36 h11B 119 120 12t 128 123 124 125

'j 28 h'?!S 117 MB 119 120 121 122 123 124

40 hIIS 117 118 119 120 121 122 123

Figure 3. Time evolution of non-dimensional vorticity contours (C//o) of a strong cy-cylonic vortex (Rov ~ 5.34) over topography of case b'. The solid lines represent thepositive value and the dashed lines are negative. The topographic height HM is 2500mand aft = 40km, bh = 120km.

It is seen that before the cyclonic vortex climbs up the top height the

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predicted paths without introducing the geometrical modification factor 8 agree very well with the shallow water results; but after that the paths predicted by the dynamic model deviate gradually from the shallow water results. The deviation is larger if the maximum mountain height is larger or the cyclonic vortex approaching the middle part of the topography for in these cases the induced circulations are expected to be stronger and modify the trajectories more significantly. The introduction of a fixed 8 for each maximum height restores the path predicted by the dynamic model to the shallow model results. It is also physically consistent that a larger value of 8 corresponds to a higher topography. The general smallness of K indicates that the variation of the effective height hE of the cyclone is much smaller than the variation of the local mountain height h B .

Figure 3 presents a series of contours plots of non-dimensional relative vorticity (C/fo) from shallow water model calculations (case b' in Fig. 2). These plots confirm the trend of increasing strength of induced circulations as the cyclonic vortex trespasses the topography. In the view of vortex symmetry, the cyclonic vortex begins to deform from circular to elliptical shape as it moves across the topography from 28h to 36h. This stage of vortex motion causes a net strength increase associated with two nagatave vorticity arms surrounding itself. Figure 4 shows similar results for the case d' corresponding to the sourthern location of vortex. As the vortex approaches the topography, two lee-side vorticity banners are cyclonically induced from 22h to 28h. These shedding vortices are thus cyclonically advected by the primary vortex as it approaches the topography. Alterna- tively, the primary vortex is heavily distorted into an irregular shape by the satellite shedding vortices from 28h to 36h.

5. Conclusion

In conclusion, this study is an attempt to propose a dynamic model that helps explain the mechanism of a cyclonic vortex encountering an isolated topography. The asumptions made here are reasonably strong while the geometrical modification factor 8 and the mean profile proportional constant IE can only be obtained by experience. Admittedly, the model is a relatively simple mathematical model; however, it does capture some of the important physical and geometrical factors. Indeed, it is noted that the paths predicted by the dynamic model do bear impotant features of some historical typhoons encountering the Island of Taiwan 14. Further development of the model that incorporate other effects such as steering flow is

679

1 - 1 '." I

Figure 4. Same as Fig. 3, but for case d'.

under investigation. The results will be reported elsewhere.

Acknowledgments

This study is supported .in part by the National Science Council of the Repulic of China, Taiwan under Contract No. NSC-91-2111-M002-022, NSC 91-2219-M-002-032 and NSC 92-2119-M-002-010-AP1.

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(1999).

(1990). 10. L.Z. Sans6n and G.J.F. van Heijst, J. Phys. Oceanogr. 30, 2141 (2000). 11. J. Pedlosky, Geophysical Fluid Dynamics (Spinger-Verlag, 1986). 12. J.A. Zehnder, J. Atmos. Sci. 50, 2519 (1993). 13. R.J. LeVeque, J. Comput. Phys. 146, 346 (1998). 14. S.T. Wang, Prediction of the behavior and intensity of typhoons in Tai-

wan and its vicinity (in Chinese) (Chinese National Science Council, Taipei, 1980).

SEA ICE FLOE TRACKING AND MOTION ANALYSIS FOR SAR IMAGERY IN THE MARGINAL ICE ZONE*

JUNW Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave


ANTONY K. LIU ONRIFO Asia, Unit 45002

PO BOX 382, APO AP 96337--500

The objective of this study is to explore the motion and interaction of ice and water masses in the marginal ice zone (MIZ). Sea ice features including thickness, type and motion have been studied using the high resolution synthetic aperture radar (SAR) imagery. The SAR images in the Bering Sea, near southeast of the St. Lawrence Island were chosen for this study. First, a segmentation technique with dynamic local thresholding (DLT) was used to segment and analyze the unstructured sea ice data. The DLT method allowed separation of the ice into thickness classes based on local intensity distributions. The initial classification was supplemented using statistical attributes and heuristic geophysical knowledge organized in expert systems. Then, statistical methods were used to derive ice motion map from the classified images and to perform ice floe tracking. With the sea ice images well classified, it is efficient to track ice floes of different sizes and to study ice motions such as translation, rotation, convergence and divergence. Finally, the image processing results and techniques for sea-ice study arc discussed and summarized.

1. Introduction

Ocean surface waves from the open sea can penetrate into the MIZ and contribute to the breakup of floes and to other processes that modify the ice cover (Liu et al. [ 19931). A variety of organized non-stationary motions, such as jets, fronts, and vortices, are observed in the MIZ in satellite images. Mesoscale features, such as upwelling/downwelling and eddy formation at the ice edge, can be enhanced owing to the nonlinear effects of wave action (Liu et al. [1994]). The St. Lawrence Island polynya (SLIP) is a commonly occurring winter phenomenon in the Bering Sea, where openings in the ice cover are recurring (Lynch et al. [1997]). These processes play important roles in the

* This work is supported by the Vermont-NASA EPSCoR Program and the University of Vermont.

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distribution of heat, mass, and momentum fluxes in Polar Regions and in the control of the ice edge and its location. An understanding of the underlying dynamics and thermodynamics, and ability to model these processes are important for studying sea ice variability. In early ice-ocean models (Clarke [1978], for example) ice cover played a passive dynamic role. Numerical models for ice edge upwelling, in which the ice is allowed to be a dynamic medium, have been developed by Ikeda [ 19851 and Hakkinen [ 19861. Seasonal variability of the ice-ocean system in the Arctic Basin and in the Norwegian- Greenland-Barents seas was studied and modeled by Hakkinen and Mellor [1992]. Using monthly climatological surface heat flux and wind stress, the seasonal simulation of the ice cover was quite realistic. The effects of wave train on ice-ocean interaction in the MIZ were studied through numerical modeling by Liu et al. [1993]. A coupled ice-ocean model which includes wave and wind stresses was used to predict ice edge dynamics and to study wave effects on the formation of ice edge meandering, ice concentration, eddies, and upwelling near the ice edge. The numerical results showed these effects from wave action quite significant, with ice edge sharpening and enhanced formation of above mesoscale ice edge phenomena.

Recent spaceborne instruments have made massive high quality oceadice observations and data for research use. Gloersen et al. [1996] reported a frequency analysis of the Arctic sea-ice concentration using the NASA’s scanning multichannel microwave radiometer (SMMR) data. A singular value decomposition method was applied to the SMMR data by Yu and Gloersen [1993] to analyze both spatial and temporal variations in the Arctic sea ice. More recently, Wu and Liu [2003] developed an automated algorithm for ocean feature detection, extraction and classification in SAR imagery, using two- dimensional wavelet analysis. A combination of these remote sensing products can produce a comprehensive picture of the ice circulation and process. These remote sensing data can also be processed to create ice motion products. Yu and Liu [2003] made a preliminary report on automated sea ice texture classifications and motion analysis using SAR imagery. Zhao and Liu [2002] used wavelet analysis of QulkSCAT and S S M data to obtain daily sea ice drift information for the Polar Regions. In the next section we describe our method for sea ice classification, motion analysis and ice floe tracking. In Section 3, we present the results. Finally, we draw some conclusion and outlook in Section 4.

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2. Method

We used S A R imagery to study sea ice motion in the MIZ. Figure 1 showed the raw S A R imagery data taken on December 14, 2001. The spatial resolution is lOOm x 100m. The outline of the St. Lawrence Island is visible in the lower- right quarter of the image, showing the geographic location of the S A R imagery.

Figure 1. SAR imagery data for the Bering Sea region taken on December 14,2001.

First, a segmentation technique with DLT was used to analyze and segment unstructured sea ice data. As was pointed out earlier, a variety of motions and mesoscale features exist in MIZ. As a result, sea ice features in S A R imagery, including thickness, type and motion vary at different locations in the MIZ. Other factors such as the time of the day when the image was taken contribute some variability in the baseline grayscale for different images. No global thresholding method can compensate for all these variations. DLT is a method of generating global thresholds through a dynamic local thresholding. Here, we adapted the method, as described in Haverkamp et al. (1995), and we subdivided the image into many smaller overlapped regions. These regions are considered small enough to be at most bimodal (containing two types or thicknesses of ice). Other steps in the DLT method included histogram computation, Gaussian curve approximation, testing for bimodality and determining global thresholds. To obtain reliable thresholds, only histograms

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which passed a bimodality test were used. One key element here was the criteria used to select those regions whose histograms were substantially bimodal. A so called "valley-to-peak" ratio was defined and used as a measure for bimodality. Regional thresholds were first determined as local thresholds. Global thresholds were then obtained by point-wise interpolation. With these global thresholds, the DLT method allowed the separation of the ice into thickness classes based on local intensity distributions.

The initial classification was supplemented using statistical attributes and heuristic geophysical knowledge organized in expert systems. The expert systems were rule-based, incorporating qualitative models including the behavior of sea ice as well as facts concern the ocean and ice dynamics, geometry, geographic location and time of year of the SAR images. Finally, a statistical method was used to derive ice motion map from the classified images and to perform ice floe tracking. Our statistical method was based on the maximum correlation of nearby pixels from the two-class segmentation results. With the sea ice images well classified, it was efficient to track ice floes of different sizes and to study ice motions. In particular, ice floe boundaries were obtained from the classified images using, for example, the Sobel edge detection method. (See Figures 7 and 8.)

3. Results

To illustrate our method, we cut out, in Figure 1, a small rectangular area near southeast of the St. Lawrence Island and obtained the raw data from the S A R images of December 11 and 14, 2001. These were shown in Figures 2 and 3, with the eastern tip of the St. Lawrence Island visible in the upper right comer. In Figures 4 and 5 we showed the two-class segmentation results using the DLT method described in Section 2. Results of three-class segmentation were also carried out and were not shown here. Our results were consistent with results from a tree-structured wavelet packet based classifier.

