NUMERICAL AND EXPERIMENTAL

INVESTIGATIONS INTO THE AERODYNAMICS OF

DRAGONFLY FLIGHT.

A Dissertation

Presented to the Faculty of the Graduate School

of Cornell University

in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

by

David Baker Russell

August 2004

This document is in the public domain.

NUMERICAL AND EXPERIMENTAL INVESTIGATIONS INTO THE

AERODYNAMICS OF DRAGONFLY FLIGHT.

David Baker Russell, Ph.D.

Cornell University 2004

Dragonflies are one of the most manueverable of the insect flyers. They are

capable of sustained gliding flight as well as hovering, and are able to change

direction very rapidly. Exactly how they use their wings to generate aerodynamic

forces remains unknown.

A new method was developed for solving 2D incompressible viscous flow prob-

lems [46] in order to numerically model the fluid response and forces generated

by multiple flapping wings. This finite difference scheme uses the streamfunction-

vorticity formulation on a regular grid, and handles multiple moving irregular

boundaries.

To test the usefulness of this model, dragonflies were tethered to a vertical force

sensor and filmed using high-speed digital video. This allowed the correlation

of specific wing kinematics to the vertical force generated, so that when these

kinematics are modeled numerically the forces calculated can be compared with

experiment.

The results include detailed descriptions of two distinct wing kinematic pat-

terns, out of four observed. These kinematics resemble motions described by pre-

vious researchers in free flight conditions except for the phase between the fore and

hind wings. The forces calculated from applying the numeric method to a 2D ap-

proximation of these movements compare well to measured forces. The differences

seen can be attributed to 3D effects and to the simplified wing cross-section used

in the model.

We show that wing inertia is a large component of the instantaneous forces

experienced by a dragonfly, and that the dragonfly generates productive force

during both the downstroke and the upstroke. The counter-stroking behavior seen

in free flight is shown to require less power than the in-phase motion observed in the

tethered dragonfly, while producing the same average vertical force. We also show

evidence suggesting that during hovering flight wing rotation is passively driven

by fluid forces, while during forward flight rotation at the end of the downstroke

is actively driven by the dragonfly. Finally, the effectiveness of applying such a 2D

model to the problem is examined, and suggestions are made for future research

to improve modeling ability.

BIOGRAPHICAL SKETCH

David Baker Russell was born on January 27, 1968 in Berkeley, California. He

was the third son of five children. From the age of three he and his family lived

in Minneapolis, Minnesota. As a child, he was fascinated with flying things of all

kinds, and spent untold hours building and flying various airborne machines.

In 1987 he went to college at the University of Minnesota Institute of Tech-

nology, where he was enrolled in the Undergraduate Honors Program and ma-

jored in Aerospace Engineering. In the summer of 1988 he took an internship at

Itasca Consulting Group, a civil engineering firm, where he mostly helped to create

demonstrations that could be run on a computer to illustrate the software they

sold and the projects they had completed. This work helped rekindle a fascination

with computer programming first aquired in high school, and soon he was helping

develop Itasca’s software as well as writing numerical simulations in the course of

his college work.

In 1992 he graduated from the University of Minnesota. Instead of going on

to graduate school as recommended by his coworkers and professors, he accepted

a full time position at Itasca Consulting. This allowed him to stay in Minneapolis

where he co-founded an income-sharing commune based on libertarian principles.

The commune failed a little more than a year later, leaving him the sole owner

of a house in the Uptown area of South Minneapolis. He continued to live there

and work for Itasca Consulting, eventually becoming a lead programmer of their

modeling software.

In 1998 his personal life left him feeling the need for a change, and at the same

time his lack of knowledge of math and theory was clearly limiting the work he

could do for Itasca. He decided to address the situation and enroll in graduate

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school. Eventually, he chose Cornell’s department of Theoretical and Applied

Mechanics under the advice of former co-workers.

Just a few months before leaving Minneapolis for Ithaca, he was introduced to

Jennifer Wall by mutual friends. They endured a long distance relationship for

the first year of his graduate school, and Jennifer moved to Ithaca at the begining

of his second year. On September 15, 2001, just four days after the attack on

the World Trade Center, Jennifer and David were married. On July 1, 2003 they

celebrated the birth of their son Emory.

At Cornell David decided to avoid finding research that was strongly related

to his previous job, instead seeking something that would broaden his experience.

Remembering his former interest in all things flight related as well as a curiosity

about the methods used in computational fluid mechanics, he choose new professor

Jane Wang as his advisor.

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This work is dedicated to my wife Jennifer and my son Emory, who have made

my graduate school experience much different than I initially imagined, and

infinitely for the better.

v

ACKNOWLEDGEMENTS

I am very grateful for the kind assistance of Professor James Marden at Penn

State University. Professor Marden generously offered me his expertise in the

capture and handling of dragonflies, and he and his graduate student spent several

days assisting me in performing the experiment in his lab.

Professor Alan Zehnder, in a display of trust for someone with no experimental

experience and over which he had no direct supervision, graciously loaned me the

equipment necessary to perform the experiment.

I would like to thank the members of my committee, especially Tim Healey,

for providing me with sage advice over the years on the perils and benefits of the

academic experience.

I am of course especially indebted to Jane Wang for having the trust to allow

me to pursue my interests, and for helping to guide those interests in a productive

direction.

I would also like to thank the entire department of Theoretical and Applied Me-

chanics for supplying an atmosphere of academic freedom that is rapidly becoming

extremely rare.

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TABLE OF CONTENTS

1 Overview and Introduction 11.1 Dragonfly wing kinematics . . . . . . . . . . . . . . . . . . . . . . . 51.2 Wing Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Numerical Method 132.1 Streamfunction-vorticity formulation . . . . . . . . . . . . . . . . . 132.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Underlying flow solver on the regular grid . . . . . . . . . . . . . . 172.4 Incorporating discontinuities into a Poisson equation solver . . . . . 20

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 A 1D Example . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 An efficient 2D implementation . . . . . . . . . . . . . . . . 232.4.4 Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Finding the velocity field . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Solving the Poisson equation . . . . . . . . . . . . . . . . . . 272.5.2 Satisfying no-penetration . . . . . . . . . . . . . . . . . . . . 30

2.6 Calculating boundary vorticity . . . . . . . . . . . . . . . . . . . . . 332.7 Integrating vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 Calculating forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.9 Summary and sequence . . . . . . . . . . . . . . . . . . . . . . . . . 452.10 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.10.1 Flow past a stationary circular cylinder . . . . . . . . . . . . 462.10.2 Flow past a moving circular cylinder . . . . . . . . . . . . . 532.10.3 Flow past two cylinders moving with respect to each other . 56

3 Experimental setup and methods 613.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 3D Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 2D Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 713.5 Inertial forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.6 Quasi-steady approximation . . . . . . . . . . . . . . . . . . . . . . 75

4 Dragonfly Kinematics 784.1 The lift stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 The thrust stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3 Stroke analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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5 Forces on Dragonfly Wings 915.1 Calculated Lift Stroke Forces . . . . . . . . . . . . . . . . . . . . . 91

5.1.1 Comparison between computed by CFD and by ODE . . . . 955.1.2 Analysis of force trace features . . . . . . . . . . . . . . . . . 975.1.3 Effects of net downwash . . . . . . . . . . . . . . . . . . . . 1005.1.4 Reynolds number dependencies . . . . . . . . . . . . . . . . 1015.1.5 Fore-hind phase relationship . . . . . . . . . . . . . . . . . . 103

5.2 Calculated Thrust Stroke Forces . . . . . . . . . . . . . . . . . . . . 1055.2.1 Analysis of force trace features . . . . . . . . . . . . . . . . . 1095.2.2 Effects of net downwash . . . . . . . . . . . . . . . . . . . . 1115.2.3 Reynolds number dependencies . . . . . . . . . . . . . . . . 113

5.3 Wing Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 Inertial Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.5 Comparison with Measured Forces . . . . . . . . . . . . . . . . . . . 120

5.5.1 Lift stroke comparison . . . . . . . . . . . . . . . . . . . . . 1205.5.2 Thrust stroke comparison . . . . . . . . . . . . . . . . . . . 121

6 Conclusions 1246.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Force Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Future Work 1287.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.3 Force Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bibliography 133

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LIST OF TABLES

2.1 Results of 2D discontinuity tests. . . . . . . . . . . . . . . . . . . . 262.2 Summary of results for Re = 20 and Re = 40. Base case results are

compared against the results of doubling the discretization densityand doubling the far field distance. . . . . . . . . . . . . . . . . . . 49

2.3 Summary of results for Re = 100 and Re = 200. Base case resultsare compared against the results of doubling the discretization den-sity and doubling the far field distance. . . . . . . . . . . . . . . . 53

2.4 Summary of results for Re = 40 moving cylinder, compared againstfixed case at the same time. . . . . . . . . . . . . . . . . . . . . . 55

