numerical study of strong free surface flow and breaking waves

NUMERICAL STUDY OF STRONG FREE SURFACE

FLOW AND BREAKING WAVES

by

Yi Liu

A dissertation submitted to The Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

January, 2013

c⃝ Yi Liu 2013

All rights reserved

Abstract

A numerical tool based on Eulerian Cartesian grid, which combines the strength of

level-set method, volume-of-fluid method, ghost fluid method, and immersed bound-

ary method, is developed for the simulation of interfacial flow and flow–structure

interaction problems.

Direct numerical simulation of two dimensional breaking waves and large eddy

simulation of wind turbulence over three dimensional steep/breaking waves are per-

formed. The relationship between the breaker type and the initial wave steepness is

investigated. Evolution of skewness, asymmetry and steepness of waves is examined.

Energy loss and energy dissipation rate are quantified. Empirical dissipation mod-

els are validated and model coefficients are quantified. Wind velocity profiles over

steep/breaking waves are studied. Wind stress and drag coefficients are quantified.

Surface current and underwater turbulence generation are studied. Airflow separation

over breaking wave is identified. Form drag during the breaking process is quantified.

Wind effect on wave breaking is also discussed.

Free surface interaction with underwater turbulence under different gravity and

ii

ABSTRACT

surface tension effects is simulated. Different flow regimes are identified. Thickness

and distribution of the intermittency layer is calculated for different Froude andWeber

numbers. Influence of gravity and surface tension effects on the blockage effect of the

free surface is studied. Turbulence statistics and flow structures such as splat are also

investigated.

A multi-scale modeling approach for the simulation of the interaction between

wind-wave and structures is developed. The large-scale is simulated through large

eddy simulation of wind on boundary fitted grid over wave field simulated by high

order spectral method. The local-scale is simulated using the numerical tool discussed

above. Inflow condition for local-scale comes from the large-scale simulation, which

makes the simulation more realistic. Wind-wave interaction with surface piercing

object is simulated with the approach and wave phase dependence of the wind drag

is observed.

Advisor:

Professor Lian Shen

Reader:

Professor Robert A. Dalrymple

Professor Tak Igusa

iii

Acknowledgments

The dissertation would not have been possible without the guidance and help of

several individuals.

First and foremost, I would like to thank my advisor Dr. Lian Shen for his unselfish

advice and help during my Ph.D study. His wise and diligence inspired my interest

in study and my passion in research. He led me into the interesting world of wave

and turbulence and trained me to become a professional researcher from a layman.

Working with him is a precious experience of my life.

I am very grateful to Dr. Robert A. Dalrymple for his advice and help in the study

of wave breaking, SPH, and GPU computing, and the inspiring discussions about my

research. I am also thankful to him for his precious time reading and revising my

thesis and paper.

I would like to thank Dr. Tak Igusa for serving as my thesis committee member.

His suggests and comments about my research are very beneficial for me to finish the

thesis.

I would also like to thank Dr. Alireza Kermani, Dr. Di yang, Dr. Xin Guo, Dr.

iv

ACKNOWLEDGMENTS

Hamid Reza Khakpour, Meilin Chen, Zhitao Li, Shengbai Xie, Guotu Li, Yi Hu,

Xinhua Lu, and Kun Liu for their friendship and help.

Most importantly, I would like to thank my wife Niannian Dun, and my par-

ents Yingbai Liu and Chengxiang Li. Their unconditional dedication and unyielding

support are the motive power for me to finish the study.

v

Dedication

This thesis is dedicated to my wife Niannian, my son Kevin, and my parents

Yingbai Liu and Chengxiang Li.

vi

Contents

Abstract ii

Acknowledgments iv

List of Tables xii

List of Figures xiii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Numerical Method for Interfacial Flow Simulation 5

2.1 Interface capturing method . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Level-set method . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Reinitialization of signed distance function . . . . . . . . . . . 11

2.2 Coupled level-set and volume-of-fluid method . . . . . . . . . . . . . 17

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CONTENTS

2.2.1 Volume-of-fluid method . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Coupled level-set/volume-of-fluid method . . . . . . . . . . . . 24

2.3 Multi-fluid flow simulation . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Interface jump condition . . . . . . . . . . . . . . . . . . . . . 34

2.3.2 Pressure Poisson equation . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Parallelization and scalability . . . . . . . . . . . . . . . . . . 41

2.4 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Zaleski problem . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.2 Two dimensional air bubble without gravity . . . . . . . . . . 46

2.4.3 Two-layer Couette flow . . . . . . . . . . . . . . . . . . . . . . 48

2.4.4 Two dimensional air bubble . . . . . . . . . . . . . . . . . . . 49

2.4.5 Three dimensional air bubble bursting on water surface . . . . 51

2.4.6 Gravity wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.4.7 Capillary wave . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Direct Numerical Simulation of Two Dimensional Wave Breaking 56

3.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Characteristics of the free surface of breaking waves . . . . . . . . . . 61

3.2.1 Wave breaking with different intensities . . . . . . . . . . . . . 61

3.2.2 Spectra of the free surface . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Steepness, skewness and asymmetry . . . . . . . . . . . . . . . 67

3.3 Velocity field under breaking waves . . . . . . . . . . . . . . . . . . . 71

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CONTENTS

3.4 Energy dissipation by wave breaking . . . . . . . . . . . . . . . . . . 76

3.5 Modeling of wave breaking . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Numerical Study of High Wind Over Steep/Breaking Water Surface

Waves 82

4.1 Simulation setup and turbulence modeling . . . . . . . . . . . . . . . 83

4.1.1 Problem setup and parameters . . . . . . . . . . . . . . . . . . 83

4.1.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1.3 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Wind over prescribed steep waves . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Wind field above prescribed waves . . . . . . . . . . . . . . . . 89

4.2.2 Wind forcing over prescribed waves . . . . . . . . . . . . . . . 93

4.3 Wind over breaking steep waves . . . . . . . . . . . . . . . . . . . . . 95

4.3.1 Wind field above the breaking waves . . . . . . . . . . . . . . 99

4.3.2 Shear stress, drag coefficient, and roughness . . . . . . . . . . 101

4.3.3 Wind pressure above breaking waves . . . . . . . . . . . . . . 106

4.3.4 Airflow separation . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.5 Turbulence and coherent structures generated by breaking . . 112

4.3.6 Surface current generated by breaking . . . . . . . . . . . . . 115

4.3.7 Effect of wind speed on wave breaking . . . . . . . . . . . . . 118

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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CONTENTS

5 Numerical Simulation of Strong Free-Surface Turbulence for Mech-

anistic Study 123

5.1 Problem setup and numerical approach . . . . . . . . . . . . . . . . . 124

5.1.1 Setup of numerical simulation . . . . . . . . . . . . . . . . . . 124

5.1.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Characteristics of the free surface . . . . . . . . . . . . . . . . . . . . 127

5.2.1 Free surface disturbed by turbulence . . . . . . . . . . . . . . 127

5.2.2 Surface spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2.3 Surface wave and roughness . . . . . . . . . . . . . . . . . . . 132

5.2.4 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3 Turbulence statistics and structures . . . . . . . . . . . . . . . . . . . 138

5.3.1 Turbulence statistics . . . . . . . . . . . . . . . . . . . . . . . 138

5.3.1.1 Horizontal velocity fluctuation u′ . . . . . . . . . . . 138

5.3.1.2 Vertical velocity fluctuation w′ . . . . . . . . . . . . 139

5.3.1.3 Phase averaged Reynolds stress . . . . . . . . . . . . 140

5.3.2 Flow structures . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3.2.1 Splat and antisplat . . . . . . . . . . . . . . . . . . . 142

5.3.2.2 Breaking surface . . . . . . . . . . . . . . . . . . . . 145

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Multi-Scale Numerical Simulation of Wind-Wave-Structure Interac-

tion 148

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CONTENTS

6.1 Large scale wind-wave simulation . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Numerical methods and simulation setup . . . . . . . . . . . . 149

6.1.2 Wind over monochromatic waves . . . . . . . . . . . . . . . . 153

6.1.3 Wind over broadband waves . . . . . . . . . . . . . . . . . . . 156

6.2 Local scale wind-wave-structure simulation . . . . . . . . . . . . . . . 160

6.2.1 Immersed boundary method for flow-structure interaction . . . 161

6.2.2 Inflow boundary condition . . . . . . . . . . . . . . . . . . . . 163

6.2.3 Outflow boundary condition . . . . . . . . . . . . . . . . . . . 163

6.3 Multi-scale simulation of wind-wave interaction with surface piercing

body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7 Summary and Future Work 168

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.2.1 Wind wave generation and growth . . . . . . . . . . . . . . . . 171

7.2.2 Coupled LS/SPH Method . . . . . . . . . . . . . . . . . . . . 175

Bibliography 181

Vita 201

xi

List of Tables

2.1 Percentages of the numerical mass loss of both pure level-set methodand coupled level-set/volume-of-fluid method for the stretching fluiddisk problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Profiling results of the coupled level-set/volume-of-fluid method codeusing Craypat on Cray-XT5 supercomputer of the High PerformanceComputing Modernization Program initiated by Department of Defense. 44

3.1 Breaking wave types for different initial wave slopes. . . . . . . . . . . 64

4.1 Simulation parameters for different cases of wind over initially steepwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Friction velocity and drag coefficient for case II-1 during the breakingprocess at different time. . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1 Values of α, θ, β, and γ for c/u∗ = 2 at different ak. Values of γ basedon the parameterization of Ref. [1], γ = 0.17(Uλ/2/c−1)2(ωρa/ρw), arelisted for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2 Values of α, θ, β, and γ at ak = 0.1 with different wave ages. . . . . . 157

7.1 Peak wave length and significant wave height from simulation and cor-responding value from JONSWAP spectrum. . . . . . . . . . . . . . . 176

xii

List of Figures

2.1 Level set function of a sphere with radius r = 1. . . . . . . . . . . . . 72.2 Level set function contours of a two dimensional ellipse: (a) initial

condition; (b) after reinitialization of 20 iterations; (c) after reinitial-ization of 40 iterations. The thick red line represents the interface withϕ = 0. The contour interval is 0.2. . . . . . . . . . . . . . . . . . . . . 16

2.3 Isosurface of ϕ = 0.1 of the level set function of a three dimensionalellipsoid: (a) Initial condition; (b) after reinitialization of 20 iterations. 17

2.4 Illustration of (a) simple line interface construction(SLIC) method; and(b) piecewise linear interface construction (PLIC) method. The thickblack line is the interface. The shadowed area is the fluid area enclosedby reconstructed line segments. . . . . . . . . . . . . . . . . . . . . . 20

2.5 Grid cell intercepted by reconstructed plane segment: (a) α < m1∆x1;(b) α < m2∆x2; (c) α < m3∆x3 and m3∆x3 < m1∆x1 +m2∆x2; (d)α < m1∆x1+m2∆x2 andm3∆x3 < m1∆x1+m2∆x2; (e) max(m3∆x3,m1∆x1+m2∆x2) < α < 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Illustration of the volume flux calculation in two dimensional volume-of-fluid method. The shadowed area is: (a) the flux contributed byhorizontal motion; (b) the flux contributed by vertical motion. . . . . 24

2.7 Illustration of the least mean square method for interface normal cal-culation in coupled level-set/volume-of-fluid method. . . . . . . . . . 25