A statistical method based on the maximum correlation of nearby pixels was applied to the two-class segmentation results, Figures 4 and 5 , to derive a motion map for the sea ice. In Figure 6, we showed the ice velocities superimposed on the two-class segmentation result, shown in Figure 4, of December 11, 2004. It can be shown that the sea ice motion is consisted of a translation and a rotation around the St. Lawrence Island located at upper right comer.

Figure 3. SAR data for the cut-out area taken on December 14,2001.

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Figure 2. S A R data for the cut-out area taken on December 11,2001.

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Figure 4. Two-class segmentation result of Figure 2, with boxed area for Figure 7.

Figure 5. Two-class segmentation result of Figure 3, with boxed area for Figure 8.

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Figure 6 . Sea ice motion map on December 11,2001, derived from Figures 4 and 5.

Finally we performed a sea ice floe tracking study and showed the results in Figures 7 and 8. Here we tracked four ice floes from December 11, 2004 to December 14, 2004 (see the boxed areas in Figures 4 and 5 for the location of these floes). By measuring the relative displacement between the two boxed areas in Figures 4 and 5 , we calculated both horizontal and vertical speeds of the translation made by the boxed area as a whole; these were about 15 km per day to the left for horizontal speed and about 1 km per day up for the vertical speed. In addition to this translation we observed, from Figures 7 and 8, a clockwise rotation for the four floes as a whole. By tracking these floes, we also observed a divergent ice flow between the left two ice floes and a convergent flow between the two lower floes.

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10

20

SO

40

50

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Figure 7. Sea ice floes on December 11,2001, see the boxed area in Figure 4 for the location.

Figure 8. Sea ice floes on December 14,2001, see the boxed area in Figure 5 for the location.

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4. Discussions and Outlook

We have presented results of ice floe tracking and motion analysis using an automated sea ice classification and statistical methods for SAR imagery. We chose to use the DLT method, supplemented with statistical attributes and geographical classification knowledge, because it does not require gray level consistency across images. Our statistical method, based on the maximum correlation of nearby pixels in the classified images, was shown to be efficient in ice floe tracking and motion analysis. With massive high quality oceanlice data, including those from S M M R , SAR, QuikSCAT and SSM/I, it is necessary to develop automated algorithm of data processing for ocean and ice studies, including feature detection, extraction and classification. These remote sensing products as a whole, as well as data products derived from them contained enormous amount of information in describing the oceadice circulation and process. Further, inter-comparisons between remotely sensed products and model simulations can in general provide additional insight into the ice and ocean dynamics and their processes. Ultimately, data assimilation techniques, such as Kalman filter and adjoint method, can be used to combine data from observation and model simulation to produce highly accurate diagnostic and predictive oceadice models.

References

1. 2.

3. 4. 5.

6. 7.

8.

9.

10. S. Y. Wu and A. K. Liu, Int. J. Remote Sens., 24 (S), 935 (2003). 11. J. Yu and P. Gloersen, The 1993 Fall Meeting abstract, published as a

supplement to Eos, Trans., AGU (ISSN 0096-3941) (1993).

A. J. Clarke, Deep Sea Rex, 25, 41 (1987). P. Gloersen, J. Yu and E. Mollo-Christensen, J. Geophys. Res., 101, NO. C3,6641 (1996). S. Hakkinen, J. Geophys. Res., 91, 819 (1986). S. Hakkinen and G. Mellor, J. Geophys. Res., 97, NO. C12,20,285 (1992). D. Haverkamp, L. K. Soh and C. Tsatsoulis, IEEE Trans. Geosci. and Rem. Sens., 33, NO. 1,46 (1995). M. Ikeda, J. Geophys. Res., 90, 9119 (1985). A. K. Liu, C. Y. Peng and T. J. Weingartner, J. Geophys. Rex, 99, NO. C11, 10,025 (1994). A. K. Liu, S. Hakkinen and C. Y. Peng, J. Geophys. Res., 98, NO. C6, 10,025 (1993). A. H. Lynch, M. F. Glueck, W. L. Chapman, D. A. Bailey and J. E. Walsh, Tellus, 49A, NO. 2,277 (1997).

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12. J. Yu and A. K. Liu, Symposium of Sensing and Mapping the Marine

13. Y . Zhao and A. K. Liu, IEEE Trans. Geosci. Rem. Sens., 40,1241 (2002). Environmentfiom near and far, London, UK, (2003).

REFLECTIONS AND RESOLUTIONS


REFLECTIONS FOR RESOLUTION TO SOME RECENT STUDIES ON FLUID MECHANICS

THEODORE YAOTSU WU California Institute of Technology, Pasadena, California 91185, USA

This work is an exposition of the course with reflections for resolution to three recent studies of fluid mechanical problems. One is to develop a unified theory for solitary waves of all heights, from the highest wave with a corner crest of 120° vertex angle down to very low ones of diminishing magnitude, with high accuracy based on the Euler model. This has been benefited from reflections on the pioneering works of Sir George G. Stokes (1880) [l] on the foundation of solitary wave theory. Another investigation is to pursue an extension of the linear unsteady wing theory of Theodore von KBrmAn and William Sears (1938)[2] to a nonlinear theory for lifting-surface with arbitrary time-varying shape, moving along arbitrary trajectory for modeling bird/insect flight and fish swimming. The original physical concept crystallized by von K&rm&n and Sears in elucidating the complete vortex system of a wing in non-uniform motion for their linear theory appears so clear that it is readily adapted here to a fully nonlinear consideration. Still another revisit is to examine the self-propulsion of ciliates, an interesting field opened by Sir G. I. Taylor (1951)[3]. Reflecting on the needs still remaining, this study has led to explore a conjecture whether the inviscid irrotational flow can be ubiquitous in the microscopic world of living micro-organisms like ciliates self-propelling at vanishing Reynolds numbers, yet still exhibiting phenomena all similar with those commonly observed in the macro-world. Here the intent is to delineate the evolving lines of thinking and deliberations rather than elaborating on substantial details.

1. Introduction

The good will and bountiful enthusiasm for scientific interaction brought together in high spirit by my long-admired friends and distinguished scholars to this 2004 OMAE Symposium on Engineering Mechanics are just overwhelming. I find myself feeling so much honored and moved in resonance that I brave intangible customs to catch the high spirit of all our participants to offer a modest return, as a token of my deep appreciation. Hopefully, I wish I could take this privileged opportunity to share with my distinguished audience my courses in taking several revisits to a few subjects of interest to me in recent years and in coming across some reflections for deeper insight for resolution to these problems and for enhancing fur-

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ther development in these fields. New findings and comprehension resulting from these studies will be discussed and illustrated in the sequel.

2. A new theory for evaluating solitary waves of all heights

First I wish to describe some reflections I have pondered on in a course of studies on solitary waves, this time devoted to exploring the main properties underlying the waves of all heights, from the highest down to low ones of diminishing magnitude. Historically, theory of solitary waves on water has grown with contributions of great significance from pioneering masters and their followers. Strong interests, however, have been largely focused on the highest and very high waves for their full resolution, leaving the very low waves virtually unattended. The fully extended scope of our recent studies is aimed at an exposition of the real richness of this wave phenomenon.

In the founding days of the theory, Sir George G. Stokes made two masterly contributions to its theoretical foundation. In one of them, Stokes (1880)[1] explored the behavior of a solitary wave of amplitude a, moving in permanent form with speed c on a layer of water of uniform quiescent depth h, and attenuating toward physical infinities at a rate which Stokes assumes to be exponential. Adopting the Euler equations, regarded by Stokes as ideal for modeling such irrotational flow of incompressible and inviscid fluid, the wave profile r](x, t ) and the velocity potential, $(x, y, t ) , which satisfies the Laplace equation, q5zz + q5yy = 0, will thus assume a functional form in 2, y, and time t for (-h I y I r](z,t), 1x1 < 00) as

as 1x1 -+ 00 in the flow region bounded below by the horizontal bottom at y = -h (at which q5y = aq5/ay = 0) and above by the water surface elevated to y = q(x, t ) from its undisturbed level at y = 0. The linearized boundary conditions on continuity and surface pressure (= 0) read

V t = q5yr q5t + gr] = 0 (y = O ) ,

both of which are satisfied by (l), for arbitrary constant a (or b) , provided

F2 = tan(p.rr)/p.rr ( F = c/&, p = kh), F beieg the Froude number (dimensionless speed) and p.rr the logarithmic decrement. And, as a stroke of genius, Stokes claimed this relation exact!

Exact it is indeed. It signifies that relation (2) holds uniformly valid for fully nonlinear and fully dispersive solitary waves of all heights. However, no further qualifications are found given for (2) by Stokes, a point which

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-1

//////////

seems left for reflection. Following Wu et al. (2005)[4], we map the infinite strip of unit width in the complex potential f = 4 +ill, plane (scaled by hc) conformally onto the unit circle I<[ 5 1 in the parametric <-plane (with the wave surface mapped onto I<I = 1, the wave crest at < = -1, and 5 = f o o onto < = 1) by (see figure 1)

&////////

1 1 + J T 2 f = ~ + i l l , = - i + - l o g - ( 1 - 8 ) .

0

I

-I

We further adopt the complex velocity w = u - iv (= d f l d z , z = x + iy), scaled by wave speed c, to propose for logarithmic hodograph w = T + i9 = log(l/w) (T = log(l/q), 6 = arctan(v/u), q = (u '+v ' )~ /~ ) the asymptotic representation about < = 1 (or the physical infinities) as

c w=o J *-$

V = - l D

where p is the principal-branch root of (2) for given F (> 1, 0 < p < 1/2), and am's are real unknown coefficients. Here, the first term (with m = 1) in the series of (3) stands for Stokes's term pertaining to (2), and its successive multiple powers (with m = 2,a.a , M ) are due to the existence of the quadratic terms of w in Bernoulli's equation (see (6) for invoking the surface pressure). Along this reflection, the crucial issue is how to determine the integral value for M ( F ) , parametrically in F , so that (3) can best represent the intrinsic wave behavior in the wave outskirts.

i

Mathematically, (3) represents a singular behavior of the flow at physical infinities since the value of p for arbitrary F , by (2), is in general an

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irrational number ( p < 0.34 for waves of all heights). By arguing that these M singular terms must invoke a smooth matching with the remaining terms of w that are analytic and regular near C = 1, Wu et a1.W have found

2Mp N 1 (with 2Mp somewhat greater than unity) (4)

as an ideal criterion for determining M . In terms of the dimensionless amplitude a = a / h and using Boussinesq-Rayleigh’s first-order approximate relation, F = J1?, so that a = tan(pr)/pr - 1, p is found to decrease monotonically from p = 0.371 (at a = 1) down to zero as a -+ 0 ( F -+ l), as shown in figure 2. Thus, criterion (4) gives the value M(a) = 2,5,10,30, and 90 at a = 1, lod1, and respectively.