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LIST OF FIGURES

1.1 An example of libellula pulchella. Image courtesy of Greg LasleyNature Photography. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Determination of velocity field by vorticity distribution and bound-ary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Simplified cartoon of the process for finding the velocity stream-function for a fixed geometry. . . . . . . . . . . . . . . . . . . . . . 16

2.3 A regular cartesian grid representing the flow. . . . . . . . . . . . . 182.4 Analytical and discrete solution to the 1-D example. . . . . . . . . 212.5 2D geometry of an irregular discontinuity. . . . . . . . . . . . . . . 232.6 Geometry of the 2D discontinuity tests. . . . . . . . . . . . . . . . 252.7 Irregular geometry overlaid on regular grid, causing a discontinuity

in ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.8 Interpolation points chosen to determine ψP . The circle marks the

point at which a value of ψP is needed, and the squares mark thegridpoints selected to determine this value. . . . . . . . . . . . . . 28

2.9 Contours of ψP for impulsively started cylinder. . . . . . . . . . . . 292.10 Contour of ψL for impulsively started cylinder. . . . . . . . . . . . 312.11 Contours of ψ for impulsively started cylinder. . . . . . . . . . . . 332.12 Interpretations of Thom’s formula. . . . . . . . . . . . . . . . . . . 342.13 Interpolation method for the determination of ∂ψP

∂n. The dark circle

indicates the location where ψP is to be determined. The blacksquared indicate the grid points on which the interpolation is con-ducted. First each row is interpolated linearly to find values at thecross points. Then the cross point are interpolated to find the valueat the dark circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.14 Local overset grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.15 Interpolation of the local grid outer node vorticity. Black squares

represent gridpoints used in the interpolation, while small whitesquares represent external irregular points. First each row (in thiscase) is interpolated linearly to the cross. Then the cross values areinterpolated to the dark circle. . . . . . . . . . . . . . . . . . . . . 39

2.16 Convergence test for n=64, 128, 256, 512, and 1024 compared ton=2048 case. Flow around cylinder at Re=40 and time=16. Curveswere normalized so that their smallest values were 1.0 for compar-ison. The solid line indicates second order convergence. . . . . . . . 41

2.17 Outer grid used to enforce global conservation of vorticity. Greysquares represent the discrete area involved. The dashed line indi-cates the surface used to track vorticity flux . . . . . . . . . . . . . 42

2.18 Geometry used for stationary cylinder examples. . . . . . . . . . . 47

x

2.19 Streamlines and vorticity contours for Re = 20 cylinder. Streamcontour values are -1:0.2:1, -0.1:0.02:0.1, and -0.01:0.002:0.01. Vor-ticity contour values are -4:0.2:4 and -0.1:0.01:0.1. . . . . . . . . . . 48

2.20 Streamlines for Re = 40 cylinder. Contour values are -4:0.2:4 and-0.1:0.01:0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.21 Vorticity for Re = 40 cylinder. Contour values are -4:0.2:4. . . . . . 492.22 Coefficient of lift vs time after a perturbation for Re = 40 and Re

= 50, showing transition to instability. . . . . . . . . . . . . . . . . 502.23 Vortex street behind cylinder at Re = 100. Contour values are

-2.5:0.1:2.5. Dotted line is zero contour. . . . . . . . . . . . . . . . 512.24 Coefficient of lift and drag vs time for cylinder at Re = 100. Os-

cillations in lift build exponentially from the start until they reachthe final amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.25 Distribution of vorticity on cylinder surface at Re = 100 and varioustimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.26 Streamlines for Re = 40 moving cylinder at a time of 32. Contourvalues are -1.4:0.1:1.4. . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.27 Vorticity for Re = 40 moving cylinder at a time of 32. Contourvalues are -4:0.2:4. Dashed lines are the contours for the fixed caseat the same time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.28 Adjusted streamlines for Re = 40 moving cylinder. Contour valuesare -4:0.2:4 and -0.1:0.01:0.1. Dashed lines are the contours for thefixed case at the same time. . . . . . . . . . . . . . . . . . . . . . 55

2.29 Comparison of drag histories for moving and fixed cylinder at Re= 40 and Re = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.30 Geometry for two moving cylinder test. . . . . . . . . . . . . . . . 572.31 Vorticity for t̂ = 16. Contour values are -4:0.2:4. . . . . . . . . . . . 572.32 Streamlines for t̂ = 16. Contour values are -1.4:0.2:1.4. . . . . . . . 582.33 Streamlines for t̂ = 32. Contour values are -1.4:0.2:1.4. . . . . . . . 582.34 Lift and Drag vs time for upper cylinder. Closest proximitity occurs

at t̂ = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.35 Stagnation angle vs time for lower cylinder. . . . . . . . . . . . . . 592.36 Pressure distribution around cylinder at t̂ = 12 and t̂ = 15.6. . . . . 60

3.1 Layout of video capture system. On the right is a closeup of thecalibration cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Sample image of a calibration frame, in this case of the back andleft faces of the calibration cube. . . . . . . . . . . . . . . . . . . . 62

3.3 A sample screenshot from the digitization program. An x marks theselected pixel in each frame. The lines are the implied projectionfrom one perspective to the other. . . . . . . . . . . . . . . . . . . 64

3.4 Diagram of a single face of the calibration cube. . . . . . . . . . . . 65

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3.5 Parameters used for the 3D description of the wing kinematics. θis the stroke plane angle, φ is the stroke plane orientation, ψ is thestroke angle, and β is the wing angle. . . . . . . . . . . . . . . . . 67

3.6 Illustration of the 2D kinematic approximation. The first image isa side view of the dragonfly and calculation cylinder. The secondview is along the axis of the calculation cylinder. The third view isan example of the 2D configuration that results after “unwrapping”the cylinder and rotating, as well as a description of the parametersused to describe the 2D wing motion. θ is the stroke plane angle,d is the distance from mid-stroke, β is the wing angle, α is theangle-of-attack, and p is the distance normal to the stroke plane. . 69

3.7 Comparison of simulations with different discretizations at Re 200using lift stroke kinematics. Forces were normalized to the maxi-mum force calculated in the base (768) discretization. . . . . . . . 71

3.8 Comparison of base (solid line) simulations with a doubled far fielddistance (dashed line) using lift stroke kinematics. Forces werenormalized to the maximum force calculated in the base case. . . . 72

3.9 Comparison of computed forces generated by the forewing usingthe lift stroke before (top) and after (bottom) smoothing . . . . . . 73

3.10 Comparison of vertical forces generated on the forewing (top) andthe hindwing (bottom) using kinematics calculated at 66% and 80%of the wing length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.11 Coordinate system and parameters for the force approximation. . . 753.12 The quasi-steady approximation used. . . . . . . . . . . . . . . . . 77

4.1 The four kinematic types observed. Dashed lines indicate hind-wing. The first (from the top) and third image in each sequencerepresents maximum and minimum forewing stroke angles. Thesecond and fourth image in the thrust sequence represents maxi-mum and minimum hindwing stroke angles. The arrows representthe direction the wing will subsequently move. . . . . . . . . . . . 79

4.2 Data from the 3D analysis of the lift stroke over five periods. Thetop graph is the stroke angle, the middle is the wing angle, andthe bottom is the deviation from the stroke plane. Vertical linesindicate stroke reversal and the shaded regions indicate forewingdownstroke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 The 2D kinematics of the ’lift’ stroke. The leading edge of eachwing is marked with a circle, the number inside represents a timeindex. The dashed lines indicate the path of the center of the wing,while the arrows indicate velocity. . . . . . . . . . . . . . . . . . . 82

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4.4 2D data from 5 stroke digitization of the lift kinematic. Verti-cal lines indicate stroke reversal and the shaded regions indicateforewing downstroke. The forewing stroke center is at position(10.7,9.6) mm from the hindwing stroke center. d is distance alongthe stroke plane, p is distance out of the stroke plane, β is wingangle relative to the stroke plane, and α is the wing angle of attack. 83

4.5 Data from the 3D analysis of the thrust stroke over five periods.The top graph is the stroke angle, the middle is the wing angle, andthe bottom is the deviation from the stroke plane. Vertical linesindicate stroke reversal and the shaded regions indicate forewingdownstroke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.6 The 2D kinematics of the ’thrust’ stroke. The leading edge of eachwing is marked with a circle, the number inside represents a timeindex. The dashed lines indicate the path of the center of the wing,while the arrows indicate velocity. . . . . . . . . . . . . . . . . . . 87

4.7 2D data from 5 stroke digitization of the thrust kinematic. Ver-tical lines indicate stroke reversal and the shaded regions indicateforewing downstroke. The forewing stroke center is at position(10.5,10.3) mm from the hindwing stroke center. d is distance alongthe stroke plane. p is distance out of the stroke plane. β is wingangle relative to the stroke plane. α is the wing angle of attack. . 88