2.8 Flow chart of the coupled level-set/volume-of-fluid method. . . . . . . 262.9 Different conditions of the redistancing of level set function ϕ from the

reconstructed interface. The point with minimum distance is located(a) on the inside; (b) on the boundary; and (c) on the vertex of theinterface segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Instantaneous interfaces of two dimensional fluid disk in a vortical flowfield simulated by pure level set method (a) t = 3, (c) t = 6; andcoupled level-set/volume-of-fluid method (b) t = 3, (d) t = 6. . . . . . 29

2.11 Sketch of the coupled air–water simulation. . . . . . . . . . . . . . . . 302.12 Schematic of the MAC grid system used in current code. . . . . . . . 33

xiii

LIST OF FIGURES

2.13 Schematics of the treatment of discontinuity for pressure and shearstress in ghost fluid method. . . . . . . . . . . . . . . . . . . . . . . . 37

2.14 Seven points stencil of the discretization of pressure poisson equation. 382.15 Illustration of the domain decomposition in current code. . . . . . . . 422.16 Result of speedup test. . . . . . . . . . . . . . . . . . . . . . . . . . . 432.17 Instantaneous interfaces of Zaleski problem calculated with (a) split-

ting scheme and (b) ENO scheme after one rotation. The dashed linesare the theoretical results. . . . . . . . . . . . . . . . . . . . . . . . . 45

2.18 Schematic of the static air bubble simulated. . . . . . . . . . . . . . . 462.19 Pressure distributions of the two dimensional static bubble simulated

with: (a,d) CSF method with ϵ = 2∆; (b,e) CSF method with ϵ =∆; (c,f) GF method. Lines in (d,e,f) are the corresponding pressuredistribution along X = 2 in (a,b,c) at the middle plane of bubble. . . 47

2.20 Schematic of the two-layer Couette flow. . . . . . . . . . . . . . . . . 482.21 Velocity profiles and error percentages of the two layer Couette flow

simulated with (a) Continuous surface force method and (b) Ghostfluid method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.22 The air–water interface of a two dimensional air bubble with radius1/3cm raising in the water at time (a) t = 0.0 s; (b) t = 0.02 s; (c)t = 0.035 s; (d) t = 0.05 s. . . . . . . . . . . . . . . . . . . . . . . . . 50

2.23 Instantaneous air–water interface of a three dimensional air bubblebursting on the free surface at time (a) t=0.0 s; (b) t=0.017 s; (c)t=0.033 s; (d) t=0.05 s; (e) t=0.067 s; (f) t=0.083 s. . . . . . . . . . . 52

2.24 Amplitude evolution of a gravity wave with initial wave slope ak = 0.1and its comparison with linear theory. . . . . . . . . . . . . . . . . . . 53

2.25 Amplitude evolution of a gravity wave with initial wave slope ak = 0.1with different resolution. . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.26 Amplitude evolution of a capillary wave with initial slope ak = 0.1(solid line) and it comparison with linear theory (dashed line). . . . . 55

3.1 Sketch of the setup of two dimensional breaking waves. . . . . . . . . 593.2 See next page for caption. . . . . . . . . . . . . . . . . . . . . . . . . 623.2 Free surface profiles for waves with different initial steepness (a) ak =

0.3; (b) ak = 0.35; (c) ak = 0.4; (d) ak = 0.44; (e) ak = 0.55. . . . . . 633.3 Surface spectra of wave surfaces for cases with: (a) (ak)0 = 0.3; (b)

(ak)0 = 0.35; (c) (ak)0 = 0.4; (d) (ak)0 = 0.44; and (e) (ak)0 = 0.55. . 663.4 Schematic of a nonlinear wave and the quantities used for definition of

skewness and asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Steepness, skewness and asymmetry evolution with time for steep non-

breaking waves with (ak)0 = 0.3. . . . . . . . . . . . . . . . . . . . . 69

xiv

LIST OF FIGURES

3.6 Surface elevation evolution with time at x = 0 for steep non-breakingwaves with (ak)0 = 0.3. The dash line enclosing the wave shows sub-harmonic with period two times the primary wave period. . . . . . . . 70

3.7 Instantaneous wave profiles at time with opposite asymmetries (hori-zontally shifted to have two zero crossing points symmetric about x=0.5). 71

3.8 Steepness, skewness and asymmetry evolution with time for spillingbreaker with (ak)0 = 0.35. . . . . . . . . . . . . . . . . . . . . . . . . 72

3.9 Steepness, skewness and asymmetry versus time for plunging breakerwith (ak)0 = 0.4 (a,d,g); (ak)0 = 0.44 (b,e,h); and (ak)0 = 0.55 (c,f,i). 73

3.10 Evolution of the maximum velocity with time for cases with (a) (ak)0 =0.35; (b) (ak)0 = 0.4; (c) (ak)0 = 0.44; and (d) (ak)0 = 0.55. . . . . . 74

3.11 Velocity contours of cases with (a) (ak)0 = 0.35; (b) (ak)0 = 0.4; (c)(ak)0 = 0.44; and (d) (ak)0 = 0.55 when the maximum velocity isachieved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.12 Evolution of (a) the total mechanical wave energy and (b) the normal-ized total mechanical wave energy with time for cases with differentinitial steepness: ————, (ak)0 = 0.3; – – – – , (ak)0 = 0.35; – · – ·– , (ak)0 = 0.4; · · · · · · · , (ak)0 = 0.44; and −−− −−− , (ak)0 = 0.55. 76

3.13 Breaking time scale versus wave steepness S and comparison withTian’s [2] model (dashed line) and data. . . . . . . . . . . . . . . . . 78

3.14 Breaking length scale versus wave steepness S and comparison withTian’s [2] model (dashed line) and data. . . . . . . . . . . . . . . . . 79

3.15 Normalized breaking length scale versus normalized breaking time scale. 803.16 Dissipation parameter b versus wave steepness S and comparison with

Drazen’s [3] model and data. . . . . . . . . . . . . . . . . . . . . . . . 81

4.1 Sketch of the setup for the wind-wave breaking problem. . . . . . . . 854.2 Phase averaged horizontal wind velocity vector field over prescribed

water waves of case: (a) II-1; (b) II-2; (c) II-3; (d) II-4. . . . . . . . 894.3 Streamline pattern of wind flow over prescribed waves for case (a) II-1;

(b) II-2; (c) II-3; and (d) II-4. . . . . . . . . . . . . . . . . . . . . . . 914.4 Mean horizontal velocity above the water surface (a) for cases I-1∼I-4

with wavelength 0.262m and wave slope ak = 0.1; (b) for cases II-1∼II-4 with wavelength 0.262m and wave slope ak = 0.35; (c) forcases III-1∼III-5 with wavelength 20m and wave slope ak = 0.55. . . 92

4.5 Phase averaged dynamic pressure field of wind flow over prescribedwaves for case (a) II-1; (b) II-2; (c) II-3; and (d) II-4. . . . . . . . . 94

4.6 Phase averaged dynamic pressure field over the wave surface of case(a) I-1∼I-4, (b) II-1∼II-4, and (c) III-1∼III-5. . . . . . . . . . . . . . 96

4.7 The instantaneous breaking water surface and streamwise velocity con-tours on two vertical planes for case II-1 at (a) t=0.29T, (b) 0.44T, (c)0.58T, (d) 0.87T, (e) 1.16T, (f) 1.45T, (g) 1.74T, (h) 2.03T . . . . . . 97

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LIST OF FIGURES

4.8 Spanwise averaged profiles of wind wave around breaking for case II-1:————, near breaking; – – – – , incipient breaking; , [4]. The errorbar represents the standard deviation of the experimental results of [4]. 98

4.9 The spanwise-averaged streamwise velocity on the air side for case II-1.(a) t=0.29T; (b) t=1.16T; (c) 2.03T. . . . . . . . . . . . . . . . . . . 100

4.10 The spanwise-averaged streamwise velocity fluctuation on the air sidefor case II-1. (a) t=0.29T; (b) t=1.16T; (c) 2.03T. . . . . . . . . . . . 101

4.11 The mean streamwise velocity above the water surface for case II-1during the breaking process at different time. . . . . . . . . . . . . . . 102

4.12 Friction velocity u∗ and drag coefficient Cd obtained in current simu-lation and presented in other literatures. . . . . . . . . . . . . . . . . 103

4.13 Roughness length scale normalized by wave height versus wave age. . 1064.14 The spanwise-averaged pressure, streamlines and vorticity at t = 1.1T

of case II-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.15 Sketch for pressure distribution over water wave before (a) and after

(b) breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.16 The form drag evolution with time for wind over breaking waves in

case II-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.17 Instantaneous flow field cut of case II-1 in the free developing stage:

(a)velocity vector; (b) horizontal velocity contour; (c) surface stream-lines. The velocities are plotted in a moving reference frame withhorizontal velocity c. Here c is the phase speed of wave. . . . . . . . . 113

4.18 Instantaneous streamwise velocity u normalized by wave phase speedc on a vertical cut for case II-1 with (ak)0 = 0.35. The time step is0.145T . T is the linear wave period. . . . . . . . . . . . . . . . . . . . 114

4.19 Spanwise averaged underwater velocity vectors for breaking wave caseIII-1 with (ak)0 = 0.55. (a) t=1.33T; (b) t=1.78T; (c) t=2.22T; (d)t=2.67T. Here T is the wave period. . . . . . . . . . . . . . . . . . . 116

4.20 Spanwise-averaged horizontal velocity on water side of case II-1: (a)t = 0.29T ; (b) t = 1.16T ; (c) t = 2.03T ; (d) t = 2.90T . . . . . . . . . 117

4.21 Horizontal plane-averaged streamwise velocity on water side of caseII-1 at different time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.22 Instantaneous flow field and the free surface at t = 0.417s ≈ 1T of case(a) II-1; (b) II-2; (c) II-3; (d) II-4. . . . . . . . . . . . . . . . . . . . . 119

5.1 Sketch of the multi-phase flow simulation setup of the free-surface tur-bulence problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Diagram of the flow regimes in the Fr − We space. Region 0: weakturbulence regime; region 1: surface tension dominated regime; region2: very strong turbulence regime; region 3: gravity dominated regime.The region between the two dash lines represents the marginal breakingregion obtained by Brochini & Peregrine (2001). [5] . . . . . . . . . . 128

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LIST OF FIGURES

5.3 Instantaneous free surface elevation for the cases of: (a) (Fr2 = 0.8,We = 40) that is in the weak turbulence regime, (b) (Fr2 = 128, We =40) that is in the surface tension dominated regime, (c) (Fr2 = 4,We = ∞) that is in the gravity dominated regime, and (d) (Fr2 = 32,We = ∞) that is in the very strong turbulence regime. . . . . . . . . 129

5.4 Surface elevation spectra of (a) the gravity dominated case of (Fr2 = 4,We = ∞) and (b) the surface tension dominated case of (Fr2 = 32,We = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.5 Normalized frequency–wavenumber spectrum of the surface elevationfor the weak turbulence case of (Fr2 = 0.8,We = 40). The solid linedenotes the dispersion relationship (equation 5.5). The dash-dot linedenotes the characteristic frequency of each wavenumber component(equation 5.6). The dashed line denotes the characteristic frequencyobtained by linearized kinematic boundary condition (equation 5.7). . 132

5.6 (a) Intermittency factors of the cases with violent free surfaces: · · ·· · · , (Fr2 = 32,We = ∞); – · · – · · – , (Fr2 = 32,We = 500);————, (Fr2 = 32,We = 40); – · – · – , (Fr2 = 8,We = ∞); (b)intermittency factors with z normalized by the equivalent thickness ησ. 135