7

. . . .

. . . .

. . ...

. ...

. . .,.

. . . .

\ . . ...

/

4

1 oa Figure 2. Variations of p(a ) and M ( a ) for solitary waves of all heights (a).

Solitary waves as small as may seem unreal, but it is generally known that earthquakegenerated tsunami waves progressing in the open Pacific Ocean (of mean depth h = 4 km) with a height commonly estimated to be merely a 21 l m or less, thus giving a = a / h 21 O(10d4). The general validity of criterion (4) for waves of various height will be discussed later.

In the other major contribution of Stokes’s concerning the crest geometry of the highest solitary wave, it is concluded by Stokes that if a wave should peak to a ridge, it must be a corner of 120°, a claim which is again proved in great simplicity by Stokes. Further, an exact solution has been

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provided by Stokes (1880)[1] for the infinite wedge flow under gravity as

f = q5 + ilc, = (2/3F)(ei"/6z)3/2 (w = d f / d z = ( l /F)ei" /4z1/2) , (5)

so that lc, = 0 and 1wI2 = lzl /F2 (or pressure p = 0) on both arg z = - ~ / 6 and = - 5 ~ / 6 , making the two wedge faces free under gravity (acting in the (-y)-direction). As the wedge faces are stationary, this is not a traveling gravity wave. The task that remains is to seek how this exact solution can be adapted (e.g. with or without additional singularities) to describe the leveling off of the wedge faces to fit the highest solitary wave profile.

We now proceed to incorporate the comprehensive properties so far acquired of solitary waves (in their outskirts in general and for corner flow in particular for the limiting wave) to develop a new unified theory for solitary waves of all heights, with intent for simplicity and high accuracy. In short, the solution will prevail under the premise that the complex variables for the coordinates, velocity, and velocity potential be analytic functions of one another under the boundary conditions:

B(o) = q2 - 1 + 271/F2 = 0 (on [ = ei", 0 < o < 2 ~ ) , (6) e = o (C real, -1 G C < l ) , (7)

w = 7 + i e + o (as C --f 1). (8)

Here, (6) is the Bernoulli equation invoking the surface pressure p = 0, (7) states the kinematic symmetry, and (8) follows from w([ = 1) = 1. Alternatively, (6) can be replaced by its gradient along the wave surface (q!~ = 0), namely by

which has the merit that the system (7)-(9) affords solutions parametrically in terms of the conjugate functions ~ ( o ) and 8(o) in a single parameter, F , whereas system (6)-(8) needs repeated quadratures for ~ ( o ) in satisfying (6). Instead of F , an alternative is p, the proportional amplitude parameter,

p = 1 - q,2 (= 2a/F2) (0 < P 5 11, (10)

which is (6) taken at the wave crest, (7 = a, w = qc), giving q: +2a/F2 = 1, or (10). It covers the range from p = 0 for the vanishing wave (a = 0) to /3 = 1 for the highest wave (with qc = 0), and has the merit that a(P) is monotonic over (0 < p 5 1) but a ( F ) is multi-valued (cf. [4]).

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2.1. The unified intrinsic functional expansion (UIFE) theory

Unifying the foregoing analysis has lent us insight to develop a simple and accurate theory for solitary waves (Wu et a1.[4]) based on two principles:

(i) This new theory first requires to establish a unified intrinsic functional expunsion (UIFE) for w(C) in terms of a set of intrinsic component functions (ICF) (by incorporating the above results for the various flow regions) to represent precisely all the intrinsic properties of the wave entity.

(ii) The unknown coefficients in the UIF-expansion for w(C) are determined by minimizing the mean-square error E& of G(a) of (9), or E i of B(a) of (6), under conditions (7)-(8) as understood, where

J% = I” G2(7(u), O(a), a)da (UIFEmethod I); (11)

Ei = I” B’ (T(~) , q(a), a)da (UIFEmethod 11). (12)

Here, UIFEmethod I permits stepwise interactive operation such that minimization of EG is optimized stepwise, starting with a few leading terms in the UIF-expansion, with a new term selected in turn by its top ranking of all the competing candidates in making a steep descent in error EG, the guideline being to find an ideal expansion with as few terms as needed to achieve an accuracy as high as attainable in practice. The UIFEmethod I1 employs numerical codes developed for automatic iteration for convergence to achieve higher accuracies, however with less versatility for optimization. Nevertheless, the two methods complement each other very well; they have been applied jointly in evaluating solitary waves of all heights. In practice, we have found the Program ’FindMinimum’ of the software Mathematica 5.0[5] very well suited for applying Method-I. Two examples are presented below to illustrate their power, simplicity, and high accuracy. Example 1. The highest solitary wave (p = 1) For the highest solitary wave, occuring at /3 = 1 with an undetermined Fkoude number, Fhst, we propose for w(C) = T +iO its unified expansion as:

M N

+ m=l C n=O x { u m n ( T ) ’ n v + b r n n < n } (13)

where umn’s and bmn’s are real unknown coefficients so that conditions (7)- (8) are fulfilled first. Here, the first term with the log function is adopted

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from Stokes’s wedge flow solution (5) to lead the expansion for modeling the cornered wave crest, the second term is added to the first to make their sum one order lower in error than the first term alone as C + 1 so as to leave the intrinsic wave outskirt properties more intact with the remaining terms. In the series with coefficients a,,, its leading term (with a,~) is a secondary singularity (with a branch point at the corner crest) which is found necessary by Grant (1973) [6] to complement the primary logarithmic singularity such that Y is the primary root of a ( 1 + 2v) = tan(vn), giving Y = 0.40134, its higher power terms admitted to n = N is a criterion (2nv 21 1) proposed by [4] similar to (4) for m. In addition, the series in n with b,, is a function regular everywhere to render the wave properties complete in the inner flow domain. Finally, the double series unifies the inner flow field with the outskirt features represented by the powers of (1 - < ) z p to make the free surface curve away from the wedge crest and level off to fit the highest wave perfectly. To this point, achieving such an insightful construction of the UIF-expansion as (13) is actually a primary step of the optimization.

Deducing T ( U ) + ie(a) = w(C = eiu) from (13) and minimizing EG of (11) by Method-I with stepwise optimization yields for the highest solitary wave a solution in terms of (13) with

Emin = 1.121 x p = 0.335056, a10 z= 0.456569, a20 = -0.102203, a30 = -0.158703, a40 = 0.0116986, all = 0.253853, a12 = 0.13121, ~ I I = -0.329156, blz = -0.0162222, b13 = 0.0014475, b14 = 0.000042136;

a13 = 0.0225912, a14 = -0.0077326,

(Yhst = 0.833121, Fhst = 1.29083, p = 2a/F2 = 1, (14)

where Emin is the minimum root-mean-square error EG found with the corresponding values of am, and b,, listed in (14), giving for the highest solitary wave the maximum height (Yhst = 0.83312, with the wave velocity a t F’roude number Fhst = .Jzcyhst = 1.29083.

In parallel, applying Method-I1 to a set of fifteen intrinsic modes yields

(Yhst = 0.8331990, Fhst = 1.2908904, local error 5 2 x lo-? (15)

So comparing (14) with (15) confirms the Method-I result accurate to four significant figures. Thus, the exact solution so attained is found, interestingly, to consist of four intrinsic component (IC) modes in each of the three

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0.2 -

-6 -5 -4 - 3 -2 -1 0 1 X

Figure 3. Distant and telescopic views of four wave profiles given by WIFE Method-I and -11 for extreme solitary waves of height a = a h s t = 0.8331990, 0.822279, 0.811386, and 0.796952, with corresponding speed Fhst = 1.290890, 1.291738, 1.293358, 1.294208. Note that the last three waves are suc- cessively lower in height but all increasingly faster than the highest wave, the last being almost exactly the fastest one (afst = 0.7959034, Ffst = 1.294211).

groups, namely, in a,o with M = 4, and in aln, bin, both with N = 4. In- deed, this distribution affords a revelation of the overall intrinsic nature of the wave entity, which is an important result of the new theory. And with this distribution, it takes only twelve IC-modes of the functional expansion to achieve a result with a high accuracy to four significant figures by Method-I, and with 15 IC modes to six significant figures by Method-11. Fi- nally, we note that with p found in (14) with M = 4, we have 2 M p = 2.68, which is about twice of that by criterion (4) in this case.

Example 2. A dwarf solitary wave ( F = 1.005, p = 0.054873) Here, w(C) = T + id assumes the expansion:

where p is given by ( 2 ) for assigned F . This unified expression for w(<) , applicable to all waves with a round crest, simply follows from (13) by dismissing all the terms pertaining to the singularities characterizing the

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- 4 -2 0 2 4 6 X

Figure 4. Wave profiles evaluated by UIFEMethod-I and -11 for five solitary waves of height a = a h s t = 0.8331990 (the highest), 0.758245,0.583690,0.407430, 0.212284, with the corresponding speed F = Fhst = 1.2908904, 1.29092, 1.24470, 1.18098, 1.09979, respectively. Note that the second wave is lower in height but faster in speed than the highest wave. It is already lower than the fastest wave.

highest wave crest. For this case, applying UIFEmethod I yields

a10 = 0.0342629, a20 = -0.0401226, a30 = 0.0218187, a40 = -0.0033137, a50 = -0.0018522, a60 = 0.0004625, a70 = 0.0003288, a80 = 0.00011619,

ago = -0.0002101, alo,o = -0.000073, a11,0 = 0.000061, all = -0.0000212;

a = 0.0114426; ETe Emin/lazol = 6.8 X lo-’; p = 2a/F2 = 0.02266. (17)

Hence this dwarf wave is shown to possess a structure consisting of eleven modes in amo (rn = 1 , 2 , . . . , M = ll), and taking only one mode in a l l (of amn with n 2 1) to reach jointly a relative error of O(10-7). Further, the corresponding value of 2Mp = 1.207, a value which is just above unity for a = 0.011, is quite a sound attestation to the proposed criterion (4) requiring M = 10 at a = (see figure 2). Similar accords have been found for the other low waves examined by Wu et al., 2005[4].

These results can therefore be regarded as numerically exact in view of their negligible error, affecting merely the sixth to eighth significant figures.