4.8 A cartoon illustrating the relationship between the lift and thruststrokes. The image on the left is a cartoon of the lift stroke, alongwith a proposed net force vector. This image is flipped horizontallyto become the center image. This image in turn is rotated clockwiseto become the right image, which is a cartoon of the thrust stroke. 90

5.1 Vorticity contours during the lift stroke. Images are equally spacedin time but do not cover a full stroke. The sequence moves fromleft to right then from top to bottom. . . . . . . . . . . . . . . . . 92

5.2 Velocity vectors during the lift stroke. Images are equally spacedin time but do not cover a full stroke. The sequence moves fromleft to right then from top to bottom. . . . . . . . . . . . . . . . . 93

5.3 Calculated total fluid forces for the lift stroke. The top graph istotal vertical force and the bottom graph is total horizontal force.Vertical lines indicate stroke reversal. The shaded region indicatesthe forewing downstroke. Re = 200. . . . . . . . . . . . . . . . . . 94

5.4 Induced velocity fields around moving objects. On top is the fieldaround a translating circle, an object at point A would see a netinduced velocity upwards. On the bottom is the field around anellipse translating at a 30◦ angle of attack, an object at point Bwould see a net induced velocity downwards. . . . . . . . . . . . . 96

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5.5 Computed vertical and horizontal forces for the first full stroke ofthe forewing and hindwing using the lift kinematic. The dashedlines are forces from simulations using only one wing. The dottedlines are starting at stroke reversal. The dash-dot lines are usingonly one wing and starting from stroke reversal. . . . . . . . . . . . 98

5.6 Calculated total fluid forces for the lift stroke assuming a back-ground fluid velocity of (−0.6,−1.05). The top graph is total ver-tical force and the bottom graph is total horizontal force. Verti-cal lines indicate stroke reversal. The shaded region indicates theforewing downstroke. Re = 200. . . . . . . . . . . . . . . . . . . . 100

5.7 Computed forces for the fourth full stroke of the forewing using thelift kinematic. The dashed lines are forces computed from simla-tions starting at stroke reversal. . . . . . . . . . . . . . . . . . . . . 101

5.8 Comparison of total forces generated by the lift stroke at a Reynoldsnumbers of 200 and 1000. The top graph is total vertical force andthe bottom graph is total horizontal force. Vertical lines indicatestroke reversal. The shaded region indicates the forewing downstroke.102

5.9 Vorticity contours during the thrust stroke. Images are equallyspaced in time but do not cover a full stroke. The sequence movesfrom left to right then from top to bottom. . . . . . . . . . . . . . 106

5.10 Velocity vectors during the thrust stroke. Images are equally spacedin time but do not cover a full stroke. The sequence moves fromleft to right then from top to bottom. . . . . . . . . . . . . . . . . 107

5.11 Calculated total fluid forces for the thrust stroke. The top graph istotal vertical force and the bottom graph is total horizontal force.Vertical lines indicate stroke reversal. The shaded region indicatesthe forewing downstroke. Re = 200. . . . . . . . . . . . . . . . . . 108

5.12 Computed vertical and horizontal forces for the first full stroke ofthe forewing and hindwing using the thrust kinematic. The dashedlines are forces from simulations using only one wing. The dottedlines are starting at stroke reversal. The dash-dot lines are usingonly one wing and starting from stroke reversal. . . . . . . . . . . . 110

5.13 Calculated total fluid forces for the thrust stroke assuming a back-ground fluid velocity of (−1.3,−0.6). The top graph is total verticalforce and the bottom graph is total horizontal force. Vertical linesindicate stroke reversal. The shaded region indicates the forewingdownstroke. Re = 200. . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.14 Comparison of total forces generated by the thrust stroke at Reynoldsnumber of 200 and 1000. The top graph is total vertical force andthe bottom graph is total horizontal force. Vertical lines indicatestroke reversal. The shaded region indicates the forewing downstroke.112

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5.15 Vorticity plots of the forewing during supination for the Re=1000simulation. A strong leading edge vortex is created that rolls overthe top of the wing during supination. These sequences correspondroughly to the numbers 8-13 in Figure 4.6 . . . . . . . . . . . . . . 114

5.16 Rotational power done by the wing on the fluid for the lift (top)and thrust (bottom) stroke. . . . . . . . . . . . . . . . . . . . . . . 115

5.17 Vertical coordinate of the wing center of mass for the lift (top)and thrust (bottom) strokes. The solid line is the forewing, thedashed line is the hindwing, and the dotted lines are the result ofthe Fourier approximation. . . . . . . . . . . . . . . . . . . . . . . 117

5.18 Vertical inertial forces for the lift (top) and thrust (bottom) strokes. 1185.19 Calculated vertical forces compared with measured forces for the

lift stroke. The top two images use computed fluid forces from aRe=200 simulation. Vertical lines indicate stroke reversal, and theshaded regions indicates forewing downstroke. . . . . . . . . . . . . 119

5.20 Calculated vertical forces compared with measured forces for thethrust stroke. All images are from a Re=200 simulation. Verti-cal lines indicate stroke reversal, and the shaded regions indicatesforewing downstroke. . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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Chapter 1

Overview and IntroductionDragonflies are one of natures most capable and acrobatic flyers. As predators of

the air, they are capable of startling feats of maneuverability. And yet they can

glide with their wings not moving for extended distances. They stand, therefore,

on a fascinating border between traditional flying and the unsteady dynamics of

insect flight.

Dragonfly flight, as with insect flight in general, is a rich field of research. Much

work is being accomplished, and yet there is a great deal that remains unknown.

A list of open questions includes:

• What are the exact wing kinematics used by the dragonfly during hovering

and cruising flight?

• How do these kinematics generate fluid forces?

• What are the most efficient and sufficiently accurate methods of modeling

the fluid forces acting on the wings?

• Can a 2D model capture the essential physics or is a full 3D model necessary?

• To what extent is it necessary to model the exact structure of a dragonfly’s

wing?

• Why do dragonflies choose the particular fore-hind wing phase relationships

they employ?

In 2000 Wang [59] showed numerically that a two-dimensional flapping motion

modeled after the kinematics observed in hovering dragonflies could generate suf-

1

2

ficient net force to sustain its own weight. It remained unclear to what extent

that lift generation mechanism applied to dragonfly flight, and more generally how

well a 2D model was capable of capturing the fluid forces generated by an actual

dragonfly.

To address these questions, we developed a 2D model of dragonfly flight based

on actual dragonfly kinematics and correlated the forces generated by the model

to forces measured on a dragonfly using those kinematics. This would offer some

validation of the idea of applying a 2D analysis to dragonfly flight.

First we needed a method of solving the incompressible Navier-Stokes equa-

tions for a system with multiple moving (and possibly deforming) wings, since the

importance of wing interaction could not be discounted. To this end, we devel-

oped a new numerical method that would theoretically be ideally suited to that

problem.

To measure the exact wing kinematics and forces acting on a dragonfly, speci-

mens were captured and tethered to a force sensor in front of a high speed digital

camera and a mirror in such a way that a 3D reconstruction of the wing motion

could be undertaken. While we do not know how the wing motions used by a

dragonfly glued to a post may differ from those seen in free flight, our goal is to

match detailed wing kinematics to measured forces in time. By modeling these

kinematics numerically in 2D and matching the force results against the measured

forces, we would have some basis for determining if a 2D model could capture the

necessary physics or if moving to a full 3D model was necessary.

We chose to start with a simple rigid model of a dragonfly wing, specifically a

rounded rectangle with an aspect ratio of 1/16. Analyzing the performance of that

model relative to measured forces will help to determine if such a simple model is

3

Figure 1.1: An example of libellula pulchella. Image courtesy of Greg LasleyNature Photography.

sufficient, or if we need to model finer structure or wing deformation to capture

the essential physics.

Chapter 2 describes in detail the numerical method developed to allow us to

model the fluid flow around both wings with second-order accuracy. The method

ultimately results in the ability to simulate five full strokes at a Reynolds number

of 200 in just under one days time on a 2.8 GHz desktop PC.

In the experiment described in chapter 3, a high-speed digital camera was used

to capture the kinematics of a tethered dragonfly at 1500 frames per second. A

mirror was placed to give two different simultaneous perspectives using a single

camera, allowing us to perform a full 3D reconstruction of the motion.

4

While three different species of dragonfly were used, the results presented are

all from a single specimen of libellula pulchella. This dragonfly displayed all of the

characteristic kinematics observed in the other two species, and most consistently

displayed kinematics that seem to resemble what would be used in actual flight. An

example of libellula pulchella can be seen in Figure 1.1. For reference, our specimen

weighed 0.585 grams and had a tip-to-tip wingspan of approximately 9 cm. The

maximum Reynolds number of the wing motions studied was approximately 3500

at the 2/3 length location, which is higher than the practical limit of 1000 for the

numerical method developed.

Chapter 4 describes the kinematic results obtained from the experiment. Of

four wing kinematics consistently observed, two were deemed both likely to be used

in flight and suitable for a 2D analysis. One is a motion associated with strong lift

and hovering behavior, and the other is associated with strong thrust and cruising

behavior.