5.7 (a) Histogram of the surface elevation of the mild surface case of(Fr2 = 32,We = 1) and the fitted Gaussian function. (b) Relation-ship between intermittency factors and the surface elevation probabil-ity density function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.8 Intermittency layer thickness for cases with: (a) the same Weber num-ber We = ∞ but different Froude numbers; (b) the same Froude num-ber Fr2 = 32 but different Weber numbers. . . . . . . . . . . . . . . . 137

5.9 Horizontal velocity fluctuations of cases with (a) the same Weber num-ber We = ∞ but different Froude numbers: · · · · · · Fr2 = 32; – – –– Fr2 = 4; – · · – · · – Fr2 = 1, (b) the same Froude number Fr = 32but different Weber numbers: · · · · · · We = ∞; – · – · – We = 40;————We = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.10 Vertical velocity fluctuations of cases with (a) the same Weber numberWe = ∞ but different Froude numbers, (b) the same Froude numberFr2 = 32 but different Weber number. (See figure 5.9 for line legend.) 140

5.11 Phase weighted horizontal turbulent normal stress < u′u′I > of caseswith (a) the same Weber number We = ∞ but different Froude num-bers; (b) the same Froude number Fr2 = 32 but different Weber num-bers. (See figure 5.9 for line legend.) . . . . . . . . . . . . . . . . . . 141

5.12 Phase weighted vertical turbulent normal stress < w′w′I > of caseswith (a) the same Weber number We = ∞ but different Froude num-bers; (b) the same Froude number Fr2 = 32 but different Weber num-bers. (See figure 5.9 for line legend.) . . . . . . . . . . . . . . . . . . 141

xvii

LIST OF FIGURES

5.13 Instantaneous flow structures of the case of (Fr2 = 32,We = 1): (a)horizontal slice close to the interface; (b) vertical slice through a splat;(c) free surface and vortex structures. . . . . . . . . . . . . . . . . . . 143

5.14 Instantaneous flow structure for the case of (Fr2 = 32,We = ∞):(a) free surface and velocity vectors; and on a vertical cross-section,distributions of (b) vertical velocity; (c) transport of horizontal turbu-lent normal stress by the vertical turbulent velocity; (d) transport ofvertical turbulent normal stress by the vertical turbulent velocity. . . 144

5.15 A surface breaking process in the case of (Fr2 = 16,We = ∞). (a) Awater sheet is brought up and begins to overturn. (b) The water sheetplunges downward to the free surface. (c) The water sheet reenters andthen splashes up. Surface elevation contours and the velocity vectorsof water are plotted. A vertical cut is extracted for analysis in figure5.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.16 Energy dissipation and turbulent Reynolds stress transport associatedwith surface breaking: (a) viscous dissipation rate; (b) horizontal trans-port of the horizontal turbulent normal stress. . . . . . . . . . . . . . 146

6.1 Illustration of wind turbulence and water wave coupled simulation.Plotted are streamwise velocity (normalized by Uλm/2) of the wind andpressure (normalized by ρau

2∗) distribution on the surface of broadband

waves (cm/u∗ = 12.3). The air domain is lifted up for better visualization.1526.2 Evolution of (a) ack and (b) atk: −−, c/u∗ = 2; − ·−, c/u∗ = 2 (from

linear wave simulation); ···, c/u∗ = 5; −··−, c/u∗ = 10; −−−, c/u∗ = 14.The time is normalized by λ/Uλ/2. . . . . . . . . . . . . . . . . . . . . 153

6.3 Surface pressure profiles over monochromatic waves: −· ·−, (c/u∗ = 2,ak = 0.05); −−−, (c/u∗ = 2, ak = 0.1); − · −, (c/u∗ = 2, ak = 0.15);· · ·, (c/u∗ = 2, ak = 0.2); −−, (c/u∗ = 2, ak = 0.25); −−, (c/u∗ = 5,ak = 0.1); −−, (c/u∗ = 10, ak = 0.1); −−, (c/u∗ = 14, ak = 0.1). (a)Comparison of simulation result with field measurement data (N) ofRef. [1]; (b) pressure profiles over waves with different steepnesses; (c)pressure profiles over waves with different wave ages. The wind andwave are from left to right. The wave phase is shown in the sketch atthe bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.4 Values of β (lines) and γ (symbols) for broadband waves: −−− and ,cm/u∗ = 5 (case I); − ·− and , cm/u∗ = 12.3 (case II); −−− and ,cm/u∗ = 16 (case III). . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.5 Wave growth rate parameter β: •, experimental results compiled inRef. 7; , numerical results of Refs. 9 and 11; , numerical results ofRef. 13; ×, current results for monochromatic waves. The lines are thecurrent broadband wave results (see the line legend in Fig. 6.4). . . . 159

6.6 Schematic of immersed boundary method (discrete force method). . . 162

xviii

LIST OF FIGURES

6.7 Illustration of multi-scale wind–wave–structure simulation. The flowcondition inside the small black window is provided to local scale wind–wave–structure simulation as inflow condition. . . . . . . . . . . . . . 164

6.8 Wind and wave fields around a surface piecing body: (a) when a wavecrest, and (b) when a wave trough arrives at the front surface of theobject. the inflow is in the x-direction. the vertical planes show thestreamwise velocity contours. the velocity field inside the small blackwindow is enlarged and shown in figure 6.9. the pressure on the objectsurface and the wave surface are shown. vortices are plotted with greycolor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.9 Enlarged streamwise velocity contours from figure 6.8: (a) above wavecrest when a crest arrives at the object; (b) above wave trough whena trough arrives at the object. . . . . . . . . . . . . . . . . . . . . . . 167

7.1 Wave field generated by the turbulent wind with 10 meter hight speed30m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.2 Evolution of the root-mean-square surface elevation with time. . . . . 1747.3 One dimensional surface spectra of the wave field generated by wind. 1747.4 Illustration of the coupled LS/SPH simulation . . . . . . . . . . . . . 1767.5 Zero-energy mode of SPH simulation with cubic spline kernel. . . . . 1787.6 Coupled LS/SPH simulation of a two dimensional linear wave with ini-

tial wave slope ak = 0.05: (a) numerical setup; and horizontal velocitycontours at (b) t = 4.25T , (c) t = 4.5T , (d) t = 4.75T , (e) t = 5T .Here T is the wave period. . . . . . . . . . . . . . . . . . . . . . . . . 180

xix

Chapter 1

Introduction

1.1 Background

To address the increasing demand of energy and the issue of global warming asso-

ciated with the use of fossil fuel, clean and renewable energy is being actively sought.

The oceans provide enormous resources for renewable energy. In addition to the wave

energy, the offshore wind power possesses many advantages over the traditional wind

power on land and has become a new frontier in wind energy. According to a report

of US Department of Energy [6], wind energy will provide 20% electricity of US de-

mand by 2030 and 18% of them will be the offshore wind energy. Compared to the

wind energy on land, the offshore wind energy is stronger and more stable, and the

convenience is sea transportation makes the installation of very large wind turbine

feasible. The increasing demand of transocean shipping coming with the integration

1

CHAPTER 1. INTRODUCTION

of world economy also brings high requirement on the safety of ocean surface vehicles.

For the development of wind and wave energy technologies and the boost of the safety

of ocean transportation, there is a critical need for the understanding and modeling

of ocean wind-waves, the lower part of marine atmospheric boundary layer at various

sea states, and wind load and wave load on offshore structures.

A lot of efforts have been devoted to explore the physics in the marine atmospheric

boundary layer and ocean boundary layer [7–9], but the complex air–sea interaction

problem is still far from being solved. Complex sea conditions make the field mea-

surement challenging and expensive. The accurate prediction of ocean surface waves

is still challenging.

The evolution of wave field is affected by wind forcing, wave breaking dissipation,

nonlinear wave interaction, wave-turbulence interaction, and etc. Wind forcing is the

major source of the wave energy in the ocean. Nonlinear wave interaction redistributes

the energy among different wave components. The wave breaking transfers energy

from wave to the surface current and the underwater turbulence. The turbulence

makes the free surface even rougher and more complicated. Most of the existing wave

prediction tools [10] calculate the evolution of directional wave spectrum with the

wind input and breaking dissipation modeled. The information of wave phases is not

contained in the wave spectrum, and the aforementioned processes are parameterized

in a phase-averaged context.

To obtain a more direct description of the wave field with finely resolved spatial

2


and temporal details, it is desirable to resolve the wave phases in the simulation.

Such information is valuable for the mechanistic study of wind-wave dynamics which

may eventually lead to improved modeling for the wave spectrum simulation. Re-

cent advancement in computing power and algorithm development has facilitated the

phase-resolved simulation of nonlinear wave interaction involving a large number of

wave modes (e.g. O(104) modes in each direction [11]), but breaking wave is modeled

by simply adding a dissipation term.

Strong free surface flow and wave breaking bring large slope and even singular

point to the ocean surface, which increase the surface roughness and induce airflow

separation. They generate spays and air entrainment which is important in the mass

exchange between atmosphere and ocean. Wave breaking also strongly affects the

backscatter of electromagnetic waves (e.g. that used by Radar) which is widely used in

the remote sensing of ocean surface motion. [12] Strong free surface flow and breaking

waves bring jeopardy to ocean surface vehicles and offshore structures such as oil rigs

and wind turbines. Rogue waves with wave heights several times the significant wave

height has capsized lots of ships in the ocean. Reviews about strong free surface flow

and wave breaking can be found in [12–16]. Detailed simulation based study of strong

free surface flow and wave breaking is the major task of current thesis.

3


1.2 Thesis overview

In current thesis, I develop a numerical tool which can address complex interfa-

cial motions on a fixed Cartesian grid system and apply it in the simulation of wave

breaking, free surface turbulence, and wind-wave-structure interaction problems. The

flow physics revealed by the simulations will be discussed. The thesis is organized

as following: Chapter 2, numerical methods for multi-fluid flow simulation; Chapter

3, direct numerical simulation of two dimensional wave breaking; Chapter 4, large

eddy simulation of high wind over steep/breaking water waves; Chapter 5, direct

numerical simulation of free surface interacting with the underwater isotropic homo-

geneous turbulence for mechanistic study; Chapter 6, multi-scale numerical simulation

of wind-wave-structure interaction; Chapter 7, summary and future work.

4

Chapter 2

Numerical Method for Interfacial

Flow Simulation

Numerical methods for interfacial flow simulation have attracted significant at-

tention in recent years. According to the grid systems used, these methods can be

classified into three categories: (1) moving grid method; (2) fixed grid method; (3)

meshless method. In moving grid method (e.g. arbitrary Lagrangian-Euler (ALE)

method [17]), the grid is conformed to interface and all quantities on the interface

are calculated directly in the simulation. Dynamic re-meshing is needed, which is

time consuming and complicated. Another way to use moving grid is to map the

physical domain and complex interface into a rectangular computational domain and

flat surface with conformal mapping or sigma mapping [18]. It usually involves com-

plex coordinate transform of governing equations, which is difficult to implement and

5

CHAPTER 2. NUMERICAL METHOD FOR INTERFACIAL FLOWSIMULATION

memory consuming for interfaces with large topological changes.

A fixed grid method is suitable for the simulation of strong free surface flow.

Based on how the interface is represented, fixed grid methods can be further divided

into interface tracking method and interface capturing method. Interface tracking

method records the exact position and velocity of discrete Lagrangian markers to track

the interface explicitly. It can accommodate large interface deformation. However,

when there are surface pinching off and merging, it becomes infeasible or difficult to

implement. Interface capturing method, which uses a global field function to represent

the interface implicitly, is robust for problems with strong interface motion. Here I

develop a multi-fluid flow solver based on the interface capturing method on a fixed

Cartesian grid system.