Regarding wave profiles, we select two figures as representative cases of the extensive results obtained by applying Method-I and -11. Figure 3 delineates profiles of four extreme solitary waves, including the highest,

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all appearing nearly coalescing together as viewed from a distance except near their crests, which are vividly differentiated in a telescopic close-up view. It also shows the three lower waves becoming increasingly faster than the highest wave toward the fastest one as specified therein. In Fig. 4, five wave profiles are exhibited to illustrate the increasing breath with decreasing height, the second highest being just lower in height than the fastest one.

Through Example 2, a new exposition of very low solitary waves appears to promise more contents of richness for new studies. In this regard, we propose to call the waves with height a 5

Summing up this topic, we have seen Stokes’s and other pioneers’ contributions reflected, studied, and extended to develop a new theory for solitary waves of all heights. Its power, simplicity, and high accuracy have been illustrated with examples. With the intrinsic wave properties over the flow regions optimally unified in composing the UIF-expansion, and with the optimization carried out stepwise aptly in evaluating the exact solution, the resulting optimum composition of the solution could shed new light on the concept about composing the initial UIF expansion to open new visions. In this light, the lecture given in this Symposium by Sunao Murashige on Nekrasov’s integral equation also affords a new outlook.

In studies on nonlinear problems, it has been often stressed on the importance of unifying theoretical, computational, and experimental efforts jointly for deeper understanding. Of the three efforts, the experimental part is in general demanding no less rigor, skill, and brilliance than the other two. The strict and rigorous techniques exhibited to the highest standard for experimentation by Joeseph Hammack in his lecture at this Symposium on his joint study with Diane Henderson et al. on solitary wave collisions has won high recognition as an outstanding role model for emulation.

dwarf solitary waves.

3. A nonlinear theory for modeling bird/insect flight and fish swimming

Here we recapitulate a nonlinear theory of a two-dimensional (actively) flexible lifting surface performing arbitrary movement (with time-varying camber shape and moving along arbitrary trajectory) for modeling aquatic and aerial animal locomotion at high Reynolds number. We opt two- dimensional theory for its simplicity to provide a foundation for further development of unsteady wing theory for general applications. For profi- ciency, we seek an existing linear theory that its conceptual structure can

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\ \ \ \

\ L a

best suit for the generalization as intended. In this respect, I recall the tale relating to a conference presentation of

Theodore von Ktirmtin and William R. Sears’s[2] linear airfoil theory for non-uniform motion when von Kkmdn expounded that their theory differs from the earlier theory of Theodore Theodorsen’s[7] in having its own physical concept and mathematical principle. Indeed, of all the linear theories for unsteady airfoil, it is the simple and clear physical concept crystallized by von K&rm&n and Sears in providing such an ingenious restructuring of the vorticity distribution over the wing surface and its trailing wake that it is readily adapted for extension by Wu (2001, Sect. 6)[8] to account fully for all possible nonlinear effects in theory. It has been further pursued to extend Herbert Wagner’s pioneering work (1925) [9] accordingly for more general applications. This nonlinear theory has been applied by Stredie (2005) [lo] to perform computations of a number of unsteady motions of bodies shedding vortex sheet(s), attaining results of high accuracy (as measured versus relative errors and experiments available) in all the cases pursued, and making valuable contributions to this subject.

The present study is more focused on the issue of arbitrary changes in wing shape and trajectory along the line discussed by Wu[8] with intent to optimize the analytical and computational efforts for attaining solutions efficiently.

G: ’c- _ _ -‘ ,/ - -.

4

\ T.E.

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Thus, we consider the irrotational flow of an incompressible and inviscid fluid generated by a two-dimensional flexible lifting surface Sb(t) of negligible thickness, moving through the fluid in arbitrary manner. Its motion can be described parametrically by using a Lagrangian coordinate system (c , 7) to identify a point X(E, t ) , Y(<, t) on the boundary surface S(t) = &(t) + S,(t) consisted of body surface s b and a wake surface S,, with S(t) lying at time t = 0 over a stretch of the &axis (at 7 = 0) and moving with time t(2 0) (see Fig. 5) as prescribed by z = z + iy = Z(E, t ) ,

z(c, t ) = x(6, t ) + iy(t, t ) on Sb(t) + s w ( t ) , (18)

where &(t) : (-1 < t < 1) and S,(t) : (1 < E < E m ) , and the complex variable Z = X + ZY (parametric in E ) is used for describing the boundary motion, E = -1 marks the leading edge and E = 1 the trailing edge of the airfoil, from the latter of which a vortex sheet is assumed to be shed smoothly (i.e. under the Kutta condition) to form a prolonging wake S,, and Em identifies the path Z(tm,t) of the starting vortex shed at t = 0 to reach Em = Cm(t) at time t. A simple choice for ( E + 27) is the initial material position of Sb(t = 0), taken to be in its stretched-straight shape such that Z(<, 0) = C (-1 < E < 1, 7 = 0), lying in an unbounded fluid initially at rest in an inertial frame of reference (see Figure 5). The flexible Sb(t) is assumed to be inextensible (lZcl = laZ/a[l = 1) and the point t on Sb(t) moves with prescribed (complex) velocity W(<, t ) = U - iV:

W(E,t) = az/at = x, - iu, ( ] E l < 1, t L 0; z = x - iY), (19)

or with tangential, Us(<, t ) , and normal component, Un(E, t ) , given by

WaZ/a( = (XcXt + yEK) - i(X<yt - YcXt) = Us - iU,, (20)

and with the same expression for the wake surface S,(t) for (1 < [ < E m ) . In the spirit of von K&rm&n and Sears, we adopt for t > 0 the following

vorticity distribution:

on Sb(t):

on S,(t): $6, t ) = To(<, t ) + r1(E, t ) T(t7 t ) = rw(E7 t )

(-1 < E < 11,

(1 < E < Em),

where yo([$) is the bound vortex distributed over sb representing the LLquasi-steady” flow past sb such that the time t in the original prescribed W(c, t ) is frozen to serve merely as a parameter in evaluating the quasi- steady 70 (by steady airfoil theory), and ?I((, t ) is the additional bound vortex induced on s b by the trailing wake vortices y,,, (<, t ) such that y1 and

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yw jointly make zero contribution to U, (but not to Us) over sb so as to reinstate the original time-varying normal velocity U, (<, t ) on Sb(t).

Thus, we represent the velocity field by a vorticity distribution, y(<, t ) , per unit length spanwise over the body and wake surfaces to give the complex velocity w(z, t ) = u - iv of the fluid at a field point z and at time t as

Applying Plemelj's formula to (21) yields for wf = limw(z(< + ir]),t) as r] -+ f O on the two sides of S the relations (for (2,Z' E S )

where Z = Z(<), Z' = Z(<'). From (22) we have y(<, t ) = (u;f - u;), and

Here, (23) shows the continuity of normal velocity = u; = u, across S and (24) gives the algebraic mean of tangential velocity us on S. From (23)-(24) we find the contributions made by "yo, 71, and yw as:

where W,,,(<, t ) = U,,,, - iUw, is the (complex) flow velocity on the wake. The problem can now be recast to describe a solution method as follows.

Equation (25) results from invoking condition that u,(<, t ) = Un(<, t ) , prescribed at sb, to give a singular integral equation for "yo which can be solved, with time t frozen, as a Riemann-Hilbert problem by applying steady airfoil theory. The velocity induced on sb by wake vorticity "yw has the normal component U1, given by (26), which is canceled out as is required of y1

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on Sb according to (27) so that the sum (26)+(27) gives an integral equation for y1 in terms of y,,,. This solution for y1, which is to be determined under the Kutta condition that the body vorticity, T b = 70 + y1 and the wake vorticity, y,, be continuous at the trailing edge, may be expressed, in principle, symbolically in the form

Finally, we apply Kelvin’s theorem that the total circulation around sb +S, must vanish Vt 2 0, i.e. ro + rl+ rw = J,, (70 + 71) dc + J,, y, d< = 0 (if it is zero initially), or, symbolically,

This in essence is the desired form of “generalized Wagner’s integral equation” for wake vorticity y, . Its original linear version has been shown by Wagner[S] and by von KArmAn and Sears[2] to play a key role in providing accurate solutions for the entire vorticity distributions and hence for the final solution to the linearized problem. For the present nonlinear theory, the kernel K(J’; <, t ) has been shown by Wu[8] to contain all possible nonlinear effects, however in a rather lengthy series expansion form; it is strongly desired to reduce it to a closed form for arbitrary wing movement. This is an outstanding outlook inspired on reflections.

In practice, it is convenient to start with the motion of sb prescribed for t 2 0. In a small time interval 6 t k at t = t k > 0, a new segment of S, is created (due to body moving forward) in the wake just beyond the trailing edge (at E = l), namely 6,z(1, t ) = r(1, t)&. The wake vorticity shed into this small segment of S, can be obtained, by analysis and numerics, from (30). Once the local y, of that fluid particle (leaving the trailing edge at t = tk) is determined, its value will remain invariant and move on with the particle at wake velocity ww((, t ) of (28) for t > t k (k = 1,2, . . -).

Summing up this theme subject, we point out that for a flat wing, dZ/d< on sb in the above equations becomes a constant slope of the plate at t , hence can be factored out, or canceled off, leaving any nonlinear effects to arise when the wing motion and its wake depart noticeably from a straight line in uniformity. However, for a flexible wing with varying shape and trajectory, such non-uniform changes in dZ/dE can give rise to nonlinear effects locally about sb. This kind of nonlinear effects are expected to play an active and important role in aerial and aquatic animal locomotion. An

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interesting case is the hovering flight of a dragonfly using a pair of fore-and- aft wings in coordinated motion discussed by Hiroshi Isshiki in his lecture at this Symposium.

4. A new approach to modeling ciliary locomotion

The third subject I have been taking a refreshing revisit is on the hydromechanics of ciliary locomotion. Biophysically, flagellates and ciliates are singlecelled protozoa, the latter being similar in function and structure with ciliary organelles existing in some mammalian organs such as the trachea, oviduct, and in other bio-tracts. The well-being or illness (e.g. cystic fibrosis) of these organelles would be of importance to human health consideration. The ciliates (e.g. paramecium) are a class of protozoa having a large number of cilia (hair-like organelles, each just like a flagellum, generally 10 - 100pm in length, 0.2pm in diameter) attached to the cell surface in a refined row formation. In locomotion, the cilia beat periodically in unison row-wise, with their phase slightly varying from row to row to form a metachronal wave somewhat reminiscent of a wheat field waving in breeze. Each cilium beats stretched straight like oar in a fast power stroke and with cilium bent in a slower recovery stroke in each cycle, beating with various patterns innate to species, so is the beating frequency, which generally ranges over 5-80 Hz. By reversing or turning in wave phase, the cell can swiftly reverse in direction or maneuver a turn and gyrate. Such maneuvering involves the inertial effects of action and reaction.

