Tesis Alvaro

Instituto Mediterraneo de Estudios Avanzados

IMEDEA (CSIC-UIB)

Departamento de Tecnologıas Marinas, Oceanografıa Operacional y

Sostenibilidad (TMOOS)

A Boussinesq-type model for wave

propagation in deep and shallow waters

and boundary layer considerations

A Doctoral Thesis Presented to the

Department of Civil Engineering of Universidad de Castilla La-Mancha

as Partial Fulfillment of Requeriments for the

Degree of Doctor of Civil Engineering

by

Alvaro Galan Alguacil

PhD supervisors:

Dr. Alejandro Orfila Forster Dr. Gonzalo Simarro Grande

Esporles, July 2011

Agradecimientos

En primer lugar me gustarıa agradecer a mis directores de Tesis, Alejandro

Orfila y Gonzalo Simarro, su gran ayuda en todo momento, la paciencia y en-

tusiasmo mostrada en cada paso del camino, y el apoyo que han supuesto para

mi en los momentos mas difıciles. Darles las gracias por el tiempo empleado,

por compartir sus conocimientos, por ser una fuente constante de ideas, por

ser algo mas que simples directores de tesis. Esta Tesis no serıa posible sin

ellos.

Agradecer a todas aquellas personas e instituciones que de una u otra man-

era han colaborado en el trabajo realizado, en especial al Consejo Superior de

Investigaciones Cientıficas (CSIC) por el soporte economico prestado para la

consecucion de esta Tesis mediante su programa JAE (Junta de Ampliacion

de Estudios) y por las ayudas para realizacion de estancias breves.

Mostrar mi mas profunda gratitud al profesor Philip L.F. Liu por brindarme

la oportunidad de pasar unos meses junto a el en la Universidad de Cornell

durante la primavera de 2010 y a Yong Sung Park por su ayuda en la real-

izacion de los ensayos. Gracias a Jose Marıa y Jorge por esas largas charlas en

las que la condicion inf-sup estaba siempre presente, y como no, a Lara y Vera,

mis companeras futboleras de mundial. Gracias a todos ellos por hacerme la

estancia en Ithaca mucho mas agradable.

v

Desearıa agradecer a todas las personas con las que he compartido experien-

cias durante esta etapa de mi vida, a Javier Gonzalez, quien me brindo la

oportunidad de comenzar en este mundo de la investigacion; al personal del

TMOOS y del Laboratorio de Hidraulica de la Escuela de Caminos de Ciudad

Real; a Marıa por su ayuda en todo momento; a Alejandro y Fernando por

abrirme las puertas de su casa; a Matıas, Tolo, Marian, Chema, etc.

Quiero agradecer a mi familia el apoyo y carino recibido: a mis padres, Va-

lentın y Luisa, un apoyo constante e incondicional imprescindible para llegar

a buen puerto; a mis hermanos, Cesar y Gema; a mis sobrinos, Daniel y Alba,

un manantial de alegrıa permanente; y por supuesto a mis abuelos.

A mis amigos, en especial a Vir y Jaime por estar siempre ahı, por esas largas

noches de “insomnio”, por todas esas risas sin motivo, por todos esos momen-

tos inolvidables vividos y por los que aun nos quedan por vivir, en definitiva,

por hacerme mas feliz; a Rober, Miren, Felipe, Mau, Val, Tania, Jorge, Julio,

Lourdes, Luci y a otros tantos que habeis aportado vuestro granito de arena.

No puedo terminar estos agradecimientos sin acordarme de una de las per-

sonas mas especiales e importantes en mi vida, Esther. Durante estos largos

cuatro anos hemos compartido buenos y malos momentos, momentos de eu-

foria por el trabajo bien hecho y momentos de decepcion cuando las cosas no

marchaban tan bien. Lo mas importante es que siempre tuve claro durante

este periodo que, al llegar a casa, siempre encontrarıa a alguien con quien com-

partir la experiencia vivida y un apoyo constante para afrontar el dıa siguiente.

Alvaro Galan Alguacil

Julio, 2011

vi

Resumen

En esta Tesis se deriva un nuevo conjunto de ecuaciones tipo Boussinesq para

la propagacion de oleaje en aguas profundas y someras. Se trata de un nuevo

conjunto de ecuaciones totalmente no lineal con propiedades dispersivas mejo-

radas respecto a los sistemas previos. Las nuevas ecuaciones son exactas hasta

O (kh)2. Se emplea un metodo de optimizacion para determinar el valor de los

coeficientes introducidos en las nuevas ecuaciones propuestas con el objetivo

de minimizar las diferencias entre el modelo y las teorıas de Airy (dispersion

lineal y asomeramiento) y de Stokes (transferencia de energıa debilmente no

lineal). Se muestra que con la adecuada eleccion de estos coeficientes el mod-

elo es aplicable hasta valores de kh = 20 con un error relativo menor del 1%

en dispersion lineal. En esta Tesis se presenta un nuevo esquema numerico

explıcito de cuarto orden para resolver y verificar el nuevo conjunto de ecua-

ciones. Ademas, se ha llevado a cabo un analisis lineal de estabilidad para

obtener una condicion tipo CFL para el paso de tiempo. La integracion tem-

poral se lleva a cabo empleando un esquema Runge-Kutta de 4o orden. El

oleaje se genera internamente en el dominio por medio de una funcion fuente.

El comportamiento tanto de las nuevas ecuaciones como del esquema numerico

es validado empleando experimentos y soluciones analıticas en 1D y 2D. El

comportamiento bidimensional del modelo se valida comprobando la evolucion

temporal de una onda inicialmente gaussiana en un dominio cerrado cuadrado.

Ademas se han simulado numericamente dos experimentos para llevar a cabo

vii

la validacion del nuevo modelo derivado. El primero de ellos es la barra 1D

sumergida de Dingemans y el segundo es la batimetrıa 2D de Vincent y Briggs.

Los resultados numericos obtenidos muestran buen ajuste para todos los casos.

Para profundizar en el conocimiento de los procesos que se dan en la capa

lımite de fondo, se ha disenado un nuevo instrumento para la medida directa

de la tension de fondo bajo distintas condiciones de oleaje periodico. Se pre-

sentan tanto casos monocromaticos como ondas en forma de diente de sierra,

comparando los resultados experimentales con un modelo numerico. El obje-

tivo final de este analisis es contar con una metodologıa para la inclusion de

la tension de fondo bajo oleajes fuertemente no lineales.

viii

Abstract

In this Thesis, a new set of Boussinesq-type of equations is derived for water

wave propagation in deep and shallow waters. The new set of equations are

fully nonlinear and the dispersive properties are improved relative to previ-

ous systems. The model equations are accurate to O (kh)2. An optimization

method is used to determine the weighting coefficients employed in the pro-

posed equations so as to minimize the differences between the model equations

and the theories by Airy (linear dispersion and shoaling) and Stokes (weakly

nonlinear energy transfer). It is shown that with the proposed choice of weight-

ing coefficients, the model is applicable up to kh = 20 with 1% relative errors

in linear frequency dispersion. A new explicit and fourth order numerical

scheme is presented in this Thesis to solve and verify the new set of equations.

Besides, a linear stability analysis is performed to obtain a CFL-type condi-

tion for the time step. The time integration is performed using a 4th order

Runge-Kutta scheme. Waves within the numerical domain are generated by

means of an internal source generation function.

The features of the new equations as well as the numerical scheme are tested

against experiments and analytical solutions in both 1D and 2D cases. The

linear 2D performance of the equations is tested modelling the evolution of an

initially gaussian wave in a squared domain. Besides, two experiments have

been simulated to test the performance of the new model equations. The first

one is the Dingemans bar (1D) and the second is the Vincent and Briggs shoal

ix

(2D). The numerical results show very good agreement in all cases.

In order to get a deeper knowledge in those processes occurring in the bot-

tom boundary layer, a new instrument built to measure directly the bottom

shear stress under different conditions of periodic waves is presented. Both

monochromatic and sawtooth shaped waves are tested and experimental shear

stress compared against a numerical model. The final goal of this analysis is

to have a methodology to further include bottom shear stress under highly

nonlinear waves.

x

To my family

xi

xii

List of Figures

2.1 Schematic picture for the range of application of different models: (a)

[Peregrine 1967], denoted by P67, is valid for weakly nonlinear and

weakly dispersive waves over arbitrary slopes; (b) [Nwogu 1993], de-

noted by N93, improves the linear frequency dispersion performance

over flat beds into deeper waters; (c) [Wei et al. 1995], W95 here, ex-

tends N93 to fully nonlinear waves for the weakly dispersive case, and

[Madsen & Schaffer 1998], denoted as M98, increases the applicability

to deeper water; (d) new model equations. . . . . . . . . . . . . . . 13

2.2 Linear dispersion: capp/cAiry−1 for sets A corresponding to κmax = 5

(); κmax = 10 (); κmax = 20 () in Table 2.3. W95 refers to Wei

et al. (1995) and M98 to Madsen & Schaffer (1998) equations. . . . . 18

2.3 Frequency dispersion: relative errors for the phase speed using Pere-

grine (1967) (denoted as P67), [2/2] Pade, Nwogu (1993) (denoted as

N93) and [4/4] Pade. . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Linear shoaling: Aappη /AAiry

η − 1 for the coefficient sets A correspond-

ing to κmax = 5 (); κmax = 10 (); κmax = 20 () in Table 2.3.

W95 refers to Wei et al. (1995) and M98 to Madsen & Schaffer (1998)

equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Eigenvalues for the truncated matrices. . . . . . . . . . . . . . . . 26

2.6 Function ν∗ = f (Π, s) . The different symbols stand for different

values of Π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

xiii

2.7 Weakly nonlinear propagation: Gapp± /GStokes

± −1 for set A and κmax = 5. 31

3.1 Stability regions for third order Adams–Bashforth (AB3), fourth or-

der Adams-Moulton (AM4) and fourth order Runge-Kutta (RK4)

time integration schemes. . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 General behaviour of function fx using coefficient from Table 2.3 for

κmax = 5 (big symbols) and κmax = 10 (small symbols). Spatial

derivatives of second order (triangles) and of fourth order (circles). . 48

3.3 Analysis of the order of convergence for a 1D linear case. . . . . . . 52

3.4 Monochromatic linear 1D propagation for h = 25m and T = 4.5 s.

Numerical results (circles) and Airy’s analytical solution (line). . . . 54

3.5 Monochromatic linear 1D propagation for T = 4.5 s. Numerical

results for the new equations (circles), Madsen & Schaffer (1998)

(squares), Wei et al. (1995) (crosses) and Airy’s analytical solution

(line) at time t = 10T. . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Experimental set-up (top panel) and free surface time histories at #A

(left) and #B (right) for periods T = 0.45s (top), T = 0.55s (middle)

and T = 0.55s (bottom). Experimental data (stars) and numerical

results (line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Free surface histories at the corner (top) and center (bottom) in a

squared basin. Analytical solution for Airy’s theory (line) and nu-

merical solution for the new Boussinesq-type equations (circles). . . 59

3.8 Dingemans’ experiments. Case A. Numerical results (lines) and ex-

perimental data (stars) for free surface elevation. . . . . . . . . . . 61

3.9 Dingemans’ experiments. Case C. Numerical results (lines) and ex-

perimental data (stars) for free surface elevation. . . . . . . . . . . 62

3.10 Dingemans’ experiments. Case A. Comparison at sections #7 and

#8 using proposed coefficients (circles), Madsen & Schaffer (1998)

(squares) and Wei et al. (1995) (crosses) with experimental data (line). 63

xiv

3.11 Bathymetry and location of gages in Vincent and Briggs’ experiment

(top) and comparison between experimental (stars) and numerical

(line) significant wave heights. . . . . . . . . . . . . . . . . . . . . 65

3.12 Snapshot for free surface elevation at time t = 20s in Vincent and

Briggs’ experiment. Values of η/η0. . . . . . . . . . . . . . . . . . 66

4.1 General overview of the new instrument designed to measure directly

bed shear stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Velocity profile in the laminar case for a steady (ub (m/s) = 1) and

unsteady (ub (m/s) = cos (ωt) with ω = π/5 s−1) cases. In the latter

case, snapshots correspond to t = 0 (—), t = T/8 (−−), t = T/4

(−·−) and t = T/2 (· · · ), where T = 2π/ω = 10 s is the period. . . . 71

4.3 Near bed velocity and bed shear stress time series in the laminar case

for a periodic motion ub (m/s) = cos (ωt) with ω = π/5 s−1. . . . . . 72

4.4 Function fw (Re, ε) according to the original model II in Simarro et

al. (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 Comparison between the turbulent boundary layer models and ex-

perimental data in Kamphuis (1975) (+), Sleath (1988) (), Jensen

(1989) () and Simons et al. (1992) (). Root Mean Square (RMS)

error takes values of 0.276, 0.185 and 0.215 respectively. . . . . . . . 76

4.6 Monochromatic and sawtooth time histories. . . . . . . . . . . . . . 79

4.7 Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2) (×,

experimental; full line, model II) time histories for smooth tests in

Table 4.1. Monochromatic waves (top panels) and sawtooth waves

(bottom panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2) (×,

experimental; full line, model II) time histories for smooth tests in

Table 4.2. Monochromatic waves only. . . . . . . . . . . . . . . . . 85

4.9 Experimental and analytical values for fw: new data (black figures)

and data available in the literature (white figures). . . . . . . . . . 86

xv

4.10 Velocity ub (m/s) (dashed line) and shear stress τb (N/m2) (×, exper-

imental; full line, model II) time histories for smooth tests in Table

4.2. Sawtooth waves only. . . . . . . . . . . . . . . . . . . . . . . 87

4.11 Shear stress τb (N/m2) time histories for tests mc T08 u08 k25 (monochro-

matic) and st T08 u08 k25 (sawtooth). Numerical results (lines) and

experimental data (symbols). . . . . . . . . . . . . . . . . . . . . . 88

A.1 Weakly nonlinear propagation: GStokes± . . . . . . . . . . . . . . . . 103

A.2 Weakly nonlinear propagation: Gapp± /GStokes

± − 1 for W95. . . . . . . 103

xvi

List of Tables

2.1 Coefficients for Wei et al. (1995) and Madsen & Schaffer (1998). . . 12

2.2 Coefficients corresponding to [4/4] Pade approximant. . . . . . . . . 15

2.3 Sets A. Simple optimization. . . . . . . . . . . . . . . . . . . . . . 16

2.4 Sets B: joint optimization. (∗) correspond to joint optimization en-

suring stability for s = 5. . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Sets C. Linearly stable for smax 5. . . . . . . . . . . . . . . . . . 29

3.1 Analysis of the order of convergence for a 1D linear case. . . . . . . 52

3.2 Dingemans’ experiments setup. . . . . . . . . . . . . . . . . . . . . 60

4.1 Smooth test cases for sinusoidal and sawtooth waves. . . . . . . . . 80

4.2 Rough test cases for sinusoidal and sawtooth shaped waves. . . . . . 81

4.3 Monochromatic tests errors with the three models. For each case, the

model with minimum is shown in bold font. . . . . . . . . . . . . 84

4.4 Sawtooth tests errors with the three models. For each case, the model

with minimum is shown in bold font. . . . . . . . . . . . . . . . . 84

xvii

xviii

Contents

Agradecimientos v

Resumen vii

Abstract ix

List of Figures xiii

List of Tables xvii

Contents xix

1 Introduction 1

1.1 Boussinesq-type models . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Bottom shear stress . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Aims and motivation . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 6

2 An improved Boussinesq-type model for wave propagation 7

2.1 New set of wave propagation equations . . . . . . . . . . . . . . 7

2.2 Improving the linear propagation . . . . . . . . . . . . . . . . . 14

2.3 Linear stability considerations . . . . . . . . . . . . . . . . . . . 21

2.3.1 Flat bed case . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 Uneven bed case . . . . . . . . . . . . . . . . . . . . . . 24

xix

2.4 Improving the weakly nonlinear propagation . . . . . . . . . . . 29

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 30

3 An explicit numerical scheme for the new model 33

3.1 Source function for wave generation and sponge layers . . . . . 33

3.1.1 Source function . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.2 Sponge layers . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Spatial discretization: matrix notation . . . . . . . . . . 41

3.2.2 Time integration: fourth-order Runge-Kutta scheme . . 44

3.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Linear tests over flat bed . . . . . . . . . . . . . . . . . 50

3.4.2 Nonlinear tests . . . . . . . . . . . . . . . . . . . . . . . 58


4 Turbulent bed shear stress 67

4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Laminar flow . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.2 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Experimental facility and setup . . . . . . . . . . . . . . . . . . 77

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


5 General conclusions and future work 91

A Features of the new equations 95

A.1 Linear propagation . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.1.1 Linear frequency dispersion . . . . . . . . . . . . . . . . 97

A.1.2 Linear wave shoaling . . . . . . . . . . . . . . . . . . . . 98

A.2 Weakly nonlinear propagation . . . . . . . . . . . . . . . . . . . 100

xx

B Stability analysis for sinusoidal bathymetries 105

References 107

Additional works 115

xxi

xxii

Chapter 1

Introduction

Water wave propagation from deep to shallow water is a crucial issue for many

scientific and engineering activities in coastal regions. Accurate modelling of

wave transformation is mandatory to study a wide range of processes such as

sediment transport or coastal flooding. Besides, any wave model for relatively

shallow waters propagation needs to properly represent not only wave shoal-

ing, refraction and diffraction, but also nonlinearity.

Early attempts to model large coastal areas were based on the wave ray the-

ory. This approximation does not allow wave energy to cross the rays and, in

fact, ray models are restricted to the study of small amplitude waves –linear

theory– over slowly varying bathymetry and in situations where diffraction

effects are not important.

A different family of wave propagation models are obtained by depth averag-

ing the vertical distribution of the velocity field. The simplest models within

this category are the shallow water equations, valid if kh 1 (where h is the

water depth and k = 2π/λ is the wavenumber). Shallow water equations allow

to account for nonlinear effects, so that no restriction is made on a/h (where

a is the wave height), which is the parameter characterizing the nonlinearity

of the waves. The Boussinesq-type equations depart from the shallow water

equations to improve their range of applicability to higher kh and they re-

1

2 Introduction

produce wave propagation processes such as shoaling, diffraction, refraction,

wave-wave interaction and nonlinear transformation.

1.1 Boussinesq-type models

The traditional Boussinesq equations were originally derived for weakly non-

linear and weakly dispersive water waves, so the accuracy of the traditional

Boussinesq equations is up to O (a/h) and O ( (kh) 2) [Peregrine 1967]. Mad-

sen et al. (1991) showed that to keep the errors of the wave celerity, estimated

by the best form of the linearised Boussinesq equations (using the depth-

averaged horizontal velocity in the wave equations), within 1% of that of Airy

waves, the water depth must be less than one-fifth of the wavelength, i.e.,

kh < 1.1. On the other hand, as the water depth decreases, wave amplitude

increases and so does the nonlinearity parameter a/h. Therefore, the tradi-

tional Boussinesq equations, strictly speaking, can only be used in a small

region in coastal zones where both frequency dispersion and nonlinearity are

relatively weak.

In the last twenty years, much of the development of wave propagation mod-

elling has been focused on extending the applicability of the depth-integrated

Boussinesq wave equations into deeper water (or shorter waves). Nwogu (1993)

derived his weakly nonlinear Boussinesq-type wave equations in terms of the

horizontal velocity on a specified elevation from the still water level, zα. He

showed that the frequency dispersion characteristics of his equations strongly

depend on the choice of zα. Nwogu suggested that, if the velocity close to the

mid-depth is used, namely zα ≈ −0.53h, the practical limit of the resulting

Boussinesq-type equations is roughly increased to kh ≈ 3.3 for 1% relative er-

ror in the wave celerity, which is a significant improvement over the traditional

Boussinesq equations (i.e., kh ≈ 1.1). Departing from Nwogu’s approach, Wei

et al. (1995) obtained the corresponding set of fully nonlinear Boussinesq-type

2

1.1 Boussinesq-type models 3

equations by including terms of Oµ2, 2, 3

.

Other researchers have recently proposed different Boussinesq-type models

that further improve the characteristics in deep waters by including highly dis-

persive terms. Perhaps the most straightforward approach is the one by Gobbi

et al. (2000), which kept the higher order terms O ((a/h)2) and O ( (kh) 4)

in their derivation of the depth-integrated wave equations. By doing so, they

increased the practical application limit to kh ≈ 6 for the linear case. The

shortcoming of the higher order Boussinesq equations is that fifth order deriva-

tives appear in the wave equations and, therefore, numerical schemes for solv-

ing these wave equations also become more complex and computationally de-

manding.

Based on the method of Agnon et al. (1999), Madsen et al. (2002) developed

a fully nonlinear model, which is accurate in very deep waters (kh ≈ 40 for

the linear dispersion). Their model requires more differential equations to be

solved, compared to other higher order models such as that by Gobbi & Kirby

(1999). The highest order of derivatives in the model is also fifth. Madsen,

et al. (2003) presented a simplified version of their original model, where the

highest order of derivative is reduced to three. However, the range of applica-

tion is also reduced to kh < 10.

Most recently, Lynett & Liu (2002, 2004) introduced a multi-layer Boussinesq-

type wave model. In their model, the water column is divided into several

layers, in which the Boussinesq approximation is employed. More specifically,

within each layer the horizontal velocity is assumed to be a quadratic poly-

nomial and the matching conditions are required along the interface of two

adjacent layers. Invoking the conservation of mass and momentum in each

layer yields a set of model equations where the highest order of derivatives

remains three. Lynett & Liu (2004) showed that the limits of application are

3

4 Introduction

kh ≈ 8, 17, 30 for 2, 3, and 4 layer model, respectively.

In summary, there already exist several Boussinesq-type wave models that can

be applied from very deep to shallow waters. However, these models contain

either more unknowns or higher order derivatives than the lower order (up to

O (kh) 2) Boussinesq-type wave models, are computationally demanding and

require artificial boundary conditions. In fact lower order equations by Wei et

al. (1995) are the most popular nowadays.

Additionally, to solve numerically the Boussinesq-type equations, implicit

schemes to step in time are usually considered [Wei & Kirby 1995]. Further,

waves are usually generated within the integration domain by means of a

“wave-source” function [Wei et al. 1999]. Albeit the schemes are implicit in

time, numerical instabilities have been reported in the literature, and the use

of numerical filters is a common practice.

1.2 Bottom shear stress

The analysis of the bottom boundary layer is crucial at least for two rea-

sons. In the one hand, the inclusion of the boundary layer in wave propa-

gation models allows to capture the wave damping due to the bed friction

[Keulegan 1949, Liu & Orfila 2004, Liu et al. 2006]. In the other, a good un-

derstanding of the boundary layer allows to properly represent the shear stress

transmitted to the bottom, which is a crucial aspect in sediment transport,

either suspended or bed load [Nielsen 1992, Fredsoe & Deigaard 1992] and,

thus, in morphological problems.

Despite the importance of the wave induced boundary layer, the knowledge of

the physical processes occurring there is far from being complete, in part due

to the difficulty to perform measurements [Kushnir 2005]. Some remarkable

4

1.3 Aims and motivation 5

experimental data for monochromatic oscillatory flows are those by Riedel

(1972), Kamphuis (1975), Jonsson & Carlsen (1976), Jensen (1989) or Mir-

fenderesk & Young (2003). However, data for non-monochromatic boundary

layers is scarce [Suntoyo et al. 2008].

Shear stress measurements under water wave fields are usually classified in

two different groups, depending on the use of direct or indirect techniques

[Sheplak et al. 2004, You & Yin 2005]. Indirect methods are supported by

the theoretical correlation between velocity field near the bottom and bed

shear stress [Jonsson 1966, Sleath 1987, Justesen 1988, Suntoyo et al. 2008].