In chapter 5 we compare the computational results with the actual forces mea-

sured during the experiment to determine how accurately the 2D computation

model reproduces the forces generated by the dragonfly. We also analyze the com-

puted forces by comparing them against the results of an ODE generated in the

quasi-steady limit [2, 40]. The differences between them give us some insight into

the source of the forces resulting from the kinematics observed. We compared

against simulations using only one wing, to probe the effects of wing interaction.

We also created simulations starting at different points in the kinematic history,

in order to explore the effects of wake interaction. The phase relationship between

the fore and hind wings were altered, and the effects on lift generated and power

expended were determined.

5

The remaining sections of this chapter review the previous work on dragonfly

kinematics, force production of flapping wings, and computational methods that

allow modeling of flows around multiple moving objects.

1.1 Dragonfly wing kinematics

As stated earlier, one of the goals of this research was to correlate specific wing

kinematic motions with force production. This would be useful both to validate

a model of the fluid forces on a dragonflies wings, but also provide insight into

the general strategy a dragonfly uses to move in the air. Any model of fluid force

production is going to be very sensitive to details of the motion of the wings. It is

especially critical to have accurate angle of attack information, as small changes

in this parameter can drastically affect the force produced.

Attempts to quantify the movement of a dragonflies wings have been performed

for many years. In 1975 Norberg [38] filmed dragonflies (Aeschna juncea L.) hover-

ing in the field using a camera capable of 80 frames per second. Since the wingbeat

frequency was 36 Hz this only captured two frames per stroke, but it was sufficient

to determine the basic features of the kinematics. In this case he noted that the

wings moved in a stroke plane angled at about 60◦ to the horizontal. During the

downstroke the wings were angled to be nearly horizontal, and during the upstroke

they rotated quickly at the bottom of the stroke (supination) to be nearly vertical.

He noted a 180◦ phase relationship between the fore and hind wings, which seems

to be typical of hovering behavior.

In 1984 Alexander [1] filmed tethered dragonflies in a wind tunnel and noted

that the phase relationship of the wings varied with activity. The general conclu-

sion reached was that the dragonflies beat their wings in-phase for short periods to

6

generate strong forces, like during takeoff or sharp turns. Otherwise they tended

to beat their wings mostly out-of-phase.

In his 1989 paper Rüppell [45] again took high-speed video of dragonflies in the

field, but captured behavior more varied than hovering. He describes kinematics

similar to those noted by Norberg and Alexander during takeoff. During forward

flight, however, he finds kinematics characterized by a low angle-of-attack down-

stroke followed by a high angle-of-attack upstroke. He associated in-phase strokes

with large force generation as well, but he also notes the existence of phase-shifted

(not completely out of phase, not in phase) strokes during forward flight.

Wakeling and Ellington [54, 55, 56] published in 1996 a comprehensive analysis

of dragonfly flight. The second part [55] investigates the kinematics used by the

dragonfly Sympetrum sanguineum. These studies again used high-speed film to

capture free-flight kinematics. Data on stroke plane angle, stroke angle, frequency,

and body orientation were generated. They used a single camera operated at 3000

frames per second. While methods were developed to determine the stroke angle

and stroke plane from a single view, insufficient detail was present to allow the

angle of attack to be specified.

Most recently, in 2002 Wang et al. [57] analyzed free-flight kinematics of dragon-

flies that included angle of attack information. They used a laser fringe technique

which required that the dragonfly fly into a specific region, and were able to take

data of forward and turning flight. Their results were very similar to the forward

flight kinematics observed by Rüppell. They were able to measure wing defor-

mation as well, and suggested that camber deformation during the stroke might

have significant effects on force production. The data covers little more than one

complete stroke, however. The material presented also made some simplifying

7

assumptions that prevented a full 3D reconstruction of the wings movements.

With the possible exception of the latest work by Wang [57], none of the pre-

vious work has resulted in a detailed kinematic description that included angle

of attack information, which is critical to the production of aerodynamic force.

The experiment described in this work resulted in a complete 3D reconstruction

of the wing motions of a tethered dragonfly, including angle of attack information.

The motions observed provide insight into the force production strategies involved,

even if one must keep in mind that the behaviour of a tethered dragonfly does not

necessarily match what occurs in free flight.

1.2 Wing Aerodynamics

There has been a great deal of research into various aspects of insect flight over

the past 40 years. For a comprehensive overview, see the recent review articles by

Sane [47], Lehmann [29], and Wang [60]. Following is a brief discussion of aspects

of this work that affect this research.

One focus of initial research was the effect of delayed stall, which allows a

transient increase in lift generation at high angles of attack due to an attached

leading-edge vortex. Ellington [15, 52] has suggested spanwise flow in a 3D flapping

motion as a stabilizing mechanism on the leading edge vortex. Subsequent work

has suggested that this effect is dependent on Reynolds number [3].

Another focus of recent work on insect flight has been the fruitfly. Dickinson [11,

13] has suggested several mechanisms for the generation of forces on flapping wings

greater than those predicted by standard steady-state aerodynamics. One is wing

rotation, or the increase in circulation and therefore lift on a translating object

that is also rotating. Another is wake capture, the mechanism of recovering the

8

fluid momentum created in the previous stroke.

Attempts have been made in the past to measure the forces generated by a

dragonfly. Unfortunately, measurement of such forces requires that the dragonfly

be tethered to a force sensor. Somps and Luttges [48] performed such a teth-

ered experiment on specimens of libellula luctuosa. The kinematics they observed

broadly resemble those associated with hovering flight, however the phase rela-

tionship was from 50◦ to 100◦ instead of the 180◦ observed in free hovering. They

noted peak forces more that 15 times the weight of the dragonfly, and reported a

force history with a single peak. They suggested that the presence of this single

peak indicated that the fore and hind wings were acting together in some way to

generate vertical force. The smoothness of the measured force history presented

has not been duplicated in the present work, however.

Wakeling & Ellington [54] investigated the steady-state aerodynamic properties

of a dragonfly wing. They determined polars of lift and drag coefficient for both the

forewing and hinding of the species sympetrum sanguineum. The also investigated

the lift and power requirements of a dragonfly in flight [56]. They lacked detailed

angle-of-attack information, however they did conclude the non-steady effects are

probably important in the generation of forces.

Most recently, Sun and Lan [49, 28] perfomed a numerical investigation of the

forces on wings in a flapping motion, both in 2D and in 3D. These simulations were

performed at a Reynolds number of about 2300, using the same assumptions as

we use later to calculate the Reynolds number over 2D flapping wings (maximum

velocity at 2/3 length location). The wing overset grid used had dimensions 27 ×

77 × 45 and the background grid had dimensions 90 × 72 × 46. The kinematics

used in this study were broadly similar to the lift stroke using a phase relationship

9

of 180◦ between the fore and hind wings. However they used simplified kinematics

which assumed a sine wave pattern for stroke angle and confined wing rotation

to regions around stroke reversal. They determined that wing interaction was

detrimental to vertical force production by about 15% overall, and that 2D results

strongly resembed 3D except for a 20% reduction in the calculated coefficient of

lift.

In this work, the actual wing kinematics of a dragonfly are modeled in two

dimensions. The computed force produced by this model is compared against

actual measurements of the force generated by a dragonfly using those kinematics.

This results in a verification of the validity of the model, as well as insight into the

source of wing forces during flight. By comparing against an ODE developed to

describe quasi-steady motion of a plate in a fluid, we can attempt to explore the

importance of unsteady effects such as wake interaction. In addition by modifying

the model, for example employing one wing only or modifying the fore-hind wing

phase relationship, we can explore the importance of wing interaction effects.

1.3 Numerical Method

The method described in chapter 2 results from our desire to model multiple objects

moving relative to each other in a 2D incompressible viscous fluid in an efficient

and accurate way. We would like to be able to model not only rigid structures in

prescribed motion, but also deformable objects moving at least partially in response

to fluid forces. Such a method would have applications beyond the analysis of

dragonfly flight, being useful anytime it is necessary to model moving boundaries

under lower to moderate Reynolds number incompressible flow.

Flow around a single moving object can be efficiently calculated by using a

10

single body fitted grid in the non-inertial reference frame, an example of which

can be seen in Wang [59]. This approach does not work, at least for a single static

discretization, for multiple objects moving relative to each other. It is possible

to use a grid mapping that varies with time as the objects move, see for exam-

ple [23]. This works well for simple geometries of fixed prescribed motions, but it

is computationally expensive for general motions.

Another method uses several overset grids, sometimes called chimera grids. A

separate grid is created that conforms to the surface of each object, and these are

overlaid on top of a general background grid. Some method of communicating

state information across grid boundaries is created. See, for example, Rogers [43],

Meakin [36], or Henshaw [21]. Sun and Lan [28, 49] have used this method to

simulate the flow around flapping wings in both two and three dimensions.