Hydromechanically, flagellar and ciliary motions are characterized by strikingly small values of Reynolds number, typically of order O(lOP3) or less, generally regarded as signifying that the overall inertial effects are negligible relative to the viscous effects. In the pioneering work of G.I. Taylor (1951) [3], the Stokes equations for incompressible viscous fluid are adopted, precisely for the reason of the inertial effects being insignificant at such low Reynolds numbers. Thus, the ciliary motion is represented by an “envelope model” in Stokes’ flow, with an “impermeable, non-slipping material sheet” enveloping the tips of the cilia beating within the ciliary layer, along which propagates a periodic metachronal wave, with the infinite flexible sheet oscillating about its flat mean position. The phenomenon is so interesting, the argument so convincing, the treatment so elegant, and the conclusion drawn so forceful that this paper of Sir G. I. Taylor’s is known to have aroused great interest to many active workers in this then brand new field. This motive force has been very strong and brisk in promoting fruitful ad-

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vances in the subject field, producing very general and powerful methods by using various sorts of singular solutions of the Stokes equations for con- structing general solutions that have resolved various important yet difficult problems of long standing. (E.g. Chwang & Wu[ll-131; Chwang[l4]; John- son & Wu[15]. For a review, see Lighthill[l6]; Wu et al. [17].) However, subsequent developments in the field introducing corrections due to restoring some of the inertia effects previously neglected, boundary effects due to the cell surface at the root of the ciliary layer, etc., as recently discussed with review by Wu (2003)[18], are still regarded as insufficient. The need of achieving a major clarification seems still remaining.

Reflecting on these needs, we find at least two issues are still outstanding. First, the requirement of having the total force vanish on a self propelling ciliate in cruise, as required by Newton’s first law, must be addressed for three-dimensional configurations proper to ciliates. This vital feature is exactly what an infinite plane ciliary sheet model is lacking. In- deed, the improvement in need has been explored ever since Taylor (1951) by Blake (1960)[19] who introduced a ciliary sub-layer model, by Brennen (1965)[20], Keller & Wu (1967)[21] and others for further possible improvements. However, all these attempts are based on using a distribution of stokeslet (a point-force singularity of the Stokes equation) to represent the power strokes of cilia under the condition that the total force be zero to satisfy Newton’s first law that any rigid body as well as animated cells of self-propelling ciliates must obey. This brings forth another basic issue, as we here argue, that modeling the flow field exterior to the ciliary layer based on any stokeslet distribution is short of having a sound physical ground. This is clear because in that approach, any error in the residual stokeslet resulting from invoking the condition of zero net force, no matter how small the error may be (unless it is always absolutely zero), would render New- ton’s law inevitably violated (even by a minute margin), therefore implying that the motion in consideration is physically impossible for manifestation.

With the foundation thus established, Wu (2003)[18] has proposed an entirely new approach. For modeling a solitary ciliate in self-propulsion, with a velocity not necessarily constant, the flow field assumes a composite structure, one exterior and the other interior to the ciliary layer in which metachronal waves propagate. The exterior flow of the surrounding fluid is assumed to be incompressible and fully viscous (just as so regarded mn- ventionally), but is completely irrotational. It is the cilium length and the mean flow velocity in the layer that makes the Reynolds number small to have the viscous forces dominate over the inertial forces, but only exclu-

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sively in this layer. We shall argue that this new model is always sufficient in fulfilling Newton’s law, regardless of any error in evaluating the solution, and that the necessity and sufficiency of the model lies only with the capability of ciliary layer in making the two flow fields precisely match over their common boundary, i.e. the surface enveloping the ciliary layer.

Along this new approach, we consider a typical ciliate performing a beating mode and self-propelling with a constant velocity U (without rotation for the moment) in a viscous incompressible fluid otherwise at rest. To address this general case in full, we adopt the Navier-Stokes model so that the continuity and momentum equations read, in conventional notation,

v * u = 0, (V = (az, a,, &)) (31) (32)

R e = veil/, 0 = w e 2 / y , (33)

gut + R e u . vu = v2u - v p + f ,

where all the variables are dimensionless, properly scaled ( R e and LT being the translational and oscillatory Reynolds numbers based on ciliate length e ) . For the absolute frame of reference, we have the boundary conditions:

u + o as 1x1 + 00, (34) (35) 4x7 Y , z , t> = UC(T Y , 2, t ) (at F ( x , Y , z, t> = 01,

where U, is the flow velocity at the geometric boundary of the ciliary envelope ( F ( z , y , z, t ) = 0 ) produced by the ciliary activities to match the exterior irrotational flow generated by the ciliate. (We note that the velocity distribution U, is unique for given body shape and body velocity U, due to the uniqueness theorem for potential flow.) This completes the formulation of the exterior flow, leaving the interior flow (due to the ciliary activities) within the ciliary layer as a separate problem to have Uc(x, y , z, t ) completely matched for the final solution. Here we present the following theorems to facilitate the final solution. Exterior potential flow theorem. A n y irrotational, incompressible, unsteady flow of a viscous fluid past a body (e.g. a ciliate) with a velocity potential d(x, t ) , such that u = Vd and V2$ = 0 in the flow field, is a solution of the Navier-Stokes equations (31)-(35) i f (a) it satisfies conditions (34)-(35), and (b) the body force f = -VZ is conservative with a potential. The proof by Wu[18] rests on that V2u = 0 follows from V2q5 = 0 to reduce (32) to Euler’s equation, which validates the potential flow solution under conditions (34)-(35), and ensures in turn that no new vorticity can be produced on the boundary or in the interior of a potential-force field.

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Z e r o t o t a l force theorem. The total stress, comprising the pressure and viscous stress given by the Navier-Stokes model, of any irrotational flow generated by a ciliate in uniform motion through a viscous incompressible (Newtonian) fluid exerts zero net hydrodynamic force on the body. In fact, the total stress tensor of a Newtonian fluid is

17 = -PI + 7, Tij = /A(dUi/dZj -I- aUj/dXi) ( 2 , j = 1 ,2 ,3 ) (36)

Tij being the viscous stress in Cartesian components, and the viscous force F, acting on a closed body surface sb is

.r .ndS = V a r d V = /A V 2 u d V = 0, (37)

since V2u EE 0 within the flow field in volume V bounded by the body surface sb and a large closed surface S, surrounding the point of infinity, the integral on S, vanishing since 11-1 = O ( I X ~ - ~ ) as 1x1 t m. As for the contribution from the pressure, p , the fact that p yields zero net force on a closed body in uniform potential flow of incompressible fluid is known as the “d’Alembert paradox.”

We note that in the proof for the viscous force component, time t is merely a parameter, hence the results hold valid even for viscous forces in unsteady flows. Therefore, for bodies moving through irrotational flow of a viscous fluid with varying body shape and velocity, it is the flow pressure that is responsible for giving rise to reaction by the fluid in terms of its added masses due to the fluid inertia. This statement applies in general, whatever the Reynolds number.

We further remark that while the viscous stress of irrotational incompressible flow does not contribute finite net force on a closed body, it does do mechanical work at the rate

s s, s, F, = Lb ~ . n d S = sb+s-

which is the dissipation function, by which the mechanical work dissipates. Summing up, it is obvious that for rendering the exterior flow irrota-

tional, it is necessary and sufficient to have a complete match between the ambient potential flow and the interior ciliary flow at the ciliary envelope. Furthermore, fulfilling this requirement always ensures zero net hydrodynamic force on the ciliate. Consequently, we can now state: Conjecture on the sufficiency and necessity c o n d i t i o n s for cil iary locomotion - For ciliary self-locomotion to manifest, it is necessary and

71 1

suficient to have the resultant ciliary flow velocity along the cilia envelope match the corresponding exterior potential flow; then the freestream velocity of the exterior flow relative to the ciliate is the opposite of ciliate’s swimming velocity through a viscous fluid otherwise at rest.

By this conjecture (when the exterior and interior flows match), the dominance of the viscous effects over the inertial effects is entirely local, all confined within the ciliary layer, stopping right at the layer’s outer edge, and keeping the exterior flow invariably irrotational at all times. So, in the ambient flow, the inertial effects at such low Reynolds numbers would play their roles exactly like theirs at the limit of high Reynolds number, leaving the viscous effects in exterior flow only to mind the viscous dissipation.

Figure 6. Streak photographs of streamlines produced by (left) a freely swimming Paramecium caudatum, and (right) an inert specimen of Paramecium caudatum sedimenting under gravity. (Photograph by Keller & Wu 1977.)

Therefore the exterior flow field can always be represented by an interior distribution of mass source-sink pairs, dipoles and higher-order poles known to potential flow theory. This dipole-like streamline pattern is convincingly displayed in the (left) photograph of Figure 6 taken by Keller and Wu (1977) [21], exhibiting its short-range effect, with the tracing particles both ahead and behind the body hardly ever moved in the time exposure of the photograph. In striking contrast, the right photograph shows the long-

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range effects in the streamlines of an inert specimen sedimenting under the external force of gravity, drawing in the fluid from far behind and expelling it to the front far ahead in a fore-and-aft symmetry, as is universal in Stokes’s flow. The two distinctly different streamline patterns are both well predicted by the potential flow theory and the Stokes model, respectively (Keller & Wu 1977[21], Keller[22]).

It is therefore remarkable to observe that potential flows do manifest at such a small scale of ciliates (about 10-100 pm in length), with such a refined fit to have the otherwise dominating viscous flow all confined within the ciliary layer. It is with this beautiful match that the (conventionally neglected) inertial effects are left intact as the only effects remaining in the exterior flow field to act in full capacity, in both steady and unsteady types of action-and-reaction, in the micro-world just as ubiquitous as we have commonly experienced in the macro-world.