The quality of these methods relies on the validity of the turbulence model

used to correlate the velocity and the shear stress (i.e., the law of the wall),

and on the quality of the measurements of the velocity. On the other hand,

direct methods usually employ a shear plate placed at the bottom of a wave

flume to measure the horizontal force that the fluid exerts on it. The force

measured by the shear plate consists on the addition of the integral of the wave

bottom shear stress and the wave pressure gradient at the ends of the plate,

which might provide inaccurate estimations if the pressure gradient is not well

estimated as mentioned in Rankin & Hires (2000) and in You & Yin (2007).

A full reference of modern measurement techniques, mostly for aerodynamics,

is found in Naughton & Sheplak (2002).

1.3 Aims and motivation

Against this background, the aim of this Thesis is to derive a set of low order

and one layer fully nonlinear Boussinesq-type equations for wave propagation

with an improved range of applicability and the development of an efficient

algorithm to solve the model. The scientific motivation is twofold: first, to

develop a numerical model able to propagate waves in areas where non linear

effects are negligible but dispersion are important and second, to develop a

5

6 Introduction

model which should be used for boundary layer and sediment transport studies.

Additionally, the Thesis pretends to get a deep insight on the wave generated

boundary layer by the measurement of the bottom shear stress.

1.4 Overview of the Thesis

The present Thesis is structured as follows. In Chapter 2, a new set of fully

nonlinear Boussinesq-type equations for wave propagation with an improved

range of applicability from deep water to shallow water is presented. A 4th

order Runge-Kutta explicit scheme including a source function for internal

wave generation and a sponge layer to deal with wave radiation is presented in

Chapter 3. The model is validated with analytical and laboratory benchmark

problems. An special treatment of the linear stability condition is performed

along the Chapter. In Chapter 4, a new instrument designed to measure

directly the bottom shear stress under symmetric and asymmetric waves is

presented. The new device is employed to validate an analytical turbulent

boundary layer model that computes the bed shear stress transmitted to the

bottom under different conditions of wave and wave-current. Finally, Chapter

5 concludes the work and outline the future areas of research related with the

Thesis.

6

Chapter 2

An improved Boussinesq-type

model for wave propagation

In this Chapter, we present a new set of low order Boussinesq-type fully non-

linear equations with improved dispersive properties accurate up to O(kh)2

,

with k the characteristic wave number and h the water depth. An optimiza-

tion method is used to determine the weighting coefficients employed in the

proposed equations so as to minimize the differences, in terms of linear fre-

quency dispersion and shoaling and weakly nonlinear energy transfer, between

the model equations and Airy and Stokes theories in deep waters. It is shown

that, with the proposed choice of weighting coefficients, the model is applica-

ble up to kh = 20 with 1.00% relative errors in linear frequency dispersion.

The results of this Chapter are under review in Journal of Waterway,

Port, Coastal, and Ocean Engineering.

2.1 New set of wave propagation equations

Dimensional and dimensionless expressions will be used at convenience through-

out this Chapter. The dimensional variables are denoted with primes, while

the dimensionless variables are unprimed. In our wave propagation problem

we consider k0, h0 and a0 as the characteristic values for the wave number,

7

8 An improved Boussinesq-type model for wave propagation

the water depth and wave amplitude respectively. Dimensionless variables are

defined as

k ≡k

k0, x, y ≡ k0 x

, y , z, h ≡z, h

h0, η ≡

η

a0,

with k the wave number, x and y the two horizontal coordinates, z the upward

vertical coordinate with z = 0 being the still water level, h the local water

depth and η the free surface elevation. We further define

t ≡ k0

gh0 t

, u ≡h0u

a0gh0

, c, cg ≡c, cg

gh0,

as the corresponding dimensionless time, horizontal fluid particle velocity vec-

tor, wave celerity and group velocity, respectively. We also define here 0 as

the characteristic horizontal length for the bathymetric changes, i.e., so that

O (∇h) = O (h0/0) . (2.1)

For later use

≡a0h0

, µ ≡ k0h0, σ ≡

1

k00

, (2.2)

are the dimensionless parameters characterizing, respectively, nonlinearity, dis-

persion and bed slope influence on the wave propagation. Note that, for given

k0 and a0, if h0 increases into deep water, the dispersive parameter µ increases

while the nonlinear parameter decreases. Also, for given k0 and a fixed bed

slope h0/0, as h

0 increases into deep water, 0 will increase and σ ≡ (k0

0)

−1

will decrease, and this will be interpreted below as the bed slope being negli-

gible in deep water.

Defining u as the velocity at z = αh, the equations by Wei et al. (1995) can

be written as

W951 = O (µ4) ,

W952 = O (µ4) ,

8

2.1 New set of wave propagation equations 9

where

W951 ≡ X−X∗ + µ2∇· [d1αh2∇X+ d2αh

3∇Y]+

+ µ2∇·

c1αh−

η

2

η∇X+

c2αh

2−

2η2

6

η∇Y

, (2.3a)

and

W952 ≡ Z− Z∗ + µ2 [c1αh∇∇· (hZ) + c2αh2∇∇·Z] −

− µ2∇η∇· (hZ) +

η2

2∇·Z

+

+ µ2∇(c1αh− η)u·∇X+

c2αh

2−

2η2

2

u·∇Y+

(X + ηY) 2

2

,

(2.3b)

in which Y ≡ ∇·u,

X ≡ ∇· (hu) , Z ≡ ut,

X∗ ≡ −ηt − ∇· (ηu) , Z∗ ≡ −

2∇ (u·u)− g∇η.

and

α ≡zαh, c1α ≡ α, c2α ≡

α2

2, d1α ≡ α+

1

2, d2α ≡

α2

2−

1

6, (2.4)

are dimensionless coefficients determined by the choice of α ≡ zα/h. The

subscript “t” above indicates that this variable appears derived respect to the

time. For future use we also define here

cα ≡ c1α + c2α, and dα ≡ d1α + d2α = cα + 1/3 . (2.5)

In the definition of Z∗ above, g = 1. This variable g is introduced so that

dimensionless equations can be easily converted into dimensional equations;

the dimensional equations are recovered from the above equations by setting

= µ = 1 and g = 9.81m/s2 and considering the rest of unprimed variables

9


as the dimensional ones.

The range of applicability of equations (2.3) is shown in Figure 2.1(c). The

only differences respect the equations presented by Nwogu (1993), as shown

in Figure 2.1, is the inclusion of terms of Oµ2, 2, 3

.

Besides, the enhancement procedure by Madsen & Schaffer (1998) leads to the

following expressions

M981 = O (µ4) ,

M982 = O (µ4) ,

where

M981 ≡ W951 + µ2 [ (δ − δσ)∇· (h2∇ (X−X∗) ) + δσ∇2 (h2 (X−X∗) ) ] ,

(2.6a)

and

M982 ≡ W952 + µ2 [ (γ − γσ)h2∇∇· (Z− Z∗) + γσh∇∇· (h (Z− Z∗) ) ] .

(2.6b)

Taking into account that, O (X−X∗) = O (Z− Z∗) = O (µ2) according to

expressions (2.3), the equations (2.6) add terms of O (µ4) to (2.3), and there-

fore remain the same order of accuracy. However, they show an improvement

of the behaviour of celerity if the coefficients δ, δσ, γ and γσ are carefully

chosen (Figure 2.1). In fact, in Madsen & Schaffer (1998) the coefficients are

chosen to mimic [4/4] Pade approximant for the dispersion expression. For

convenience, we have renamed the original variables α1,α2,β1,β2 in Madsen

& Schaffer (1998) as

β1 ↔ −δ, β2 ↔ −δσ,

α1 ↔ −γ α2 ↔ −γσ,

10


and we also note that α in this work by Madsen & Schaffer corresponds to our

cα.

The expression for M981 above introduces two new terms with respect to

W951, which are

µ2 (δ − δσ)∇· (h2∇ (X−X∗) ) , and µ2δσ∇2 (h2 (X−X∗) ) ,

that in the flat bed case reduce to µ2δ∇· (h2∇ (X−X∗) ) . Thus, δ is intro-

duced to improve the linear frequency dispersion and δσ to improve the linear

shoaling. We note also that the new terms have the same structure of the

linear nondispersive term in W951

µ2∇· [d1αh2∇X] ,

but changing the constant and replacing X with X−X∗, which is O (µ2) . The

same arguments hold for M982.

We will here extend the above arguments to introduce new nonlinear terms.

Because we are to compare the new results with the weakly nonlinear theory

over flat beds, we will only introduce one more term in each equation, and not

two as for the linear case. Further, the terms to be added to M981 and M982

have the same structure of terms of O (µ2) in Wei et al. (1995), i.e.

µ2∇· [hη∇X] , and − µ2∇ [η∇· (hZ)] ,

but, again, we will substitute X and Z by X−X∗ and Z−Z∗ respectively. In

summary, the equations proposed here are

M981 + µ2δ∇· [hη∇ (X−X∗) ] = O (µ4) , (2.7a)

M982 − µ2γ∇ [η∇· (h (Z− Z∗) )] = O (µ4) , (2.7b)

where the coefficients δ and γ will allow us to improve the nonlinear perfor-

mance.

11


Wei et al. (1995) Madsen & Schaffer (1998)

α −0.53096 −0.54122

δ 0 −0.03917

γ 0 −0.01052

δσ 0 −0.14453

γσ 0 −0.02153

δ 0 0

γ 0 0

Table 2.1: Coefficients for Wei et al. (1995) and Madsen & Schaffer (1998).

Obviously, the expressions (2.3) and (2.6) are particular cases of equations

(2.7). Table 2.1 shows the coefficients corresponding to those expressions.

We are concerned in finding the free coefficients α, δ, γ, δσ, γσ, δ and γ to

improve the performance of the equations (2.7) within the range described in

Figure 2.1(d). The equations are derived using an asymptotic expansion in

the dispersive parameter µ and, regardless of the choice of the above coeffi-

cients, their behaviour converges to the exact solution in shallow waters for

arbitrarily high nonlinear and slope effects. We will find the parameters so

as to improve, in deep water, its performance regarding linear dispersion over

flat bed, linear shoaling over mild slopes and weakly nonlinear behaviour over

flat beds.

Hereinafter the truncation error O (µ4) will not be written out, and µ is con-

sidered to be arbitrarily large.

12


P67(a)

N93(b)

M98

W95(c) (d)

fully dispersive

arbitrary slope

fully nonlinear

Figure 2.1: Schematic picture for the range of application of different models:

(a) [Peregrine 1967], denoted by P67, is valid for weakly nonlinear and weakly dis-

persive waves over arbitrary slopes; (b) [Nwogu 1993], denoted by N93, improves

the linear frequency dispersion performance over flat beds into deeper waters; (c)

[Wei et al. 1995], W95 here, extends N93 to fully nonlinear waves for the weakly

dispersive case, and [Madsen & Schaffer 1998], denoted as M98, increases the appli-

cability to deeper water; (d) new model equations.

13


2.2 Improving the linear propagation

In this Section we will focus on the improvement of the linear dispersion be-

haviour to very deep waters, namely kh 20. The linear (and nonlinear)

features of the new equations, which are to be used in this Section, are fully

described in Appendix A. Hereafter, the results using Airy’s theory are de-

noted with “Airy”, while those from the Boussinesq-type equations (2.7) with

“app” (from approximation).

Linear frequency dispersion: α, δ and γ

From expression (A.10) in Appendix A, the celerity corresponding to our lin-

earised Boussinesq-type equations is given by

c2appgh

=1− (mcη +mdu) ξ2 +mdumcηξ4

1− (mdη +mcu) ξ2 +mcumdηξ4, (2.8)

where ξ ≡ kh = µkh and the coefficients mdu, mdη, mcu and mcη, which

are defined in expression (A.2), depend exclusively on α, δ and γ. Comparing

the above expression to the [4/4] Pade approximation of the Airy expression,

which is [Dingemans 1997]

c2Airy

h=

1 + 19 ξ

2 + 1945 ξ

4

1 + 49 ξ

2 + 163 ξ

4+O (ξ10) , (2.9)

one finds out the corresponding values for mdu, mdη, mcu and mcη. As already

well known [Madsen & Schaffer 1998], the solutions are the four possible com-

binations of

mdu = −1

18±

√805

630, mdη = −mcu −

4

9, (2.10a)

mcu = −4

18±

10√133

630, mcη = −mdu −

1

9. (2.10b)

i.e., the results for α, δ and γ in Table 2.2.

14

2.2 Improving the linear propagation 15

I II III IV

α −0.54122 −0.37500 −0.02907 0.05965

δ −0.03917 −0.03917 −0.40528 −0.40528

γ −0.01052 −0.10059 −0.01052 −0.10059

Table 2.2: Coefficients corresponding to [4/4] Pade approximant.

The four solutions correspond to a basic solution, namely solution I, which is

mdu = −0.10059, mdη = −0.03917, mcu = −0.40528, mcη = −0.01052,

and the combinations obtained by switching mdu ↔ mcη and mcu ↔ mdη

which, according to equation (2.8), will give the same expression.

Madsen & Schaffer (1998) noted that the solutions II and III are bad condi-

tioned to nonlinear behaviour. Further, solutions III and IV largely differ from

the coefficients in Wei et al. (1995) (α = −0.531, δ = γ = 0) while solutions

II and, particularly, I, are close to them. We remark that Madsen & Schaffer

(1998) considered the solution I.

Here, we try to further reduce the error in linear frequency dispersion, relative

to the Airy theory, by finding other values for α, δ and γ. We note that the

comparisons should avoid the use of the variable ξ ≡ µkh to measure the

water depth because k itself, and consequently ξ, depends on whether we use

the approximate expression (2.8) or the exact one (Airy’s expression (A.11)

in Appendix A). Instead, we favour the use of a k-independent dimensionless

variable, κ, defined as [Nwogu 1993]

κ ≡µ2ω2h

g

=

c2

ghξ2, (2.11)

which, in the case of Airy’s theory, is κ = ξ tanh ξ, so that κ ≈ ξ for κ 3,

i.e., in deep waters.

15


κmax 5 10 20

α −0.54663 −0.54587 −0.53523

δ −0.03270 −0.02411 −0.01502

γ −0.00732 −0.00439 −0.00211

δσ −0.15138 0.10169 0.08463

γσ −0.07523 0.01919 0.01907

δ −0.26154 −0.34459 −0.47055

γ 0.11840 0.13106 0.14687

εc 0.0080% 0.17% 1.0%

εs 1.2% 12% 19%

ε (κmax = 2) 15% 14% 13%

smax 2.9 4.6 4.6

Table 2.3: Sets A. Simple optimization.

We now will minimize, for a given range of κ, 0 ≤ κ ≤ κmax, the error in

celerity, εc, defined as

εc ≡ max0κκmax

cappcAiry

− 1

. (2.12)

Table 2.3 provides three sets of coefficients (Sets A), α, δ and γ, by minimizing

the error εc for different values of κmax. The errors εc are also included in the

lower part of the Table. Analogously to the [4/4] Pade coefficients, for each

κmax there are three other possible solutions yielding the same frequency dis-

persion. Following similar arguments as those by Madsen & Schaffer (1998),

and also for stability considerations, we consider only the sets in Table 2.3,

which are small perturbations from Pade’s solution I or those in Wei et al.

(1995).

Figure 2.2 shows capp/cAiry − 1 as a function of κ for the three κmax in set

A, together with the results for Wei et al. (1995) and Madsen & Schaffer

16


(1998). The new sets allow a substantial reduction of the errors. We note

that the errors for Madsen and Shaffer’s equations (2.6) at κ = 5, κ = 10 and

κ = 20 are, respectively, 0.73%, 8.7% and 33% (for Wei et al.’s equations the

errors are 9.2%, 36% and 81%). Only for very small values of κ, [4/4] Pade

approximant [Madsen & Schaffer 1998] beats the three proposed sets, just the

same way [2/2] Pade is better, for very small values of κ, than Nwogu’s choice

(Figure 2.3). However, the new coefficients allow to reduce the error to just

0.008% for κ 5. At this point, we remark that if the maximum expected

value of κ in a particular problem is, e.g., κ ≈ 3, one should consider the

coefficients for κmax = 5 , and not those for κmax = 20 (which should be used

only when very high values of κ are expected).

Linear wave shoaling: δσ and γσ

To optimize the shoaling behaviour we consider the minimization of the error

εs, given by

εs ≡ max0κκmax

Aapp

η

AAiryη

− 1

, (2.13)

where Aη is the propagated wave amplitude. According to the results in Ap-

pendix A, depends on α, δ, γ, already determined by optimizing the linear

frequency dispersion, but also on δσ and γσ.

Table 2.3 shows the values of δσ and γσ minimizing the linear shoaling error

in equation (2.13) for different values of κmax, assuming in each case the cor-

responding values of α, γ and δ. We remark here that the linear frequency

dispersion behaviour depends only on α, δ and γ, and is not affected by δσ

and γσ.

Figure 2.4 shows Aappη /AAiry

η − 1 for all three sets in Table 2.3, together with

the results from expressions (2.3) and (2.6). Specially remarkable is the fact

that, for values of κ > 5, both Wei et al. (1995) and Madsen & Schaffer (1998)

17


0 5 10 15 20!0.1

0

0.1

0.2

!

c app/cAiry!1 W95 M98

0 1 2 3 4 5!0.01

0

0.01

!

c app/c

Airy!1

W95

M98

zoom in

Figure 2.2: Linear dispersion: capp/cAiry − 1 for sets A corresponding to κmax = 5

(); κmax = 10 (); κmax = 20 () in Table 2.3. W95 refers to Wei et al. (1995) and

M98 to Madsen & Schaffer (1998) equations.

18


0 5 10!0.05

0

0.05

0.1

0.15

+0.01

!0.01

kh

c/c Airy!1

P67

[2/2]N93

[4/4]

Figure 2.3: Frequency dispersion: relative errors for the phase speed using Peregrine

(1967) (denoted as P67), [2/2] Pade, Nwogu (1993) (denoted as N93) and [4/4] Pade.

0 5 10 15 20

!0.2

!0.1

0

0.1

0.2

!

A"app /A"Airy!1

W95

M98

Figure 2.4: Linear shoaling: Aappη /AAiry

η − 1 for the coefficient sets A corresponding

to κmax = 5 (); κmax = 10 (); κmax = 20 () in Table 2.3. W95 refers to Wei et

al. (1995) and M98 to Madsen & Schaffer (1998) equations.

19


κmax 5 5(∗) 10 20

α −0.55590 −0.56209 −0.58520 −0.58347

δ −0.01071 −0.02726 −0.03687 −0.01945

γ −0.00260 −0.00407 −0.00542 −0.00166

δσ −0.07368 −0.03380 0.07650 0.02705

γσ −0.04447 −0.03124 0.01235 0.00349

δ −0.51184 −0.29739 −0.15052 −0.33143

γ 0.14924 0.13304 0.13304 0.13063

εc 0.29% 1.3% 2.4% 7.5%

εs 0.29% 1.3% 2.4% 7.5%

ε (κmax = 2) 13% 15% 17% 15%

smax 3.5 5.0 10 25

Table 2.4: Sets B: joint optimization. (∗) correspond to joint optimization ensuring

stability for s = 5.

yield to errors in linear shoaling tend to 99% asymptotically, while, with the

new equations, they are considerably reduced.

Because the errors εc and εs are unbalanced in Table 2.3, i.e., εs εc, we

secondly consider the joint search of α, δ, γ, δσ and γσ that minimize

max εc (κmax) , εs (κmax) ,

The results, Sets B, shown in Table 2.4, are again small perturbations around

Wei et al. (1995). Compared to those in Table 2.3, the new set sacrifices

dispersion performance in order to improve the shoaling.

20

2.3 Linear stability considerations 21

2.3 Linear stability considerations

So far, different values for coefficients α, δ, γ, δσ and δσ have been found, and

those for δ and γ, which appear only in the nonlinear terms, are still required.

Before proceeding with the nonlinear optimization, it is worth to mention that

not all the combinations of α, δ, γ, δσ and δσ give useful expressions. Some of

them, as we will show below for the linearised problem, can have instabilities.

We remark that in the linear case, the new proposed equations (2.7) coincide

with those in Madsen & Schaffer (1998).

The linear 1D versions of equations (2.7) read

(hu) x + ηt + µ2 [d1αh2 (hu) xx + d2αh

3uxx] x+

+ µ2 [ (δ − δσ) (h2 ( (hu) x + ηt) x) x + δσ (h

2 ( (hu) x + ηt) ) xx] = 0, (2.14a)

and

ut + gηx + µ2 [c1αh (hut) xx + c2αh2uxxt] +

+ µ2 [ (γ − γσ)h2 (ut + gηx) xx + γσh (h (ut + gηx) ) xx] = 0, (2.14b)

where subscript “x” indicate the spatial derivative of variables.

2.3.1 Flat bed case

In the flat bed case, the above equations reduce to

ηt + hux + µ2h2 (mduhuxxx +mdηηxxt) = 0, (2.15a)

ut + gηx + µ2h2 (mcuuxxt + gmcηηxxx) = 0. (2.15b)

Now, let us consider a spatially periodic problem, where 2π/k is the principal

length, so that η =

ηn exp (inkx) and u =

un exp (inkx) . The equations

(2.15) read

ηn,t

un,t

= A·

ηn

un

, A ≡ −i

0 a12

a21 0

, (2.16)

21


where, being ξn ≡ µknh,

a12 ≡ knh1−mduξ2n1−mdηξ2n

, a21 ≡ gkn1−mcηξ2n1−mcuξ2n

.

The solutions for ηn and un can be straightforwardly obtained from equation

(2.16) by diagonalizing the matrix A, being the behaviour of the solution

governed by the eigenvalues of A, which are

νn,± = ±√−a12a21 = ±ikn

gh(1−mduξ2n) (1−mcηξ2n)

(1−mcuξ2n) (1−mdηξ2n). (2.17)

The argument in the square root is, recalling the expression (2.8), the squared

celerity c2app for the wave number kn, and it must be a positive value for

any n. Otherwise, there would be positive real eigenvalues, indicating the

amplification of some harmonics. Therefore, we must require

(1−mduξ2n) (1−mcηξ2n)

(1−mcuξ2n) (1−mdηξ2n)> 0, for any ξn. (2.18)

It can be easily shown that

max mdu,mcu,mcη,mdη 0, (2.19)

is a necessary and sufficient condition for the numerator and denominator in

equation (2.18) to be positive, therefore ensuring the stability for the flat bed

case. Recalling the definitions of mdu, mdη, mcu and mcη in equation (A.2),

Appendix A, we remark that all the sets A obtained above (see Table 2.3)

verify the stability condition for flat beds. In fact, they verify

max mdu,mcu,mcη,mdη < 0,

so that

limn→∞

νn,± = ±ikngh

mdumcη

mcumdη= O (n

√h) .

For future use, we note that the characteristic value for νn,± is, according to

expression (2.17)

ν 0 = k0

gh0,

22


and we define

Ωn (h) = kngh

mdumcη

mcumdη, (2.20)

which in turn is the frequency associate to the nth component for a given depth

for big values of n.