An alternative is to employ a fixed Cartesian grid method and adjust for the

presence of the boundary; this has the advantage of enabling the use of fast solvers

that take advantage of the regularity of the grid. One drawback of using such a

method is that there is no grid refinement in the area of greatest interest, near

the boundary of the objects. In fact, in general the boundaries are where the

errors associated with the method are usually the largest. Since we are interested

primarily in low to moderate Reynolds number ranges (about 10 to 5,000), we felt

it was possible for us to afford the computational cost of ensuring that there is

sufficient boundary discretization. Another difficulty lies in how to incorporate the

boundary conditions on the immersed surfaces in an accurate way.

Many existing methods have followed this philosophy, the Immersed Boundary

Method [41, 42] is one example. In this case the object in the flow is replaced

by forcing terms which represent the forces imposed on the fluid by the presence

11

of the object. Originally, the boundary representation was first-order accurate.

Recent refinements, for example Minion and Cortez [8] and Lai and Peskin [27],

increase the accuracy to second-order and above at the boundary. However such

a method runs into difficulty when the object is rigid, as the rigid boundary must

be approximated by stiff springs to generate forcing terms. The stiffer the springs

the better the model represents a rigid surface, but the smaller the stable timestep

becomes.

Mohd-Yusof [65] has devised an alternate means of calculating the forcing

terms, which has subsequently been refined and applied [16, 53]. This method

appears to work very well for rigid bodies in prescribed motion, but it is not clear

how to apply it to a streamfunction-vorticity formulation.

A number of other Cartesian methods have been applied to problems of irreg-

ular geometry[24, 18]. Udaykumar et al. [51, 64] has presented a finite-volume

Cartesian method, using local geometry and quadratic interpolation functions to

calculate flux across the elements that intersect the immersed boundary. The re-

sulting linear system is solved using a multi-grid method, and has been shown to

be applicable to moving geometry problems.

Recently Calhoun [5] put forward a method of modeling irregular shapes in a

Cartesian grid using a streamfunction-vorticity formulation. The irregular bound-

ary is represented on the underlying flow solver using discontinuity conditions.

This is an extension of a method developed by Mayo and McKenney[35] to quickly

solve a Poisson equation on an irregular region using a regular Cartesian under-

lying grid. To calculate the values of the discontinuities, the method requires a

solution to a rather large linear system which is eventually condensed into a man-

ageable matrix. For a static geometry, once this system has been LU decomposed,

12

the discontinuity values for each step may be found via backsubstitution. For

a moving geometry, the linear system must be regenerated every timestep. The

computational cost associated with this is prohibitive; as a result, this method is

very efficient for static geometry but not practical for moving geometry. Here we

develop a method of calculating discontinuity values that does not suffer such a

penalty if the geometry is moving with respect to the regular grid.

In the method presented in chapter 2, the underlying flow solver is also based

on a streamfunction-vorticity formulation calculated on a regular Cartesian grid.

Our method also corrects for the presence of the irregular moving objects by using

embedded discontinuities in the streamfunction. Instead of using a linear system

to couple all the variables involved, we calculate the values of the discontinuities

in separate steps. Streamfunction discontinuity values associated with the no-

penetration condition are calculated using a superposed homogeneous solution.

Boundary vorticity is generated to satisfy the no-slip condition by using an in-

terpretation of Thom’s formula, which is a method of converting the Neumann

boundary condition on the streamfunction into a Dirichlet boundary condition on

the vorticity. The net result is a method that asymptotes to a computational cost

of O(N ln(N) + M) operations per time step where N is the number of nodes in

the regular grid and M is the total surface discretization of all immersed objects.

The computations efficiency thus aquired enables a computer model to be run

on standard desktop PC’s in a reasonable time frame. This allows multiple pa-

rameters to be explored, as well as allowing the creation of modified simulations

to test force production sources.

Chapter 2

Numerical MethodIn this chapter the algorithm developed is presented as well as initial examples of

its use. Section 2.1 briefly describes the streamfunction-vorticity formulation of

the 2D Navier-Stokes equations for incompressible fluids. Section 2.2 provides an

overview of the method used to solve this formulation, the details of which are

presented in sections 2.3 to 2.9. Section 2.10 provides examples of the algorithm,

primarily against the problem of 2D incompressible flow past a circular cylinder.

We also provide an example of multiple moving objects, two cylinders moving past

one another. Finally, section 6.1 contains our conclusions and possible future work.

This work has been published in an earlier form in the Journal of Computational

Physics [46].

2.1 Streamfunction-vorticity formulation

In a streamfunction-vorticity formulation, the equations governing incompressible

2D flow can be written as

∂ω

∂t+ u · ∇ω = ν∆ω (2.1)

∆ψ = ω (2.2)

u = ∇⊥ψ = (−∂ψ

∂y,∂ψ

∂x) (2.3)

ω = ∇× u (2.4)

where ω is the vorticity (a scalar in 2D flow), u is the velocity, ν is the kinematic

viscosity, and ψ is the streamfunction.

The equation governing vorticity evolution (2.1) is derived from the full Navier-

13

14

Stokes equations by assuming incompressiblity and taking the curl. The incom-

pressibility assumption (∇ · u = 0) is implicitly satisfied by the use of a stream-

function to determine the velocity, as can be easily seen by taking the divergence

of equation (2.3). Note that some authors use ∆ψ = −ω for equation (2.2) and

use the negative of our choice for the perpendicular gradient operator ∇⊥. The

effective difference is a change in the sign of calculated ψ.

One interesting aspect of this formulation is the nature of the boundary con-

ditions. For an object moving in the flow, the boundary conditions applied are

derived from velocity conditions. Namely, that there is no penetration (normal

velocity) and no slip (tangential velocity). This translates into conditions on ∂ψ∂n

and ∂ψ∂s

where n represents a direction normal to the boundary and s represents a

direction tangential to the boundary.

While other boundary conditions are generally used for the far field, in this

chapter they will be presented as also using velocity-prescribed conditions for sim-

plicity. On the far field, the ∂ψ∂s

condition may be integrated around the boundary

since the streamfunction can have an arbitrary constant added without affecting

the solution. This results in both Dirichlet and Neumann boundary conditions for

the Poisson equation for velocity (2.2) and none for the vorticity evolution equa-

tion (2.1). It is important to note that, for the full system of both equations, the

proper number of boundary conditions exist to make this a well-posed problem.

2.2 Overview

One of the major difficulties that must be overcome is solving for a velocity field

that is both consistent with the vorticity distribution and satisfies the boundary

conditions. Lighthill [31] introduced a line of reasoning that addresses this issue.

15

Figure 2.1: Determination of velocity field by vorticity distribution andboundary conditions.

Consider a particular vorticity distribution that specifies a velocity field via the

Biot-Savart law. A cartoon of a velocity distribution resulting from a simple vor-

ticity distribution is shown in the left side of Figure 2.1. The resulting velocity field

is not unique in the sense that any irrotational velocity field may be superposed

upon it without changing the vorticity distribution. The irrotational field can be

determined by the kinematic constraint representing the no-penetration boundary

condition and the imposition of the far-field boundary condition. The superposi-

tion of these fields uniquely determines the velocity field for the problem subject

to the no-penetration condition, as shown in the right side of Figure 2.1.

However in general the resulting velocity field will have non-zero tangential

velocities at the solid boundaries. In order to satisfy no-slip conditions, Lighthill

proposes the existence of a discontinuity in the velocity field. This discontinuity

would manifest itself as a singular distribution of vorticity along the boundary.

The vorticity created by this ’vortex sheet’ would then be diffused and convected

into the surrounding fluid.

In some sense the presence of an impenetrable object is communicated instantly

16

PSfrag replacements

ψP = f(s) ψL = 0

∂ψL∂s

= −∂ψP∂s

ψ = f(s)

∂ψ

∂s= 0

∆ψP = ω ∆ψL = 0∆ψ = ω

+ =

Figure 2.2: Simplified cartoon of the process for finding the velocity stream-function for a fixed geometry.

throughout the whole of the fluid and introduces a global adjustment to the flow

field. This adjustment does not create any additional vorticity in the system.

Satisfying the no-slip condition, on the other hand, is a local adjustment to the flow

field involving the introduction of vorticity into the system along the boundaries.

In terms of differential equations, the particular solution is given by the distri-

bution of the right-hand-side field while a homogeneous solution is given by the

Dirichlet boundary conditions. A much simplified cartoon of the sequence we follow

is presented in Figure 2.2, where f(s) represents the assigned Dirichlet boundary

conditions. The difference between the resulting streamfunction normal gradient

at the boundary and the desired Neumann boundary condition implies a disconti-

nuity in the normal gradient, which can be enforced by a singular distribution of

vorticity along the boundary.

Following this sequence allows us to satisfy the no-penetration and no-slip con-

ditions in separate distinct adjustments, rather than being coupled in one large

system.