The problem of interior ciliary flow is very interesting but challenging. Just like what we have pursued for the basic principles in resolving the exterior flow, it is useful to form a sound physical concept about the basic mechanism underlying the ciliary activities to generate the required matching flow. In this regard, it is important to retain the unsteady effects and explore for their active roles. The complexity of the interior flow in the ciliary layer can be exemplified by the dense array of row cilia beating in unison to render the row plane partly porous to flow and partly sustaining a pressure jump across, basically a phenomenon which is well addressed by the lecture delivered in this Symposium by Daniel Weihs. For the dynamics within cilium and three-dimensional ciliary beating activities, recent developments may be referred to the contributions of Gueron, Liron, Levit-Gurevich [23-25] from the Technion School of biological mathematics led by Nadav Liron. In addition, the fundamental solutions called unsteady stokeslet and unsteady oseenlet recently found by Chan & Chwang (2000)[26] as singular solutions of the unsteady Stokes equation and unsteady Oseen equation together with their use by Shu & Chwang (2001)[27] may furnish sharp tools for such studies.

5. Conclusion

I am very pleased to have this opportunity to present some new results I found from revisiting some exciting subject fields where I had worked before and find myself benefited by reflections to gain more insight in pursuing these interesting new studies. These pursuits are being held mostly for

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continuing studies and are very much open for creative ideas and interaction for promoting further advances in engineering mechanics and engineering science.

Epilogue

In introducing a new basic law of mechanics, Karl Friedrich Gauss (1829)[28] takes a far-reaching view on the philosophy and the art of pursuing scientific research and intellectual development, saying that

“In advancement of science and development of individuals, the easy generally precedes the difficult, the simple precedes the complicated, and the special cases precede the general ones. Still, once the advance has reached a higher level, it opens the opposite outlook, such as in the view that the field of statics is a special case of the theory of mechanics.” To this viewpoint perhaps we may add, for mutual encouragement, that in scientific research seeking to discover the basic principles underlying a specific phenomenon in question, the ultimate goal is to achieve a thorough understanding of the theme subject and a resolution in optimum simplicity, elegance, accuracy, and generality. This course can be greatly enhanced by first grasping a clear and sound physical concept for deep insight to accomplish an effective method of solution for the phenomenon under study.

Acknowledgment Heartfelt gratitude and warm thanks are due from me to all those for their bountiful encouragement, creative intellectual interactions, profound friendship in high spirit, and much more that have benefited me beyond measure in my lifetime. In particular, I wish to congratulate the Organizing Committee for this Symposium, especially Allen Chwang, Michelle Teng, Daniel Valentine, Chiang C. Mei, and others on their leadership and expertise in achieving a full success of the Symposium which we greatly enjoyed attending and benefit from. Finally, I have pleasure thanking my dear wife, Dr. Chinhua Shih Wu, Fonda B. Wu, Melba B. Wu and family for their untiring encouragement, great care, and their generous donations to the common cause. I further wish to acknowledge having the graceful sponsorship of the Chinese-American Scholarship Foundation for my recent studies.

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References 1. G.G. Stokes, Math. and Phys. Pap. 1, 197-229, 314;

Math. and Phys. Pap. 5, 146-159, (1880) Cambridge U. Press. 2. T. von KArmAn and W. R. Sears, J. Aero. Sci. 5, 379-390 (1938). 3. G.I. Taylor, Proc. R. SOC. A209, 447-461 (1951). 4. T.Y. Wu, J . Kao and J.E. Zhang, Acta Mechanica Sanica. 21 1-15 (2005). 5. S. Wolfram, Mathernatica Version 5.0 (2003) Wolfram Media, Cambridge. 6. M.A. Grant, J. Fluid Mech. 59, 257-262 (1973). 7. T. Theodorsen, N A C A Tech. Report 496 (1935). 8. T.Y. Wu, Advances in Appl. Mech. 38, 291-353 (2001) Academic Press. 9. H. Wagner, Z A M M 5 , 17-35 (1925). 10. V.G. Stredie, Mathematical modeling and simulation of aquatic and aerial

animal locomotion. Ph.D. Thesis, California Institute of Technology, Pasadena, CA (2005)

11. A.T. Chwang and T. Y. Wu, J. Fluid Mech. 63, 607-622 (1974). 12. A.T. Chwang and T. Y. Wu, J. Fluid Mech. 67, 787-815 (1975). 13. A.T. Chwang and T. Y. Wu, J. Fluid Mech. 75, 677-689 (1974). 14. A.T. Chwang, J. Fluid Mech. 72, 17-34 (1975). 15. R.E. Johnson and T. Y. Wu, J. Fluid Mech. 95, 263-277 (1979). 16. M.J. Lighthill, Mathematical Biojluiddynamics, SIAM 17 (1975) SIAM. 17. T.Y. Wu, C.J. Brokaw and C. Brennen, Swimming and Flying in Nature

(1975) Plenum Press. 18. T.Y. Wu, A new approach to ciliary locomotion. In Fkontiers in

Biomedical Engineering, 177-184 (2003) Kluwer Acad. Publ. 19. J.R. Blake, J. FZuid Mech. 55, 1-23 (1972). 20. C. Brennen, J. Fluid Mech. 65, 799-824 (1974). 21. S.R. Keller and T. Y. Wu, J. Fluid Mech. 80, 259-278 (1977). 22. S.R. Keller, Fluid mechanical investigations of ciliary propulsion.

Ph. D. Thesis, Calif. Inst. of Technology, Pasadena, CA (1975). 23. S. Gueron and N. Liron, Biophysical Journal 63, 1045-1058 (1992). 24. S. Gueron and N. Liron, Biophysical Journal 65, 499-507 (1993). 25. S. Gueron and K. Levit-Gurevich, Biophysical J. 74, 1658-1676 (1998). 26. A.T. Chan and A.T. Chwang, Proc. Inst. Mech. Engrs. J. Mech. Eng. Sci.

27. J.J. Shu and A.T. Chwang, Phys. Rev. E 63 051201.1-6 (2001). 28. C.F. Gauss, Crelle’s J. Reine u. Angew. Math. VI, 232-235 (1829).

214 175-179 (2000).

APPENDIX THE ORIGIN OF THIS BOOK OF SCIENTIFIC REFLECTIONS:

THE OTHER SIDE OF DOING ENGINEERING SCIENCE'

DANIEL T. VALENTINE Clarkson University, Potsdarn, NY 13699, USA

This paper presents an anecdotal description of the proceedings of Track 8 of the 23" International Conference on Offshore Mechanics and Arctic Engineering held in Vancouver, British Columbia, CANADA in June 2004. Track 8, B e Zheodore Y.-T Wu Symposium on Engineering Mechanics, was held June 21-22, 2004. It was a collegial celebration of Professor Wu's 80" birthday. It was a scientific marathon of sorts with 50 presentations on topics influenced directly and indirectly by Professor Wu. The papers in this book are original works based on the talks given at this special symposium. They provide reflections and outlooks on the latest advances and the future of engineering mechanics. Hence, they are useful to graduate students and researchers. The exchange of ideas at a gathering known as a symposium is an important part of the practice of engineering mechanics. Such an event is particularly successful when it leads to further dissemination of the ideas discussed via published manuscripts of original writing. The forgoing chapters fulfill this requirement. This appendix describes the other features of the symposium that makes doing science and engineering so much fun and also provides further insight into the exemplary researcher and teacher we honored.

1. Introduction

The organization of this symposium was a collaborative effort of the Ocean Engineering Committee of the Ocean, Offshore and Arctic Engineering Division of ASME and the Fluids Technical Committee of the Engineering Mechanics Division of ASCE. The author of this paper considers the beginning of this symposium the keynote presentation by John Nick Newman at the end of the Plenary Session held in the morning of 21 June of the 23rd International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2004). After lunch of the same day, at 14:00, opening remarks by the organizers initiated the one-half and one-full day scientific marathon of fifty 10-minute (APS-Division of Fluid Dynamics style) talks.

I The contributions of Marshall P. Tulin, Theodore, Y.-T. Wu, and Howard Stone to this appendix

are gratefully acknowledged. The help given by A. Chwang, M. Teng and Professor Wu in preparing the biographical sketch is also gratefully acknowledged.

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The symposium was organized and held to honor the significant accomplishments of Professor Theodore Yao-Tsu Wu in the fields of biofluiddynamics, marine hydrodynamics, cavitation, fluid-structure interaction, nonlinear waves, and other advanced topics of fluid mechanics that have guided our community in our investigations of fluid mechanical phenomena. The symposium presentations also reviewed the present state of engineering mechanics, and provided insights into the charting of the future of this area of investigation. This book, which is a collection of original papers based on the presentations made at this symposium, provides guidance and inspiration for those of us interested in continuing to advance engineering mechanics as we begin the 2 1 St century.

During the evening meal of the first day Professor Wu gave his paper to a mixed audience of participants and their spouses. We were very grateful to have Chin-Hua (Wu’s wife), Melba (Wu’s daughter) and Fonda (Wu’s son) in attendance along with two grandsons and a son in law. I had the good fortune and pleasure to sit with the Wu family. Immediately to my right sat Professor John Chia-Kun Chu from Columbia University (he is a long time friend of our honored guest and an elder statesman in computational mechanics; he earned his Ph.D. in 1959 under the mentorship of Kurt Otto Friedrichs at the Courant Institute, New York University). After Professor Wu’s very instructive and enlightening presentation, we presented him with the Theodore von K i i d n Medal awarded the previous week by the ASCEiEMD but presented at OMAE 2004. Brief speeches were made; a letter was read and an embrace exchanged. The Acknowledgement for Acceptance written by Professor Wu was read by Tin-Kan Hung; it is reproduced in the next section of this paper. In addition to this medal, the Ocean, Offshore and Arctic Engineering Division of ASME presented the 2004 OOAE DIVISION-ASME LIFE-TIME ACHIEVEMENT AWARD to Professor Theodore Y.-T. Wu. It was presented in grateful recognition of significant lifetime contributions to Hydrodynamics and Fluid Mechanics. He is the third person to receive this award (the two previous awardees are John V. Wehausen, 2002, and John N. Newman, 2003). These two awards are new additions that should be added to his biographical sketch, the last section of this paper.

The section following the next section presents remarks by Marshall P. Tulin, a letter to Professor Wu from Howard A. Stone and other interesting events at this symposium. The purpose of presenting this mformation in this paper is to provide insight into the kind of mentor and teacher Professor Wu is. He encourages the continuing education of all who have come in contact with him. The next three sections provide the best guidance for anyone interested in

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the improvement of scientific, technical, engineering and mathematics education. Professor Wu, the likes of Professor Wu and Caltech are very special people at a center of excellence in scholarship. The inspirations provided by the legacy of Milliken and von K h d n provide examples from which we can all learn the fruits of scholarship and the skills of life-long learning.