In the 2D case the linearised equations (2.7) reduce, for the flat bed case, to

ηt + h∇ · u+ µ2h2 mduh∇2∇ · u+mdη∇

2ηt = O (µ4) , (2.21a)

ut + g∇η + µ2h2 mcu∇∇ · ut + gmcη∇∇2η = O (µ4) , (2.21b)

so that, proceeding as in the 1D case, now with

η =

ηnm exp (inkxx) exp (imkyy) ,

u =

unm exp (inkxx) exp (imkyy) ,

the equations read, u and v being the x and y components of the velocity,

b11 0 0

0 b22 b23

0 b32 b33

ηnm,t

unm,t

vnm,t

= −i

0 a12 a13

a21 0 0

a31 0 0

·

ηnm

unm

vnm

, (2.22)

where ξn ≡ µkxnh, ξm ≡ µkymh, ξ2nm ≡ ξ2n + ξ2m, b11 ≡ 1−mdηξ2nm,

b22 ≡ 1−mcuξ2n, b33 ≡ 1−mcuξ

2m, b23 = b32 ≡ −mcuξnξm,

and

a12 = hnkx (1−mduξ2nm) , a13 = hmky (1−mduξ

2nm) ,

a21 ≡ gnkx (1−mcηξ2nm) , a31 ≡ gmky (1−mcηξ

2nm) .

The three eigenvalues of the system (2.22) are

νnm,± = ±ik2xn

2 + k2ym2

gh(1−mduξ2nm) (1−mcηξ2nm)

(1−mcuξ2nm) (1−mdηξ2nm), (2.23)

and zero. Therefore, the stability condition remains the same.

23


2.3.2 Uneven bed case

In general, no analytical solution can be found for arbitrary bathymetries, even

in the simplest 1D linearised problem. In order to show the above anticipated

stability problems, let us consider the simple sinusoidal bathymetry

h = hc + h1 exp (+ikx) + h1 exp (−ikx) = hc + 2h1 cos (kx) . (2.24)

Assuming periodic solutions in space

η =

ηn exp (inkx) and u =

un exp (inkx) ,

equations (2.14) read

2

j=−2

aη,jηn+j,t =3

j=−3

bu,jun+j , (2.25a)

2

j=−2

au,jun+j,t =2

j=−2

bη,jηn+j , (2.25b)

for −∞ < n < +∞. The coefficients aη,j , bu,j , au,j and bη,j are in Appendix

B. The above equations provide the information of the interaction between

harmonics due to the uneven bathymetry.

Denoting here η and u as the column vectors including all Fourier components

ηn and un for −∞ < n < ∞, the above equations (2.25) can be written as

L11 0

0 L22

·

ηt

ut

=

0 R12

R21 0

·

η

u

, (2.26)

where L11, L22, R21 and R12 are banded matrices with the bathymetric in-

formation. For the flat bed case the matrices become diagonal, recovering the

results presented above for flat bed case.

The system in equation (2.26) can be rewritten as

ηt

ut

= A·

η

u

.

24


To study the stability of the problem, the eigenvalues νj of A have to be

analyzed. The above matrices L11, L22, R12 and R21 are infinite and will be

truncated to finite squared matrices for the analysis. Let m be the maximum

harmonic of the Fourier components considered in the column vectors, so that

the size of these matrices is (2m+ 1) × (2m+ 1) . The Figure 2.5 displays

the eigenvalues of A for the coefficients corresponding to κmax = 10 in Table

2.3 with

k =2π

250m, hc = 500m, h1 = 225m,

i.e., a sinusoidal bathymetry ranging between 50m and 950m with a wave-

length = 250m. According to the expression (2.24), the maximum slope is

2kh1 ≈ 11.3, a huge value.

Two different values for the truncation m are considered in Figure 2.5: m = 50

and m = 100. As depicted from the Figure, the higher the number of harmon-

ics considered, the higher frequencies are achieved. This is an expectable result

since higher wavenumbers have higher corresponding frequencies, as shown in

equation (2.20). In fact, as shown in Figure 2.5, the maximum values of the

imaginary parts of the eigenvalues of A are controlled by Ωm (h) defined in

equation (2.20), for the given coefficients and the maximum depth, in this

case h = 950m. For the lower frequencies, which will be shown to be the

most important to analyze stability, the eigenvalues for m = 50 and m = 100

coincide, showing that the truncation procedure does not destroy the analysis.

Besides, from Figure 2.5, for these coefficients and bathymetry, the eigenvalues

νj have real positive parts for low frequencies, thus indicating instability of the

equations. For high frequencies (wavenumbers) the wave does not feel bathy-

metric changes and, provided that condition (2.19) is satisfied, the eigenvalues

are all pure imaginaries.

The existence of positive real parts is to be avoided, since they destroy the

25


!1 !0.5 0 0.5 1!50

0

50

Real!" (Hz)

Imag

!"

(Hz)

#"100(h"=950m)

#"50(h"=950m)

m = 50m = 100

!1 !0.5 0 0.5 1

!2

!1

0

1

2

Real!" (Hz)

Imag

!"

(Hz)

zoom inm = 50m = 100

Figure 2.5: Eigenvalues for the truncated matrices.

26


solution. Amongst the different sets of coefficients presented so far (Wei et al.

(1995), Madsen & Schaffer (1998) and set A for the new equations in Table

2.3), only coefficients for Wei et al. has shown to give pure imaginary eigenval-

ues even for the most steep sinusoidal bathymetries. For the rest of sets, the

trend is that the equations are stable for “mild” slopes and become unstable

for “steep” ones.

Thinking specifically in a dimensional way, given a set of coefficients, the max-

imum real part of the eigenvalues of A depends on g, k, hc and h1, provided

the influence of m is null for m sufficiently large. Applying dimensional anal-

ysis

ν∗ ≡maxj ( ν j )

k

ghc= f (Π, s) , Π ≡

h1hc

, s ≡ 2kh1, (2.27)

where Π, which satisfies 0 Π < 1/2 according to equation (2.24), represents

the ratio of the maximum to minimum depth and s is the maximum slope of

the bathymetry.

The above function has been evaluated numerically, and its behaviour is shown

in Figure 2.6 for the coefficients corresponding to κmax = 5 in the Table 2.3.

For the given set of coefficients, ν∗ = 0 for small slopes, while ν∗ > 0 if a given

value of s, herein smax, is surpassed. The value of smax will depend on Π, but

the influence of Π has shown to be rather small, having most unfavourable

values of smax for Π in its middle values (Figure 2.6). The trend has shown

to be similar for the rest of coefficient sets, except for Wei et al.’s, for which

ν∗ = 0 for any pair Π, s , i.e., smax → ∞. For the computation of the

maximum admissible slope, smax, we always used Π = 0.25 and m = 50, and

the variable ν∗ is considered positive when ν∗ > 10−8. The values of smax for

sets A are included in Table 2.3. From Table 2.3, smax decreases for sets with

smaller κmax to smax ≈ 2.9, which might be an insufficient value. Despite, for

Madsen & Schaffer (1998), smax ≈ 2.5 ≤ 2.9.

27


0 2 4 6 80

1

2

3

4

5

6

s

! *

0.050.100.200.300.400.450.49

Figure 2.6: Function ν∗ = f (Π, s) . The different symbols stand for different values

of Π.

In order to ensure linear stability for steeper slopes, i.e., to increase smax, we

consider here two different options. We first propose to keep the coefficients

α, δ and γ as in Table 2.3 and recompute δσ and γσ to optimize the linear

shoaling performance but so that smax 5, since s = 5 is a sufficiently steep

slope to be considered (Sets C). The results are shown in Table 2.5.

From Tables 2.3 and 2.5, it can be seen that we are now sacrificing shoaling

performance to obtain stability. However, the errors in shoaling are still re-

duced relative to works by Wei et al. (1995) and Madsen & Schaffer (1998).

Further, the new values for δσ and γσ are closer to zero, and the overall co-

efficients are very close to those in Wei et al. (1995), which has shown to be

very stable.

Following the same argument given above about the unbalanced errors in

shoaling and linear dispersion we consider the joint search of α, δ, γ, δσ and

28

2.4 Improving the weakly nonlinear propagation 29

κmax 5 10 20

α −0.54663 −0.54587 −0.53523

δ −0.03270 −0.02411 −0.01502

γ −0.00732 −0.00439 −0.00211

δσ −0.03334 0.09314 0.07096

γσ −0.03754 0.01630 0.01419

δ −0.26154 −0.34459 −0.47055

γ 0.11840 0.13106 0.14687

εc 0.0080% 0.17% 1.0%

εs 2.6% 12% 20%

ε (κmax = 2) 15% 14% 13%

Table 2.5: Sets C. Linearly stable for smax 5.

γσ ensuring stability for s = 5. The results are shown in Table 2.4 for κmax = 5,

the only one which yields to values of smax < 5 with joint optimization proce-

dure is not used.

2.4 Improving the weakly nonlinear propagation

Taking into account the weakly nonlinear properties of the equations, fully

described in Appendix A, to obtain the coefficients δ and γ we will minimize

the error

ε = max0κi,κjκ

max

Gapp

+

GStokes+

− 1

,Gapp

−GStokes

−− 1

, (2.28)

where G± (κi,κj) are the super- and sub-harmonic nonlinear energy trans-

fer functions for two waves with κi and κj for the Boussinesq-type equations

(“app”) and for the Stokes theory.

The value of κmax to compute ε must be that value of κ where nonlinear effects

29


are indeed negligible. This will depend, of course, on the wave amplitude. If a

is the deep water wave amplitude, T its period and we consider that = 0.02

is the threshold for the nonlinearity to be negligible, recalling the definition of

κ in equation (2.11)

κmax =(2π/T) 2 (a/0.02)

g=

200π2a

gT2,

which yields, for instance, for a = 0.5m and T = 10 s, a value κmax ≈ 1.0.

We will here consider κmax = 2 to obtain the coefficients, but other values of

κmax could be considered in particular problems. Tables 2.3 to 2.5 include, for

the given values of α, δ, γ, δσ and γσ, the corresponding values for δ and γ

and the errors ε (κmax = 2) . We note that the corresponding errors for Wei

et al. (1995) and Madsen & Schaffer(1998) equations are, respectively, 59%

and 53%. The relative errors Gapp± /GStokes

± − 1 for the A set with κmax = 5 are

plot in Figure 2.7.

No stability analysis for the nonlinear equations has been performed, but the

obtained coefficients have not shown any numerical stability problems.

2.5 Concluding remarks

A new set of fully nonlinear Boussinesq-type equations for the wave propa-

gation problem with an improved range of applicability from deep water to

shallow water has been presented in this Chapter. The model equations are

modifications and improvements of those by Madsen & Schaffer (1998). An

optimization procedure is used to minimize the errors compared to those for

Airy and Stokes waves, and a linear stability analysis of the resulting equa-

tions is presented for uneven sinusoidal bathymetries. The relative errors for

each scenarios are also provided. For example, if it is desirable to model very

short waves (deep water) with kh = 20, the model has a maximum relative

error equal to 1% in linear dispersion.

30

2.5 Concluding remarks 31

!0.1

!0.1

!0.05

!0.05

!0.02

!0.02

!0.020.02

0.02

0.02

0.02

0.05

0.05

0.1

0.1

G+

G!

!i

! j

0 1 20

1

2

Figure 2.7: Weakly nonlinear propagation: Gapp± /GStokes

± −1 for set A and κmax = 5.

31


32

Chapter 3

An explicit numerical scheme

for the new model

In this Chapter an explicit numerical scheme is implemented to solve the model

equations (2.7) presented in Chapter 2. A 4th order Runge-Kutta scheme is

employed to step in time, while the spatial derivatives are differenced to an

accuracy of O∆x4

. An internal source function is derived for internal wave

generation and a CFL-type condition is obtained for the time step. The new

numerical scheme and model equations are validated using some experimental

and analytical tests. Along this Chapter all variables are dimensional and

primes are neglecting for convenience.

The results of this Chapter are under review in Journal of Waterway,

Port, Coastal, and Ocean Engineering.

3.1 Source function for wave generation and sponge

layers

Two practical problems need to be resolved before a new numerical scheme is

developed. We need to be able to generate waves within the computational

domain avoiding problems derived from the treatment of waves as boundary

conditions. Secondly, we need to radiate the waves generated locally out of

33

34 An explicit numerical scheme for the new model

the computational domain if not reflective boundary conditions are required.

Other phenomenon such as wave breaking, runup or wave dissipation due to

boundary layer are not considered in the model equations. Nevertheless, they

can be included following, e.g.., the same treatment described by Kennedy et

al. (2000) and Liu & Orfila (2004).

3.1.1 Source function

It is well known that the generation of waves into the computational domain

through the boundary conditions is difficult in wave propagation models. Fol-

lowing Wei et al. (1999) we consider the inclusion of a source function, s, in

the continuity equation so as to generate the waves within the computational

domain. The function is built to generate the desired linear waves in the far

field. We consider the linearized version of the equations (2.7) over a flat bed

[Wei et al. 1999]

ηt + h∇·u+ h2 mduh∇2∇·u+mdη∇

2ηt = s, (3.1a)

ut + g∇η + h2 mcu∇∇·ut + gmcη∇∇2η = 0, (3.1b)

where

mdu ≡ d1α + d2α + δ, mdη ≡ δ, (3.2a)

mcu ≡ c1α + c2α + γ, mcη ≡ γ. (3.2b)

For flat bed there exists a potential function φ such that ∇φ = u. There-

fore, taking spatial integration and temporal differentiation of the momentum

equation(3.1b), the above equations can be written for the scalar variables η

and φ as

ηt + h∇2φ+ h2 mduh∇2∇

2φ+mdη∇2ηt = s, (3.3a)

φtt + gηt + h2 mcu∇2φtt + gmcη∇

2ηt = 0. (3.3b)

34

3.1 Source function for wave generation and sponge layers 35

The solution of the homogeneous version of equations (3.3) is

η = η0 exp (i (kxx+ kyy − ωt) ) , (3.4a)

φ = φ0 exp (i (kxx+ kyy − ωt) ) , (3.4b)

where i ≡√−1 is the imaginary constant and k2 ≡ k2x + k2y must satisfy the

frequency dispersion relation derived in Chapter 2 given by

ω2

gk2h=

(1−mduξ2) (1−mcηξ2)

(1−mcuξ2) (1−mdηξ2), (3.5)

where ξ = kh. Besides, η0 and φ0 in equation (3.4) must also satisfy

η0 = iΠφ0, (3.6)

with

Π ≡k2h (1−mduξ2)

ω (1−mdηξ2)

≡

ω (1−mcuξ2)

g (1−mcηξ2)

. (3.7)

The dispersion equation (3.5) can be written as a cubic equation for k2

c3 (k2)

3+ c2 (k

2)2+ c1 (k

2)1+ c0 (k

2)0= 0, (3.8)

with

c3 ≡ −gmdumcηh5,

c2 ≡ ω2mdηmcuh4 + g (mdu +mcη)h

3,

c1 ≡ −ω2 (mcu +mdη)h2− gh,

c0 ≡ ω2.

The equations by Wei et al. (1995) have mdu = mcη = 0 so that the equation

(3.8) reduces to a second order equation for k2, which is the case analyzed in

Wei et al. (1999). Here we will focus on the extended cubic case. However,

for completeness, the solution for mdu = mcη = 0 is included at the end of

this Section.

35


The equation (3.8) has, for any of the combinations of mdu,mdη,mcu,mcη

obtained in the previous Chapter, three real roots for k2: one positive (de-

noted here r21, with r1 > 0) corresponding to the progressive waves, and two

negatives (i.e., −r22 and −r23, with r2, r3 > 0) corresponding to evanescent

modes. Therefore, the six solutions for the wave number k are

k1 ≡ +r1, k2 ≡ +ir2, k3 ≡ +ir3, (3.9)

and their opposite (−r1, −ir2 and −ir3). Following Wei et al. (1999) we

assume that, being ψ any of η, φ or s, for the inhomogeneous case we can

write

ψ (x, y, t) = ψ (x) exp (i (kyy − ωt) ) ,

with |ky| < k1, so that equations (3.3) read

Aφ,1φ + Bφ,1φ

+Cφ,1φ+ Bη,1η +Cη,1η = s, (3.10a)

+Bφ,2φ +Cφ,2φ+ Bη,2η

+Cη,2η = 0, (3.10b)

where the primes stand for x derivatives, and

Aφ,1 ≡ h3mdu,

Bφ,1 ≡ h1− 2mduk

2yh

2, Bφ,2 ≡ −ω2mcuh

2,

Cφ,1 ≡ −hk2y1−mduk

2yh

2, Cφ,2 ≡ −ω2

1−mcuk

2yh

2,

Bη,1 ≡ −iωmdηh2, Bη,2 ≡ −iωgmcηh

2,

Cη,1 ≡ −iω1−mdηk

2yh

2, Cη,2 ≡ −iωg

1−mcηk

2yh

2.

To find a particular solution for the problem in equations (3.10) we first con-

sider the Green functions Gη (x, ) and Gφ (x, ) , solutions for a pulse at

x = , i.e., solutions of

Aφ,1Gφ + Bφ,1G

φ +Cφ,1Gφ + Bη,1G

η +Cη,1Gη = δ (x− ) , (3.11a)

Bφ,2Gφ +Cφ,2Gφ + Bη,2G

η +Cη,2Gη = 0. (3.11b)

36


In order to automatically satisfy the above equations at every point except at

x = , the solutions Gη and Gφ of the above problem can be written

Gη =

3j=1 aη,j exp (ikx,j (x− ) ) , if x > ;

3j=1 aη,j exp (ikx,j (− x) ) , if x < ,

(3.12a)

and

Gφ =

3j=1 aφ,j exp (ikx,j (x− ) ) , if x > ;

3j=1 aφ,j exp (ikx,j (− x) ) , if x < ,

(3.12b)

where kx,j must be so that k2x,j = k2j − k2y, with kj given in equation (3.9).

More specifically

kx,1 = +r21 − k2y, kx,2 = +i

r22 + k2y, kx,3 = +i

r23 + k2y,

so that the evanescent modes cancel at x → ±∞, where, according to equations

(3.12), we get waves travelling rightwards at x → ∞ and leftwards at x → −∞.

Further, in a result consistent with equation (3.6), to satisfy the homogeneous

equation (3.11), the coefficients aη,j and aφ,j in equations (3.12) must satisfy

aη,j = iΠjaφ,j , (3.13)

where

Πj ≡k2jh (1−mduk2jh

2)

ω (1−mdηk2jh2)

. (3.14)

The above Green functions Gη and Gφ satisfy the equation (3.11) at every

point except x = , i.e., satisfy the homogeneous equations. We still require

to take into account the point x = . Integrating equations (3.11) in x from

− to +

Aφ,1Gφ

+−

+ Bφ,1Gφ

+−

+Cφ,1

+

−Gφdx+ Bη,1G

η

+−

+Cη,1

+

−Gηdx = 1,

+Bφ,2Gφ

+−

+Cφ,2

+

−Gφdx+ Bη,2G

η

+−

+Cη,2

+

−Gηdx = 0.

37


Equations (3.12) satisfy +

− Gφ dx = +

− Gη dx = 0, so that we further impose

Gη

+−

=Gφ

+−

= 0, and Aφ,1Gφ

+−

= 1, (3.16)

i.e., recalling expressions (3.12),

kx,1aη,1 + kx,2aη,2 + kx,3aη,3 = 0,

kx,1aφ,1 + kx,2aφ,2 + kx,3aφ,3 = 0,

k3x,1aφ,1 + k3x,2aφ,2 + k3x,3aφ,3 = i/ (2Aφ,1) ,

which, taking into account the expression (3.13), allows to solve for aφ,1, aφ,2

and aφ,3. We anticipate that we will be only interested in aφ,1, which is

aφ,1 =Π2 −Π3

k2x,1 (Π2 −Π3) + k2x,2 (Π3 −Π1) + k2x,3 (Π1 −Π2)

i

2kx,1Aφ,1.

Once Gφ in equation (3.12b) has been determined, the solution of φ in equation

(3.10) is given by the convolution integral

φ (x) =

+∞

−∞Gφ (x, ) s () d, (3.17)

where, following Wei et al. (1999), we consider

s () = D exp (−β2) , (3.18)

where D is still to be determined. The value of the parameter β can be

chosen following the recommendations in Wei et al. (1999), i.e., β = 15 k21 for

monochromatic wave generation. Substituting the equations (3.12) and (3.18)

into (3.17) we get that, for x → +∞, in the far field, the terms corresponding

to aφ,2 and aφ,3, i.e., the evanescent modes, drop off and

limx→+∞

φ (x) = DIaφ,1 exp (ikx,1x) , (3.19)

where

I ≡

π

βexp

−k2x,14β

.

38


Since we want φ to behave as φ0 exp (ikx,1x) at x → +∞, from the expression

(3.19) we get

D =φ0

Iaφ,1=

η0iΠ1Iaφ,1

, (3.20)

and the source function can be written simply as

s (x, y, t) = |D| exp (−βx2) sin (kyy − ωt) . (3.21)

When mcη = mdη = 0, as for the case of Wei et al. (1995), the above expres-

sions (3.20) and (3.21) hold, but now

aφ,1 =i

2kx,1Aφ,1 (k2x,1 − k2x,2), Π1 ≡

k21h (1−mduk21h2)

ω,

with

kx,1 = +r21 − k2y, kx,2 = +i

r22 + k2y, k1 = +r1.

Here, similar to the general case, r1, r2 > 0 are so that r21 and −r22 are,

respectively, the positive and negative roots of the parabola for k2

gmduh3 (k2)

2− (gh+mcuh

2ω2) (k2)1+ ω2 (k2)

0= 0,

which is the expression (3.42) in this case.

3.1.2 Sponge layers

Following Wei & Kirby (1995), an artificial damping term d will be added to

the momentum equation in a so-called sponge layer so as to allow the radiating

waves propagating outside the domain and to avoid possible reflection effects

within the domain due to the boundaries. This additional term is given by

d = −w1u+ w2∇∇·u+ w3

g/h η, (3.22)

i.e., the addition of three different damping forms: Newtonian cooling, viscous

damping and sponge filter respectively [Israeli & Orszag 1981].

39


In the equation (3.22), wi are space dependent functions that can be writ-

ten in terms of incident wave frequency, ω, and a smooth monotonically in-

creasing function, f (x, y) which varies from 0 to 1 within the sponge layer

[Wei & Kirby 1995]. For instance, in a problem where the wave propagates in

the x-direction, for a sponge layer from xa to xb we consider

wi = ciωf (x) , f (x) =exp ( (x− xa) / (xb − xa) )

2− 1

exp (1)− 1, (3.23)

so that f (xa) = 0 and f (xb) = 1. The empirical coefficients ci govern

the magnitude of all three different damping mechanisms. We have obtained

satisfactory results using c1 = 7.5, c2 = c3 = 0, and a sponge layer length,

xb − xa, equal to two characteristic wave lengths.

3.2 Numerical model

Boussinesq-type models often employ implicit predictor-corrector iterative nu-

merical schemes in the time marching process [Wei & Kirby 1995], where a

3rd-order Adams-Bashford method is used in the prediction and a 4th-order

Adams-Moulton method in the corrections until convergence is achieved. Al-

though the implicitness, numerical instabilities have been reported and the

use of filters is frequent. In this Section, an explicit 4th−order Runge-Kutta

scheme to solve the fully nonlinear equations for wave propagation presented

in previous Chapter is developed.