To summarize, the sequence is as follows:

17

• Solve Poisson’s equation for velocity including the discontinuity represented

by the body (vorticity zero on the inside) (ψP ).

• Solve a homogeneous inviscid problem using boundary methods in such a

way that when it is superposed upon the previous solution the no-penetration

condition is satisfied (ψL).

• Distribute vorticity around the boundaries to satisfy the no-slip condition.

• Integrate the vorticity equation in time, including the effects of those singular

sources.

Following are the details of the implementation of each step in this sequence.

2.3 Underlying flow solver on the regular grid

For the underlying flow solver on the regular grid, we use a method recently put

forward by E and Liu in a paper on vorticity boundary conditions [14] .

Using second order centered finite differencing in space, equation (2.1) can be

written in discrete form as

∂ωi,j∂t

≈ −uxDx[ωi,j] − uyDy[ωi,j] + ν(Dxx[ωi,j] +Dyy[ωi,j]) (2.5)

where

Dx[ωi,j] =1

2∆x(ωi+1,j − ωi−1,j) (2.6)

Dy[ωi,j] =1

2∆y(ωi,j+1 − ωi,j−1) (2.7)

Dxx[ωi,j] =1

∆x2(ωi+1,j + ωi−1,j − 2ωi,j) (2.8)

Dyy[ωi,j] =1

∆y2(ωi,j+1 + ωi,j−1 − 2ωi,j) (2.9)

18

PSfrag replacements

ψ & ∂ψ∂n

Figure 2.3: A regular cartesian grid representing the flow.

This equation is marched forward in time using fourth-order Runge-Kutta integra-

tion. In addition the Poisson equation for velocity can be represented by

Dxx[ψi,j] +Dyy[ψi,j] = ωi,j (2.10)

ui,j = (−Dy[ψi,j], Dx[ψi,j]) (2.11)

As described by E and Liu, the calculation sequence for each Runge-Kutta

substep is as follows:

• Calculate ωn+1 from ωn using equation (2.5) explicitly on all nodes not on

the far field boundary.

• Calculate ψn+1 by solving a Poisson equation (2.10) on the discrete regular

grid using Dirichlet boundary conditions. Note that the boundary values of

ωn+1 are not needed for this step.

• Calculate velocities using ψn+1 and equation (2.11).

• Calculate boundary values of ωn+1 by using Thom’s formula (described im-

mediately following), which also satisfies the Neumann condition for ψn+1.

19

Using the left-hand-side boundary as an example, Thom’s formula can be writ-

ten as

ω0,j =2

∆x2

(

ψ1,j − ψ0,j − ∆x∂ψ

∂x 0,j

)

+Dyy[ψ0,j] (2.12)

where ∂ψ∂x

represents the prescribed Neumann boundary conditions. This formula

can be easily derived by assuming a “ghost point” value at ψ−1,j , using both the

discrete equations for vertical velocity and vorticity at (0, j), and solving for vor-

ticity. Thom’s formula has several possible interpretations, which will be discussed

later. It was for many years considered to be only first-order accurate, but it has

recently been shown [22] to be second-order accurate.

The important thing to note about the above calculation sequence is that the

values of boundary vorticity are calculated last using the streamfunction informa-

tion from that step. This is possible because the Poisson equation for the stream-

function can be solved without far field boundary values of vorticity. Also, note

that while the use of fourth-order Runge-Kutta integration in time adds additional

calculations per time step, it also relaxes the time-step restrictions associated with

stability in an explicit method.

The Poisson equation can be solved using any available “black box” solver,

here, and where it appears elsewhere in this chapter. In our case we used an FFT

solver, which takes O(N ln(N)) time.

20

2.4 Incorporating discontinuities into a Poisson equation

solver

2.4.1 Introduction

An important tool used in the complete numerical method ultimately developed is

the ability to solve a Poisson equation on a regular grid while including the effect of

known discontinuities in the solution on an irregular boundary. Poisson’s equation

can be written as

∆ψ = ω (2.13)

and in this case we assume Dirichlet boundary conditions on the regular exterior.

Using a regular underlying grid, we overlay an irregular boundary with known

discontinuities in [ψ], [∂ψ∂n

], and [∂2ψ

∂n2]. The goal is to find the effects of these

discontinuities on the discrete ω distribution to second-order accuracy, allowing a

subsequent Poisson equation solution to be second-order accurate.

Mayo [35, 33] put forward a method of solving the Poisson equation numer-

ically on an irregular region using a regular underlying grid. The presence of

the irregular boundary was included with second-order accuracy by treating the

boundary as a discontinuity in the values of vorticity at those locations, resulting

in a discontinuity in the second derivative of the streamfunction, as well.

Leveque [30] later generalized the approach, showing a method of maintaining

second-order accuracy for equations of the form ∇ · (β∇ψ) + κψ = f including

discontinuities in β, κ, and f . Leveque’s method requires the solution of a 6 × 6

system of equations for each irregular point on the boundary.

In our particular case, there is no need for a discontinuity in β, so we can

21

PSfrag replacements

xj xj+1x∗

dψ

ψj

ψ̃j+1

ψj+1

Figure 2.4: Analytical and discrete solution to the 1-D example.

automatically simplify Leveque’s formulation. In practice, we use a method that

allows an explicit correction for the presence of the discontinuity. There is no need

to solve a system of equations.

2.4.2 A 1D Example

To better understand the concepts involved, it is instructive to look at a simple

1D system. Specifically, consider the problem

∂2ψ

∂x2= ω (2.14)

ω = δ(x∗)[∂ψ

∂x

]

(2.15)

ψ(0) = 0 (2.16)

ψ(L) = 0 (2.17)

The solution to this problem is a sequence of two lines joined at x∗ whose difference

in slope is [∂ψ∂x

], as seen in Figure 2.4.

A center-difference-scheme formulation of this problem could be given as

ψi+1 − 2ψi + ψi−1∆x2

= ωi (2.18)

ψ0 = 0 (2.19)

22

ψN = 0 (2.20)

Thus, given ψi, the above difference scheme gives zero ωi’s everywhere except

for those locations whose stencil crosses the discontinuity at x∗, which we call xj

and xj+1. Specifically for point xj,

ωj =d[

∂ψ

∂x

]

∆x2(2.21)

This immediately suggests that when solving the inverse problem, in order to

recover ψi correctly using the same finite difference scheme, ωj has to be modified

from its continuous value by the amount specified in equation (2.21). The discrete

value of ω at point xj+1 must also be adjusted in a similar manner.

This result can be generalized to discontinuities in ψ and any of its derivatives,

resulting in a Taylor-like expansion

ωj = ω̃j +[ψ] + d

[

∂ψ

∂x

]

+ 12d2

[

∂2ψ

∂x2

]

+ · · ·

∆x2(2.22)

It is worth noting that an alternative method for handling the discontinuity

is expressing it in terms of a delta function and approximating the delta function

by a distribution function of a finite width, as is done, for example, in Peskin’s

Immersed Boundary Method [41, 42]. A difficulty there is to find an appropriate

distribution, which, in general, is unknown apriori. In a sense, our method uses

the finite difference scheme to determine the distribution function, which turns

out to be a hat function for the centered-difference scheme. Also, note that the

above approach can be extended in a straightforward way into multiple dimensions

(as is shown in Section 2.4.3 ), while finding the appropriate multi-dimensional

distribution functions can be cumbersome.

23

PSfrag replacements

n̂

d

= irregular point

= nearest normal

Figure 2.5: 2D geometry of an irregular discontinuity.

2.4.3 An efficient 2D implementation

The points affected by the presence of the boundary are illustrated in Figure 2.5.

Points whose standard five-point finite difference stencil crosses the boundary are

called irregular points, following the convention introduced by Mayo [33]. It is

these points whose values of discrete vorticity must be modified.

Referring to Figure 2.5, we would like to find the correction to the value of

ψ4 that a Dxx[ψ�] + Dyy[ψ�] operation would see if the discontinuity were not

present. In other words, if the values of ψ+, ∂ψ+

∂n, and ∂

2ψ+

∂n2continued on as they

were without interruption on the side. Armed with this correction to ψ4, which

we will call ψc4, we can calculate the effective ω� using standard finite-difference

formulas.

In our approach we find the nearest normal to the boundary for every irregu-

lar point, (the point marked as • in Figure 2.5). In some sense this is the most

accurate location to choose, since we know the discontinuities in terms of the nor-

mal derivative. Also, it simplifies expressions for ψc4

since cross terms in the s

direction disappear. This does, however, introduce problems if there is no nearest

normal (as in the case of a sharp corner) so in the current formulation smooth-

24

ness assumptions are made. The final correction formula is a simple Taylor series

expansion

ψc4 = [ψ•] + d[∂ψ•∂n

]

+1

2d2

[∂2ψ•∂n2

]

(2.23)

where d is the distance to the nearest normal point. Note that very careful attention

must be paid to signs, and to which direction the normal is defined. Our sign

conventions are that [ψ] = ψ+ − ψ−, where + represents the positive normal side.