2.1 Interface capturing method

In this section, two interface capturing methods (i.e. level-set method and volume-

of-fluid method) and their coupling that are adopted in current solver are elaborated

and corresponding numerical tests are performed.

2.1.1 Level-set method

Level-set (LS) method is invented by Osher and Sethian [19] to simulate the

motion of a surface with curvature dependent speed. It has been widely used in

6


Figure 2.1: Level set function of a sphere with radius r = 1.

applications such as breaking waves, bubble dynamics, combustion and reacting flows,

and computer graphics. [20,21]

In the LS method [22,23], free surface is represented implicitly by a signed distance

function (also called level-set function)

ϕ(x, t) =

d in water,

0 on surface,

−d in air.

(2.1)

Here d is the distance from point x to free surface. The points with zero LS func-

tion values lie on the surface. An example of level set function representing a three

dimensional sphere with radius r = 1 is given in figure 2.1.

The LS function is advected by the flow according to the Lagrangian-invariant

level-set equation

∂ϕ

∂t+ u · ∇ϕ = 0. (2.2)

7


Here u is the velocity vector.

For incompressible flow, divergence free condition ∇ · u = 0 can be incorporated

and the above LS equation can be written as

∂ϕ

∂t+∇ · (uϕ) = 0. (2.3)

A fixed Cartesian grid is used in the current solver and complex mesh generation

is avoided. The LS equation is integrated to obtain the evolution of the interface.

The advection term can be discretized by different numerical schemes. The central

difference scheme is non-diffusive but encounters instability (i.e. Gibbs phenomenon)

when interface is not smooth. A stable second order ENO scheme and second order

operator splitting scheme are implemented in current solver.

ENO scheme was invented by Harten et al. [24]. It chooses the smoothest inter-

polation polynomial to calculate the derivatives. It is widely used for problems with

contact discontinuity and is able to avoid numerical instability. It can be constructed

to arbitrary high order. Here a five point stencil that can achieve second order is

implemented. A table of divided differences [25]

ϕI,k =ϕk+1−ϕk

xk+1−xk,

ϕII,k =ϕI,k+1−ϕI,k

xk+2−xk.

(2.4)

is used to construct the upwind ENO scheme. Here ϕII,k is used as the smoothness

indicator. When ui ≤ 0, the derivative

D−x ϕi = ϕI,i−1 +Minmod(ϕII,i−2, ϕII,i−1)(xi − xi−1), (2.5)

8


and when ui < 0,

D+x ϕi = ϕI,i +Minmod(ϕII,i−1, ϕII,i)(xi − xi+1). (2.6)

Here

Minmod(a, b) =

a when |a| ≤ |b| and ab > 0,

b when |a| > |b| and ab > 0,

0 when ab ≤ 0.

(2.7)

A second order conservative operator splitting advection scheme is also imple-

mented for the level set equation [26] as

ϕi,j,k =ϕni,j,k + (∆t/∆x)(Gi−1/2,j,k −Gi+1/2,j,k)

1− (∆t/∆x)(ui+1/2,j,k − ui−1/2,j,k), (2.8)

ϕi,j,k =ϕni,j,k + (∆t/∆y)(Gi,j−1/2,k − Gi,j+1/2,k)

1− (∆t/∆y)(vi,j+1/2,k − vi,j−1/2,k), (2.9)

ϕi,j,k =ϕni,j,k + (∆t/∆z)(Gi,j,k−1/2 − Gi,j,k+1/2)

1− (∆t/∆z)(wi,j,k+1/2 − wi,j,k−1/2). (2.10)

ϕn+1i,j,k = ϕ−∆t

(ϕi,j,k

∆x(ui+1/2,j,k − ui−1/2,j,k)

+ϕi,j,k

∆y(vi,j+1/2,k − vi,j−1/2,k) +

ϕi,j,k

∆z(wi,j,k+1/2 − wi,j,k−1/2)

) (2.11)

Here, G = uϕ is the flux of ϕ. A scheme based on the predictor-corrector method [27]

is used to calculate ϕ on the cell boundaries (the grid system used is demonstrated

in figure 2.12). When ui+1/2,j,k > 0,

ϕi+1/2,j,k = ϕni,j,k +

∆x

2

(1− ui+1/2,j,k

∆t

∆x

)ϕni+1,j,k − ϕn

i−1,j,k

∆x, (2.12)

else

ϕi+1/2,j,k = ϕni+1,j,k −

∆x

2

(1 + ui+1/2,j,k

∆t

∆x

)ϕni+2,j,k − ϕn

i,j,k

∆x. (2.13)

9


Here the Strang splitting [28] method which alternates the sweep direction each time

step between x-y-z and z-y-x is used to alleviate the possible asymmetry induced by

splitting.

In §2.4.1, the Zaleski problem (rotation of a notched disk) is simulated with both

the ENO scheme and the operator splitting scheme. The latter shows smaller numer-

ical diffusion than ENO scheme.

The unit surface normal vector can be calculated from ϕ as

n = ∇ϕ, (2.14)

and |∇ϕ| = |n| = 1 is the property of the signed distance function. To avoid numerical

error, we use

n =∇ϕ

|∇ϕ|. (2.15)

The surface curvature is calculated as

κ = ∇ · n = ∇ · ∇ϕ

|∇ϕ|. (2.16)

In a Cartesian grid system with three coordinates x, y, z, the curvature can be ex-

pressed as

κ = (ϕ2xϕyy − 2ϕxϕyϕxy + ϕ2

yϕxx + ϕ2xϕzz − 2ϕxϕzϕxz + ϕ2

zϕxx

+ϕ2yϕzz − 2ϕyϕzϕyz + ϕ2

zϕyy)/|∇ϕ|3(2.17)

The above equations for interface normal and curvature are discretized with central

difference scheme. In under-resolved regions, the curvature is truncated to grid size

to avoid instability [29].

10


2.1.2 Reinitialization of signed distance function

The signed distance function ϕ is not a conserved quantity. Away from the inter-

face, ϕ and its variation are independent of local flow field and they are completely

decided by the location of the interface. Equation 2.2 and 2.3 can not guarantee

the signed distance property of ϕ as time evolves. Near the interface, contours of ϕ

may become too dense or sparse (e.g. figure 2.2(a)), which may incur large error in

the calculation of ϕ’s derivatives and make the interface thickness nonconstant. A

reinitialization procedure is needed.

The reinitialization of ϕ is equivalent to have ϕ satisfying |∇ϕ| = 1 without moving

the interface. Rouy & Tourin [30] proposed a method to reinitialize ϕ by solving

ϕt + |∇ϕ| = 1. (2.18)

This equation alone could move the interface which should be fixed during reinitial-

ization. So the distance function near the interface need to be calculated by hand in

advance to provide boundary condition.

The following equation proposed by Sussman, Smereka and Osher [22]

∂ϕc

∂τ+ sign(ϕ)(|∇ϕc| − 1) = 0 (2.19)

can be used to correct ϕ without calculating distance explicitly. Here τ is an artificial

time. Initial condition is ϕc(x, 0) = ϕ(x). After equation 2.19 is solved to a steady

state, ϕ takes the value of ϕc. In equation 2.19, the second term can be transformed

11


to a convection-like form as

∂ϕc

∂τ+

(sign(ϕ)

∇ϕc

|∇ϕc|

)· ∇ϕc − sign(ϕ) = 0. (2.20)

Here, the term in the parenthesis is the advection velocity of the level set function

from the interface. The absolute value of the velocity is 1 for a perfect signed dis-

tance function with |∇ϕ| = 1. For CFL condition to be satisfied, we need ∆τ <

min(∆x,∆y,∆z). In current code, CFL number 0.8 is chosen.

A smoothed sign function [22]

Sϵ(ϕ0) =ϕ0√ϕ20 + ϵ2

(2.21)

is used in the reinitialization equation. Here, ϵ is a small number which is usually

1 ∼ 2∆x. With the smoothed sign function, the advective velocity is damped to zero

towards the interface to minimize the spurious move of the interface.

For some extreme conditions, the level set function near the interface is far beyond

a signed distance function and its spatial gradient is much larger than one. It could

induce numerical instability to the reinitialization equation. To avoid instability, a

more stable expression for the sign function is proposed as

S(ϕ) =ϕ√

ϕ2 + |∇ϕ|2(∆x)2(2.22)

by Peng et al. [31].

Equation 2.19 may slightly move the interface in one grid which can affect the

mass conservation of each fluid. Correction of ϕ for mass conservation is needed

especially for problems involving small scale interface structures.

12


A global area-preserving reinitialization is proposed by Chang et al. [32]. A per-

turbed Hamilton-Jacobi equation

∂

∂tϕ+ (A0 − A(t))(−P + κ)|∇ϕ| = 0 (2.23)

is solved to steady state. Here A0 is the initial mass at t = 0; A(t) is the mass at

time t; P is a positive constant; κ is the local curvature. This method works well for

global conservation, but local conservation is not tested.

A local correction method is proposed to equation 2.19 [23] by applying a local

constraint

∂

∂τ

∫Ωijk

H(ϕ) = 0 (2.24)

at grid points near the interface. Here Ωijk is the volume of cell (i, j, k) and H(x) is

the Heaviside step function. Equation 2.19 is then modified to

∂ϕc

∂τ+ sign(ϕ)(|∇ϕc| − 1) + λijkf(ϕ) = 0. (2.25)

Here λijk is constant in each cell and

f(ϕ) = δ(ϕ)|∇ϕ|. (2.26)

f(ϕ) is nontrivial only near the interface. Substituting it into equation 2.24, there is

∂∂τ

∫Ωijk

H(ϕ) =∫Ωijk

H ′(ϕ)∂ϕ∂τ

= −∫Ωijk

δ(ϕ)(sign(ϕ)(|∇ϕc| − 1) + λijkf(ϕ))

= 0,

(2.27)

13


and the constant

λijk =−∫Ωijk

sign(ϕ)(|∇ϕc| − 1)∫Ωijk

f(ϕ)(2.28)

is obtained. Here δ(x) is the Dirac delta function and its smoothed form (equation

2.71) is used here.

Russo and Smereka [25] modified Sussman’s method by using a upwind scheme

without using information from the other side of the interface. The modified reini-

tialization scheme becomes

ϕn+1ijk =

ϕnijk − δt

δx

(sign(ϕ0

ijk)|ϕnijk| −Dijk

)if (i, j, k) ∈ Σ∆

ϕnijk −∆t sign(ϕ0

ijk)G(ϕ)ijk otherwise

(2.29)

Here Dijk is the distance between node (i, j, k) and the interface. It can be calculated

as

Dijk =ϕ0ijk

max([ϕ2x + ϕ2

y + ϕ2z

]1/2ijk

, ϵ). (2.30)

Here ϵ is a small positive number to avoid singularity; and

Σ∆ = (i, j, k): ϕ0i,j,kϕ

0i−1,j,k < 0 or ϕ0

i,j,kϕ0i+1,j,k < 0 or ϕ0

i,j,kϕ0i,j−1,k < 0 or

ϕ0i,j,kϕ

0i,j+1,k < 0 or ϕ0

i,j,kϕ0i,j,k−1 < 0 or ϕ0

i,j,kϕ0i,j,k+1 < 0,

(2.31)

is the union of grid points within one grid size from the interface. This method could

avoid interface movement during reinitialization process and is adopted in current

solver.

14


Reinitialization tests

A two dimensional ellipse is tested with current reinitialization code. Its initial

level set function is disturbed (i.e. not a signed distance function) [25] to be

ϕ(x, y, 0) =(ϵ+ (x− x0)

2 + (y − y0)2)(√(x2

a2+

y2

b2

)− 1

). (2.32)

The length of the semi-major axis is a = 4. The length of the semi-minor axis is b = 2.