Taking into consideration the source function, s, for wave generation and the

damping function, d, for the sponge layers, and being x = x, y , the equa-

tions (2.7) can be written as

Υ0 (ηt) +Υ (η, ηt) = E0 (u) + E (η,u) + s (t,x) (3.24a)

U0 (ut) +U (η,ut) = F0 (η) + F (η,u) + d (η,u) (3.24b)

40

3.2 Numerical model 41

where the linear terms are

Υ0 ≡ η + (δ − δσ)∇· (h2∇η) + δσ∇2 (h2η) , (3.25a)

U0 ≡ u+ (c1α + γσ)h∇X+ (c2α + γ − γσ)h2∇Y, (3.25b)

E0 ≡ −X−∇· (d1α + δ − δσ)h2∇X+ d2αh

3∇Y − δσ∇2 (h2X) , (3.25c)

F0 ≡ −g∇η − g (γ − γσ)h2∇∇

2η + γσh∇∇· (h∇η) , (3.25d)

s ≡ |D| exp (−βx2) sin (kyy − ωt) , (3.25e)

d ≡ −w1u+ w2∇Y+ w3

g/h η, (3.25f)

and the nonlinear terms read

Υ ≡ δ∇· hη∇ηt , (3.26a)

U ≡ −∇ (1 + γ) η∇· (hut) + η2∇·ut/2 , (3.26b)

E ≡ −M−∇· [ (c1α + δ)hη − η2/2]∇X+ (c2αh2η − η3/6)∇Y−

− (δ − δσ)∇· (h2∇M) + δσ∇2 (h2M) + δ∇· (hη∇M) , (3.26c)

F ≡ −N+ γ∇ η∇· [h (N+ g∇η) ] −

−∇ (c1αh− η)u·∇X+ (c2αh2− η2/2)u·∇Y+ (X + ηY) 2/2−

− (γ − γσ)h2∇∇·N+ γσh∇∇· (hN) , (3.26d)

with

M ≡ ∇· (ηu) , N ≡1

2∇ (u·u) .

Assuming the solution to be smooth, we solve the equations (3.24) using finite

differences schemes in space.

3.2.1 Spatial discretization: matrix notation

Following the so-called method of the lines, we first semidiscretize the equa-

tions in space considering a uniform unstaggered grid with n = nx·ny nodes

and using finite differences.

41


We consider η, ux, uy and h being the column vectors (i.e., n× 1) containing

the n nodal values of η, the x and y components of horizontal velocity and h

respectively. We define here the column vector u as

u ≡

ux

uy

.

Let further Dx and Dy be the n×n matrices for the finite differences approx-

imations of the first derivatives in x and y, i.e., so that, for instance, Dx·η

yields the vector containing the nodal values of the approximations of ∂η/∂x.

The construction of these matrices comes easily from using either

∂ψ

∂x

n

=ψn+1 − ψn−1

2∆x+O

∆x2

, (3.27a)

or∂ψ

∂x

n

=−ψn+2 + 8ψn+1 − 8ψn−1 + ψn−2

12∆x+O

∆x4

, (3.27b)

i.e., second or fourth order finite difference approximations.

The divergence, gradient and Laplacian matrices are defined, respectively, as

dv ≡ (Dx Dy) , gr ≡

Dx

Dy

, lp ≡ dv·gr ≡ D2

x +D2y . (3.28)

We will further consider

Dxx ≡ D2x, Dxy ≡ Dx·Dy ≡ Dy·Dx , Dyy ≡ D2

y, (3.29)

as the matrices for the second derivatives, having the same order of accuracy

than those for first order derivatives.

We denote (·)∗ as a diagonal matrix constructed with the elements of the vector

(·) in its diagonal. For later use, we define H ≡ h∗, Θ ≡ η∗ and U ≡ (u∗x u∗

y) .

42


Finally, we introduce the (·) operator operating on a squared matrix as

(·) =

(·) 0

0 (·)

, (3.30)

where 0 denotes the null matrix with the same size of (·).

Using the above notation, equations (3.24) can be written as

(L0 + L (η,u)) ·

ηt

ut

= R0·

η

u

+ r (η,u) + χ (t,η,u) . (3.31)

In the above equation matrices L0 and R0, which contain linear terms, are

from equation (3.25),

L0 ≡

L0,11 0n×2n

02n×n L0,22

, R0 ≡

0n×n R0,12

R0,21 02n×2n

, (3.32)

where

L0,11 = In + (δ − δσ)dv·H2·gr+ δσlp·H

2,

L0,22 = I2n + (c1α + γσ)H·gr·dv·H+ (c2α + γ − γσ)H2·gr·dv,

R0,12 = −dv·H− dv· (d1α + δ − δσ)H2·gr·dv·H+ d2αH

3·gr·dv−

− δσlp·H2·dv·H,

R0,21 = −ggr− g (γ − γσ)H2·gr·lp+ γσH·gr·dv·H·gr .

The nonlinear matrix L and nonlinear vector r in equation (3.31) are

L ≡

L,11 0n×2n

02n×n L,22

, r ≡

r,1

r,2

, (3.33)

43


where

L,11 = δdv·H·Θ·gr,

L,22 = −gr· (1 + γ)Θ·dv·H+Θ2·dv/2 ,

r,1 = −m− dv· [ (c1α + δ)H·Θ−Θ2/2]gr·x−

− dv· (c2αH2·Θ−Θ

3/6)gr·y−

− (δ − δσ)dv·H2·gr+ δσlp·H

2 + δdv·H·Θ·grm,

r,2 = −N+ γgr·Θ·dv·H· (N+ ggr·η)−

− gr (c1αH−Θ)U·gr·x+ (c2αH2−Θ2/2)U·gr·y−

− gr· ( (x+Θ·y) · (x+Θ·y)T)/2−

− (γ − γσ)H2·gr·dv + γσH·gr·dv·Hn,

and

m ≡ dv·Θ·u, x ≡ dv·H·u, y ≡ dv·u, n = gr·U·UT

/2,

with the operator (·) giving a column vector with the diagonal elements of

matrix (·).

Finally the vector χ, standing for source function, s, and damping terms, d,

is

χ =

χs

0n×1

+

0n×n 0n×2n

χd,21 χd,22

·

η

u

, (3.34)

where the matrices χs, χd,21 and χd,22 are easily obtained taking into consid-

eration equations (3.21) and (3.22).

3.2.2 Time integration: fourth-order Runge-Kutta scheme

The expression (3.31) represents a nonlinear system of Ordinary Differential

Equations (ODE’s) for the nodal values of our unknowns. By defining

f ≡

η

u

, (3.35)

44


equation (3.31) can be rewritten as

ft = (L0 + L (f) )−1

·r (t, f) , (3.36)

with r (t, f) ≡ R0·f + r (f) + χ (t, f) .

Due to the reasons to be argued in Section 3.3, fourth order Runge-Kutta

(hereinafter RK4) or third order Adams–Bashforth (AB3) methods are rec-

ommended. Note that both schemes are explicit in time. Here we focus on

RK4 method.

According to the usual (i.e., Kutta) version of this method with fn being the

solution at time tn = t0 + n∆t, we can write

fn+1 = fn +∆tk1 + 2k2 + 2k3 + k4

6, (3.37)

where

k1 = (L0 + L (fn) )−1

·r (tn, fn) ,

k2 = (L0 + L (fn +∆tk1/2) )

−1·r (tn +∆t/2, fn +∆tk1/2) ,

k3 = (L0 + L (fn +∆tk2/2) )

−1·r (tn +∆t/2, fn +∆tk2/2) ,

k4 = (L0 + L (fn +∆tk3) )

−1·r (tn +∆t, fn +∆tk3) ,

so that, in each time step we are required to solve four linear systems or the

form

(L0 + L) ·k = r, (3.39)

with the matrices being block-diagonal and sparse.

To solve the above systems, we consider two options: Gaussian elimination

for sparse systems, which requires O (n3/2) operations, or the Jacobi iterative

method, with numerical complexity O (nitn) , where nit represents the number

of iterations for an acceptable convergence, generally less than 50 for all tests

45


carried out. Above, n is the size of the matrix system (twice the number of

nodes in the 1D case and three times in the 2D case). For n 2500 Jacobi

method is preferred.

3.3 Linear stability

Before we present, in the following Section, some numerical examples, we

introduce here a linear stability analysis of the numerical scheme. Ignoring

the source terms, equation (3.36) for the semidiscretized equations reads, in

the linear case,

ft = A·f , (3.40)

with A ≡ L−10 ·R0, where L0 and R0 are defined in equation (3.32). Most

importantly, we first note that, using sinusoidal bathymetries and the coeffi-

cients and slope limitations defined in Tables from 2.3 to 2.5, the eigenvalues

of the matrix A have shown to be pure imaginary values, so that the system

is intrinsically stable.

If νj are the eigenvalues of the matrix A and ∆t is the numerical time step,

Figure 3.1 shows the regions where the values νj∆t must fall so that three

different ODE solvers will be stable. The regions for the third order Adams–

Bashforth (AB3, which is explicit), fourth order Adams–Moulton (AM4, im-

plicit) and fourth order Runge-Kutta (RK4, explicit) are shown, together with

a qualitative illustration of the values νj∆t. We note that stability region for

AM4 does not include the imaginary axis, and it is therefore discarded. On

the contrary, AB3 and RK4 do include parts of the imaginary axis.

As depicted from the Figure, defining νmax ≡ max |νj |, the AB3 method gives

linearly stable schemes as long as

νmax∆t 0.7236,

46

3.3 Linear stability 47

!4 !3 !2 !1 0 1 2!3

!2

!1

0

1

2

3

AM4

AB3

RK4

Re(! " t)

Im(!

" t)

Figure 3.1: Stability regions for third order Adams–Bashforth (AB3), fourth or-

der Adams-Moulton (AM4) and fourth order Runge-Kutta (RK4) time integration

schemes.

and the RK4 method if

νmax∆t 2.8278. (3.41)

Because RK4 gives a wider stability region than AB3 does, and also a higher

accuracy, the RK4 is the time integration scheme of choice hereinafter.

In the one dimensional flat bed case, the value of νmax depends on the chosen

coefficients α, δ and γ, on g, h and ∆x, on the order of accuracy used to

compute the spatial derivatives (“o”), and, also, on the number of nodes n.

Applying dimensional analysis

νmax∆x√gh

= fx

α, δ, γ,Πx ≡

∆x

h, o, n

. (3.42)

In Figure 3.2 we present the behaviour of function fx using two different sets

of coefficients presented in Table 2.3: κmax = 5 (big symbols) and κmax = 10

(small symbols). In all cases n = 50, since the influence of n has shown to

be negligible for high values (n 20). Also, in Figure 3.2 we consider second

47


10!2 100 1020

0.5

1

1.5

4th

2nd

!x

f x

Figure 3.2: General behaviour of function fx using coefficient from Table 2.3 for

κmax = 5 (big symbols) and κmax = 10 (small symbols). Spatial derivatives of second

order (triangles) and of fourth order (circles).

and fourth order accuracies in space derivatives, i.e., o = 2 and o = 4. From

the Figure, for high values of the group Πx, i.e., in “shallow waters” (recall

that ∆x ∝ k−1 so that Πx ∝ (kh)−1), fx not only becomes independent on

the coefficients, which was expected since the coefficients affect the dispersive

performance, but it is a function of the order of accuracy only. The limits are

limΠx→∞

fx =

1.3722, for 4th order in space,

1.0000, for 2nd order in space,(3.43)

further indicating that√gh seems to be the proper velocity to construct the

dimensionless group in the LHS of equation (3.42).

On the other hand, as depicted in Figure 3.2, after a transition zone (10−1 Πx 100), whenever Πx 1, in deep water, the function fx decreases and

48

3.3 Linear stability 49

again becomes Πx independent. Now, however, the limits, which have shown

to be always smaller than those in expression (3.43) for Πx 1, will depend

on the coefficients used in the equations (the higher κmax, the lower the limits

of fx in deep waters). This comes as no surprise, for these coefficients affect

the deep water performance of the equations.

According to expressions (3.41) and (3.42), using a RK4 time integration, ∆t

must be chosen so that √gh∆t

∆x 2.8278

fx, (3.44)

and, recalling the shallow water upper bounds in equation (3.43)

√gh∆t

∆x


2.828, for 2nd order in space.(3.45)

which is consistent with the results by Baldauf (2008) for non dispersive equa-

tions.

The linear stability analysis for the 2D case can be performed similarly to the

1D. Encouraged by the 1D results over uneven beds, we consider here the flat

bed case only. The maximum absolute value of the eigenvalues (which are

always pure imaginaries), satisfies, for n sufficiently high,

νmax∆s√gh

= fs

Πs ≡

∆s

h,Πxy ≡

∆x

∆y, o

, (3.46)

where ∆s2 ≡ ∆x2 + ∆y2. Further, it has been empirically checked that the

function fs is related to fx in (3.42) through the simple expression

fs (Πs,Πxy, o) = (Πxy +Π−1xy ) fx

Πs

Πxy +Π−1xy

, o

, (3.47)

so that, recalling expression (3.43), if, e.g., ∆x = ∆y, we get

limΠs→∞

fs =


2.0000, for 2nd order in space,(3.48)

49


which yields the corresponding CFL-type condition for RK4

√gh∆t

∆s 2.8278

fs=


1.414, for 2nd order in space.(3.49)

which is valid for the case ∆x = ∆y. If ∆x = ∆y, different stability conditions

can be readily obtained from the above results.

3.4 Numerical results

In this Section some numerical results are introduced to show the capabilities

of i) the equations (2.7) with the coefficients in Table 2.3 and ii) the numerical

scheme presented above1. Linear cases over flat beds are considered first, so

as to compare the numerical results with the analytical solutions, which are

readily available for simple contours. Secondly, 1- and 2-dimensional nonlinear

numerical results are compared to experimental data available in the literature.

In all cases we consider 4th-order accuracy for the spatial derivatives.

3.4.1 Linear tests over flat bed

In order to perform preliminary verifications, we consider the comparison of

the numerical results with the exact solutions, either for the model equations

or for the Airy theory, obtained for linear cases.

1D case: order of convergence

A first 1D example is meant to check the order of convergence of the proposed

numerical scheme. The convergence rate is expected to be four since the

spatial derivatives considered are fourth order accurate and we use a 4th order

Runge-Kutta scheme for time integration. We consider the comparison of

the numerical results to the exact solution of the linearized model equations

1source code can be downloaded at http://erkwave.pbworks.com.

50

3.4 Numerical results 51

obtained for a simple case where the domain is defined by −5m ≤ x ≤ 5m.

The initial velocity field is null and the initial free surface is

η (x, t = 0) = η0 exp

−

x2

1m2

,

so that η 0 at the boundaries initially.

The exact solution of the linearized Boussinesq-type equations, without sources,

can be easily obtained using Fourier analysis. A 1-meter deep (h = 1m) basin

with η0 = 0.05m is considered. Both for the numerical and the analytical

solution, the coefficients in Table 2.3 for κmax = 5 are considered.

Five numerical tests with decreasing grid size ∆x were run (Table 3.1). Re-

calling the 1D stability condition, the time step must satisfy

∆t 2.061∆x√gh

≈ 0.658m−1s∆x,

so that in each case we simply set ∆t = 0.5m−1s∆x. Denoting ηn and ηa,

respectively, as the numerical and analytical solutions at x = 0 and t = 10 s,

the error

ε ≡

ηn − ηa

ηa

,

is also shown in Table 3.1.

Figure 3.3 shows, in a log-log plot, the error ε against the grid size∆x, showing

that the slope is nearly 4, as expected.

1D case: wave generation and dispersion performance

In this second 1D example, the purpose is to show that i) the source function

allows to reproduce the desired wave train and, ii) the good dispersive prop-

erties of the equations. Monochromatic wave trains propagating over a flat

bottom have been simulated and compared with Airy’s solution.

51


# of cells ∆x (m) ε

50 0.2000 3.22·10−2

100 0.1000 2.00·10−3

200 0.0500 1.25·10−4

400 0.0250 7.78·10−6

800 0.0125 4.86·10−7

Table 3.1: Analysis of the order of convergence for a 1D linear case.

10!3 10!2 10!1 10010!10

10!8

10!6

10!4

10!2

100

! x (m)

"

Figure 3.3: Analysis of the order of convergence for a 1D linear case.

52


Taking T = 4.5s and g = 9.81m/s2, three tests with different depths h (m) =

25, 50, 100, i.e., κ = 4.97, 9.94, 19.87 are considered. They all correspond

to deep water conditions (κ 3). In fact, the exact (or Airy’s) wavelength L

is 31.61m and the celerity is c = 7.026m/s in all cases. We want to generate

wave trains with amplitude η0 = 0.05m and we always consider ∆x = 1.25m.

In the first case κ 5 and we consider the coefficients corresponding to

κmax = 5 in Table 2.3. The celerity for the Boussinesq model is in this case

7.026m/s, i.e., indistinguishable from Airy’s result to this accuracy (error in

celerity is in this case bounded to 0.008%). Figure 3.4 shows three snapshots

of the numerical and Airy solutions. The model represents well the dispersion

(Figure 3.4, bottom), and the source function is yielding the desired wave am-

plitude. Figure 3.5 (top panel) shows, for the same test, the numerical results

obtained using the coefficients by Wei et al. (1995), denoted hereinafter as

W95, and for coefficients by Madsen & Schaffer (1998), denoted as M98. The

results are consistent with the fact that the linear dispersion errors for W95

and M98 are, respectively, 9.2% and 0.73% at κ = 5.

Figure 3.5 does also show the numerical and Airy’s results for the test with

h = 50m (κ 10) and h = 100m (κ 20) at t = 10T = 45s. This results have

been obtained using the coefficients in Table 2.3 corresponding to κmax = 10

(for the case h = 50m) and κmax = 20 (for h = 100m).

For the test with h = 50m the celerity of the new coefficients corresponding to

κmax = 10 is 7.035m/s (0.13% error relative to Airy’s) and the comparison is

again very good, as shown in the middle panel of Figure 3.5. The errors using

W95 and M98 are now, respectively, 36% and 8.7%. Finally, for the test with

h = 100m (bottom panel in Figure 3.5) the celerity obtained with κmax = 20

coefficients is 7.085m/s (0.84% error) and a small lag is noticed relative to

Airy’s solution. For this very deep case the coefficients by W95 and M98 have

53


!101 t = T

!/!

0

!101 t = 3T

!/!

0

0 1 2 3 4 5!1

01 t = 10T

x/L

!/!

0

Airy New model equations

Figure 3.4: Monochromatic linear 1D propagation for h = 25m and T = 4.5 s.

Numerical results (circles) and Airy’s analytical solution (line).

54


!101 h = 25m

!/!

0

!101 h = 50m

!/!

0

0 1 2 3 4!1

01 h = 100m

x/L

!/!

0

Airy New model equations M98 W95

Figure 3.5: Monochromatic linear 1D propagation for T = 4.5 s. Numerical results

for the new equations (circles), Madsen & Schaffer (1998) (squares), Wei et al. (1995)

(crosses) and Airy’s analytical solution (line) at time t = 10T.

55


errors of 81% and 33% respectively, and are clearly unable to represent the

linear dispersion.

Besides, three new experiments have been carried out in the DeFrees Hy-

draulics Laboratory at Cornell University which are also considered for the 1D

linear case. The experimental setup, which is shown in Figure 3.6, consists of

a 9-meter long basin filled to a constant depth of 0.50m, and with a dissipative

1:10 (H:V) beach at its end to avoid the reflection. Three different wave peri-

ods were generated, T = 0.45 s, 0.55 s, 0.65 s , so that κ = 9.93, 6.65, 4.76

respectively. Contrary to the above examples, in this case κ is modified by

changing the period instead of the water depth. Wave amplitude in all the

cases was η0 = 0.002m, so that nonlinear effects were negligible. The free

surface elevation η was measured at two gages (Figure 3.6). The first one

(#A) was located at x = 3m, to measure the incident generated wave, and the

second (#B), was located 4.9m from the first gage, to measure the propagated

wave.

For the numerical computations, the coefficients in 2.3 corresponding to κmax =

10 were used in the first two experiments, while those for κmax = 5 were con-

sidered for the third experiment. Further

∆x = 0.02m and ∆t = 0.01 s

<

2.061∆x√gh

,

so that the stability condition is satisfied. The comparisons between experi-

mental data and numerical results are shown in Figure 3.6. The agreement in

the phase is very good in all three experiments. The small modulation in the

experimental results, in the order of 0.1mm, can be explained as a seiche of

the flume. Similar to what it is seen in Figure 3.5, the results of new equations

substantially improve those by W95 and M98.

56


!2!1

012

!/!

0

!2!1

012

!2!1

012

!/!

0

!2!1

012

0 1 2 3 4!2!1

012

t/T

!/!

0

0 1 2 3 4!2!1

012

t/T

T = 0.45s

T = 0.55s

T = 0.65s

#A #B

0 3 7.9 9

!0.5

0#A #B

1:10

x (m)

z (m

)

New model equations Experimental data

Figure 3.6: Experimental set-up (top panel) and free surface time histories at #A

(left) and #B (right) for periods T = 0.45s (top), T = 0.55s (middle) and T = 0.55s

(bottom). Experimental data (stars) and numerical results (line).

57


2D case: 2D dispersion performance

The last linear example is aimed to check the 2D performance of the equations

and numerical scheme. We consider a 1-m deep squared basin defined by

−5m ≤ x ≤ 5m and −5m ≤ y ≤ 5m. The initial velocity field is null and the

initial free surface elevation is given by

η (x, y, t = 0) = η0 exp

−x2 + y2

1m2

with η0 = 0.05m.

Figure 3.7 shows the time history of the free surface elevation η at corner

and at the center of the basin, for the numerical scheme solving the linearized

Boussinesq equations (using here the coefficients for κmax = 5 and ∆x = ∆y =

1m/15 and ∆t = 0.02s) and for the exact solution of the Airy equations, which

can be easily found for this simple case using Fourier analysis. The comparison

is very good and allows to check, for the 2-dimensional multiharmonic case

which includes a whole range of κs, the good dispersive performance of the

model equations as well as the proposed numerical scheme. Indistinguishable

results are obtained using the coefficients for κmax = 10.

3.4.2 Nonlinear tests

Two well-known experiments related to nonlinear waves are considered herein

to test the performance of the model equations and the numerical scheme. The

first is the 1-dimensional Dingemans’ bar [Dingemans 1994], and the second

is the 2-dimensional Vincent and Briggs’ shoal [Vincent & Briggs 1989].

1D: the Dingemans’ bar

The bathymetry of Dingemans’ experiments is a bar (see, e.g., Figures 3.8

and 3.9) situated in a flume where the reference depth is 0.86m. The most

significant aspects of the geometry are i) a minimum depth of 0.20m at the

58


!1.0!0.5

0.00.51.0

!/!

0

corner

0 5 10 15!1.0!0.5

0.00.51.0

t(s)

!/!

0

center

Airy New model equations

Figure 3.7: Free surface histories at the corner (top) and center (bottom) in a squared

basin. Analytical solution for Airy’s theory (line) and numerical solution for the new

Boussinesq-type equations (circles).

59


Case η0 (m) T (s) κ ∆x (m) ∆t (s)

A 0.020 2.02√2 0.43 0.05 0.025

C 0.041 1.01√2 1.70 0.05 0.020

Table 3.2: Dingemans’ experiments setup.

top of the bar where, depending on the input wave, nonlinear effects are ex-

pected to be important, and ii) slopes up to 1:10, so that mild slope conditions

are violated. Therefore, if short waves are used as inputs, all three relevant

aspects in wave propagation models (dispersion, nonlinearity and bed slope)

are to be properly accounted for in the model. This fact makes this series of

experiments a useful benchmark problem.

Two out of the three experiments reported by Dingemans are analyzed here.

They are cases A and C; case B represents a wave breaking case and is, there-

fore, avoided. Table 3.2 summarizes the experimental and numerical condi-

tions for cases A and C. In Table 3.2, η0 is the incident wave amplitude and κ

is computed using the initial depth h = 0.86m. Nonlinear wave decomposition

is expected over the bar, and the values of κ associated to the decomposed

waves will be higher than the original. However, the coefficients corresponding

to κmax = 5 in Table 2.3 have been considered in both cases. The results using

the coefficients for κmax = 10 would be indistinguishable in a figure.