Also, the distance to the interface d is positive for points on the positive normal

side and negative for points on the other side. In our example in Figure 2.5 the

distance d has a negative value.

For our particular implementation we do not directly know[

∂2ψ•∂n2

]

, instead we

know [ω]. If we assume a local curvature coordinate system we can write

[ω] =[∂2ψ

∂n2

]

+ κ[∂ψ

∂n

]

+[∂2ψ

∂s2

]

(2.24)

where κ is the local curvature at that point. And since[

∂2ψ

∂s2

]

is uniquely determined

by the distribution of [ψ] we can ultimately write

ψc4

= [ψ•] + d[∂ψ•∂n

]

+1

2d2

(

[ω] − κ[∂ψ•∂n

]

−∂2[ψ•]

∂s2

])

(2.25)

The final correction to the vorticity is then

ω� = ω� −ψc4

∆x2(2.26)

In practice we use the following sequence. We first calculate and store ψc for

all irregular nodes using equation (2.25). We then calculate the correction to ω for

each irregular point using the values of ψc for those nodes in its stencil that cross

the boundary.

The geometric information required can be found in O(M) time, where M is

the number of points discretizing the boundary. This assumes that the length of

a boundary segment is the same order of magnitude as the grid spacing.

25

PSfrag replacements

-1-1

1

1

R = 12

Figure 2.6: Geometry of the 2D discontinuity tests.

2.4.4 Convergence Test

In the description of his method for calculating discontinuity correction values,

Leveque shows second order accuracy analytically. This calculation has not been

done for our alternate method, but we have performed tests on the same example

problems performed by Leveque [30]. Figure 2.6 shows the geometry used for all

the examples, a circle inside a simple regtangular region. Different equations for

ψ are assumed to exist on the inside and the outside of the cylinder, resulting in

discontinuities along the circular boundary.

Table 2.1 summarizes the results for those problems that were tested by Leveque

as well. The first column displays the equation for ψ used on the inside of the circle,

and the second that used on the outside. The third column shows the resulting

discontinuity conditions on the circular boundary for that problem. The problems

were calculated using an n×n grid, where n is listed in the fourth column. The fifth

and sixth columns present the errors calculated from the analytical solutions on the

inside and outside, with Leveque’s results shown for comparison. The error norm

used was ‖En‖∞ = max|u(xi, yi) − ui,j|, or the maximum absolute error over any

gridpoint. The seventh column shows in ratio of the previous error to the error with

26

Table 2.1: Results of 2D discontinuity tests.

‖En‖∞

Inside Outside Discontinuities n Leveque [30] Present Ratio

1 1 + ln(2r)[

∂ψ

∂n

]

,[

∂2ψ

∂n2

]

20 2.3908e-3 6.1254e-3 -

40 8.3461e-4 1.2600e-3 4.87

80 2.4451e-4 3.1379e-4 4.01

160 6.6856e-5 7.7495e-5 4.05

320 1.5672e-5 1.8041e-5 4.21

ex cos y 0 [ψ],[

∂ψ

∂n

]

20 4.37883e-4 2.3669e-4 -

40 1.07887e-4 6.6066e-5 3.98

80 2.77752e-5 1.5251e-5 3.95

160 7.49907e-6 3.8565e-6 4.09

320 1.74001e-6 9.4390e-7 3.97

x2 − y2 0 [ψ],[

∂ψ

∂n

]

20 - 2.5535e-15 -

40 - 2.7478e-15 -

80 - 1.1546e-14 -

160 - 1.2812e-14 -

320 - 2.6298e-14 -

27

a doubling of the discretiztion. We clearly show second-order convergence, with

our errors being either slightly better or slightly worse than Leveque depending on

the problem. For a problem that can be represented by a quadratic polynomial

(as seen in the third system shown), we show convergence to numerical limits for

all discretizations, as expected.

2.5 Finding the velocity field

2.5.1 Solving the Poisson equation

The domain of our solution will always be a regular rectangular region, chosen

around the full range of motion of the objects immersed in the fluid. The underlying

grid will always be regular but the discretization need not be the same in the x

and y directions.

Let us assume for present purposes that we are imposing velocity boundary

conditions on both the far field boundaries and on the boundaries of the immersed

object. We can then describe the boundary conditions as being in both ∂ψ∂s

and ∂ψ∂n

.

We can integrate ∂ψ∂s

along the far field boundary and choose an arbitrary value

(generally 0) for the integration constant, producing standard Dirichlet boundary

conditions for the far field.

Let us also assume that we start with a known vorticity distribution in the

fluid, as well as known values of the vorticity on the immersed object boundaries.

For the first timestep both are generally zero. For subsequent steps, the vorticity in

the regular grid outside the immersed object is given by integrating the vorticity

evolution equation in time, and the vorticity on the boundary of the object is

calculated to satisfy the no-slip condition as described in detail later.

28

PSfrag replacements

ω = 0

Figure 2.7: Irregular geometry overlaid on regular grid, causing a disconti-nuity in ω.

Figure 2.8: Interpolation points chosen to determine ψP . The circle marksthe point at which a value of ψP is needed, and the squares markthe gridpoints selected to determine this value.

We would like to calculate the velocity field created by that vorticity distribu-

tion by solving the equation

∆ψP = ω (2.27)

where ω = 2Ω inside the object if it is has an angular velocity of Ω.

We can now use the discontinuity method presented in section 2.4 to solve the

Poisson equation to second-order accuracy including the effects of the discontinuity

in ω along the irregular boundary. We call the streamfunction that results ψP .

The ψP field is interpolated onto the irregular boundary to calculate∂ψP∂s

on

29

-2. 5 -2 -1. 5 -1 -0. 5 0 0.5 1 1.5 2 2.5-2. 5

-2

-1. 5

-1

-0. 5

0

0.5

1

1.5

2

2.5

Figure 2.9: Contours of ψP for impulsively started cylinder.

the object boundaries. To perform this interpolation, six points on the inside of

the object are selected. Using points entirely on the inside of the object was found

in practice to produce the best results, as in theory the stream function should be

either constant or linearly varying within the body. An example of this selection

is seen in Figure 2.8. These points are used to determine the constants in a 2D

quadratic interpolation using the equation

f(x, y) = Ax2 +By2 + Cxy +Dx+ Ey + F. (2.28)

The solution matrix for the constants is inverted ahead of time for a list of possible

non-singular choices of points on a regular grid. This allows the efficient interpo-

lation/extrapolation of ψP on the boundary without involving points across the

boundary. As described later, it is sufficient to determine values of ψP instead of

using differences to calculate values of ∂ψP∂s

.

Figure 2.9 illustrates the solution found for ψP in the case of flow past an

impulsively started circular cylinder at time 0+. Since there is no vorticity, either

30

in the fluid or the boundary in the initial state, the solution found in this case is

simply a uniform flow to the right.

2.5.2 Satisfying no-penetration

In section 3.3 we calculated values of ∂ψP∂s

(actually ψP ) on the immersed object

boundaries. Note that the velocity field specified by ψP does not satisfy either the

no-penetration or no-slip condition on the boundaries of the immersed objects. We

know that ultimately we need our solution to match a given ∂ψ∂s

which is determined

by the no-penetration condition. We solve for a value ∂ψL∂s

such that

∂ψP∂s

+∂ψL∂s

=∂ψ

∂s. (2.29)

Now we can integrate ∂ψL∂s

to find Dirichlet conditions for the immersed object

boundary and solve the system

∆ψL = 0 (2.30)

with far-field boundary conditions

ψL = 0 (2.31)

and boundary conditions on the object of

ψL(s) =∫ s

0

∂ψL∂s

ds+ C (2.32)

where C is a constant of integration.

To determine C, we appeal to Lighthill’s interpretation of the streamfunction-

vorticity formulation. One can describe the homogeneous correction we are seeking

as an inviscid correction to the flow field. Any circulation created around an object

in the flow by this correction implies a source of vorticity created somewhere within

31

-2. 5 -2 -1. 5 -1 -0. 5 0 0.5 1 1.5 2 2.5-2. 5

-2

-1. 5

-1

-0. 5

0

0.5

1

1.5

2

2.5

Figure 2.10: Contour of ψL for impulsively started cylinder.

the object. We assert that this correction should not add any vorticity to the

system.

This gives us a method for determining C for each object immersed in the

flow. We determine what value of C will result in zero circulation added in the ψL

correction. To do this, we require that

∮

∂ψL∂n

ds = 0 (2.33)

This adds one equation and one unknown for each object immersed in the flow.

In practice (as mentioned in the previous subsection) we do not actually calcu-

late ∂ψP∂n

and then integrate it as that would accumulate unneccessary numerical

error. Instead we interpolate to find values of ψP on the boundary, and integrate

∂ψ

∂sthe velocity boundary conditions to find values of ψ on the boundary accurate

to some arbitrary constant datum. Since our method solves for a constant offset

to the Dirichlet conditions on the boundary, we simply assign ψL = ψ − ψP .