The multiplier ϵ + (x − x0)2 + (y − y0)

2 determines the significance of disturbance.

Here ϵ = 0.1; x0 = 3.5; and y0 = 2. The resolution is Nx ×Ny = 200× 200. Initially,

the absolute value of gradient of ϕ is larger than one except at the upper right corner

where it is smaller than one (figure 2.2(a)). After 20 iterations of reinitialization,

the six contour lines adjacent to the interface become equal-spaced ellipses. After 40

iterations, all contour lines become equal-spaced ellipses as that of the signed distance

function.

A three dimensional ellipsoid is also tested. The initial level set function is dis-

turbed to be

ϕ(x, y, z, 0) =(ϵ+ (x− x0)

2 + (y − y0)2 + (z − z0)

2)(√(x2

a2+

y2

b2+

z2

c2

)− 1

)(2.33)

Here, the length of the semi-major axis a = 4; the length of the semi-minor axis

b = 2 and c = 2; ϵ = 0.1; x0 = 3.5; y0 = 2; and z0 = 2. The resolution is

Nx ×Ny ×Nz = 200× 200× 200.

In this case, the initial isosurfaces of the level set function are severely clustered

15


X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

(a)

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

(b) (c)

Figure 2.2: Level set function contours of a two dimensional ellipse: (a) initial condi-tion; (b) after reinitialization of 20 iterations; (c) after reinitialization of 40 iterations.The thick red line represents the interface with ϕ = 0. The contour interval is 0.2.

16


X

-4-2

02

4

Y

-4-2

02

4

Z

-4

-2

0

2

4

Y

X

Z

X

-4-2

02

4

Y

-4-2

02

4

Z

-4

-2

0

2

4

Y

X

Z

(a) (b)Figure 2.3: Isosurface of ϕ = 0.1 of the level set function of a three dimensionalellipsoid: (a) Initial condition; (b) after reinitialization of 20 iterations.

towards the interface on the lower left and stretched away from the interface on the

upper right corner (i.e. the bulge in figure 2.3(a)). After 20 iterations, the isosurface

of ϕ = 0.1 becomes a perfect ellipsoid as that of the signed distance function (figure

2.3(b)).

2.2 Coupled level-set and volume-of-fluid

method

The level set function is not a conserved quantity. The solution of the level set

equation can not guarantee the mass conservation of each fluid, which could deterio-

rate the simulation result of problems such as wave breaking involving small droplets

and bubbles. A global correction proposed by Chang et al. [32] does not guarantee

17


local conservation and a local correction proposed by Sussman et al. [23] only adjusts

the error generated by reinitialization equation. Volume-of-fluid (VOF) method [33],

another interface capturing method, is good at conserving mass. However, the ac-

curate calculation of surface normal and curvature is challenging. The coupling of

the VOF method with the level-set method can utilize the advantages of both meth-

ods [26].

The coupled level set/volume-of-fluid (CLSVOF) method is implemented in our

model to further improve the mass conservation and capture fine-scale interfacial

structures such as water droplets and bubbles.

2.2.1 Volume-of-fluid method

The volume-of-fluid method was invented in 1980s and is implemented in com-

mercial codes such as SOLA-VOF [34], NASA-VOF2D [35], Flow-3D [36], etc. Com-

prehensive review of VOF method is given by Scardovelli & Zaleski [37].

In VOF method, the volume fraction of fluid 1 (suppose there are only two fluids)

F =

1 with only fluid 1

V1/Vcell with both fluids

0 with only fluid 2

(2.34)

in each grid cell is introduced as phase indicator. Here V1 and Vcell are the volume

of fluid 1 and the total volume of the grid cell respectively. The volume fraction F is

a conserved quantity. The evolution of the volume fraction is governed by the VOF

18


equation

∂F

∂t+ u · ∇F = 0. (2.35)

For incompressible flow, the VOF equation can be written in a conservative form as

∂F

∂t+∇ · (uF ) = 0. (2.36)

The volume fraction F has a sharp jump across the interface. Discretizing the VOF

equation directly will smear the interface, so the interface needs to be reconstructed

explicitly. Simple line interface construction (SLIC) method [34, 38] constructs the

interface with piecewise segments aligned with the grid lines as in figure 2.4(a). This

method is first order accurate and its would generate large amount of flotsam. The

piecewise linear interface construction (PLIC) method [37,39] constructs interface in

each grid cell with linear plane segments for 3D problems and linear line segments in

2D problems as in figure 2.4(b). These segments are not required to be connected.

The volume fraction F can be updated in three steps: reconstructing the interface;

calculating volume flux; and updating F . The relationship between the interface

segment and the volume fraction F is the key part of the first two steps. Suppose

the interface normal vector is (m1,m2,m3) (how to obtain it will be discussed in

§2.2.2), the volume fraction and the interface segment have an one-on-one relation.

The plane segment is determined by its distance to the origin α. Assuming m1∆x1 ≤

m2∆x2 ≤ m3∆x3, the intersection of the grid cell by the reconstructed plane can

have 5 conditions according to the distance from the origin to the plane α as shown

19


(a) (b)

Figure 2.4: Illustration of (a) simple line interface construction(SLIC) method; and(b) piecewise linear interface construction (PLIC) method. The thick black line isthe interface. The shadowed area is the fluid area enclosed by reconstructed linesegments.

in figure 2.5. The intersection can be triangles (figure 2.5(a)), quadrilaterals (figure

2.5(b,d)), or pentagons (figure 2.5(c,e)).

The fluid volume enclosed by the plane and the grid cell can be calculated from

the analytical relation [40]

F =1

6m1m2m3∆x1∆x2∆x3

[α3 −

3∑i=1

f3(α−mi∆xi) +3∑

i=1

f3(α− αmax +mi∆xi)

](2.37)

Here ∆xi are the grid space in the ith direction;

αmax =3∑

i=1

mi∆xi; (2.38)

and function

fn(y) =

yn when y > 0,

0 when y <= 0.

(2.39)

20


(a) (b)

(c) (d)

(e)

Figure 2.5: Grid cell intercepted by reconstructed plane segment: (a) α < m1∆x1; (b)α < m2∆x2; (c) α < m3∆x3 andm3∆x3 < m1∆x1+m2∆x2; (d) α < m1∆x1+m2∆x2

and m3∆x3 < m1∆x1 +m2∆x2; (e) max(m3∆x3,m1∆x1 +m2∆x2) < α < 1/2.

21


The reconstruction of the interface is now equivalent to finding α when F is known.

This is the inverse problem of equation 2.37 and

α =

(6m1∆1m2∆2m3∆3F )1/3 when 0 ≤ F < F1,

12(m1∆1 +

√(m1∆1)2 + 8m2∆2m3∆3(F − F1)) when F1 ≤ F < F2,

α|aα3+bα2+cα+d=0 when F2 ≤ F < F3.

(2.40)

Here ∆i = ∆xi for simplicity; ∆12 = m1∆x1 +m2∆x2; and

F1 = F |α=m1∆1 = m21/max(6m2m3, ϵ)

F2 = F |α=∆12 = F1 + (m2∆2 −m1∆1)/max(2m2∆3, ϵ)

(2.41)

The coefficients for the third order polynomial are a = −1, b = 3∆12, c = −3((m1∆1)2+

(m2∆2)2), and d = (m1∆1)

3 + (m2∆2)3 + 6m1∆1m2∆2m3∆3F . For F3, two different

conditions [40]

F3 =

F31 = [m23(3m12 −m3) +m2

1(m1 − 3m3) +m22(m2 − 3m3)]/(6m1m2m3)

when m1∆1 +m2∆2 > m3∆3

F32 = m12/2m3∆3

when m1∆1 +m2∆2 < m3∆3

(2.42)

need to be considered. When F3 ≤ F ≤ 1,

α =

α|a′α3+b′α2+c′α+d′=0 when F31 ≤ F < 1/2

m3∆3F +∆12/2 when F32 ≤ F < 1/2

(2.43)

Third order polynomial equation is solved by the root formulation

α =√−p0(

√3 sin θ − cos θ)− b/3. (2.44)

22


Here a = 1; p0 = c/3− b2/9; q0 = (bc− 3d)/6− c3/27; and θ = 13arccos(q0/

√−p30).

These relations are applied when F ≤ 1/2 and α ≤ (m1∆1 + m2∆2 + m3∆3).

When F > 1/2, the above relation holds for 1−F and (m1∆1 +m2∆2 +m3∆3) < α.

After the interface is reconstructed, the volume flux f across each cell face can be

calculated. The volume fraction F is updated in a direction splitting way as

Fi,j,k =ϕni,j,k + (∆t/∆x)(fi−1/2,j,k − fi+1/2,j,k)

1− (∆t/∆x)(ui+1/2,j,k − ui−1/2,j,k), (2.45)

Fi,j,k =F ni,j,k + (∆t/∆y)(fi,j−1/2,k − fi,j+1/2,k)

1− (∆t/∆y)(vi,j+1/2,k − vi,j−1/2,k), (2.46)

Fi,j,k =F ni,j,k + (∆t/∆z)(fi,j,k−1/2 − fi,j,k+1/2)

1− (∆t/∆z)(wi,j,k+1/2 − wi,j,k−1/2). (2.47)

F n+1i,j,k = F −∆t

(Fi,j,k

∆x(ui+1/2,j,k − ui−1/2,j,k)

+Fi,j,k

∆y(vi,j+1/2,k − vi,j−1/2,k) +

Fi,j,k

∆z(wi,j,k+1/2 − wi,j,k−1/2)

) (2.48)

In each splitting step, the volume flux is the interception of the volume enclosed by

the reconstructed interface and the hexahedron (rectangle for 2D) volume flowing

into the cell as illustrated in figure 2.6. It can be calculated using equation 2.38.

Lopez et al. [41] improved the PLIC-VOF method by using markers at the mid-

dle of the reconstructed interface segment. With the help of the markers, interface

structures thinner than the grid space can be captured. However, it increases the

complexity and computational cost and is not adopted in current model.

23


(a) (b)

Figure 2.6: Illustration of the volume flux calculation in two dimensional volume-of-fluid method. The shadowed area is: (a) the flux contributed by horizontal motion;(b) the flux contributed by vertical motion.

2.2.2 Coupled level-set/volume-of-fluid method

In CLSVOF method, level set function is used to calculate the surface normal for

the reconstruction in VOF method. Here a weighted least square method is used to

calculate the surface normal. In figure 2.7, the 9 points reconstruction scheme for

two dimensional problems is presented. For three dimensional problems, a 27 points

scheme is used. The interface in cell (i, j, k) can be represented by

ai,j,kx+ bi,j,ky + ci,j,kz = di,j,k. (2.49)

The coefficients can be obtained by minimizing the weighted integral

Ei,j,k =∫ xi+1/2

xi−1/2

∫ yj+1/2

yj−1/2

∫ zk+1/2

zk−1/2δ(ϕ)(ϕ− ai,j,k(x− xi)− bi,j,k(y − yj)

−ci,j,k(z − zk)− di,j,k)2dxdy.

(2.50)

24


Figure 2.7: Illustration of the least mean square method for interface normal calcu-lation in coupled level-set/volume-of-fluid method.

Discretizing it on the 27 points stencil, it becomes

Ei,j,k =∑i′=i+1

i′=i−1

∑j′=j+1j′=j−1

∑k′=k+1k′=k−1 wi′−i,j′−j,k′−kδϵ(ϕi′,j′)(ϕi′,j′,k′ − ai,j,k(xi′ − xi)

−bi,j,k(yj′ − yj)− ci,j,k(zk′ − zk)− di,j,k)2.