Figures 3.8 and 3.9 show the time history comparison between numerical re-

sults and experimental data at different reported gages. Section #1 has been

used as control section, allowing to synchronize model and experimental time.

From Figures 3.8 and 3.9, the comparison between numerical and experimen-

tal results is fair, and particularly good for case A (where dispersive effects

are smaller than in case C).

60


0 5 10 15 20 25 30 35 40 45 50

!0.50

0.5#1 #2 #3 #4 #5 #6 #7 #8

z (m

)

x (m)

1:20 1:10

!2.00.02.0

#1!/!

0

#2

!2.00.02.0

#3!/!

0

#4

!2.00.02.0

#5!/!

0

#6

0 1 2 3 4 5!2.0

0.02.0

#7

t/T

!/!

0

0 1 2 3 4 5#8

t/T


Figure 3.8: Dingemans’ experiments. Case A. Numerical results (lines) and experi-

mental data (stars) for free surface elevation.

61


0 5 10 15 20 25 30 35 40 45 50

!0.50

0.5#1 #2 #3 #4 #5 #6 #7 #8

z (m

)

x (m)

1:20 1:10

!2.00.02.0 #1

!/!

0 #2

!2.00.02.0 #3

!/!

0 #4

!2.00.02.0 #5

!/!

0 #6

0 1 2 3 4 5!2.0

0.02.0 #7

t/T

!/!

0

0 1 2 3 4 5

#8

t/T


Figure 3.9: Dingemans’ experiments. Case C. Numerical results (lines) and experi-

mental data (stars) for free surface elevation.

62


0 1!2.0

0.0

2.0

t/T

!/!

0#A7

0 1!2.0

0.0

2.0

t/T

!/!

0

#A8

Experimental data New model equations M98 W95

Figure 3.10: Dingemans’ experiments. Case A. Comparison at sections #7 and #8

using proposed coefficients (circles), Madsen & Schaffer (1998) (squares) and Wei et

al. (1995) (crosses) with experimental data (line).

Figure 3.10 shows comparisons between experimental data and numerical re-

sults at gages #7 and #8 for case A using the proposed coefficients and also

the coefficients corresponding to M98 and W95. The new model equations

produce better agreement with experimental data than previous models.

2D: Vincent and Briggs’ shoal

The last test case corresponds to the Vincent and Briggs’ “M2” experiment

[Vincent & Briggs 1989], where nonlinear wave propagation, diffraction and

refraction effects over a submerged shoal are analyzed. The experiment was

carried out in a 35m-wide, 29m-long wave basin. The shoal, 30.48 cm high,

is located over a flat bottom with a depth of 45.72cm (see the plan view in

Figure 3.11, top panel). The location of the gages where the significant wave

height Hs was measured are also shown in Figure 3.11 (top panel).

We shall only focus in test case “M2”, where a monochromatic wave train

63


with a period T = 1.3 s and amplitude η0 = 4.80 cm is propagated along the

x-direction. Since κ ≈ 1.0 for the incident wave, the coefficients employed in

the numerical results are, again those for κmax = 5 in Table 2.3. Further, in

the numerical simulations ∆x = ∆y = 0.10m and ∆t = 0.025 s (satisfying the

CFL-type condition).

Figure 3.11 shows the numerical results for Hs at the different transects. In

general, good agreement between experimental and numerical results are ob-

tained. For illustrative purposes, Figure 3.12 shows a snap shot at t = 20 s,

where the top and minimum value of vertical displacement are located just

before the shoal and diffraction and refraction effects due to bathymetry can

be observed.


A fourth order explicit scheme has been presented to solve the modified set

of fully non-linear Boussinesq-type equations with improved dispersion per-

formance. Using the so-called method of the lines, the equations are first

semidiscretized in space and then integrated in time using a 4th-order Runge-

Kutta scheme. A CFL-type condition is given and a specific source function

for wave generation derived. The model has been tested to show the linear

dispersion performance against the Airy theory and experimental data as well

as the nonlinear behaviour tested with the experiments of Vincent and Briggs

and Dingemans bar.

64


x(m)

y(m

)

0 5 10 15 20 25 30

5

10

15

20

0.01.02.03.0 x = 18.15 m

Hs/!

0

0.01.02.03.0 x = 21.20 m

Hs/!

0

10.67 12.20 13.72 15.25 16.770.01.02.03.0 x = 24.25 m

Hs/!

0

y(m)

0.01.02.03.0 y = 10.67 m

0.01.02.03.0 y = 13.72 m

18.15 19.68 21.20 22.73 24.250.01.02.03.0 y = 16.77 m

x(m)


Figure 3.11: Bathymetry and location of gages in Vincent and Briggs’ experiment

(top) and comparison between experimental (stars) and numerical (line) significant

wave heights.

65


x(m)

y(m)

10 15 20 250

5

10

15

20

25

!2.5

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

Figure 3.12: Snapshot for free surface elevation at time t = 20s in Vincent and

Briggs’ experiment. Values of η/η0.

66

Chapter 4

Turbulent bed shear stress

In this Chapter, we introduce a new instrument to measure directly the wall

shear stress under multiharmonic periodic waves avoiding some of the difficul-

ties for shear plates. The instrument is conformed by two concentric cylinders,

the gap between them filled with water (see Figure 4.1). The inner cylinder is

fixed and the outer one, where the boundary layer is to be developed, rotates

with a given velocity time history. A torque transducer attached to the rota-

tion axle allows to measure the stress induced by the water boundary layer at

the outer cylinder. Due to the sensitivity of the torque meter, only turbulent

boundary layers are analyzed. The experimental results for monochromatic

waves are compared to those in the literature for checking purposes. Exper-

imental results for both monochromatic and sawtooth waves are compared

with a simple turbulent boundary layer model by Simarro et al. (2008).

The results of this Chapter are published in Journal of Hydraulic

Engineering [Galan et al. 2011a]

4.1 Theory

In this Section we present some theoretical considerations about the structure

of the flow between the cylinders for periodic movements of the outer cylinder.

67

68 Turbulent bed shear stress

As it will be justified below, we consider that the flow is essentially 2D (Taylor-

Couette flow).

Figure 4.1: General overview of the new instrument designed to measure directly

bed shear stress.

4.1.1 Laminar flow

Since the laminar case has an analytical solution and gives a good insight of

the flow within the cylinders, we will consider it firstly. In polar coordinates,

r, θ , the momentum equation in the θ direction is, being ∂/∂θ = ur = 0

68

4.1 Theory 69

since the flow is axisymmetric,

ρ∂u

∂t=

∂τ

∂r+

ρν

r

∂u

∂r−

u

r

, (4.1)

where ρ and ν are the fluid density and viscosity, respectively, u is the tan-

gential velocity (usually denoted uθ) and τ is the shear stress, given by

τ = ρν∂u

∂r. (4.2)

Taking into account equation (4.2), the equation (4.1) is a PDE for velocity

u = u (r, t) . The boundary conditions are the velocity at the inner side of the

outer cylinder (i.e., ub at r = b = 196mm) and at the outer side of the inner

cylinder (ua at r = a = 100mm). Since the inner cylinder is fixed, ua = 0.

Above we have anticipated some dimensions of the device.

Considering hereafter 2π/ω periodic motions in time, the Fourier expansion

for the velocity reads

u =1

2

∞

−∞un exp (inωt) =

u02

+

n>0

un exp (inωt) , (4.3)

where the Fourier components un = un (r) are un ≡ ω/π π/ω−π/ω u exp (−inωτ) dτ .

In this laminar case, since ν and ρ are constants, the problem is linear and

equation (4.1) reads, for un,

inωun = ν∂

∂r

∂un∂r

+

ν

r

∂un∂r

−unr

, (4.4)

and the boundary conditions are

un (r = a) = 0, and un (r = b) = ub,n, (4.5)

where ub,n is known from the given function ub (t) . The two-point boundary

value problem defined by equations (4.4) and (4.5) can be solved analytically

for un and τn. For n = 0 (mean current) the solution, u0 and τ0, is

u0 =ub,0b

b2 − a2

r −

a2

r

, τ0 = ρν

∂u0∂r

=ρνub,0b

b2 − a2

1 +

a2

r2

. (4.6)

69


For n = 0 (oscillatory components), the solution to equations (4.4) and (4.5)

is given by

un = C1 J1rδ

√−i

+C2K1

−i

r

δ

√−i

, (4.7)

where J1 and K1 are Bessel functions and Ci can be found from the boundary

conditions in equation (4.5). In the above expression

δ ≡

ν

nω, (4.8)

which is a well-known result of the characteristic thickness of the boundary

layer.

Figure 4.2 shows the velocity profile for the steady case (n = 0) and for a

monochromatic one (n = 1 with ω = π/5 s−1). In the second case, four signif-

icant snapshots are shown. From the Figure it is clear that, for waves, a thin

boundary layer develops at the outer cylinder; the thickness of the boundary

layer is in the order of the centimeter, consistently with equation (4.8) taking

into account that the viscosity is chosen to be ν = 10−5m2/s.

For the steady case the velocity gradients are small and the stresses at both

cylinders are very small. However, for the oscillatory case, huge velocity gradi-

ents appear at the outer cylinder, indicating the existence of significant stress,

τb, at r = b. The time histories of ub and τb are shown in Figure 4.3. As we

can observe, there is a lag between velocity and stress of nearly T/8.

The above laminar analysis allows us to introduce, in an analytical manner,

the problem under consideration. Before introducing a turbulent boundary

layer model in the next Section, it is convenient to make some remarks. First

of all, in the present problem the boundary is moving, while in the real world

the sea bed is fixed, i.e., we are solving the dual problem. The two problems

are, however, perfectly analogous, since here we are computing the shear stress

transmitted “from the wall to the water” while in the sea one is interested in

70

4.1 Theory 71

0.1 0.15 0.2!1

!0.5

0

0.5

1

r (m)

u (m

/s)

ub(t)=1 m/s

ub(t)=cos(! t) m/s

Figure 4.2: Velocity profile in the laminar case for a steady (ub (m/s) = 1) and

unsteady (ub (m/s) = cos (ωt) with ω = π/5 s−1) cases. In the latter case, snapshots

correspond to t = 0 (—), t = T/8 (−−), t = T/4 (−·−) and t = T/2 (· · · ), where

T = 2π/ω = 10 s is the period.

the shear transmitted “from the water to the wall”.

Secondly, the thickness of the layer of fluid, which is 96mm, is orders of mag-

nitude smaller than usual water depths. If the flow is oscillatory, as it will

be the case, this aspect is not crucial as long as the boundary layer thickness

δ ∼ν/ω does not reach the inner cylinder. Obviously, this is the case shown

in Figure 4.2. The distance between the cylinders, b − a, has been chosen so

that (b− a) δ for all experimental conditions.

Third, the problem is in this case cylindrical. This difference can be shown

to be rather small (below 1%), since the solution of the equation (4.1) for our

radii is very similar to the equation for the plane case (r → ∞)

ρ∂u

∂t=

∂τ

∂r.

A simple scaling analysis of the equations shows that the curvature is of

71


0 2 4 6 8 10!4

!2

0

2

4

t (s)

ub (m/s)

!b (N/m2)

Figure 4.3: Near bed velocity and bed shear stress time series in the laminar case

for a periodic motion ub (m/s) = cos (ωt) with ω = π/5 s−1.

minor influence if the characteristic radius of the instrument, rc, satisfies

rc ν/ωδ ∼ δ, as it is in our case.

Finally, we can point out that the experimental facility has a finite height (1

meter), and the bottom part of it rotates with the outer cylinder, therefore

generating a boundary layer at the bottom. As long as no mean velocity (i.e.,

current) is present, the thickness of this boundary layer will be very small

compared to the height, and will not significantly affect the results. We will

restrict to the case of pure oscillatory flows, i.e., no wave-current experiments

will be carried out, and we will work in 2D.

4.1.2 Turbulent flow

We anticipate that the flow in all the experiments carried out is, as in most

engineering problems, turbulent. The laminar case has only been presented in

order to gain, in an analytical way, a better understanding of the boundary

layer developed in the facility. The results obtained in the experiments will

72

4.1 Theory 73

be compared to an existing turbulent wave-current boundary layer model pre-

sented in Simarro et al. (2008), which was originally developed for a flat bed

boundary layer. Although the differences between the boundary layer over a

flat bed and that in the instrument are small, it is easy, and it seems conve-

nient, to transpose the original model to our instrument geometry.

Assuming the eddy viscosity concept to model turbulent shear [Wilcox 2004],

the momentum equation reads now

ρ∂u

∂t=

∂

∂r

ρνt

∂u

∂r

+

ρνtr

∂u

∂r−

u

r

. (4.9)

Using the mixing length closure type of model, we can write [Grant & Madsen

1979, Orfila et al. 2007, Simarro et al. 2008]

νt = κu∗c ξ, (4.10)

where κ ≈ 0.4 is the von Karman universal constant, u∗c is a constant charac-

teristic friction velocity and ξ is the distance to the wall. Since the boundary

layer develops in the outer cylinder, both u∗c and ξ refer to the outer cylinder.

Further, since νt is considered constant in time, we can write again the above

expression (4.9) in Fourier components as

inωun =∂

∂r

νt∂un∂r

+

νtr

∂un∂r

−unr

, (4.11)

with boundary conditions

un (r = a) = 0, and un (r = b− ξ0) = ub,n, (4.12)

where

ξ0 ≡ks30

+ν

9.2u∗c, (4.13)

is the (small) distance to the outer cylinder where the velocity of the fluid

is equal to that of the cylinder. In the equation (4.13), ks stands for the

equivalent roughness and ν is the molecular kinematic viscosity of the water.

73


The equation (4.11) with boundary conditions in equation (4.12) does not have

an analytical solution, and it must be solved numerically. Further, the Fourier

components of the wall shear stress at the outer cylinder are given by

τb,n =ρνt∂un∂r

r=b−ξ0

, (4.14)

and τb can be computed using equation (4.3), τb = 12

∞−∞ τb,n exp (inωt) .

Finally, the model is closed by an extra relation between the characteristic

friction velocity, u∗c, and the stress time history. Three different models are

considered following Simarro et al. (2008)

u2∗c =|τb,max|

ρ, (4.15a)

u2∗c = |τb|

ρ, (4.15b)

u∗c = τb ub

ρ u∗ ub, (4.15c)

where τb,max is the maximum value for τb within a period, · stands for the

average over a period and u∗ = u∗ (t) is so that τb = ρu∗ |u∗|. The last closure

equation corresponds to the energetic model by Kajiura (1964). The above

options are denoted hereafter, as models I, II and III respectively.

The above models allow for the computation of boundary layers with several

harmonics (namely, sawtooth shapes). For pure monochromatic waves, i.e.,

ub = umax sin (ωt) , both the original model for flat bed and the presented

here for cylindrical geometry allow to find the friction coefficient, fw, defined

as [Jonsson 1966]

fw ≡2τb,max

ρu2max.

Being ab ≡ umaxω−1 the near bed particle semi-excursion and ks the equivalent

roughness, dimensional analysis shows that friction coefficient is a function of

two groups, namely

Re ≡umaxab

ν

=

u2max

νω

, and ε ≡

abks

=

umax

ksω

. (4.16)

74

4.1 Theory 75

103 104 105 106 10710!3

10!2

10!1

100

! = 100

! = 101

! = 102

! = 103

! = 104

Re

f w

Figure 4.4: Function fw (Re, ε) according to the original model II in Simarro et al.

(2008)

.

Figure 4.4 shows fw (Re, ε) , meaningful only for monochromatic waves, ob-

tained from the original model II. Models I and III yield slightly higher values

(up to 20% higher than model II). In fact, recalling expression (4.10) for the

eddy viscosity, νt, and the closure expressions in equations (4.15), it is intu-

itive that model I will provide larger values for the shear stress than model II.

From Figure 4.4, and as a general trend, for high Reynolds numbers, i.e., in

the rough case, fw (Re, ε) → fw (ε) , while for small Reynolds, in the smooth

case, fw (Re, ε) → fw (Re) . Simarro et al. (2008) have shown that in the

rough case the model works well for ε 30, while for ε 30, which in the

rough case is equivalent to fw 0.05, all three models tend to underpredict

the friction coefficient (relative to available experimental data).

Anticipating some results from later Sections, we focus only in the turbulent

75


10!3 10!2 10!110!3

10!2

10!1

model I

com

pute

d f w

10!3 10!2 10!110!3

10!2

10!1

model II

com

pute

d f w

10!3 10!2 10!110!3

10!2

10!1

model III

experimental fw

com

pute

d f w

Figure 4.5: Comparison between the turbulent boundary layer models and experi-

mental data in Kamphuis (1975) (+), Sleath (1988) (), Jensen (1989) () and Simons

et al. (1992) (). Root Mean Square (RMS) error takes values of 0.276, 0.185 and

0.215 respectively.

76

4.2 Experimental facility and setup 77

region. For turbulent flows the model II is the one with the best agreement

with experimental data in the literature, as shown in Figure 4.5.

According to the expression (4.16), both Re and ε depend on wave conditions.

For instance, a monochromatic wave train with period T = 12 s and wave

height H = 0.4m travelling over a depth h = 3m has a near bed semiexcursion

ab ≈ 0.67m, and therefore Re ≈ 2.4·105. If the bed has an equivalent rough-

ness of one millimeter, then ε ≈ 6.7·102.

Due to the scarcity of experimental data, the above model has not been checked

for the case of sawtooth shapes. The aim of this Chapter is to perform this

comparison. Therefore, experiments will be focused on monochromatic waves

(to check the new facility) and on sawtooth shaped waves.

4.2 Experimental facility and setup

The device is conformed by two 4-mm thick concentric cylinders with exter-

nal diameters of, respectively, 200mm and 400mm. The space between the

two cylinders, 96mm thick, can be filled up with water or simply left empty.

Besides, the inner side of the outer cylinder can be roughened gluing sand of

a given size. Both cylinders are 1m-high (Figure 4.1).

The outer cylinder rotates by means of an electronically commutated DC

motor following a desired velocity time history ub (t) . At any given time, the

desired velocity ub is transmitted by a positioning controller, which compares

the actual velocity to the desired one so as to adjust the motor driver for

the next time step. The time step for the signal control is fixed at ∆t =

0.125 s and the maximum velocity and acceleration are, respectively, 2.0m/s

and 3.4m/s−2 for the motor used. The obtained velocity was checked to be

indistinguishable to the targeted one in all cases.

77


To measure the shear stress at the wall, a torque meter with a capacity of

±20Nm and a sensitivity of ±0.05Nm was coupled to the rotary axle. Ana-

log output data are acquired by the torque transducer each time step, and

transmitted by USB connection to the computer, where the torque (in volts)

and time are stored for later analysis. The measured torque will depend both

on the instrument itself, i.e., the mass of the cylinder and the friction of the

system, and on the shear stress produced by the water boundary layer. The

component of the torque that does not depend on the water boundary layer

can be obtained reproducing the experiment with the gap between the cylin-

ders left empty. The wall shear stress time history for a given velocity time

history is therefore the difference between the torque measurements with and

without water.

In summary, the general procedure for each experiment is this: the gap be-

tween the two cylinders is filled to the top with water, and the velocity signal

is sent to the motor controller during 70 periods to obtain, at least, 50 valid

wave periods after a start-up transient [Sleath 1987]. The torque signal is

post-processed summarizing the data from the 50 valid periods in a single

period that considers, at each point (or phase within the period), the “most

probable value” of the signal obtained through a histogram which considers

ten groups between the minimum and maximum signal. The same process is

then repeated without water, so as to measure the torque required to move

the system. The difference is the torque corresponding to the water boundary

layer, which allows the direct computation of the shear stress at the wall of

the outer cylinder.

As mentioned in previous Section, two different sets of experiments are pre-

sented, corresponding to monochromatic and sawtooth waves. For monochro-

matic waves the near bed velocity ub can be expressed as

ub (t) = umax sin (ωt) , (4.17)

78

4.2 Experimental facility and setup 79

where umax is the maximum velocity and ω ≡ 2π/T is the wave angular ve-

locity.

For sawtooth waves, the velocity ub is given here by

ub (t) = umax3

4

20

i=1

exp (i(n− 1/2)π)

2n−1exp (inωt)

, (4.18)

where it can be checked that umax is indeed the maximum velocity (Figure 4.6).

0 T/4 T/2 3T/4 T

!1

!0.5

0

0.5

1

u b/umax

monochromaticsawtooth

Figure 4.6: Monochromatic and sawtooth time histories.

In the laminar case, the signal is the same order of magnitude of torque meter

sensitivity (0.05Nm). According to Nielsen (1992), in the laminar flow case

(Re 104), the maximum stress is given by τmax = ρ√νω umax. Further, for

monochromatic waves the wave angular velocity is ω = amax/umax being amax

the maximum acceleration. Taking into account device’s limitations for veloc-

ity and acceleration, and imposing Re 104, the maximum expected torque

for the laminar case can be shown to be 0.10Nm, i.e., which is in the order

of the device’s sensitivity. Therefore, we will focus only on turbulent boundary

layers, which are the most important for engineering purposes.

79


test T (s) umax (m/s) Re

mc T08 u125 k00 8 1.25 1.99·106

mc T13 u135 k00 13 1.35 3.77·106

st T08 u125 k00 8 1.25 —

st T13 u135 k00 13 1.35 —

Table 4.1: Smooth test cases for sinusoidal and sawtooth waves.

Two different roughnesses were employed in the experiments. The first one

corresponds to the material of the cylinders (methacrylate, ks ≈ 10−2mm),

yielding smooth boundary layers. This is not an interesting condition for

engineering purposes and, therefore, only few tests were carried out for this

condition (Table 4.1). In Table 4.1, for example, test “mc T08 u125 k00” cor-

responds to a monochromatic (“mc”, while “st” stands for sawtooth) wave

with T = 8 s, umax = 1.25m and a roughness ks = 0mm (for the roughness is

negligible).

The rest of the tests were performed gluing a very uniform (σg < 1.1) 1mm-

diameter sand to the inner side of the outer cylinder. Table 4.2 shows the

conditions tested considering that ks ≈ 2.5 d50 (proposed by Nielsen, 1992)

with d50 = 1.0mm.

4.3 Results

Figure 4.7 shows the velocity ub and the bed shear stress τb time histories for

monochromatic and sawtooth waves for smooth test cases (Table 4.1). The

experimental results for τb (crosses) compare well, in general, with the pro-

posed model II. The above is in spite of the fact that in smooth boundary layer

the stress transmitted to the water is small, the noise of the system becoming

more important.

80

4.3 Results 81

test T (s) umax (m/s) Re ε

mc T04 u04 k25 4 0.4 1.02·105 1.02·102

mc T08 u04 k25 8 0.4 2.04·105 2.04·102

mc T12 u04 k25 12 0.4 3.06·105 3.06·102

mc T16 u04 k25 16 0.4 4.07·105 4.07·102

mc T04 u08 k25 4 0.8 4.07·105 2.04·102

mc T08 u08 k25 8 0.8 8.14·105 4.07·102

mc T12 u08 k25 12 0.8 1.22·106 6.11·102

mc T16 u08 k25 16 0.8 1.63·106 8.15·102

mc T04 u12 k25 4 1.2 9.17·105 3.06·102

mc T08 u12 k25 8 1.2 1.83·106 6.11·102

mc T12 u12 k25 12 1.2 2.75·106 9.17·102

mc T16 u12 k25 16 1.2 3.67·106 1.22·103

mc T08 u16 k25 8 1.6 3.26·106 8.15·102

mc T12 u16 k25 12 1.6 4.89·106 1.22·103

mc T16 u16 k25 16 1.6 6.52·106 1.63·103

st T08 u04 k25 8 0.4 — —

st T08 u06 k25 8 0.6 — —

st T08 u08 k25 8 0.8 — —

st T08 u10 k25 8 1.0 — —

Table 4.2: Rough test cases for sinusoidal and sawtooth shaped waves.