Figure AI . LeA to right: Daniel T. Valentine, Theodore Y.-T. Wu, and Allen T. Chwang at the end of the first-night dinner gathering.

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Figure A2. Professor Wu accepting the OOAE Life-time Achievement award from the author.

Figure A3. Professor Wu’s former students and postdocs with Professor Wu and Chin-Hua.

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2. The 2004 Theodore von KBrmBn Medal Acceptance

Acknowledgement for Acceptance of the 2004 Theodore von KBrmBn Medal conferred by The American Society of Civil Engineers: This letter is by Theodore Y. Wu.

Dear Chairman and Professor Iwan, Colleagues and fiends, Ladies and Gentlemen:

This is an overwhelming occasion. It dates back to the wisdom and foresight of the Engineering Mechanics Division of ASCE in establishing the Theodore von K h d n award. To this shinning beginning, I wish to share with you my reflections with deep appreciation in several aspects.

I shall begin with the illustrious appointment of Professor von Khrrnhn by Robert A. Millikan to build up a Graduate School of Aeronautics at Caltech. Determined to build Caltech into a leading institution in the world, Millikan, then the newly arrived first President of Caltech, asked at first for three names of top aerodynamicists in the world for choice. In response, Paul Epstein and Harry Bateman on the faculty provided the names von K h d n of Aachen, Ludwig PrandtI of Germany and Sir G. I. Taylor of the United Kingdom, with a footnote that von K a d n be their first choice and the most hopeful to win over. To thls end, as I heard from von K h d n himself during one of my visits to his winter residence in Pasadena, it happened one day in 1926 summer, he received a telegram; it read: “What is the first boat you can take to come here?” And it was signed: Millikan. “Then I found myself aboard the next ship to New York,” said von K a d n .

Von K A d n was notably known for his talent at bringing out the very best in his students. He made no secret that he always prepared class lectures with great care. He would stress the emphasis on makmg a crisp clear physical concept of the phenomenon in question to come fust, followed by the underlying basic principles, then invariably exhibiting his amazing talent to express a seemingly complicated problem in terms of simple mathematical relations. Concerning mathematics, he used to comment that the tools in his math-lab are all rather simple, but none rusty. “I never could learn the new methods of modem mathematics,” he would say, “so I solved complex problems by resorting to fundamental principles.” He never hesitated to extend a complex problem into an ideal case, so much so as to eliminate minor details, come to grips with only the essentials, draw out conclusions as profound as possible, and finally to explain it all in the simplest way for others to understand.

In this spirit, von K A d n educated students at Caltech to become distinguished scholars and professors for other universities and institutions in the

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world, captains and chief engineers in industry, and even a line of military generals endowed with hi-tech expertise. He was noted for his natural inclination toward informality to approach students himself and give them his focused attention. The more dedicated a student in self learning and developing, the stronger attention from the mentor. This was amply illustrated through his joint works with Bill Sears, H. S. Tsien, C. C. Lin, Pol Duwez, and others.

It is therefore no wonder that these all-embracing achievements in advanced education and learning by the master finally led to the famed ceremony in 1963 when President Jack Kennedy presented the Inaugural National Medal of Science to Professor Theodore von Kirmin, the unique solitary recipient, with a statement “I know of no one else who so completely represents all the areas involved in the medal --- science, engineering, and education.”

Personally, this medal I have the great fortune to accept means especially significant to me for it brings me to a line of memorable reflections and appreciations, dating back to the great pleasure in meeting and visiting with von Kirmin earlier in Pasadena, to receiving from him a few searching queries on my graduate studies that really shed bright light on the ensuing course of my further work, to being greatly benefited by his principles of learning and research, and to assuming my years of editorship for the series Advances in Applied Mechanics following the spirit of the two great founders, Theodore von K h i n and Richard von Mises, of the serial. For all the pleasure and encouragement coming with the medal, my warm thanks are due to the ASCE Engineering Mechanics Division, its Advisory Board and Executive Committee, under the dynamic leadership of Professors Bill Iwan, Stein Sture, Hayley Shen, Alex Cheng and others.

For an enlightening outlook, we might reflect on the tradition that solid, fluid as well as classical mechanics have invariably been cultivated in an invisible field, with diligence, by the pioneering giants and followers. As exhibited in his Collected Works, von K h i n contributed equal efforts to solid as well as to fluid mechanics, such as his development of the nonlinear theory of buckling with Professor H. S. Tsien. He had uncanny abilities to provide new theories ahead of experiments. But he never failed to reiterate that experiments always have the final say. With the prevailing interests nowadays emerging deeper into micron and nanometer scales, the new scope will cover phenomena involving molecular mechanics, chemistry, biology, medicine, etc. The micro-world, with and without life, can be exceedingly remarkable to behold. Collaborative studies in the spirit of Theodore von K h i n will undoubtedly be endeavors endlessly exciting and richly rewarding.

Thank you very much.

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3. Tributes

3.1. Dedication by Marshall P. Tulin (read in Vancouver)

During T.Y. Wu's studies at Caltech, 1949-1952, and for some years afterwards, Engineering Education in the US was in ferment almost everywhere, bringing great changes and very great postwar opportunities.

Theodore von K h B n , in the 40s and 50s, with his great vision, had preached of the necessity to deal directly and realistically with non-linear problems in Engineering. This required a much higher mathematical ability than was common among University Professors of Engineering in the US. Nowhere was this more true than in the case of Fluids. And nowhere in Academia was there a greater response to this mathematical challenge in engineering education than at Caltech, and particularly in GALCIT, the Guggenheim Aeronautical Laboratory, founded by von Ktirm6n. Important early Faculty names were Hans Liepmann, H. S. Tsien, and Paco Lagerstrom. And there were remarkabIe students. In Aeronautics: Wallie Hayes, Stanley Corrsin and the Clauser brothers. In Lagerstrom's group there were: Julian Cole, Leon Trilling, Milton Van Dyke, Gordon Latta and Saul Kaplun. It was in this rich soil that Ted Wu studied and trained, received his PhD in 1952, and began to do research as a Professor.

I first met Ted about 1955, when I traveled from Washington, DC to Caltech to become better acquainted with our Contractors for ONR, where I then worked. As a young Faculty member, Ted was part of a major effort in Naval Hydrodynamics, which had been conducted in Pasadena during WWII, for about 15 years at that time. Some of the senior investigators were Robert Knapp, Hollander, and Milton Plesset. Milton, a physicist who had spent time in Copenhagen with Niels Bohr, was an urbane and brilliant man, who had become involved in wartime problems in Hydrodynamics and had published theoretical papers on cavity flows as early as 1948. Partly under his influence a very energetic young group in Pasadena was hard at work on cavitating and supercavitating flows; these included Ted Wu, Alan Acosta, Blaine Parkin, Bob Kermeen, and Andy FabuIa.

Now, 48 years later, Ted stands as the leading long-time practitioner of theory in our field, both by virtue of the very high quality of his mathematical studies, but also by virtue of the variety of problems, which he has investigated, and by his wonderful productivity.

This remarkable success is even eclipsed by his dedication and great success in the teaching and training of graduate students. It is not an exaggeration to say

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that he created an Academic Dynasty, since a number of his earlier students have themselves gone on elsewhere to great success in Teaching and Research in Hydrodynamics, of which his second PhD, Professor C. C. Mei is a particularly distinguished example.

So, Ted, we have at least to imagine that you are wearing a royal crown, and we toast you: Long Live King Ted!

The rise to royalty of a young Chinese student immigrating here fust to Iowa and then to California, from Chiao-Tung University in Shanghai, can be better understood in the context of old and great Chinese traditions in their devotion to Scholarship and Teaching, and especially in their influence on the formation of this very talented person who still carries in himself the essential wisdom and culture, personal graciousness, and good humor, and the innate politeness of the best of those traditions.

I have throughout my own career been fortunate to have had colleagues and companions who were also products of those great traditions, including S.S. Chin at MIT, Jin Wu, C. C. Hsu, and C. F. Chen at Hydronautics, and then my wonderful doctoral students at UCSB from Chiao-Tung: Ms. Pei Wang, Yi Tao Yao (her husband), J. J. Li, Ming Wu, and H. Ma. It was not entirely a coincidence that the latter students came to study with me; it eventually traces back to King Ted. During my stay at Berkeley in the 70s, Ted asked me to look after a Chiao-Tung Professor then visiting Berkeley, Ying Zhong Liu. This led to a warm personal and professional relationshp, and eventually when I arrived in Santa Barbara he sent several of his students to me. All of my Chiao-Tung students have broadened my life and enlightened me.

Finally, Ted, it has been for me a great pleasure to have had you as a companion in our Long March in Naval Hydrodynamics. Thank you!

Marshall P. Tulin Emeritus Professor & former Director of the Ocean Engineering Laboratory University of California at Santa Barbara

3.2. Letter from Howard A. Stone dated 30 May 2004

Professor C. C. Mei read this letter to Professor Wu at dinner on the 21 June 2004.

To: Professor Theodore Y . Wu Dear Professor Wu: It gives me great pleasure to write this letter to you in honor of your

receiving the von K i d n medal form ASCEEMD. Back in the early 1980s I was a graduate student at Caltech in Chemical Engineering and I had great

723

interest in fluid dynamics. It was my good fortune to take your introductory course and then to have the opportunity to continue to interact with you during my graduate years. Your great insights into fluid dynamics and the care and rigor with which you developed the many foundational ideas of our subject made a lasting impression on me. Many of these concepts and ideas I repeat, generally in the form I learned them in your class, each fall when I teach the introductory graduate course here!

I always enjoyed going to lectures, and as I write this letter I still have many fond memories of your lectures, your beautiful handwriting and clear, organized presentations, and the patient way you answered my questions. In addition, over the years I have worked on a variety of research problems that have led me to some of your research papers. In particular, your work on hydrodynamics of swimming, singular solutions for low-Reynolds-number flows, and even an early paper on generalizations of the “Rayleigh problem” have all spent considerable time on my desk. I hope that you can look back with pride at this large body of valuable research that stemmed from your original insights and mathematical analyses, and also recognize the great value that you have brought to the international fluid dynamics community through the many graduate students you trained, and others, llke me, that had the good fortune to attend your lectures.