(2.51)

Here wr,s,t is the weight and δϵ(x) is the smoothed Dirac delta function (equation 2.71)

with smoothing length ϵ. For two dimensional problems, we use 16 for the center point

and 1 for others. [26] In three dimensional problems, we use w = 52 for the center

point and w = 1 for others. The large weight for the center point is necessary when

the grid space and the local curvature become comparable. To minimize Ei,j,k,

∂Ei,j,k

∂ai,j,k=

∂Ei,j,k

∂bi,j,k=

∂Ei,j,k

∂ci,j,k=

∂Ei,j,k

∂di,j,k= 0, (2.52)

25


Figure 2.8: Flow chart of the coupled level-set/volume-of-fluid method.

and we have∑∑

wδX2∑∑

wδXY∑∑

wδXZ∑∑

wδX∑∑wδXY

∑∑wδY 2

∑∑wδY Z

∑∑wδY∑∑

wδXZ∑∑

wδY Z∑∑

wδZ2∑∑

wδZ∑∑wδX

∑∑wδY

∑∑wδZ

∑∑wδ

ai,j,k

bi,j,k

ci,j,k

di,j,k

=

∑∑

wδΦX∑∑wδΦY∑∑wδΦZ∑∑wδΦ

. (2.53)

Here X = xi′ − xi; Y = yj′ − yj; and Z = zk′ − zk.

The flow chart of the CLSVOF method is presented in figure 2.8. In each time

step, the surface normal n and curvature κ are calculated from ϕ and are given to the

VOF method for interface reconstruction. After the volume fraction F is updated,

the reconstructed new interface is used for the correction of ϕ to improve the mass

conservation.

The points within n grid size from the interface are involved in the correction.

Sussman [26] use n = 5. In our tests, n = 2 can give the same accurate result as n = 5

for uniform grid. Smaller n makes the code more scalable for parallel computing. For

26


(a) (b) (c)

Figure 2.9: Different conditions of the redistancing of level set function ϕ from thereconstructed interface. The point with minimum distance is located (a) on the inside;(b) on the boundary; and (c) on the vertex of the interface segment.

each point involved, the minimum distance to adjacent plane segments is used as the

absolute value of the new level set function. The interface segments are not connected

with those of adjacent cells. The point with the minimum distance can be located on

the interface as in 2.9(a). It can also be located on the boundary or vertex as shown

in figure 2.9(b,c). This is why di,j,k is not used as the new level set function as in [42].

After the reinitialization by the reconstructed interface segments from VOF, a

classic level set reinitialization (i.e. equation 2.19) is applied to assure that property of

signed distance is satisfied in the entire domain. It also eliminates possible oscillations

which could be induced by the disconnected interface segments in highly stretched

grid system.

27


Stretching of two dimensional fluid disk

A two dimensional fluid disk with radius r = 0.15 stretched by a periodic vortical

flow field is simulated with both pure level-set method and the CLSVOF method.

The computational domain size is Lx × Ly = 1× 1. The velocity components are

u = −sin2(πx)sin(2πy)cos(πt/T ), (2.54)

v = sin(2πx)sin2(πy)cos(πt/T ). (2.55)

Here u and v are the velocities in x and y direction; T is the period of the velocity

variation. The center of the fluid disk is located at (0.5, 0.75).

The simulation results are presented in figure 2.10. At t = 3, the interface is

stretched to maximum and the width of the tail becomes comparable with the grid

size. With pure LS method (figure 2.10(a)), the tail is thin and loses some mass

compared to CLSVOF (figure 2.10(b)). At t = 6, the interface should return to its

original position, which is the case for CLSVOF(figure 2.10(d)). For pure LS method,

the circle becomes flat and distorted. In table 2.1, the percentage of mass loss is listed

for both methods and CLSVOF method demonstrates better conservation than that

of pure level-set method.

2.3 Multi-fluid flow simulation

In current model, coupled air-water system is simulated on a fixed Cartesian grid

as a one-fluid flow system and the coupled level-set/volume-of-fluid method is used

28


X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

(a) (b)

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

(c) (d)

Figure 2.10: Instantaneous interfaces of two dimensional fluid disk in a vortical flowfield simulated by pure level set method (a) t = 3, (c) t = 6; and coupled level-set/volume-of-fluid method (b) t = 3, (d) t = 6.

29


E% T/2 T

LS 0.69% 0.37%

CLSVOF 0.0058% 0.0033%

Table 2.1: Percentages of the numerical mass loss of both pure level-set method andcoupled level-set/volume-of-fluid method for the stretching fluid disk problem.

Figure 2.11: Sketch of the coupled air–water simulation.

to capture the air–water interface (figure 2.11).

The density and viscosity in the multi-fluid flow system can be written asρ(ϕ) = ρwH(ϕ) + ρa(1−H(ϕ)),

µ(ϕ) = µwH(ϕ) + µa(1−H(ϕ)).

(2.56)

Here ρw, ρa and µw, µa are the densities and viscosities of water and air respectively;

and H(x) is the Heaviside step function.

The compressibility of water is very small. Its motion can be described by the

30


incompressible Navier-Stokes equations as

∂uw

∂t+ (uw · ∇)uw = −∇p

ρw+∇2uw + g,

∇ · uw = 0.

(2.57)

Air is compressible. When the wind speed is low (i.e. less than 10% of the sound

speed and the Mach number is less than 0.1), the incompressible assumption is still a

good approximation. We use the incompressible Navier-Stokes equations to describe

its motion as

∂ua

∂t+ (ua · ∇)ua = −∇p

ρa+∇2ua + g,

∇ · ua = 0.

(2.58)

Utilizing equation 2.56, the Navier-Stokes equations for both air and water can

now be combined into equations

ρ(ϕ)(∂u∂t

+∇ · (uu)) = −∇p+∇ · (2µ(ϕ)D) + g + σκδ(ϕ)n.

∇ · u = 0

(2.59)

Here D = 12(∇u+∇uT ) is the strain rate tensor; σ is the surface tension coefficient.

Compared to equations of single fluid, equation 2.59 has one extra term σρ(ϕ)

κδ(ϕ)n

which represents the surface tension.

The physical domain is mapped into a computational domain. Length scale L,

velocity scale U , water density ρw, and water kinematic viscosity µw are used to

non-dimensionalize the NS equation to be

∂u∂t

= −∇ · (uu)− 1ρ(ϕ)

∇p+ 1Re

1ρ(ϕ)

∇ · (2µ(ϕ)D)

+ 1Fr2

k + 1We

1ρ(ϕ)

κδ(ϕ)n.

(2.60)

31


All symbols in equation 2.60 are the nondimensionalized counterparts of that in equa-

tion 2.59. The Reynolds number is defined as

Re =ρwUL

µw

; (2.61)

the Froude number is

Fr =U√gL

; (2.62)

and the Weber number is

We =ρwU

2L

σ. (2.63)

The primary variables u and p are defined on a staggered Marker-And-Cell(MAC)

type grid. Velocities are defined at the center of cell surfaces as in figure 2.12, and all

other quantities are defined in the center of the grid cell.

A second order Runge-Kutta method is used for time integration and a fractional

step method is used to solve the NS equation. The projection method [43] is used to

ensure the divergence-free requirement of incompressible flow. The following are the

four steps of current solver:

step 1,

u∗p − un

∆t= RHSn; (2.64)

step 2,

∇ · ∇ppρ

= −∇ · u∗

p

∆t, (2.65)

up − u∗p

∆t= −∇pp

ρ; (2.66)

32


Figure 2.12: Schematic of the MAC grid system used in current code.

step 3,

u∗c − up

0.5∆t= (RHSp −RHSn)− (−∇pp

ρ); (2.67)

step 4,

∇ · ∇pcρ

= −∇ · u∗c

0.5∆t, (2.68)

un+1 − u∗c

0.5∆t= −∇pc

ρ. (2.69)

Here RHS represents all the terms on the right hand side of equation 2.60.

The convective term (u∇) · u is nonlinear and can be discretized with different

schemes. Central difference scheme is non-dissipative but is not stable for problems

involving discontinuities. ENO scheme is stable but is dissipative and will kill high

frequency wave and turbulence components. A hybrid central difference and ENO

33


scheme is used in current model. In the vicinity of the interface within a range

of five grid points, the ENO scheme is used and it can avoid the instability (i.e.

Gibbs phenomenon) induced by the discontinuity across the interface. Away from the

interface, central difference scheme is used and it can avoid the numerical dissipation

brought by ENO scheme. Numerical tests show that the use of ENO scheme around

the interface has only negligible effect to the decay rate of the water wave. At the

same time, the central difference scheme used in the bulk flow on both air and water

sides assures the high fidelity turbulence (in both DNS and LES) and wave simulation.

2.3.1 Interface jump condition

For the air–water coupled system, the density, viscosity, pressure (when surface

tension is present), and velocity gradient are discontinuous across the interface. To

solve the unified equation 2.60, we will encounter spatial derivatives of discontinuous

quantities. Calculating the derivatives directly across the interface will generate nu-

merical oscillation near the interface (Gibbs phenomena). The Dirac delta function

in surface tension is singular on the interface and can not be implemented directly.

One way to address these discontinuities is to use a smooth transitional region

to replace the discontinuities and every discontinuous physical quantities are transi-

tioned smoothly from one fluid to the other (continuous surface force (CSF) method).

The Heaviside function and the Dirac delta function are replaced by their smoothed

34


counterparts

H(ϕ; ϵ) =1

2(1 +

ϕ

ϵ+

1

πsin(

ϕπ

ϵ)) (2.70)

δ(ϕ; ϵ) =1

2ϵ(1 + cos(

πϕ

ϵ)) (2.71)

Here ϵ is the smoothing width on each side of the interface. It is chosen as 2∆ and

∆ is the grid space. With the smooth transition, all derivatives across the interface

can be done as that in the region away from the interface.

Ghost fluid method

In CSF method, the physical quantities in the transition region can have large

error. The pressure gradient in the transition zone may generate spurious current

and contaminate the simulation. Ghost fluid (GF) method [29,44] is incorporated to

treat the interface in a sharp fashion. It addresses the contact discontinuity without

numerical smearing.

In GF method, the following interface jump condition [29]

N

T1

T2

(pI− τ)NT

=

σκ

0

0

(2.72)

needs to be implemented explicitly. Here N is the unit normal vector of the free

surface; T1 and T2 are the two unit tangent vectors; τ is the stress tensor; and

[ · ] denotes the jump across the interface. Combined with the velocity continuity

35


condition at the interface

[u] = 0, (2.73)

we have the stress jump condition

[µux] [µuy] [µuz ]

[µvx] [µvy] [µvz ]

[µwx] [µwy ] [µwz ]

= [µ]

∇u

∇v

∇w

0

T1

T2

T 0

T1

T2

+ [µ]NT N

∇u

∇v

∇w

NT N

−[µ]

0

T1

T2

T 0

T1

T2

∇u

∇v

∇w

NT N,

(2.74)

and the pressure jump condition

[p] = 2[µ](∇u · N ,∇v · N ,∇w · N) · N + σκ. (2.75)

With the gravity term absorbed into the pressure, the dynamic pressure jump condi-

tion becomes

[pd] = 2[µ](∇u · N ,∇v · N ,∇w · N) · N + σκ+ [ρ]gz. (2.76)

In the GF method, density and pressure are discontinuous and the weighted pres-

sure gradient 1ρ

∂p∂xi

is approximately continuous as shown in figure 2.13(a). To imple-

ment the jump condition explicitly, linear interpolation of level set function across the

interface is used to get the zero level set point (interface) first. Suppose the interface

passes between points i and i+1 (here we use x direction as an example, i is in water

and i+ 1 is in air), the position of the interface x0 is obtained from

xi+1 − x0

x0 − xi

=−ϕi+1

ϕi

. (2.77)

36


(a) (b)

Figure 2.13: Schematics of the treatment of discontinuity for pressure and shear stressin ghost fluid method.