81


!5

!2.5

0

2.5

5

u b, !b

0 2 4 6 8!5

!2.5

0

2.5

5

t(s)

u b, !b

T = 8s, umax = 1.25 m/s T = 13s, umax = 1.35 m/s

0 3.25 6.5 9.75 13t(s)

Figure 4.7: Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2) (×, ex-

perimental; full line, model II) time histories for smooth tests in Table 4.1. Monochro-

matic waves (top panels) and sawtooth waves (bottom panels).

Figure 4.8 shows the equivalent results for the monochromatic tests in Table

4.2, corresponding to ks = 2.5mm. Again, there is, in general, a very good

agreement between experimental and modelled (model II) results regarding

the shape, the phase lag and the absolute values.

For the experiments in Figure 4.8, Table 4.3 shows the error , defined as

≡

τmodelmax

−τ experimental

max

|τmodel|max

, (4.19)

obtained by using the three models. As for the experimental data in the lit-

erature, the model II gives, in general, better results (Figure 4.5), and the

82

4.3 Results 83

models I and III tend to overpredict the shear stress. Only in the tests with

smaller velocity the model I seems to work better. However, those tests cor-

respond to smaller values of ε, and we already know that all the models tend

to underpredict the shear stress for small values of ε.

By construction, all three models give shear stress times histories with the

same phase. To analyze the errors in the shear stress phase relative to the

experimental results, we consider that the error in the phase, φ (relative to

T), is the one minimizing the function

j

[τ experimental (t = tj) − τmodel (t = tj + φT) ]2, (4.20)

where tj indicates time values where experimental data are available. Table

4.3 includes the phase errors for the monochromatic tests (|φ| 0.078).

To further check both the validity of the new instrument and of the proposed

models, Figure 4.9 shows, in terms of fw, our experimental results together

with those in the literature already presented in Figure 4.5, comparing them

with the above models. The new experimental data follow the same trend as

those in the literature, confirming the validity of the new experimental ap-

proach.

Finally, Figure 4.10 shows the results for the sawtooth waves presented in

Table 4.2. The corresponding errors appear in Table 4.4. We remark that

here |φ| 0.062 and that, again, model II is the one yielding better results

when compared to the experimental data. The new experimental results show,

similar to the experimental results by Suntoyo et al. (2008) using indirect mea-

suring techniques, that shear stress time history has a “flat” zone which the

models are unable to represent. Besides, the experimental results in Figure

4.10 clearly show that the maximum positive stress has a higher absolute value

than the negative one. In sawtooth shaped time histories (dashed line in Figure

83


error

test model I model II model III |φ|

mc T04 u04 k25 0.343 0.133 0.248 0.061

mc T08 u04 k25 0.016 -0.316 -0.131 0.031

mc T12 u04 k25 0.097 -0.214 -0.038 0.031

mc T16 u04 k25 -0.036 -0.387 -0.188 0.031

mc T04 u08 k25 0.383 0.173 0.288 0.061

mc T08 u08 k25 0.279 0.032 0.173 0.046

mc T12 u08 k25 0.194 -0.068 0.081 0.052

mc T16 u08 k25 0.187 -0.067 0.076 0.047

mc T04 u12 k25 0.405 0.198 0.313 0.061

mc T08 u12 k25 0.318 0.095 0.223 0.062

mc T12 u12 k25 0.288 0.066 0.190 0.062

mc T16 u12 k25 0.286 0.062 0.187 0.070

mc T08 u16 k25 0.297 0.076 0.202 0.062

mc T12 u16 k25 0.316 0.101 0.220 0.072

mc T16 u16 k25 0.313 0.085 0.210 0.078

Table 4.3: Monochromatic tests errors with the three models. For each case, the

model with minimum is shown in bold font.

testerror

model I model II model III |φ|

st T08 u04 k25 0.5536 0.2372 0.3911 0.062

st T08 u06 k25 0.4891 0.1147 0.2997 0.031

st T08 u08 k25 0.4605 0.0605 0.2607 0.031

st T08 u10 k25 0.4572 0.0557 0.2586 0.046

Table 4.4: Sawtooth tests errors with the three models. For each case, the model

with minimum is shown in bold font.

84

4.3 Results 85

!2!1

012

u b, !b

!4!2

024

u b, !b

0 1 2 3 4!8!4

048

t(s)

u b, !b

0 2 4 6 8!10!5

05

10

t(s)

u b, !b

0 3 6 9 12t(s)

0 4 8 1216t(s)

T = 4 s T = 8 s T = 12 s T = 16 s

u max

= 1

.6m

/s2

u max

= 1

.2m

/s2

u max

= 0

.8m

/s2

u max

= 0

.4m

/s2

Figure 4.8: Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2) (×, ex-

perimental; full line, model II) time histories for smooth tests in Table 4.2. Monochro-

matic waves only.

4.6), the maximum positive acceleration, which corresponds to the ascending

steep slope of ub (t) , is higher, in absolute value, than the negative one, which

corresponds to the descending gentle slope. Because the bed shear stress de-

pends on the local acceleration [Liu & Orfila 2004, Simarro et al. 2008], the

above behaviour of the bed shear stress can be explained. In the monochro-

matic case (full line in Figure 4.6) there is symmetry, i.e., the shape is the

same running the time rightwards or leftwards, and therefore, the positive and

negative shear stresses have the same absolute values.

85


10!3 10!2 10!110!3

10!2

10!1

experimental fw

com

pute

d f w

model Imodel IImodel III

Figure 4.9: Experimental and analytical values for fw: new data (black figures) and

data available in the literature (white figures).

The asymmetric behaviour of the bed shear stress under sawtooth waves, which

is in fact very well captured by the model, is of major importance in sediment

transport processes, since the sediment transport rate has a nonlinear depen-

dence on the shear stress.

In order to show in a more clear fashion the influence of velocity time history

shape on the bed shear stress, Figure 4.11 shows the numerical and experimen-

tal bed shear stress time histories for tests mc T08 u08 k25 (monochromatic)

and st T08 u08 k25 (sawtooth), which share the same period, maximum ve-

locity and roughness. From the Figure, the sawtooth time history gives, rel-

ative to the monochromatic, slightly (≈ 10%) bigger maximum shear stress,

while the differences in the minimum values are more significant (≈ 30%).

86

4.3 Results 87

!3

!1.5

0

1.5

3

umax=0.4m/s

u b, !b

umax=0.6m/s

0 2 4 6 8!5

!2.5

0

2.5

5

umax=0.8m/s

t(s)

u b, !b

0 2 4 6 8umax=1.0m/s

t(s)

Figure 4.10: Velocity ub (m/s) (dashed line) and shear stress τb (N/m2) (×, exper-

imental; full line, model II) time histories for smooth tests in Table 4.2. Sawtooth

waves only.

87


0 2 4 6 8!3

!2

!1

0

1

2

3

4

u b, !b

t(s)

Figure 4.11: Shear stress τb (N/m2) time histories for tests mc T08 u08 k25

(monochromatic) and st T08 u08 k25 (sawtooth). Numerical results (lines) and ex-

perimental data (symbols).

88



In this Chapter we have presented an instrument to measure the bottom shear

stress under symmetric (monochromatic) and asymmetric (sawtooth shaped)

waves in a direct way. The instrument is similar to Taylor’s viscosimeter, but

introduces a transducer to measure the torque required for the outer cylinder

to follow a given velocity time history. The experimental results have been val-

idated against experimental results for monochromatic waves in the literature.

Further, the experimental results have been compared to an existing simple

multiharmonic turbulent boundary layer model. Experiments have considered

both smooth and rough turbulent boundary layers as well as monochromatic

and sawtooth waves, and the agreement between experimental and numerical

results is fair. The closure option in model II has shown to perform the best.

89


90

Chapter 5

General conclusions and

future work

In this Thesis a new set of low order Boussinesq-type equations, where the

maximum order of the spatial derivatives is three, has been derived. The

equations have been derived as a modification from the original set by Mad-

sen and Schaffer (1998) by adding some weighting coefficients which improve

the dispersion characteristics of the original model. To obtain the values of

the coefficients, an optimization method has been applied so as to minimize

the errors in linear dispersion and linear shoaling respect to the Airy’s the-

ory (deeper waters) and weakly nonlinear energy transfer between different

harmonics respect to the Stokes’ theory (shallower waters). The new set of

equations are able to accurately propagate water waves from deep to shallow

waters.

After the optimization process, the new set of equations provides errors in

wave celerity below 1% for κ ≤ 20 whereas the original equations the errors

in celerity is 33%. Besides, the new equations present errors in linear shoaling

bounded by 12% in the region where κ ≤ 10 whereas for the original set the

error is 99%. Finally, the derived equations have an error bounded by 15% in

nonlinear energy transfer for κ ≤ 2, while the Madsen & Shaffer’s equations

91

92 General conclusions and future work

is 59%.

A linear stability analysis of the linearized Boussinesq-type equations has been

performed over uneven beds, which have been characterized by the maximum

slope of the bathymetry, smax. Some stability problems have been reported

when Madsen & Schaffer’s equations are employed for smax ≥ 2.5. All sets of

coefficients presented in this Thesis yield more stable equations than those by

Madsen & Schaffer (1998).

To solve the modified Boussinesq-type equations a new numerical scheme has

been developed. An explicit 4th order Runge-Kutta method with a source func-

tion for wave generation within the domain has been implemented. Numerical

tests have been carried out to validate both the new set of equations and the

numerical scheme. In all tests, the model provides very good agreement with

both analytical solutions and experimental data. A CFL-type stability con-

dition has been derived for the time step taking into account the definition

of stability region for the 4th order Runge-Kutta method. This general pro-

cedure avoids the use of numerical filtering during the simulations, which to

a greater or lesser extent introduces artificial numerical diffusion in the wave

propagation.

In this Thesis the effect of boundary layer under different wave conditions has

been studied experimentally. A new experimental device similar to Taylor’s

viscosimeter has been designed and constructed in order to measure directly

the bottom shear stress under monochromatic and non-monochromatic waves.

The new measuring device has been tested by some experiments available in

the literature providing very good agreement in the comparisons. The device

has been also used to validate a turbulent bed shear stress model which will

be used to couple the Boussinesq type of equations with a sediment transport

module.

92

93

Future work

During the development of this Thesis the author identified several scientific

issues, not yet properly solved. Some of these points have been already ex-

plored by the author and they will constitute some of the future research

lines of the candidate in the next years. First of all, and of great importance

when dealing with non linear wave models in shallow waters, is the inclusion

of the wave breaking. Nowadays, wave breaking models require aprioristic

knowledge of breaking location. The author will investigate the inclusion on

breaking through depth-integration of a k − turbulence model, which has

been used successfully in Navier-Stokes models [Lin et al. 1999]. Moreover

the dry and wet algorithm to deal with the wave run-up in beaches will be

implemented in the numerical code in the next months.

Bottom boundary layer effects which are of major concern for sediment trans-

port models will be coupled with the new set of Boussinesq-type equations.

The boundary layer effects will be incorporated by modifying the bottom

boundary condition for the core region (potential flow) in order to incorpo-

rate the irrotational velocity induced by the boundary layer. This approach

has been already incorporated in traditional weakly non linear as well as in

highly non linear Boussinesq models by several authors [Liu & Orfila 2004,

Liu et al. 2006, Orfila et al. 2007] but the enormous computational cost of

their approaches restrict current models to the study of small scale processes.

Once the bottom boundary layer is treated in the new Boussinesq model, the

next problem to be solved is the sediment transport module. Available models

for sediment transport are usually derived from the quadratic law of the veloc-

ity to obtain the bottom shear stress that drives de sediment. This approach

leads to innacuracies not only in the sediment transported but also in the

phase lag. In fact, one of the main problems in coastal morphodynamics is the

93

94 General conclusions and future work

generation, destruction and movement of submerged sandbars and this prob-

lems constitutes today one of the milestones to be solved [Simarro et al. 2008].

This problem will be treated not only under a numerical perspective but also

flume and field experiments will be carried out in the next year.

Dynamics and morphodynamics in very shallow areas are in many cases the

result of the nonlinear interaction of oscillatory flow and mean current. A

common and not well resolved example is the generation of rip currents as

the result of this interaction. In order to study such coastal processes the

candidate pretends to include in the new formulation presented in the Thesis

the interaction of the oscillatory flow with the mean current by performing an

accurate multiple scale analysis and obtaining a new set of equations.

94

Appendix A

Features of the new equations

In this appendix the linear and weakly nonlinear behaviour of the expres-

sions (2.7) are analyzed. Without loss of generality, we consider here the

one-dimensional case only. Although some of the results below are already

expressed in Madsen & Shcaffer (1998), they will be presented below for com-

pleteness and clarity.

A.1 Linear propagation

Assuming mild slope conditions, i.e., O (σ) 1, the one-dimensional linear

versions of equations (2.7) are

ηt + hux + µ2h2 (mduhuxxx +mdηηxxt)+

+hx u+ µ2h (mσduhuxx +mσ

dηηxt) = O (σ2) , (A.1a)

ut + gηx + µ2h2 (mcuuxxt + gmcηηxxx)+

+hxµ2h mσ

cuuxt + gmσcηηxx = O (σ2) , (A.1b)

where

mdu ≡ dα + δ, mdη ≡ δ, (A.2a)

mcu ≡ cα + γ, mcη ≡ γ, (A.2b)

95

96 Features of the new equations

are the coefficients corresponding to the terms O (σ0) and

mσdu ≡ 2d1α + 3 (dα + δ) + 2 (δ + δσ) , mσ

dη ≡ 2 (δ + δσ) , (A.2c)

mσcu ≡ 2c1α + 2γσ, mσ

cη ≡ 2γσ, (A.2d)

those in terms O (σ1) .

The solutions of equations (A.1) can be expressed as [Dingemans 1997]

η = (Aη + iAησ) exp(iS), (A.3a)

u = (Au + iAuσ) exp(iS), (A.3b)

where i ≡√−1 is the imaginary constant, S ≡

kdx − ωt is the wave phase

angle and Aη, Au, Aησ and Auσ are real valued amplitudes. Both Aησ and

Auσ are O (σ) and, besides, k and all four amplitude functions are slowly

varying in space. Introducing the expressions (A.3) into (A.1), we can collect

the following two leading problems

ωgdη −khgdu

−gkgcη ωgcu

·

Aη

Au

=

0

0

, (A.4)

and ωgdη −khgdu

−gkgcη ωgcu

·

Aησ

Auσ

=

qη

qu

, (A.5)

where

gdu ≡ 1−mduξ2, gdη ≡ 1−mdηξ

2, (A.6a)

gcu ≡ 1−mcuξ2, gcη ≡ 1−mcηξ

2, (A.6b)

with ξ ≡ µkh, and

qη ≡ − (1− 3mduξ2)hAu,x + µ2h2 (3mdukhkxAu −mdηω (kxAη + 2kAη,x) )−

− hx (1−mσduξ

2)Au + µ2mσdηkhωAη , (A.7a)

qu ≡ −g (1− 3mcηξ2)Aη,x + µ2h2 (3gmcηkkxAη −mcuω (kxAu + 2kAu,x) )−

− hx µ2mσ

cukhωAu − gµ2mσcηk

2hAη . (A.7b)

96

A.1 Linear propagation 97

For future use, we define

gd ≡gdηgdu

, gc ≡gcηgcu

. (A.8)

A.1.1 Linear frequency dispersion

To have a non-trivial solution for (A.4), the determinant of the coefficient

matrix of these equations must vanish, i.e.

ω

khgd =

gk

ωgc

≡ m =

Au

Aη

, (A.9)

which yields the following frequency dispersion relation for the wave system

ω2

gk2h=

c2

gh=

gcgd

≡ gcd , (A.10)

allowing the computation of k and the celerity c as a function of ω and h.

The goodness of the above dispersion relation is usually measured by compar-

ing the result to Airy’s wave celerity given by

c2

gh=

tanh ξ

ξ. (A.11)

Following the notation in the main text, the results using Airy’s theory are

denoted with “Airy”, and hereafter those from the Boussinesq-type equations

(2.7) with “app”. Since ξ ≡ µkh depends on the whether we use the approxi-

mate expression (A.10) or the exact one, we favour the use of a k-independent

variable, κ, defined as [Nwogu 1993]

κ ≡µ2ω2h

g

=

c2

ghξ2, (A.12)

which, recalling the expressions (A.10) and (A.11), in the approximate case

can be rewritten as κ = gcdξ2 and in the exact one as κ = ξ tanh ξ, so that

κ ≈ ξ for κ 3, i.e., in deep water.

97


Rewriting ξ2 as κ/gcd in equations (A.6), the expression (A.10) gives

−gcd (gcd −mdηκ) (gcd −mcuκ) + (gcd −mduκ) (gcd −mcηκ) = 0,

allowing to solve gcd, i.e. capp, directly as a function of κ.

A.1.2 Linear wave shoaling

Following Chen & Liu (1995), we consider the linear shoaling performance in

terms of the propagated wave amplitudes as

Aappη

AAiryη

= exp

κ

0

αAiryAη

− αappAη

κ∗dκ∗

, (A.13)

where αAiryAη

and αappAη

are the shoaling gradients for both approaches.

The “α-operator” [Madsen & Sorensen 1992] operating on a variable “a”, which

depends on water depth h, is defined as αa ≡ −ha−1∂a/∂h, so that αa mea-

sures the percentage change of the variable “a” caused by a unit percentage

change in water depth h. It is of interest to note that αab = αa+αb and that,

being b = b (a (h) ) , αb = αaab−1db/da.

The shoaling gradient αAiryAη

within equation (A.13) is obtained from the Green’s

law, i.e., A2ηcg = ctt, and expressed as

αAiryAη

= −αcg

2=

G

(1 + G) 2

1 +

G

2[1− cosh (2ξ) ]

, (A.14)

with

G ≡2ξ

sinh 2ξ.

Above, αcg is obtained applying the α-operator to the group celerity cg, given

from Airy’s dispersion equation (A.11) as cg ≡ dω/dk = c (1 + G) /2.

98

A.1 Linear propagation 99

To obtain αappAη

we remark that the solvability of the system (A.5) requires the

condition qη = −nqu to be satisfied, with

n ≡ωgdηgkgcη

=

khgduωgcu

,

and qη and qu given in equations (A.7). Recalling that Au = mAη, after some

manipulation of the condition qη = −nqu we get

αappAη

=zh − zkα

appk − zmαm

zAη

, (A.15)

where

zh ≡1−mσ

duξ2

gdu+

mσdηξ

2

gdη+

mσcuξ

2

gcu−

mσcηξ

2

gcη,

zk ≡ −3mduξ2

gdu+

mdηξ2

gdη+

mcuξ2

gcu−

3mcηξ2

gcη,

zm ≡1− 3mduξ2

gdu+

2mcuξ2

gcu,

zAη ≡1− 3mduξ2

gdu+

2mdηξ2

gdη+

2mcuξ2

gcu+

1− 3mcηξ2

gcη.

and

αappk =

Γ

1 + Γ,

αm =1

2+

1

gdu−

1

gdη+

1

gcu−

1

gcη

(αapp

k − 1) ,

which have been obtained by applying the α-operator to dispersion equation

(A.10) and to the definition of m in equation (A.9). Above

Γ ≡ 1 + 2

−

1

gdu+

1

gdη+

1

gcu−

1

gcη

.

99


A.2 Weakly nonlinear propagation

We now consider the nonlinear behaviour of the equation sets (2.7) over flat

beds and for the one dimensional case. In this case we get

ηt + hux + µ2h2 (mduhuxxx +mdηηxxt) + pη = O (2) , (A.17a)

ut + gηx + µ2h2 (mcuuxxt + gmcηηxxx) + pu = O (2) , (A.17b)

where

pη ≡ (ηu) x + µ2h2 (mdη (ηu) xx +mcuηuxx) x + µ2h (m

cηηηxt) x, (A.18a)

pu ≡ uux + µ2h2 (mcη (uux) x + cαuuxx + u2x/2) x−

− µ2h (η (muuxt + gm

ηηxx) ) x, (A.18b)

with

mcu ≡ cα + δ, m

cη ≡ δ, (A.19a)

mu ≡ 1 + γ, m

η ≡ γ. (A.19b)

the parameters involved in the nonlinear terms.

Following, e.g., Schaffer (1996), we consider here the asymptotic expansions

η = η1 + η2 + . . . and u = u1 + u2 + . . . , and two leading order, i.e., O (0) ,

waves with different frequencies (ωi and ωj), so that

η1 = ηi1 + ηj1 = Aiη cos Si +Aj

η cos Sj (A.20a)

u1 = ηi1 + ηj1 = Aiu cos Si +Aj

u cos Sj , (A.20b)

where Aau = maAa

η with m’s given in equation (A.9). Also, the phases are

written as Sa = kax− ωat+ θa. Substituting (A.20) into equations (A.18) for

100

A.2 Weakly nonlinear propagation 101

pη and pu we can write

pη = ρi+jri+jη sin Si+j + ρi−jr

i−jη sin Si−j+

+ ρi+iri+iη sin Si+i + ρj+jr

j+jη sin Sj+j ,

pu = ρi+jri+ju sin Si+j + ρi−jr

i−ju sin Si−j+

+ ρi+iri+iu sin Si+i + ρj+jr

j+ju sin Sj+j ,

where Sa±b ≡ Sa ± Sb = ka±bx − ωa±bt + θa±b, being ka±b ≡ ka ± kb, ωa±b ≡

ωa ± ωb and θa±b ≡ θa ± θb. In the above expressions for pη and pu

ρa±b ≡δabka±bAa

ηAbη

2,

with

δab =

1/2 if ωa = ωb,

1 if ωa = ωb,

and, finally

ra±bη = −ma+bg

a±bdη + µ2h (m

cuh k2m a+b −m

cη ωk a+b) ,

ra±bu = −mambg

a±bcη +

+ µ2h (mu ωkm a+b + (cαhmamb − gm

η) k2 a+b ± hkakbmamb) ,

where ma+b ≡ ma + mb and so on, while ga±bdu , ga±b

dη , ga±bcu , ga±b

cη follow the

definitions in equation (A.6) but using ξa±b ≡ µka±bh instead of ξ.

Recalling the expression (A.17), the second order terms η2 and u2 will neces-

sarily be of the form

η2 = Ai+jη cos Si+j +Ai−j

η cos Si−j +Ai+iη cos Si+i +Aj+j

η cos Sj+j ,

u2 = Ai+ju cos Si+j +Ai−j

u cos Si−j +Ai+iu cos Si+i +Aj+j

u cos Sj+j ,

and, substituting, it must hold

ωi±jgi±jdη −ki±jhg

i±jdu

−gki±jgi±jcη ωi±jg

i±jcu

·

Ai±j

η

Ai±ju

= −ρi±j

ri±jη

ri±ju

. (A.21)

101


Finally, the transfer function is defined as

Gapp± ≡

hAi±jη

AiηA

jη

= δijhωi±jki±jg

i±jcu ri±j

η + k2i±jhgi±jdu ri±j

u

2gk2i±jhgi±jdu gi±j

cη − 2ω2i±jg

i±jdη gi±j

cu

,

having used the solution of the system (A.21) above.

The transfer function for the Stokes theory is [Schaffer 1996]

GStokes± = δijh

µωi±j

Hi±j

Di±j− Li±j

, (A.22)

where, following the above notation,

Hi±j ≡ ωi±j

±µ2ωiωj

g−

gkikjωiωj

+

µ2

g

ω3 i±j

2−

g

2

k2

ω

i±j

,

Di±j ≡ gki±j tanh (µki±jh) − µω2i±j ,

Li±j ≡1

2

gkikjωiωj

∓µ2ωiωj

g−

µ2 ω2 i+j

g

.