To find the full ψL field, we first solve the Dirichlet-Neumann map. Meaning,

32

using the Dirichlet boundary values of ψL, we can solve an integral relation result-

ing in unique values of ∂ψL∂n

. For ease of development, we currently use a simple

Boundary Element method modified to solve for the extra unknown and constraint

for each object. The underlying integral equation used is

Θ(~p)ψL(~p) =∫

SψL(~x)

cos(θ(~p, ~x))

r(~p, ~x)dS −

∫

S

∂ψL(~x)

∂nln(r(~p, ~x))dS (2.34)

where Θ(~p) = 2π on the interior of the region and represents the interior angle if

~p falls on the boundary, θ(~p, ~x) is the angle between the surface normal and the

vector from the surface to ~p, and r(~p, ~x) is the distance from the surface point to ~p.

This equation is discretized, converted into a linear system on the boundary points,

and solved using LU decomposition and back-substitution. This results in values

of ∂ψL∂n

at the discrete boundary points. The code currently is capable of solving the

discrete boundary equation using either constant or linear discontinuous elements.

Linear elements provide second order accuracy, while constant elements provide a

working solution much more efficiently.

It is possible to use a multipole method to solve this system in times ap-

proaching order M where M represents the number of boundary elements. See for

example [19, 6]. We intend to implement this alternate solver at a future date,

however the necessity of enabling mixed Dirichlet Neumann boundary conditions

complicates matters somewhat. The ψL field that results from such a computation

is displayed in Figure 2.10.

We now have the boundary values of ψL and∂ψL∂n

, which can be used in the

discrete form of equation (2.34) to fully describe the interior ψL field. Actually

evaluating the boundary integral equations for every point in the field would be

computationally expensive, even with a multipole evaluator. However, we can

once again use the concept of embedded discontinuities presented in section 2.4.

33

-2. 5 -2 -1. 5 -1 -0. 5 0 0.5 1 1.5 2 2.5-2. 5

-2

-1. 5

-1

-0. 5

0

0.5

1

1.5

2

2.5

Figure 2.11: Contours of ψ for impulsively started cylinder.

We treat the boundary values of ψL and∂ψL∂n

as discontinuities in the ψP problem.

Now we include the effects of discontinuities in all three of [ψ],[

∂ψ

∂n

]

, and [ω] in a

discrete Poisson equation problem. This has the effect of solving for the ψL field

and superposing it on the ψP solution at the same time. The result is the discrete

solution of ψ which now conforms to both the far field and the immersed object

Dirichlet boundary conditions.

An example of the solution (to this point) for the impulsively started circular

cylinder is shown in Figure 2.11.

2.6 Calculating boundary vorticity

At this point, the no-penetration condition on the far field and immersed object

boundaries have been satisfied. It remains to satisfy the no-slip condition. On

the far field boundary, we can simply use Thom’s formula exactly as for the pure

cavity flow case described in [14]. It seems logical, then, to seek a generalization of

34

PSfrag replacements

∆x2

ψ1,jψ1,j

ψ0,jψ0,j

ψ−1,j

Geometric Vortex Sheet

Figure 2.12: Interpretations of Thom’s formula.

Thom’s formula for the irregular immersed boundaries. First, however, we explore

Thom’s formula and pose some interpretations of it.

In a previous section, we discussed Lighthill’s view of the no-slip condition being

enforced by the existence of vortex sheets. These sheets represent discontinuities

in the velocity field from some finite relative tangential velocity with the surface

to no-slip. For consistency, if nothing else, we would like to be able to relate these

vortex sheets to Thom’s formula.

Thom’s formula for the left boundary can be written as

ω0,j =2

∆x2

(

ψ1,j − ψ0,j + ∆xV0,j)

+Dyy[ψ0,j ] (2.35)

where V0,j now represents the desired tangential velocity at the surface point (0, j),

and Dyy is a standard second-order centered finite difference formula. The left side

of Figure 2.12 shows a geometric interpretation of the derivation of Thom’s formula

35

using a “ghost point” outside the computational domain. The “ghost point” (−1, j)

is positioned in such a way that the slope between it and the point (1, j) matches

the desired Neumann boundary conditions. This determines the difference in slope

between (−1, j) to (0, j) and (0, j) to (1, j), which in turn determines the effective

boundary vorticity.

There is another available interpretation of Thom’s formula, one which can

in fact be used to derive it. If we desire a discontinuity in tangential velocity of

[V ], the vorticity sheet strength γ necessary to cause that discontinuity is simply

γ = −[V ]. Looking at equation (2.35), we see that the ψ1,j−ψ0,j can be written as

a first-order forward finite-difference approximation of the slope at the boundary.

We can now re-write equation (2.35) as

ω0,j = −2

∆x[V ] (2.36)

where [V ] now represents the difference between desired and actual tangential

velocity.

We can see that the formula for boundary vorticity is now different from the

vortex sheet strength by a factor of 2∆x

. It seems that Thom’s formula represents

the value of vorticity a boundary node should have to represent a sheet strength of

-[V ]. It is pointless to use the value of vorticity at the sheet location, since that is

infinite by virtue of it being a singular source. However, in a discrete system, the

vorticity at a node can be thought of (in a very approximate sense) as representing

a volume average of the total vorticity in a region it occupies. If we consider the

occupying region of a boundary node to be half the distance to the next node

inwards (since the other direction is outside the computational domain) we arrive

at exactly Thom’s formula.

It is troubling that the value of vorticity at the boundary is grid-dependent in

36

Figure 2.13: Interpolation method for the determination of ∂ψP∂n

. The darkcircle indicates the location where ψP is to be determined. Theblack squared indicate the grid points on which the interpolationis conducted. First each row is interpolated linearly to findvalues at the cross points. Then the cross point are interpolatedto find the value at the dark circle.

the sense that it contains a ∆x term. This brings up the issue of how to generalize

Thom’s formula for an irregular boundary, since there is no clear consistent ∆x. It

is also important to realize that any arbitrary distance will not suffice; whatever

∆x is chosen at a point on the irregular boundary must be consistent with the

finite difference equations used to evolve the vorticity.

The choice of ∆x and its integration into the vorticity evolution equation is

left to the next section. For now, assuming we have a value of ∆x, we still need

to calculate an effective boundary vorticity. Since the desired tangential velocity

is known, we need to calculate an accurate value of actual tangential velocity or,

equivalently, the normal slope of the streamfunction. We do this by calculating ∂ψP∂n

at the same time as ∂ψP∂s

is calculated in the previous section. Since ∂ψL∂n

is directly

returned by the Dirichlet-Neumann map solved on the boundary, the superposition

gives us ∂ψ∂n

or the tangential velocity at the surface.

The determination of ∂ψP∂n

is accomplished by finding a value of ψP at a cer-

37

tain distance from the surface and then using first order differencing, as shown in

Figure 2.13. The distance d is the maximum of the grid spacing in the x and y

direction. The interpolation is done using five linear third order polynomial inter-

polations on a four by four grid of selected gridpoints. First four interpolations

are done on the horizontal columns, and then a fifth interpolation is done in the

vertical direction. The position of the four by four grid is determined by finding

the position where the point would be in the center cell and then moving one

grid in the x and y direction away from the surface. Any selected gridpoints that

still fall across the boundary have their values corrected for the presence of the

discontinuity.

2.7 Integrating vorticity

We have now shown how to solve the Poisson equation to determine the velocity

field and how to generate vorticity on the boundaries to satisfy the no-slip con-

dition. We have not yet shown how to introduce the singular vorticity and the

discontinuities in the flow field into the discrete vorticity evolution equation:

∂ωi,j∂t

= −u

2∆x(ωi+1,j − ωi−1,j) −

v

2∆y(ωi,j+1 − ωi,j−1) (2.37)

+ν

∆x2(ωi−1,j − 2ωi,j + ωi+1,j) +

ν

∆y2(ωi,j−1 − 2ωi,j + ωi,j+1)

The problem lies in how to handle the exterior irregular points, marked as an

× on figure 2.14. These are points whose values must be calculated each timestep

but at which the finite-difference stencil crosses the discontinuity. Our goal is to

find a way to calculate values for these points at each step without drastically

reducing our stable time step.

On possibility would be to make use of the same discontinuity tools used in

38

PSfrag replacements

outlier

middle

inside

∆n ∆s

∆n = max(∆x,∆y)

∆s = min(∆x,∆y)

Figure 2.14: Local overset grid.

previous sections. However, to do this with second order accuracy would require

knowing the values of [∂ω∂n

] on the boundary (and preferably [∂2ω∂n2

] as well). We

do not have a method for determining this without performing finite-differencing

on the surrounding values of vorticity, which would introduce serious stability

problems.

Gibou [18] provides a met