With the location of the interface known, the pressure jump condition can be written

as

p+0 − p−0 = [p],

1ρa

pi+1−p+0∆x+ = 1

ρw

p−0 −pi∆x− ,

(2.78)

and both p+0 and p−0 can be obtained. Here ∆x− = x0 − xi; ∆x+ = xi+1 − x0; p+0 is

the pressure on the right side of the interface and p−0 is that on the left side. The

first order derivative ∂p∂x

is then calculated asp−0 −pi∆x− , which is equivalent to use a ghost

point gi+1 on the air side in figure 2.13(a). The second order derivative can then be

calculated as

∂2p

∂x2=

p−0 −pi∆x− − pi−pi−1

xi−xi−1

(xi+1 − xi−1)/2(2.79)

For velocity derivatives in the stress jump condition, similar procedure as that for

pressure is applied. Velocity is continuous at the interface and its derivative is not

37


Figure 2.14: Seven points stencil of the discretization of pressure poisson equation.

(figure 2.13(b)). The stress jump condition now becomes

µaui+1 − u0

∆x+− µw

u0 − ui

∆x− = [µux], (2.80)

and the velocity on the interface u0 is obtained. The second order derivative becomes

∂2u

∂x2

∣∣∣∣∣i =u0−ui

∆x− − ui−ui−1

xi−xi−1

(xi+1 − xi−1)/2. (2.81)

All other derivatives are calculated in the same way.

2.3.2 Pressure Poisson equation

The pressure Poisson equation is discretized with a seven point stencil as in figure

2.14 and linear algebra system

Apijk = bijk (2.82)

is obtained. Here, A is a square matrix with dimension N×N and N is the number of

total grid points in the computational domain. Coefficient matrix A is a sparse matrix

38


with most of its elements equal to zero and only the nonzero ones are stored. Because

of the density difference in the multi-fluid flow, the resulting coefficient matrix A is

not symmetric and it is solved with a preconditioned Bi-CGSTAB method [45]. The

preconditioned Bi-CGSTAB algorithm is briefly described as follows:

Initialize:

r0 = b− Ax0, x0 is the initial guess and r is the residual;

r0 = r0

ρ0 = α0 = ω0 = 1

v0 = p0 = 0

Iteration: (for i=1,2,3,...)

ρi = (ri, ri)

βi−1 = (ρi/ρi−1)/(αi−1/ωi−1)

pi = ri + βi−1(pi−1 − ωi−1vi−1)

solve p in Kp = pi

vi = Ap

αi = ρi/(vi, r0)

s = ri − αivi

Kq = s ⇒ q

u = Aq

ωi = (u, s)/(u,u)

xi+1 = xi + αip+ ωiq

39


ri+1 = s− ωiu

if ||ri+1||/||b|| ≤ ϵ return; else i = i+ 1 iterate.

Here K is the preconditioner to reduce the condition number of the resulting linear

algebra system. In current code, a ADI type tridiagonal factorization [46]

K = (D + ALx + AU

x )D−1(D + AL

y + AUy )D

−1(D + ALz + AU

z ) (2.83)

is adopted as the preconditioner. To utilize the capability of large scale pipe line

structure of modern supercomputers and get perfect parallelism, Xiao [47] proposed

an improvement by making use of the Jacobi splitting in the direction of domain

decomposition. Kim and Moin [48] proposed a method to utilize constant coefficient

Poisson equation. In our code, two dimensional transposing is used when the tridiag-

onal matrix needs to be solved in the direction of domain decomposition and is found

to be efficient and fast in the numerical tests.

With periodic boundary condition, the resulting matrix Asi (can be AL

x , AUx , A

Ly ,

AUy , A

Lz , or A

Uz )

Asi =

a1,1 a1,2 b

a2,1 a2,2 a2,3

..

.

an−1,n−2 an−1,n−1 an−1,n

c an,n−1 an,n

(2.84)

is not fully tridiagonal. Sherman-Morrison [49–51] method is adopted. The matrix

Asi is split into a tridiagonal matrix B and the dyadic of two vector u and v, and

(Asi )

−1 = (B + uvT )−1 = B−1 − B−1uvTB−1

1 + vTB−1u. (2.85)

40


Here

B =

2a1,1 a1,2

a2,1 a2,2 a2,3

..

.

an−1,n−2 an−1,n−1 an−1,n

an,n−1 an,n + bca1,1

(2.86)

and

u =

−a1,1

0

...

0

c

v =

1

0

...

0

−b/a1,1

(2.87)

2.3.3 Parallelization and scalability

Parallel computing on large-scale computers is needed for high resolution simu-

lation. Current code is parallelized using Message Passing Interface (MPI) [52, 53]

based on domain decomposition as illustrated in figure 2.15.

Speedup tests are performed. Figure 2.16(a) shows the scaling test obtained on

the SGI computer, in which the total problem size is fixed and the relation between

simulation time and the number of processing elements (PEs) NPE is examined. In

the figure, the speedup is defined as

TrefNref/TNPE. (2.88)

Here, Nref is the PE number of a reference simulation (set to be 32 here; because

the problem is large, it cannot run on one PE), Tref is the corresponding reference

41


Figure 2.15: Illustration of the domain decomposition in current code.

wall-clock simulation time, and TNPEis the simulation time using NPE PEs.

As shown in figure 2.16(a), tests result for problems with different sizes are per-

formed. For NPE = 256 and smaller, good scaling is obtained and super-linear

speedup is obtained for some PE numbers. As NPE increases to 512, there is a drop

in the speedup for the case with smaller size. This drop can be explained by the fact

that the simulation are three dimensional and domain decomposition is performed in

the y-direction only. The larger case has 4 times grid points in the y direction as the

smaller case. As NPE becomes large, the grid number per PE for the smaller case

gets close to one in the y-direction and thus the communication overhead increases

and speedup drops.

The load balance, communication and synchronization overhead, and I/O are ana-

lyzed using the profiling tool CrayPat [54,55] on Cray XT series supercomputers. An

example result shown in table 2.2 is discussed below. The imbalance time percentage

of user functions is defined as

Imbalance% = 100× Timbalance

Tmaximum

× NPE

NPE − 1(2.89)

42


NP

Spe

edup

100 101 102 103100

101

102

103

17M Grids67M GridsIdeal Speedup

Figure 2.16: Result of speedup test.

Here, Tmaximum is the maximum time among the NPE PEs, and its difference from the

PE-averaged value is Timbalance. As shown in table 2.2, the imbalance percentage is

small. The communication and synchronization time percentage increases with NPE

but is still a small portion. The I/O overhead is also very small.

With the rapid developing of computer technology, more PEs are available and

larger problems can be attacked. The code can be further optimized for even larger

problems. The blocking communication and global MPI operations can be further

reduced. The hybrid MPI/OpenMP programming model can be used on computers

that use multi-core processors, each computing node is a shared memory system,

and different nodes are interconnected to form a distributed memory system. The

hybrid MPI/OpenMP model [56, 57] uses OpenMP within the node and MPI across

the different nodes. This hybrid model is expected to reduce the communication and

synchronization overhead, especially for large NPE. The I/O overhead may become a

bottleneck if NPE becomes large. The MPI I/O [58,59] implemented in MPI2 can be

43


# of cores MPI+MPI SYNC IO Imbalance%

16 1.7% 1.1% 0.5

32 4.5% 1.0% 0.8

64 9.4% 1.3% 1.8

128 8.6% 2.3% 1.5

256 15.2% 4.1% 2.4

Table 2.2: Profiling results of the coupled level-set/volume-of-fluid method code usingCraypat on Cray-XT5 supercomputer of the High Performance Computing Modern-ization Program initiated by Department of Defense.

used to optimize the noncontiguous data read/write. With the parallel, non-blocking,

and collective read/write, the total simulation time is expected to be reduced.

2.4 Test cases

2.4.1 Zaleski problem

The Zaleski problem [37] concerns a notched disk (dashed line in figure 2.17) in a

rotational flow field. The computational domain is Lx×Ly = 100×100. The velocity

field is that of a point vortex located at the center of the domain with angular velocity

Ω = 0.01. The center of the disk is located at (50, 75) and its radius r = 15 initially.

The notch is located at the bottom of the disk with width 10 and ends at y = 85.

In figure 2.17, the instantaneous interfaces simulated with both splitting scheme

44


X

Y

0 20 40 60 80 1000

20

40

60

80

100

X

Y

0 20 40 60 80 1000

20

40

60

80

100

(a) (b)

Figure 2.17: Instantaneous interfaces of Zaleski problem calculated with (a) splittingscheme and (b) ENO scheme after one rotation. The dashed lines are the theoreticalresults.

and ENO scheme are presented. Solid lines are the interfaces after one rotation and

the dashed lines represent the exact interface. As shown in figure 2.17(b), the notch

simulated with ENO scheme gradually disappears after one rotation. With splitting

scheme, the length of the notch is almost the same as the exact interface although

a little bit asymmetry is observed, which demonstrates that splitting scheme has

less numerical diffusion than ENO scheme and is able to handle slender interface

structures.

45


bubble

σκ

Figure 2.18: Schematic of the static air bubble simulated.

2.4.2 Two dimensional air bubble without gravity

A two dimensional air bubble as in figure 2.18 is simulated using both CSF method

and GF method without considering the gravity effect. The radius of the bubble is

r = 1; the domain size is 4× 4; and the surface tension coefficient is 1. The pressure

inside and outside the bubble should be constant and the difference ∆p = σκ = 1.

As shown in figure 2.19(c,f), the pressure obtained by GF method has a sharp jump

across the interface with pressure difference approximately 1. For interfaces obtained

by CSF method (figure 2.19(a,b,d,e)), transition zones are observed and pressure

oscillations are observed along the interface. When the smoothing length becomes

smaller (ϵ = ∆), larger oscillation is generated near the interface.

46


(a) (b) (c)

Y

P

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Y

P

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Y

P

0 1 2 3 4-0.5

0

0.5

1

1.5

2

(d) (e) (f)

Figure 2.19: Pressure distributions of the two dimensional static bubble simulatedwith: (a,d) CSF method with ϵ = 2∆; (b,e) CSF method with ϵ = ∆; (c,f) GFmethod. Lines in (d,e,f) are the corresponding pressure distribution along X = 2 in(a,b,c) at the middle plane of bubble.

47


ρ2=1µ2=0.1

ρ1=1µ1=1

U

Figure 2.20: Schematic of the two-layer Couette flow.

2.4.3 Two-layer Couette flow

A two-layer Couette flow as in figure 2.20 is simulated with both CSF and GF

methods. The fluids have the same density but different viscosity. The domain size

is 2 × 2 domain and the heights of both layers are 1. The top boundary is moving

with a constant speed U = 1.1. The steady horizontal velocity profile should be

u(y) =

0.1y when y <= 1,

0.1 + (y − 1) when y > 1.