Since ω defines κ for a given h, GStokes± and Gapp

± are functions of κi and κj .

Further, they are symmetric, so that G+ and G− can be presented in a single

plot [Madsen et al. 2003]. Figure A.1 shows the shape of GStokes± in the range

0 κi,κj 2. Figure A.2 shows the relative errors of Wei et al. (1995)

equations relative to the Stokes theory. The maximum error is 59% in the

considered range (0 κi,κj 2). The results for equations in Madsen &

Schaffer (1998) are similar.

If both leading order waves are equal we get

hAappη

A2η

=gcgd4

g4cum−1r+η + gdg4dum−2r+ugdg4dug4cη − gcg4dηg4cu

, (A.23)

with

r+η ≡ −2m 1− (4mdη +mcu −m

cηg−1d ) ξ2 ,

r+u ≡ −m21− (4mcη + 2m

ug−1d + 2cα − 2m

ηg−1c g−1

d + 1) ξ2 ,

102

A.2 Weakly nonlinear propagation 103

!10 !10!5 !5

!3 !3

!2!2

!1

3

3

5

5

10

10

G+

G!

!i

! j

0 1 20

1

2

Figure A.1: Weakly nonlinear propagation: GStokes± .

!0.2

!0.1

!0.05 !0.05

!0.02

!0.02

!0.01

!0.01

0.01

0.01

0.01

0.02

0.02

0.050.05

0.05

0.10.1

0.1

G+

G!

!i

! j

0 1 20

1

2

Figure A.2: Weakly nonlinear propagation: Gapp± /GStokes

± − 1 for W95.

103


while the transfer function for the Stokes theory in (A.22) reduces to

hAStokesη

A2η

=1

2

ξ

2(3 coth2 ξ − 1) coth ξ

. (A.24)

104

Appendix B

Stability analysis for

sinusoidal bathymetries

Defining mσσdu ≡ d1α + δσ + δ, mσσ

cu ≡ c1α + γσ, and ha+b ≡ ah2c + bh21, and

denoting n+ j as n+j , the coefficients for the continuity equation are

aη,−2 ≡ h21k2µ2

mdηn2−2 +mσ

dηn−2 + 4δσ ,

aη,+2 ≡ h21k2µ2

mdηn2+2 −mσ

dηn+2 + 4δσ ,

aη,−1 ≡ h1hck2µ2

2mdηn2−1 +mσ

dηn−1 + 2δσ ,

aη,+1 ≡ h1hck2µ2

2mdηn2+1 −mσ

dηn+1 + 2δσ ,

aη,0 ≡ − 1− k2µ2mdηh1+2n2

,

and

−bu,−3 ≡ ih31k3µ2

mdun3−3 +mσ

dun2−3 + (7mσσ

du + δσ)n−3 + 3 (mσσdu + δσ) ,

−bu,+3 ≡ ih31k3µ2

mdun3+3 −mσ

dun2+3 + (7mσσ

du + δσ)n+3 − 3 (mσσdu + δσ) ,

−bu,−2 = ih21hck3µ2

3mdun3−2 + 2mσ

dun2−2 + 10mσσ

dun−2 + 4mσσdu ,

−bu,+2 = ih21hck3µ2

3mdun3+2 − 2mσ

dun2+2 + 10mσσ

dun+2 − 4mσσdu ,

−bu,−1 ≡ ih1k3µ2

3mduh1+1n3

−1 +mσduh

1+1n2−1+

+ ih1k3µ2

(3mσσdu − δσ)h

1+0 + 5 (mσσdu − δσ)h

0+1n−1+

+ ih1k3µ2

+(mσσdu − δσ)h

1+0 + (3mσσdu − 5δσ)h

0+1 + ih1k −n−1 ,

105

106 Stability analysis for sinusoidal bathymetries

−bu,+1 ≡ ih1k3µ2

3mduh1+1n3

+1 −mσduh

1+1n2+1+

+ ih1k3µ2

(3mdu − δσ)h1+0 + 5 (mσσ

du − δσ)h0+1

n+1+

+ ih1k3µ2

− (mσσdu − δσ)h

1+0− (3mσσ

du − 5δσ)h0+1

+ ih1k −n+1 ,

−bu,0 ≡ +ihck3µ2

mduh1+6n3 + 4 (mσσ

du − 2δσ)h0+1n − ihckn.

For the momentum equation, the coefficients are

au,−2 ≡ h21k2µ2

mc12n2−2 +mσ

c12mn−2 +mσσcum

2 ,

au,+2 ≡ h21k2µ2

mc12n2+2 −mσ

c12n+2 +mσσcu ,

au,−1 ≡ h1hck2µ2

2mcun2−1 +mσ

cun−1 +mσσcu ,

au,+1 ≡ h1hck2µ2

2mcun2+1 −mσ

cun+1 +mσσcu ,

au,0 ≡ k2µ2mc12h

1+2n2 + 2mσσcu h

0+1 − 1,

and

−bη,−2 ≡ ih21gk3µ2

mcηn3−2 +mσ

cηn2−2 + γσn−2 ,

−bη,+2 ≡ ih21gk3µ2

mcηn3+2 −mσ

cηn2+2 + γσn+2 ,

−bη,−1 ≡ ih1ghck3µ2

2mcηn3−1 +mσ

cηn2−1 + γσn−1 ,

−bη,+1 ≡ ih1ghck3µ2

2mcηn3+1 −mσ

cηn2+1 + γσn+1 ,

−bη,0 ≡ +igk3µ2mcηh

1+2n3 + 2γσh0+1n − igkn.

106

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115

Bottom friction effects on linear wave propagation

G. Simarro a,*, A. Orfila b, A. Galán a,b, G.A. Zarruk b,1

a E.T.S.I. Caminos, Canales y Puertos, Universidad de Castilla–La Mancha, 13071 Ciudad Real, Spainb Institut Mediterrani d’Estudis Avançats (IMEDEA), C. Miquel Marques 21, 07190 Esporles, Spain

a r t i c l e i n f o

Article history:Received 18 March 2008Received in revised form 12 January 2009Accepted 2 June 2009Available online 7 June 2009

Keywords:Linear theoryBoundary layerWKB aproximation

a b s t r a c t

Bottom boundary layer effects on the linear wave propagation over mild slope bottoms areanalyzed. A modified WKB approximation is presented including boundary layer effects.Within the boundary layer, two cases are considered: laminar (constant viscosity) and tur-bulent. Boundary layer effects are introduced by coupling the velocity inside the boundarylayer to the irrotational velocity in the core region through the bottom boundary condition.This formulation properly accounts for the phase between near bed velocity and bed shearstress. The resulting differential equation for the energy conservation introduces a newterm accounting for the energy losses due to the boundary layer effects.

! 2009 Elsevier B.V. All rights reserved.

1. Introduction

The bottom boundary layer in water wave propagation is important, at least, in two fundamental aspects related to coast-al and environmental engineering. First, it determines the stress that the water transmits to the bottom, which is importantin the near shore morphodynamics and ecosystems, since bottom shear stress is responsible for sediment transport [4,9].

Secondly, the energy dissipation in the boundary layer is responsible for wave damping, modifying not only the waveamplitude but also wave celerity and phase. Boundary layer effects can be of first order importance if propagation over longdistances is considered. Therefore, in order to obtain accurate models for water waves propagation, the energy dissipationwithin the boundary layer has to be taken into account in their formulation. This is often done by introducing the bed shearstress in the horizontal momentum equation. The shear stress transmitted to the bottom is usually expressed as the squareof the near bottom velocity as [3,13],

s ! qCfujuj;

where s is the bottom shear stress, q is the water density, u is the near bed velocity and Cf is a dimensionless friction coef-ficient, which is a function of a relative bed roughness and a Reynolds number [4]. As long as Cf is properly chosen, the abovefrictional model is appropriate if the primary concern is the amount of energy dissipated in a time scale bigger than one waveperiod. However, this bottom stress model does not correctly describe the phase of the bottom stress relative to the bottomvelocity, since it is well known, for instance, that the bottom stress is p=4 out of phase with the bottom velocity for an oscil-latory laminar boundary layer [9]. This phase lag is smaller in the turbulent case [4]. A bottom stress such as the above de-scribed is not adequate when computing sediment transport rate, unless an empirical phase shift is introduced.

For the laminar boundary layer case and linear water wave propagation, some results introducing the proper phase havealready been obtained [1,2]. More recently [7] introduced the result of integrating the linearized and laminar boundary layer

0165-2125/$ - see front matter ! 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.wavemoti.2009.06.009

* Corresponding author. Tel.: +34 926295300.E-mail address: [email protected] (G. Simarro).

1 Present address: Institut for Energy Technology, P.O. Box 40, N-2027, Kjeller, Norway.

Wave Motion 46 (2009) 489–497

Contents lists available at ScienceDirect

Wave Motion

journal homepage: www.elsevier .com/locate /wavemoti

into the bottom boundary condition for the potential in the core, obtaining a set of Boussinesq-type equations including thebottom laminar boundary layer effects. The boundary layer effects were introduced in the continuity equation as a convo-lution integral. The extension for uneven bottoms was derived by [8] for mild slope conditions. The above derivations weremade under the assumption of a laminar or constant viscosity.

Sometimes the model of constant eddy viscosity is used to describe the bottom boundary layer and, further, the eddy vis-cosity is modeled as a linear function of the distance to the bottom [6,5,12]. Following this concept, Orfila et al. [10] extendedthe formulation by [8] to include the effects of a fully developed turbulent bottom boundary layer in a nonlinear wave prop-agation model. In this work, we estimate the effects of the bottom friction within the framework of such a model leavingaside the discussion of the accuracy of its performance. In reality the turbulent boundary layer is certainly nonlinear.

In many real situations, water wave linear theory is accurate enough for propagating waves from deep to shallow water,so that simulations can be done in a much more efficient way. The aim of this paper is to introduce the above mentionedboundary layer formulations (laminar and turbulent) in linear models. Since the goal is to present the influence of the bound-ary layer, numerical results will only be presented in one dimensional cases.

The paper is structured in the following manner. For completeness, we summarize the governing equations, boundaryconditions and the Fourier expansion for periodic waves in Section 2. The equations for the WKB approximation are derivedin Section 3. The leading order solution provides the modified equation for the energy conservation where the dissipationeffects are included. These set of equations are expressed in terms of the Fourier (harmonic) components of the free surfaceamplitude. A brief discussion of the results is presented in Section 4.

2. Governing equations

Hereinafter, dimensional variables are primed and dimensionless variables are unprimed. We will consider the domainbetween the sea bed, which is located at z0 ! "h0#x0; y0$, and the free surface at z0 ! g0#x0; y0$ (Fig. 1). The mean water level(MWL) is considered to be at z0 ! 0.

For convenience, here $0 will stand for #@=@x0; @=@y0$ and a0 for #a0x; a0y$. The no flux boundary conditions at the bottom and

the free surface are obtained considering that they are material surfaces. These conditions read

@h0

@t0% u0 & $0h0 %w0 ! 0; z0 ! "h0; #1a$

@g0

@t0% u0 & $0g0 "w0 ! 0; z0 ! g0; #1b$

where u0 is the horizontal velocity, w0 is the vertical component and t0 is time.In water wave propagation problems, it is usually assumed that the fluid is inviscid, so that it is allowed to slip over the

contours. In that case, the no flux conditions in expressions (1) are the only kinematic conditions to be used. Moreover, thevelocity is considered to be irrotational, i.e., there exists a velocity potential U0 so that

u0 ! $0U0; and w0 ! @U0

@z0: #2$

For incompressible fluids, the continuity equation for the potential reads

r02U0 % @2U0

@z02! 0; "h0 6 z0 6 g0; #3a$

and the boundary conditions (1) for rigid bed (i.e., @h0=@t0 ! 0) are,

$0U0 & $0h0 %@U0

@z0! 0; z0 ! "h0

; #3b$

@g0

@t0% $0U0 & $0g0 " @U0

@z0! 0; z0 ! g0: #3c$

Fig. 1. Illustration of the variables and the boundary conditions for the two dimensional wave propagation problem.

490 G. Simarro et al. /Wave Motion 46 (2009) 489–497

The dynamic free surface boundary condition can be obtained from Bernoulli’s equation imposing the continuity of the pres-sure field at the water–air interface. Assuming constant atmospheric pressure, it reads

@U0

@t0% 12

$0U0 & $0U0 % @U0

@z0@U0

@z0

! "% g0g0 ! 0; z0 ! g0 #3d$

being g0 the acceleration of gravity.

2.1. Dimensionless equations

The above equations are scaled by considering usual dimensionless variables

fz;hg ' 1h00

fz0; h0g; g ' g0

a00; fx; yg ' k00fx

0; y0g;

and

U ' k00h00

a00#########g0h0

0

q U0; u ' h00

a00#########g0h0

0

q u0; w ' k00h020

a00#########g0h0

0

q w0; t ' k00#########g0h0

0

qt0;

with a00 and h0

0 being characteristic lengths for the wave amplitude and the water depth, respectively, and k00 is the charac-teristic wave number. The boundary value problem defined in Eq. (3) reads, in dimensionless form

r2U% 1l2

@2U@z2

! 0; "h 6 z 6 !g; #4a$

$U & $h% 1l2

@U@z

! 0; z ! "h; #4b$

@g@t

% !$U & $g" 1l2

@U@z

! 0; z ! !g; #4c$

@U@t

% !2

$U & $U% 1l2

@U@z

@U@z

! "% g ! 0; z ! !g; #4d$

where

! ' a00h00

; and l ' k00h00;

are the dimensionless parameters representing, respectively, the nonlinear and the dispersive effects. Throughout this paperonly linear waves are to be analyzed and, hence, hereafter, it will be considered that ! ' 0. The dispersive parameter, l, issmall in shallow water conditions, but throughout this work no assumption about its value will be made.

2.2. The boundary layer

In real fluids, viscous effects become important within a thin layer attached to the bottom (Fig. 1), where velocity gradi-ents are large. Therefore, the hypothesis of inviscid fluid is no longer valid in this region (boundary layer) and rotational andirrotational velocity components have to be considered, i.e.,

u ! $U% ur; and w ! @U@z

%wr; #5$

with ur and wr standing, respectively, for the horizontal and vertical components of the rotational velocity. The no fluxboundary condition at the bottom in (4b) reads now

#$U% ur$ & $h% 1l2

@U@z

%wr

l2 ! 0; z ! "h: #6$

Following [8], to solve the rotational velocity at the seabed we will consider a coordinate system locally parallel to the bed(hats in Fig. 2). We note that the rotational velocity component normal to the bottom #wr$ is

wr ! wr#1% O##l$h2$$$ % l2ur & $h;

where O#$h$ ( O#1$ since mild slope conditions will be assumed. Therefore, the expression (6) is also

$U & $h% 1l2

@U@z

% wr

l2 ! 0; z ! "h: #7$

Above expression (7) introduces the term wr=l2 which accounts for the boundary layer effects (compare to expression (4b)).

G. Simarro et al. /Wave Motion 46 (2009) 489–497 491

The result for wr at z ! "h is, for the laminar case [7]

wr#z ! "h$ ! "vl2####p

pZ t

0

r2U#z ! "h; t ! n$###########t " n

p dn;

where v is a dimensionless parameter accounting for the boundary layer strength and defined as

v ' 1h00

##################m0

k00#########g0h0

0

qvuut ( 1; #8$

with m0 being the kinematic viscosity, which is constant for the laminar case.

2.2.1. Turbulent boundary layer for periodic wavesAs mentioned, boundary layer is usually turbulent in real cases. In the case of turbulent boundary layer, the problem is

much more complex. The constant eddy viscosity linear model [6,5] has been used often to describe the bottom boundarylayer. In this paper we use such model leaving aside the boundary layer nonlinearity issue.

Considering the flow periodic in time, any time dependent function n can be expressed as

n ! 12

X

n

nn exp#"inxt$ ! n02%X

n>1

Rfnn exp#"inxt$g; #9$

withx being the main frequency and nn the Fourier components. The solution for wr can be written in compact form for boththe laminar and the turbulent cases as [11]

wr;n#z ! "h$ ! "vl2r2Un#z ! "h$ 1% i##########2nx

p #n; #10$

where

#n !1; laminar BL;#####z0

p K1 2###########"inxz0

p$ %

K0 2###########"inxz0

p$ % ; turbulent BL;

8<

: #11$

and with z0 defined as

z0 ' 1v

z00h00;

where z00 is the elevation over the seabed where the velocity cancels.The elevation z0 depends on the bed roughness and a Reynolds number. Besides, the parameter v is given in Eq. (8): in the

case of turbulent boundary layer (which is the usual one), m0 must be replaced by a characteristic kinematic viscosity, which

Fig. 2. Schematic view of the coordinate system locally parallel to the bed.


depends on the own solution. The solution is, therefore, to be obtained in a iterative way [10]. The performance of thisboundary layer model in terms of the friction factor can be found in [12].

2.3. Potential equations for periodic waves

Since the above boundary layer results were presented in terms of the Fourier components, the equations for the coreregion will be hereinafter presented in terms of Fourier components (assuming that the movement is periodic).

Recalling expression (9), the above Eq. (4) read now, replacing (4b) by (7) and linearizing (i.e., setting ! ' 0)

r2Un %1l2

@2Un

@z2! 0; "h 6 z 6 0; #12a$

$Un & $h% 1l2

@Un

@z" vr2Un

1% i##########2nx

p #n ! 0; z ! "h; #12b$

" inxgn "1l2

@Un

@z! 0; z ! 0; #12c$

" inxUn % gn ! 0; z ! 0: #12d$

We remark that the boundary layer effects appear only at the bottom boundary condition (12b). Besides, combining theboundary conditions at the free surface we get

1l2

@Un

@z" n2x2Un ! 0; z ! 0: #13$

Since the problem is linear, there is no interaction between different harmonics and, hereafter, only n ! 1 will be considered(and n will be omitted). Further, in order to solve the potential, only Eqs. (12a), (12b) and (13) are to be used, i.e.,

r2U% 1l2

@2U@z2

! 0; "h 6 z 6 0; #14a$

$U & $h% 1l2

@U@z

" vXr2U ! 0; z ! "h; #14b$

1l2

@U@z

"x2U ! 0; z ! 0; #14c$

with

X ' 1% i#######2x

p #: #15$

Once the potential is solved, the free surface elevation can be computed from Eq. (12d), i.e.,

g ! ixU; z ! 0: #16$

The above Eq. (14) are solved in the following section using the WKB approximation.

3. The WKB approximation

The WKB is usually employed in water wave propagation to handle with the fact that there are two characteristic lengthscales: one corresponding to the wave length and another corresponding to the bottom variations.

Here, three different horizontal characteristic lengths will be present: the two above mentioned plus the one correspond-ing to the boundary layer effects on the wave damping.

For the sake of clarity, we consider first the case when the bed is flat. As above mentioned, in this case there are two dif-ferent scales (the wave length, which is order 1 according to the scaling, and the characteristic damping scale, which will beorder v"1). The WKB approximation can be here performed as usual, except for the boundary layer effects playing the roleusually played by the bed being uneven.

The governing Eq. (14) for the potential are, in this case

r2U% 1l2

@2U@z2

! 0; "h 6 z 6 0; #17a$

1l2

@U@z

" vXr2U ! 0; z ! "h; #17b$

"x2U% 1l2

@U@z

! 0; z ! 0: #17c$


As usual in WKB, we first consider the potential U#x; z$ written as

U#x; z$ ! A#x; z$ exp#iS#x; z$$; #18$

with A and S real functions. Substituting (18) into (17) we get

r2A" A$S & $S% 1l2

@2A@z2

" A@S@z

@S@z

!

! 0; "h 6 z 6 0; #19a$

Al2

@A@z

% vIfXg$ & #A2$S$ " vRfXg#Ar2A" A2$S & $S$ ! 0; z ! "h; #19b$

"x2A% 1l2

@A@z

! 0; z ! 0; #19c$

from the real parts, while from the imaginary parts we get

$ & #A2$S$ % 1l2

@

@zA2 @S

@z

& '! 0; "h 6 z 6 0; #20a$

A2

l2

@S@z

" vRfXg$ & #A2$S$ " vIfXg#Ar2A" A2$S & $S$ ! 0; z ! "h; #20b$

Al2

@S@z

! 0; z ! 0: #20c$

In the above expressions, ‘‘I” and ‘‘R” stand, respectively, for ‘‘imaginary part of” and ‘‘real part of”.The essence of the WKB approximation is to consider two different horizontal scales, usually one corresponds to the wave

and the other to the bottom variations. In this case, the latter will correspond to the boundary layer effects (which are orderv). Following usual WKB procedure (see, e.g., [3] for details), we consider the asymptotic expansions

A ! a0 % v2a1 % v4a2 % & & & ; #21a$S ! v"1#b0 % v2b1 % v4b2 % & & &$: #21b$

The slow variable is now defined as x ! vx, so that $ ! v$. We remark that functions ai and bi in the above expansions areslowly varying, i.e., O#$ai$ ! O#$bi$ ! O#1$.

Introducing expansions (21) into Eq. (20), the leading order implies the well known result that @b0=@z ! 0, i.e., b0 ! b0#x$.Taking this into account, and substituting (21) into Eq. (19) we get, to the leading order

" a0$b0 & $b0 %1l2

@2a0

@z2! 0; "h 6 z 6 0; #22a$

a0

l2

@a0

@z! 0; z ! "h; #22b$

"x2a0 %1l2

@a0

@z! 0; z ! 0; #22c$

which, at it is well known, implies

a0#x; z$ ! aU#x$f #x; z$; f #x; z$ ' cosh#lk#z% h$$cosh#lkh$ ; #23$

where k satisfies the Eikonal equation k2 ! $b0 & $b0. Further, the expression (22c) implies

x2 ! kl tanh#lkh$; #24$

which is the dispersion relationship (in dimensionless form) allowing to compute k as a function of x and h.In order to know the spatial evolution of a0#x; z$, i.e., aU#x$, the terms order O#v2$ are to be analyzed. Introducing expan-

sions (21) into Eq. (20) we get, to the following order

$ & #a20$b0$ %

1l2

@

@za20@b1

@z

& '! 0; "h 6 z 6 0; #25a$

a20

l2

@b1

@z% IfXga2

0$b0 & $b0 ! 0; z ! "h; #25b$

a0@b1

@z! 0; z ! 0: #25c$


Depth integrating the continuity Eq. (25a), using the boundary conditions at z ! "h and z ! 0, and recalling expression (23),we get

$ & a2U$b0

Z 0

"hf 2dz

& '! "vIfXg a2Uk

2

cosh2#lkh$; #26$

where the boundary layer effects, order v, appear in the right hand side.It is well known (see, e.g., [3]) that

R 0"h f

2dz ! ccg , where c and cg are the wave and group celerities, respectively, given by

c ' xk; and cg !

@x@k

! c2#1% G$;

with

G ' 2lkhsinh#2lkh$ !

2k0h0

sinh#2k0h0$: #27$

The analysis for the mild slope case is similar. However, the bottom boundary condition introduces an extra term accountingfor the bed slope (see expression (14b)). The leading order yields the same results in Eqs. (23) and (24), and now the expres-sion (25b) becomes

a20$b0 & $h% a2

0

l2

@b1

@z% IfXga2

0$b0 & $b0 ! 0; z ! "h; #28$

but, depth integrating, the expression (26) remains valid.