Again, many congratulations upon receiving the von K A d n medal. I can think of no one more deserving. It seems most fitting that you, who like von K A d n , spent so many productive years at Caltech, should receive this honor.

With best regards and best wishes for many more years of health and happiness,

Howard A. Stone Gordon McKay Professor of Chemical Engineering and Applied Mechanics Division of Engineering and Applied Sciences, Harvard University

3.3. The closing dinner cruise

After the second full day of presentations the proceedings closed with relaxation and socializing on a dinner cruise up the fjords. As a principal organizer I, most unfortunately, could not attend this final event --- I gave up at least three opportunities to more deserving colleagues; these are the duties and responsibilities of an organizer and host. I did, however, enjoy the evening with a new friend, Celso Pesce, a former student of Professor Wu and the other Brazilians attending OMAE 2004 including my friend Antonio Fernandes, a former student of John N. Newman.

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The cruise was very well attended, the capacity of the boat allowed for no more cruisers. As a principal organizer, this pleased me highly. All who were able to attend indicated that the event was most enjoyable.

Figure A4. From left to right, the Ertekin’s, Joe Hammack and Touvia Miloh on the boat cruise.

4. Biographical sketch of Professor T. Y.-T. Wu

Professor Wu’s contributions to engineering mechanics illustrate the richness and productive hstory of this field and Professor Wu’s significance to its development. He has made major contributions in fluid physics and stability theory, nonlinear water waves, and geophysical and biophysical fluid mechanics. He earned his B.S. in 1946 fkom Chiao-Tung University, his M.S. in 1948 from Iowa State University, and his Ph.D. in 1952 from the California Institute of Technology (Caltech). As described to the editors by Professor Wu: “My Ph.D. thesis advisor at Caltech was Professor Paco A. Lagerstrom, who officially so acted throughout my doctoral study (1949-1952). About then, Paco produced such outstanding Ph.D. students as Julian D. Cole (1949), Milton Van Dyke (1949), Saul Kaplun (1954), and more. We collaborated like a closely knit team on developing the so-called matched asymptotic expansions as a principal mathematical method, with various applications like that to the Navier-Stokes equations and free-surface flows, leading to four books (by Van Dyke, 1964; Kaplun 1967; Cole, 1968; Lagerstrom, 1988) and more. The impact of this

725

thrust has been tremendous as reflected by the inner- and outer-expansion technique broadly applied in the literature for decades (my joint contribution with Peter Rispin being the nonlinear theory of planning surface with a spray sheet of arbitrary thickness, 1967). In this period, Julian Cole generously provided advice and encouragement as a truly caring mentor to me. We wrote joint papers on heat conduction in compressible fluids. Thus, with give and take in sharing knowledge and innovative ideas, intellectual interaction soon became transcendental in significance, heart and soul. In this sense, I regard both Lagerstrom and Cole as my wise counsel casting more light on the course of my development. After my graduation, I joined the Engineering Science Department and have stayed ever since.”

Professor Wu is Professor Emeritus of Engineering Science at Caltech (1996-present). His honors are numerous. Some of them are as follows. In 1981 he was named the UC Berkeley Russell Springer Honorary Professor. He was a John Simon Guggenheim Fellow (1964), an Australian CSIRO and Universities Fellow (1976), a Japan JSPS Fellow (1981), a Member of the U.S. Academy of Engineering (1982), a Member of the Academia Sinica (1984), an Honorary Fellow of the Institute of Mechanics (1988), and a Foreign Member of the Chinese Academy of Sciences (2002). He is an American Physical Society Fellow and in 1993 the American Physical Society awarded him the Fluid Mechanics Prize.

With hls basic training in aeronautics, mathematics, and fluid physics, Dr. Wu has taught at Caltech since h s graduation and pursued research in engineering science, a newly evolving field that forges frontiers bridging engineering with different branches of science. Enjoying teachmg, he has been called upon to teach undergraduate core courses and advanced graduate core courses, for which endeavor he was accorded a Caltech Distinguished Teaching Award. He is in full conviction that all students can be inspired, with due stimuli, to learn how to learn best by themselves. In his career, Dr. Wu has shown a broad and versatile interest in research, including waves of compressible, viscous and heat-conducting fluids; free-streamline theory of cavities, jets and wakes; water waves and free-surface waves, mechanics of swimming and birdinsect fight; wind and ocean-current energy, internal waves in the ocean, mathematics of nonlinear evolution equations, etc. With students and visiting colleagues, he collaborated on studies of low-Reynolds-number hydrodynamics, microorganism locomotion and related biophysical phenomena. The recent research interest of Wu‘s group is focused on forced generation of nonlinear waves at resonance of soliton-bearing systems, three-dimensional vortex dynamics, and coastal oceanography. His new theoretical model for l l l y

726

nonlinear dispersive waves in water of variable depth is ideal for investigating mitigation and control of such natural hazards as tsunami, storm surge, and hurricanes. The new idea and concept, based on devising means for active and passive control and mitigation of devastating waves and fluid waves, may germinate growth of a new field. Revisiting biophysics, he has developed a fully nonlinear lifting theory for modeling bird flying and fish swimming. In collection, Dr. Wu has authored and co-authored over 150 papers published in archive journals and book chapters. His lifetime contributions have been recognized by winning the Lifetime Achievement Awards of the Chinese- American Faculty Association of Southern California (1993), of the Chinese Engineers and Scientists Association of Southern California (1995), and of the North American Chiao-Tung University Alumni Association (2000).

Professionally, Dr. Wu pursues, with conviction, the goal of enhancing academic interaction between individuals, groups, societies and nations. This is the best road, he believes, to promote science, disseminate knowledge and share wisdom. He is always sincere in responding to discussions and to giving plenum lectures at scientific meetings. He has made the U.S. Southwest Universities Lecture Tour (1968), U.S. Midwest Universities Lecture Tour (1985), Herbert Wagner Memorial Lecture in Munich (1984), Distinguished William Mong Lecture in Hong Kong (1 997), Distinguished Israel Pollack Lecture in Haifa (1999) and other invited lectures. He has accepted appointments as Honorary Professor in several universities abroad. He is a co-editor of the serial of Advances in Applied Mechanics that was founded by Theodore von K h d n and Richard von Mises; he has edited scores of books and Proceeding Volumes of international conferences he organized. In teaching, he has guided a good number of students to doctorate, an endeavor he has conducted with dedication, great expectation, and pride.

5. Conclusion

Thank you, Professor Theodore Yao-Tsu Wu, for accepting our request and honoring us with your presents to celebrate your 80" buthday in such a grand manner. I enjoyed meeting you again, meeting your family and befriending them all. Over the years I read and studied your papers on cavity flows, propellers, nonlinear waves, and swimming and flying in nature. I have had the distinct pleasure of being the fust to have read your paper as well as the other contributions in this book. Finally, the poem by Professor Wu, located below his picture at the beginning of this book, is a most beautiful note on which to end this project and, hence, to offer this book to our community to now enjoy.

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We s M m t cease from e;\glbration M t h endof d o u r e q h i r g

W 6 e to arrive whre we started andk?ww tfie p h fm t f ie jkt time.

T. S . Eliot, Little Gidding

ZIiis is a perfectfluid hvirg M age nor ~ T S ,

survivirg suahs, udtered, Ibuirg rest.

Muriel Rukeyser, The Dam


AUTHOR INDEX

Bao, J.Y. 493 Bao, Weiguang 355 Ban, Brian C. 195 Barta, Efrath 5 15

Camassa, Roberto 222 Caulkins-Pennell, C. 493 Chang, Chien-Cheng 669 Chen, Bang-Fuh 625 Chen,Chen 70 Chen, Hung-Cheng 669 Chen, Xiao-Bo 371 Cheung, Anthony T. 423 Chiang, Wen-Son 247 Choi, W. 94 Chow, K. W. 11 1 ,Chu, Chin-Chou 669 Chwang, A. T. 239,633,65 1

Dai, Shiqiang 2 13 Deng, L. M. 239

Feng, Kelie 257

Gao, S. 60 Guyenne, Philippe 173

Hammack, Joseph 173 Hancock, Mathew 29 Henderson, Diane 173 Hsu, Chin-Tsau 660 Hsu, Ming-Kuang 297 Hu, Wenrong 557

Huang, Lingyan 633 Huang, Norden E. 150 Hung, Tin-Kan 446,625 Hwung, Hwung-Hweng 247

Isshiki, Hiroshi 539

Kent, C. P. 94 Kinoshita, Takeshi 355 Kuang, Jun 660

Lam, Horace H. 484 Lee,Long 222 Lee, S. J. 408 Li, Hsien-Wen 3 14 Li,Yile 29 Liao, Shijun 70 Lin, S. P. 140 Ling, S. C. 440 Liu, Antony 297,68 1 Liu, Hao 557 Liu,Hua 128 Liu, Philip L.-F. 265 Long, Steven R. 327 Lu , Dong-Qiang 2 13

Mei,C. C. 29 Miloh, Touvia 603 Mohseni, Kamra 574 Murashige, Sunao 84 Murray, A. Brad 117

Ng, Chiu-On 633

729

730

Oishi, Shin’ichi 84 Ong,C. T. 60

Pao, Hsien P. 286 Pesce, Celso P. 390 Pihl, Jorgen H. 29 Poon, C. K. 1 1 1

Schillinger, C. J. 94 Serebryany, Andrey N. 286 Shen, S. S. P. 60 Su, Xiao-Bing 2 13 Sun,Ren 651

Teng, Michelle H. 257 Tong, Binggang 557 Tulin, Marshall 3

Valentine, Daniel T. 195,7 15

Wang, Benlong 128 Wang,Chun 70 Wei, Gang 213

Weihs, Daniel 515 Winet, H. 493 Wu, Theodore Y.-T. 693 Wu, Yung-Ching 3 14

Xu, Z. 60

Yang,Yi 613 Yates, George T. 474 Yi,Ming 173 Yoshida, Motoki 355 Yu, Jie 117 Yu, Jun 613,681 Yu, Yi-Hsiang 625

Zhang, D. H. 1 I1 Zhang, JinE. 48 Zhang, Qinghai 265 Zhao, Yunhe 297 Zheng, Q. 60 Zhou, Hongqiang 257

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Advances in Engineering Mechanics - At Chwang MH Teng DT Vale (2005 World Scientific)

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