4. Discussion of the boundary layer effects

Summarizing the above results, the WKB approximation yield

U ! aUcosh#lk#z% h$$

cosh#lkh$ exp#iv"1b0$;

where k#x$ is given by the dispersion expression (24), b0#x$ satisfies

$b0 & $b0 ! k2 i:e: $b0 & $b0 ! v2k2;

and the amplitude aU#x$ satisfies the Eq. (26). Besides, recalling expression (16), the free surface satisfies

g ! iag exp#i#v"1b0$$;

with ag ' xaU.In order to show the influence of the boundary layer on the wave propagation, let us focus on the behavior of ag in the one

dimensional case. According to the expression (26), and recalling Eq. (15),

$ & a2gccg$b0

( )! "v

a2gk2

cosh2#lkh$I

1% i#######2x

p #

! ";

so that in the one dimensional case, being $b0 ! @b0=@x ! k and ck ! x,

@

@xa2gcg

( )! "v

a2gk2

xcosh2#lkh$I

1% i#######2x

p #

! ";

or, alternatively

@

@xa2gcg

( )! "v

a2gG#####x

p

hI

1% i###2

p #

! "; #29$

with G defined in Eq. (27). In dimensional form, expression (29) reads

@

@x0a02g c

0g

( )! "

##########m00x0

p a02g Gh0 I

1% i###2

p #

! "; #30$

where m00 is the already mentioned characteristic kinematic eddy viscosity (or the viscosity if it is considered constant). Weremark that the RHS stands for the damping. For the laminar case, # ! 1, the above expression is equivalent to the resultspresented in [2]. According to the definition of G, this term rapidly decreases in deep water #k0h0 ) 1$. If viscous effectsare ignored, the well know energy conservation expression a02

g c0g ! ctt is recovered.


Further, potential and free surface Fourier components read, in dimensional form and expressed using the wave ampli-tude a0

g,

U0 ! g0

x0 a0gf exp i

Zk0dx0 % d

& '& '; #31a$

and

g0 ! ia0g exp i

Zk0dx0 % d

& '& ': #31b$

From the potential, the horizontal velocity is

u0 ' @U0

@x0! g0

x0

@ a0gf( )

@x0% ik0a0gf

0

@

1

A exp iZ

k0dx0 % d& '& '

: #32$

Comparing expressions (32) and (31b), velocity and free surface elevation are slightly out of phase due to the term

@ a0gf( )

@x0;

that stands for both, depth variations and bottom boundary layer effects.To illustrate some of the above results, we shall focus on the constant viscosity case (i.e., laminar or constant eddy vis-

cosity). For constant eddy viscosity, m00 is a constant given value and, according to expression (11), # ! 1.We consider a monochromatic wave train with a period of 8 s propagating over a 500 m distance (Fig. 3), the depth vary-

ing linearly from 10 to 2 m. Fig. 3 shows also the evolution of c0g , since it is important in shoaling aspects according to expres-sion (30). Note that, because c0g decreases in the propagation direction, shoaling is expected.

Fig. 4 compares the results for the wave amplitude obtained with and without boundary layer. A constant eddy viscositym00 ! 10"4 m2=s has been used. As depicted from the figure, boundary layer reduces the wave amplitude as long as wavespropagates: in this case, however, shoaling due to depth variations are stronger than boundary layer effects.

0 100 200 300 400 500!10

!5

0

5

10

x position (m)

bottom (m)group celerity c

g (m/s)

Fig. 3. Evolution of the group celerity across 500 m propagation. The incident wave is composed only by one component with a period of 8 s.

0 100 200 300 400 5001

1.1

1.2

1.3

1.4

x position (m)

norm

aliz

ed a

mpl

itude

(!)

without boundary layerwith boundary layer

Fig. 4. Normalized wave amplitude along the distance considering the damping term in Eq. (30) (solid line) and neglecting the viscous effects (dashed line).


5. Concluding remarks

A formulation for linear wave propagation with boundary layer effects has been presented. The boundary layer effects areintroduced in the wave propagation boundary value problem through a modification of the bottom boundary layer. Themodel can be applied with either laminar or turbulent eddy viscosity. The boundary layer effects appear as a modificationof the energy equation where a new term accounts for the effects of the bottom friction. The results have been here pre-sented in a one dimensional case, but it can be easily implemented to two dimensional linear wave propagation problemsin an efficient computational way.

Acknowledgement

The authors thank financial support from MEC thought Project CTM2006-12072.

References

[1] N. Booij, Gravity Waves on Water with Non-Uniform Depth and Current. Ph.D. Thesis, Technical University of Delft, The Netherlands, 1981.[2] R.A. Dalrymple, J.T. Kirby, P.A. Hwang, Wave diffraction due to areas of energy dissipation, J. Waterway Port Coast. Ocean Eng. 110 (1984) 67–79.[3] M.W. Dingemans, Water Wave Propagation over Uneven Bottoms, World Scientific, Singapore, 1997.[4] J. Fredsoe, R. Deigaard, Mechanics of Coastal Sediment Transport, World Scientific, New York, New York, 1992.[5] W.D. Grant, O.S. Madsen, Combined wave and current interaction with a rough bottom, J. Geophys. Res. 84 (C4) (1979) 1797–1808.[6] K. Kajiura, On the bottom friction in an oscillatory current, Bull. Eqrthquake Res. Int. 42 (1964) 147–174.[7] P.L.-F. Liu, A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech. 520 (2004) 83–92.[8] P.L.-F. Liu, G. Simarro, J. Vandever, A. Orfila, Experimental and numerical investigations of viscous effects on solitary wave propagation in a wave tank,

Coast. Eng. 520 (2-3) (2006) 181–190.[9] P. Nielsen, Coastal Bottom Boundary Layers and Sediment Transport, World Scientific, Singapore, 1992.[10] A. Orfila, G. Simarro, P.L.-F. Liu, Bottom frictional effects on periodic long wave propagation, Coast. Eng. 54 (11) (2007) 856–864.[11] G. Simarro, A. Orfila, Boundary Layer Effects on the Propagation of Weakly Nonlinear LongWaves. In Nonlinear Wave Dynamics, World Scientific, 2009.[12] G. Simarro, A. Orfila, P.L.-F. Liu, Bed shear stress under wave-current turbulent boundary layer, J. Hydraul. Eng. 134 (2008) 2, 225–23.[13] I.A. Svendsen, Introduction to nearshore hydrodynamics, Advanced Series on Ocean Engineering, World Scientific, Singapore, 2005.


GEOPHYSICAL RESEARCH LETTERS, VOL. ???, XXXX, DOI:10.1029/,

Mixing rise in coastal areas induced by waves

A. Galan,1 A. Orfila,2 G. Simarro,3 C. Lopez4, and I. Hernandez-Carrasco5

We study the horizontal surface mixing and the transportinduced by waves using local Lyapunov exponents and highresolution data from numerical simulations of waves and cur-rents. By choosing the proper spatial (temporal) parameterswe compute the Finite Size and Finite Time Lyapunov ex-ponents (FSLE and FTLE) focussing in the local stirringand diffusion inferred from the Lagrangian Coherent Struc-tures (LCS). The methodology is tested by deploying a setof eight lagrangian drifters and studying the path followedin front of LCS derived under current field and wave andcurrents.

1. Introduction

Transport, dispersion and mixing of coastal waters are ofcrucial interest due to the ecological and economical impor-tance of these areas. Despite the increasing advance in thescientific description of the physical processes that take placein the coastal ocean a high degree of uncertainty still remainswhen predicting trajectories of particles advected by the flowin coastal environments. On one hand coastal dynamics isinfluenced by deep water conditions over a complex topog-raphy and driven at the surface by highly variable (spatiallyand temporally) wind conditions. On the other hand the ef-fects of wind generated waves modify the current field by theexcess of momentum flux induced by waves. In summary,flows in coastal areas are the combination of currents withvariations of hours or days and wave oscillatory flows withperiods of seconds to tens of seconds. This interaction isusually accounted in coastal ocean models by the radiationstress concept [Longuet-Higgins and Stewart , 1964].

Trajectory of water particles have been studied exten-sively under a lagrangian point of view [Ozgokmen, 2000;Molcard et al. , 2006]. Moreover, the stretching by advec-tion is usually analyzed by means of the Lyapunov exponents[Haller and Yuan, 2000; Mancho et al., 2008; Hernandez-Carrasco et al., 2011]. Local finite-time Lyapunov exponentsare the exponential rate of separation, averaged over infinitetime, of fluid parcels separated infinitesimally [d’Ovideo etal., 2004]. These lagrangian descriptors provide the exis-tence of patterns that are a proxy of the whole flow [Wig-gins, 1992]. Mixing properties of passive tracers in timedependent flows, depend on the chaotic nature of the La-grangian particle trajectories [Lapeyre, 2002]. These struc-tures are known as Lagrangian Coherent Structures (LCS).

1ETS. Caminos. Universidad de Castilla la Mancha,13072 Ciudad Real, SPAIN. IMEDEA(CSIC-UIB), 07190Esporles, SPAIN

2IMEDEA(CSIC-UIB), 07190 Esporles, SPAIN.Corresponding author

3Institut de Ciencies del Mar ICM (CSIC), 08003Barcelona, SPAIN.

4IFISC (CSIC-UIB). 07122 Palma de Mallorca, SPAIN.5IFISC (CSIC-UIB). 07122 Palma de Mallorca, SPAIN.

Copyright 2011 by the American Geophysical Union.0094-8276/11/$5.00

Finite Time Lyapunov Exponent (FTLE) for a fluid particlelocated initially in a position x at time t0 with a finite timeintegration T is given by,

λT (x, t) =1|T | ln

δx(t0 + T )

δx(t0), (1)

where δx(t0 + T ) is the distance between the particle anda neighboring particle advected by the flow after time T .Analogous, the Finite Size Lyapunov Exponent (FSLE) fortwo particles initially separated a distance δ0 is,

λδf (x, t, δ0, δf ) =1|τ | ln

δfδ0

, (2)

being τ the average time it takes the two particles sepa-rated by an initial distance δ0 to reach a separation of δf .In FTLE, one has to fix a finite integration time while inFSLE the initial and final distance between two adjacentparticles (δ0, δf ) are fixed. This two magnitudes have tobe equivalent if they are measured with the correspondingtime T and distance δf . Both exponents can be integratedforward and backwards in time providing information of thebarriers, boundaries or lines of strong stretching, by meansof LCS. Under a transport perspective these LCS provideinformation about those areas where trajectories of initiallyclose particles are quickly separated or particles of differentorigin attracted.

In this work we analyze the influence of waves on themodification on the LCS in a coastal area. FSLE are de-rived from ocean circulation model as well as from wavemodel and results are compared with available data fromdrifters.

2. Data and Methods

The study has been performed in a semi enclosed baylocated in the southern side of the Island of Mallorca, West-ern Mediterrenean Sea (Figure 1). Velocity data was ob-tained from the Regional Ocean Model System (ROMS),a free-surface, hydrostatic, primitive equation ocean modelthat uses stretched, terrain-following coordinates in the ver-tical and orthogonal coordinates in the horizontal [Song andHaidvogel , 1994].

Three different domains were implemented in order toobtain high resolution currents in the area of study. Thecoarser mesh with a resolution of dθ = dλ = 1/74 (e.g.dx dy = 1500m) take boundary conditions from an oper-ational general ocean circulation model (MFSTEP). Thisdomain, is nested to a second domain with a mesh ofdx = dy = 300 m and the later to a third domain coveringthe study area which has a grid resolution of dx = dy = 75m (Figure 1, right). This area is around 18 km wide withdepths at its open boundary around 80 m. The Bay isopen to southerly to southwesterly swells. The final gridis 348×260 nodes with 10 vertical levels . All domains wereforced using wind provided by the PSU/NCAR mesoscalemodel MM5 [Grell et al., 1995].

The study was done from November 10th to November24th 2009. During this period two different simulations

1

Under review in GeophysicalResearch Letters

X - 2 GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS

2oE 20’ 40’ 3oE 20’ 40’

39oN

10’

20’

30’

40’

#1500m

#300m

#75m

10

10

10

20

20

30

30

40

40

50

50

607080

#75m

33’ 36’ 2oE 39.00’

42’ 45’

26’

28’

39oN 30.00’

32’

34’

Figure 1. Area covering the domains for the wave andcurrent models (left). Fine resolution grid detail (right).

were performed by forcing the ocean model with realisticwind fields (hereinafter set I) and forcing the model withthe same wind fields plus the additional gradients of the ra-diation stress tensor (hereinafter set II). The vertical struc-ture of temperature and salinity was obtained from Levitusdatabase [Locarnini et al., 2006; Antonov et al., 2006]. Theradiation stresses were obtained in the same domains of thecirculation model by integrating WAM model, a third gen-eration spectral wave model specifically designed for globaland shelf sea applications [Komen et al., 1994]. Boundaryconditions for the first domain were taken from the oper-ational model for the Western Mediterranean operated bythe Spanish Harbor Authority. Although the wave model isnot appropriate for very shallow waters since typical phys-ical processes at those areas such as diffraction, triad-waveinteractions or depth-induced wave breaking are not consid-ered, the model provides for typical wavelengths the correctwave field for depths higher than 10m.

For set II, the effects of waves are included in the threedomains by adding the gradient of the radiation stresses asan additional forcing term acting on the surface. Stressesare updated every three hours since it is a reasonable inter-val to characterize the wave climate. For both simulations,velocity fields were? stored every 5 minutes.

On November 19th eight drifting buoys were deployed inthe area of study for three days. The buoys were specifi-cally designed for coastal studies and provide the positionthrough a GPS position via GSM transmission every 5 min-utes. Drifters were deployed at the vertex of a square ingroups of 2 buoys. Deep water significant wave heights dur-ing this period reach 1m from the southwest.

3. Results

We have performed simulations with the circulationmodel for the period considered storing surface velocityfields every 5 minutes. For both set I and set II the windforcing and waves were updated every 3 hours. A grid of par-ticles is launched every 3 hours for all the period. Particlesare advected using a first order Euler algorithm which inte-grate velocity data from the numerical model in time. TheFSLE at each point is defined as the maximum Lyapunovexponent of the 4 neighboring particles. On the other hand,the FTLE is computed using the spatial gradient of the flowat the four closest neighbors once the final position of eachparticle at the desired T has been reached [Shadden et al.,2005].

LCS are dynamical patters organizing the flow which cannot be crossed by particles [Joseph and Legras , 2002]. Be-sides, LCS obtained from Lyapunov exponents integratedforward in time, characterize repelling material lines (un-stable structures) while LCS provided by Lyapunov expo-nents integrated backward in time, identify lines of attract-ing material (stable structures) [d’Ovideo et al., 2004]. To

Figure 2. Finite Time Lyapunov Exponents for time integration T = 3h, T = 6h, T = 12hand T = 24h (top panel).Finite Size Lyapunov Exponents for final separation of δf = 150m, δf = 250m,δf = 500m and δf = 750m (bottom panel)(November 23, 2009; 18 : 00h). Positive values of the exponents correspond to forward integration and negative values tobackward integration.

GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS X - 3

show the general behavior of the Lyapunov exponents weperformed for a given day the FTLE and FSLE forwardand backward in time using the numerical simulations ofset I. The FTLE for November 23rd 2009 at 18.00h com-puted for time integration of T = 3h, T = 6h, T = 12h andT = 24h are are shown in Figure 2 (top panels) where onlythe LCS defined as the maximum values of the FTLE aredisplayed. The structures with positive values correspondto FTLE integrated forward in time (hereinafter FTLEf)while structures with negative values to those exponentscomputed backwards in time (hereinafter FTLEb). As seenwhen time integration in small, many LCS are present inthe flow which are filtered when longer times are used. In-tersection of LCS forwards and LCS backwards correspondto hyperbolic points where stable and unstable directions ofparticles coexist. Analogous, the FSLE computed for finalseparation of particles of δf = 150m, δf = 250m, δf = 500mand δf = 750m are shown in Figure 2 (bottom panels) start-ing at November 23rd at 18.00h. Similarly the LCS of theFSLE with positive values correspond to those exponentscomputed forward in time (hereinafter FSLEf) and thosewith negative values to exponents computed backward intime (hereinafter FSLEb). As seen, FTLE for time inte-gration T = 12h present a pattern of LCS similar to FSLEintegrated for final separation of δf = 250m. Maximum val-ues for the Lyapunov exponent are of the order of 2 days−1

corresponding to mixing times of 12h. It is noticeable thatLCS from FSLE are more defined than those obtained byFTLE since for the latter, all particles of the domain willhave a finite Lyapunov exponent but only those particlesthat reach the final δf will have a value of the Lyapunov ex-ponent when FSLE are computed. Since we are interested inthe mixing and presence of barriers of transport induced bywaves, we will show only the LCS obtained from the FSLEb.The final distance considered is δf = 250m. which is roughly3 mesh points.

The influence of waves in the stirring of the surface layerin the study area is assessed by analyzing the LCS from pe-riods with different wave conditions. Starting on November15th relatively mild wave conditions were present in the area.For this period, we compute the LCS of the FSLEb for bothset I ( Figure 3, left panel) and for set II (Figure 3, rightpanel). LCS are shown for November 15th to November 18th

at 00.00h. Not surprisingly the structure of the LCS displaya similar pattern for both set of data in this 4-days analyzed.A barrier parallel to the coast is almost present during allthe period separating two areas in the shallow zone. Shal-low areas are characterized by stronger mixing as seen bythe complex pattern of the LCS. Mean values of the FSLEbfor currents are −0.13, −0.17, −0.31 and −0.31 days−1 forNovember 15th, 16th, 17th and 18th respectively. For thesame period these values for waves and currents are −0.18,−0.17, −0.29 and −0.31 days−1.

To show the utility of LCS in the study of the dispersionof surface material such as pollutants as well as the indi-cation of such structures to be zones of attracting flow, wedeployed 296 virtual neutral particles at four different loca-tions. For the period comprised between November 15th-18th, the path followed general behaviour of lagrangian par-ticles are similar for wind conditions and wind and wavesconditions. It is noticeable that those particles located overLCS remain there and those particles located at differentsides of the structures do not cross the LCS tending to themore attractive structures. LCS provide therefore a dynam-

ical picture of the organization of the flow that is a powerful

tool when analyzing areas of strong converge (divergence) of

the flow.

Figure 3. Daily snapshots of Finite Size LyapunovExponents backward in time for a final separation ofδf = 250m computed by currents (left panel) and withwaves and currents (right panel) starting on November15th. Virtual particles at four different areas are markedby circles, diamonds, squares and stars.

A different situation is obtained for November 21st to

November 24th. During this period, significant wave height

(Hs) measured at deep waters reached 1m. The effect of

waves is to increase mixing and therefore to modify the

transport at the surface. The LCS for this period are dis-

played in Figure 4 for set I (left) and for set II (right). LCS

are displayed each day at 00.00h. Mean values for the wind

stress during this period is 0.1N/m2 which is increased by

the effects of waves up to 0.25N/m2. The barrier that was

located near the coast in mild wave conditions is moved on-

shore appearing new areas of strong mixing in the middle of

the Bay.

X - 4 GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS

Figure 4. Daily snapshots of FSLEb for a final sepa-ration of δf = 250m computed by currents (left panel)and with waves and currents (right panel) starting onNovember 21th. Virtual particles at four different areasare marked by circles, diamonds, squares and stars.

Contrarily to previous situation, the LCS from set I andthe LCS from set II show a very different pattern. In general,the effect of waves is to break the LCS generating chaotictangles that increase the horizontal stirring and mixing. Themean values of the FSLE for set I are −0.27, −0.31, −0.27and −0.40 days−1 for currents at November 21th, 22th, 23th

and 24th respectively, being −0.31, −0.33, −0.30 and −0.42days−1 for set II. Those data suggest that in this analizedcase the effects of waves is to increase mixing around a 10%respect the simulations made only with wind forcing.

Again, we deployed 296 virtual particles at four differentareas selected so as to be at both sides of the LCS of the ini-tial day. On November 22nd particles launched at the flowfrom set I are located on areas characterized by low valuesof FSLEb implying small dispersion (Figure 4 left). The ini-tial shape of the particles deployed are maintained after 24hours. Nevertheless particles within the flow driven by set IIare distributed in deep waters over a LCS that is the resultsof the waves that broke the original shape (Figure 4 right).

Figure 5. 6 hour snapshot of FSLEb for a final separa-tion of δf = 250m computed by currents and trajectoriesof lagrangian drifters.

Moreover two different sets collapsed to this line and finallygot mixed (see diamonds and stars on Figure 4). One day af-ter, on November 23rd, differences become more important.The two set of particles originally deployed at the east, havebeen moved to the center of the Bay since they have beenattracted by the LCS which cross the bay in a south-northdirection. At evening on November 23th wave energy de-crease which can be inferred from the LCS displayed on the24th. Both LCS fields present the same structure of themanifolds (Figure 4, bottom). However, due to the historyof the dynamics, particles are disposed over different lines ofattraction being particles from different sets totally mixedin set II (see circles, diamonds, squares and star at Figure4, bottom left and right).

To elucidate the role of the radiation stress from waveswaves on the computation of LCS in coastal areas, eight la-grangian drifters were deployed on November 19th at 12.00.Figure 5 and Figure 6 display the LCS every 6 hours, start-ing the date of the deployment for set I and for set II, respec-tively, and trajectories followed by the buoys. The first 18

GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS X - 5

hours waves were below 30 cm and it is reflected in the pat-tern of the LCS. Along this period, there is a clear separationbetween two areas of the Bay being the deeper part morediffusive than the shallow part due to the fact that dynamicswithin the Bay ia mainly driven by shelf slope dynamics andrelatively mild small scale wind driven circulation acts in theBay. Drifters at the shallow part travel small distances indi-cating high residence time of waters for this period. Thesebuoys fall in small diffusive areas in both, set I (Figure 5)and set II (Figure 6). Drifters deployed at deeper waters, areattracted by the strong structure that cross the Bay formEast to West moving along it. On November 20th at 6.00 Hs

increased and differences between the two patterns of LCSbecame evident. A new line of attraction appeared for set IIparallel to the main structure. This line is fully developedon November 20th at 18:00 not being present when com-puting the LCS in absence of waves (Figure 5). This newstructure continues evolving and moving onshore increasing

Figure 6. 6 hour snapshot of FSLEb for a final separa-tion of δf = 250m computed by currents and waves andtrajectories of lagrangian drifters.

mixing in set II. The path of the lagrangian drifters followthe LCS from FSLEb provided by set II.

4. Conclusions

We have shown that either FTLE and FSLE can be usedby choosing the proper scale to analyze the dynamical fieldof coastal areas. The LCS integrated backward in time,provide valuable information when trying to response andmitigate possible spills in these areas. However we haveshown that a proper characterization of currents includingthe effects induced by waves is necessary in order to obtaina correct description of the mixing activity and the coherentstructures that control the transport at the scale of interest.

Acknowledgments. Authors would like to thank financialsupport from Spanish MICINN thought project CTM2010-16915and from Med Project TOSCA (G-MED09-425).

References

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d’Ovideo, F., V. Fernandez, E. Hernandez-Garcia and C. Lopez(2004), Mixing structures in the Mediterranean sea fromFinite-Size Lyapunov Exponents, Geophys. Res. Lett, 31,L17203, doi:10.1029/2004GL020328.

Haller, G. and G. Yuan (2000), Lagrangian coherent structuresand mixing in two-dimensional turbulence, Physica D, 147,352-370. How reliable are Finite-Size Lyapunov Exponents forthe assessment of ocean dynamics.

Hernandez-Carrasco, I., C. Lopez, E. Hernandez-Garcia and A.Turiel (2011), Ocean Modelling,36, 208-218.

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