(2.90)

The simulated velocity profiles together with the exact solution are plotted in figure

2.21. The difference between the simulated profiles and the exact profiles looks in-

discernible. The relative error is also presented in figure 2.21. Around the interface,

the error of the CSF simulation is as large as 12 percent and that of GF simulation is

almost zero, which shows that GF method can effectively avoid the spurious current

48


U

Error%

Y

0 0.2 0.4 0.6 0.8 1

0. 5.% 10.% 15.% 20.%

0

0.5

1

1.5

2

CSFExact SolutionError%

U

Error%

Y

0 0.2 0.4 0.6 0.8 1

0 5% 10% 15% 20%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

GFMExact SolutionError%

(a) (b)

Figure 2.21: Velocity profiles and error percentages of the two layer Couette flowsimulated with (a) Continuous surface force method and (b) Ghost fluid method.

that is encountered by CSF method.

2.4.4 Two dimensional air bubble

A two dimensional circular air bubble with radius 1/3cm initially static in the

water is simulated. The computational domain is 2cm × 3cm. The center of the

bubble is located at y = 1cm. Nondimensionalizing the NS equation with length scale

L = 1cm and gravitational acceleration g = 9.8m/s2, we have the Reynolds number

Re = 3.13 × 103, the Froude number Fr = 1.0, and the Weber number We = 13.6.

The surface of the bubble at different time are plotted in figure 2.22. The bubble

moves upward because of buoyancy force. The shape of the bubble changes from

circle to meniscus when rising up and the interface keeps coherent.

49


x (cm)

y(c

m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

x (cm)

y(c

m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

(a) (b)

x (cm)

y(c

m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

x (cm)

y(c

m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

(c) (d)

Figure 2.22: The air–water interface of a two dimensional air bubble with radius1/3cm raising in the water at time (a) t = 0.0 s; (b) t = 0.02 s; (c) t = 0.035 s; (d)t = 0.05 s.

50


2.4.5 Three dimensional air bubble bursting on

water surface

A three dimensional air bubble interacting with the free surface is simulated. Here

I consider a spherical air bubble with radius r = 5mm located 1mm under the water

surface initially. The domain size is 3cm × 6cm. Choosing length scale L = 5mm

and velocity scale U = 0.12m/s, the Reynolds number Re = 600, Froude number

Fr2 = 0.29, and the weber number We = 1.

In figure 2.23, the instantaneous air–water interfaces at different time are plotted.

As the bubble rises up, its bottom becomes flat and concave. After the bubble

bursting on the surface, a water jet splashes up. When the jet returns, a water

droplet is pinched off at the tip. The droplets generated by bubble bursting is one of

the major source of the water spray over ocean surface.

2.4.6 Gravity wave

A two dimensional sinusoidal gravity wave is simulated with the numerical tool

we developed. The domain size is 2π × 2π and the mean water depth is π. The

wave length is 2π, so the wave number is 1. The Froude number Fr = 1 and the

Reynolds number Re = 100. No surface tension force is included. The wave slope

ak = 0.1 is used (here k is the wavenumber), so linear wave theory is still valid and

corresponding velocity field is used as initial condition. The wave decays with time

51


(a) (b)

(c) (d)

(e) (f)

Figure 2.23: Instantaneous air–water interface of a three dimensional air bubble burst-ing on the free surface at time (a) t=0.0 s; (b) t=0.017 s; (c) t=0.033 s; (d) t=0.05 s;(e) t=0.067 s; (f) t=0.083 s.

52


t/T

a

0 5 10 15 20 25 30 35

0.02

0.04

0.06

0.08

0.1

SimulationTheory

Figure 2.24: Amplitude evolution of a gravity wave with initial wave slope ak = 0.1and its comparison with linear theory.

because of viscous dissipation. According to Lamb [60], the wave amplitude

a(t) = a(0) e−1Re

k2t. (2.91)

The amplitude evolution of the simulation with resolution Nx × Nz = 128 × 128 is

plotted in figure 2.24 together with the theoretical solution. The simulation result

collapses with the theory very well during the 36 wave periods simulated.

The same problem is simulated with different resolutions for convergence test. The

results are plotted in figure 2.25 and the wave amplitudes for resolution 128 × 128

and 256× 256 almost collapse, which indicate the convergence of our simulation.

2.4.7 Capillary wave

Surface tension effect is important for small scale interface structures. Here a

capillary wave is simulated with the numerical tool we developed. No gravity effect

53


t/T

a

0 1 2 3 4 50.02

0.04

0.06

0.08

0.1

64x64128x128256x256

Figure 2.25: Amplitude evolution of a gravity wave with initial wave slope ak = 0.1with different resolution.

is included. The initial wave surface elevation is given as

η(x) = a cos(kx). (2.92)

Here wave amplitude a = 0.1 and wave number k = 1. The initial water side velocity

field is given as

u(x) = ak3

ωWeekz cos(kx),

w(x) = ak3

ωWeekz sin(kx).

(2.93)

Here the Weber number We = 1; the Reynolds number Re = 100; and the angular

frequency satisfies dispersion relationship ω =√

k3

We. The amplitude evolution is

plotted in figure 2.26. The dashed line represents the decaying amplitude according

to linear theory. The decaying wave amplitude of our simulation collapses with that

of the theory very well. The oscillation is caused by the standing wave with wave

length the same as the domain size.

54


t/T

a

0 2 4 6 80.02

0.04

0.06

0.08

0.1

Figure 2.26: Amplitude evolution of a capillary wave with initial slope ak = 0.1 (solidline) and it comparison with linear theory (dashed line).

55

Chapter 3

Direct Numerical Simulation of

Two Dimensional Wave Breaking

Wave breaking is a ubiquitous phenomenon in the ocean. It generates surface

current and underwater turbulence, dissipates wave energy, and enhances heat and

moisture exchange between air and water. The study of wave breaking is difficult

because of the strong nonlinearity and wide scale range involved in the breaking

process.

Strong breaking waves can be identified from the overturning crests or air en-

trainment visually. Breaking can also happen without those characteristics and is

difficult to be distinguished from non-breaking waves. Breaking criteria based on

geometry, velocity, and acceleration have been proposed, but universal ones have not

been found. The criterion based on local mean wave energy and momentum densi-

56

CHAPTER 3. DIRECT NUMERICAL SIMULATION OF TWO DIMENSIONALWAVE BREAKING

ties [61] seems promising but it needs the full knowledge of the flow field under the

free surface and equivalent expression with only the surface information is not found.

Further exploration of the surface characteristics associated with breaking waves is

still needed.

Wave breaking is the major energy sink in the ocean, so most of the models

currently in use, such as WAM, SWAN, and WAVEWATCH III, model the effects of

wave breaking as a dissipation term in a phase-averaged context. For phase-resolved

wave models currently in development, wave breaking also need to be modeled. Tian

et al. [2,62,63] proposed an eddy viscosity model into the high order spectral (HOS)

method to model the effect of wave breaking. Liu et al. [64] coupled wind LES and

high-order spectral wave simulation. The wave breaking effect is represented by a

numerical dissipation. Further understanding of the energy dissipation process and

quantification of model coefficients are expected to validate these models.

Various techniques have been used to study breaking waves. In the field, statistic

study based on visual effects of breaking waves gives the probability density function

in the wave model of [65]. Experimentally, Rapp & Melville [66] studied the wave

breaking generated by linear focusing. Chang & Liu [67, 68] investigated the break-

ing generated by strong monochromatic waves. Perlin et al. [69] studied the steep

breaking waves generated by a modified Davis and Zarnick technique [70]. Melville

et al. [71] measured the velocity field under breaking waves through a “mosaic” tech-

nique on the images obtained by digital particle image velocimetry.

57


Numerical simulation of wave breaking is favorable because of the detailed in-

formation it provides. Longuet-Higgins & Cokelet [72, 73] simulated the breaking of

steep progressive waves with boundary integral method based on potential theory.

The simulation can proceed until the jet touches down on the front face of the wave.

Chen et. al. [74] performed direct numerical simulation of progressive breaking waves

by solving the Navier-Stokes equations with the help of volume-of-fluid (VOF) method

to capture the air-water interface. Viscous effect is included and the simulation is per-

formed for the entire breaking process including the jet reentry and splash-up. Large

amplitude third-order stokes wave is used as initial condition. Increased air-water

density ratio 0.01 and viscosity ratio 0.4 are adopted. Hendrickson [75] studied the

kinematics and dynamics of wave breaking in detail with level-set method. Different

initial conditions are tested and energy dissipation mechanisms and transfer between

air and water are discussed. Iafrati [76] pursued detailed study of the breaking inten-

sity effect on the wave breaking process using level-set method for interface capturing.

The real air-water density ratio 0.00125 is used but an increased viscosity ratio 0.4 is

adopted. Large eddy simulations of wave breaking on the beach have been performed

by Watanabe et al. [77] and Lakehal & Liovic [78].

Here we perform direct numerical simulation of wave breaking with real air–water

density ratio and real air–water viscosity ratio to study the physics of the flow asso-

ciated with breaking waves in deep water.

58


X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

air

water

3rd order Stokes wave

mean water level

Figure 3.1: Sketch of the setup of two dimensional breaking waves.

3.1 Problem setup

The sketch of the setup of our simulation is presented in figure 3.1. A steep 3rd

order Stokes wave is simulated in a domain of size Lx × Ly = 1 × 1 with periodic

boundary conditions in horizontal directions. The mean water level is located at

y = 0.5. Free slip boundary conditions are applied at top and bottom boundaries.

The initial free surface elevation is

η(x) = a cos(kx) +1

2a2k cos(2kx) +

3

8a3k2 cos(3kx). (3.1)

Here a is the wave amplitude and k is the wave number. The initial velocity compo-

59


nents on the water side are given as

u = F31 cosh(k(h+ y)) cos(kx) + F32 cosh(2k(h+ y)) cos(2kx)

+F33 cosh(3k(h+ y)) cos(3kx)

v = F31 sinh(k(h+ y)) sin(kx) + F32 sinh(2k(h+ y)) sin(2kx)

+F33 sinh(3k(h+ y)) sin(3kx)

(3.2)

Here h is the water depth and

F31 = aσ/ sinh(kh)− a2kσ cosh2(kh)(1 + 5 cosh2(kh))/8/ sinh5(kh),

F32 = 34a2kσ/ sinh4(kh),

F33 = 364a3k2σ(11− 2 cosh(2kh))/ sinh7(kh).

(3.3)

On the air side, we damp the velocity from the free surface to zero at the top boundary

exponentially as

ua = use−10(y−η(x)) (3.4)

va = vse−10(y−η(x)) (3.5)

to avoid the initial velocity discontinuity across the interface. Here us and vs are the

velocity components at the wave surface.

The Reynolds number Re = 5000, Froude number Fr = 1, and Weber number

We = 10000. The wave length in physical domain is around 0.271 m without con-

sidering the Reynolds number. Different initial wave slopes as listed in table 3.1 are

considered.

60


3.2 Characteristics of the free surface of

breaking waves

3.2.1 Wave breaking with different intensities

Wave breaking can be classified into three categories [79]: plunging breaking,

spilling breaking, and surging breaking. In plunging breaking, a plunging jet is formed

and impinges on the trough of the wave and entrains large amount of air. Spilling

breaking is milder than plunging breaking. It usually occurs with parasitic capillary

waves riding on the front face of the wave. Surging breaking usually occurs on very

steep beaches. We mainly focus on the wave breaking in deep water and surging

breaking is out of the scope of current thesis. The types of breaking for cases with

different initial slopes are listed in table 3.1 and they are consistent with that of

Iafrati [76].

In figure 3.2, the evolutions of the wave surface for cases with different initial

slopes are plotted. For the case of (ak)0 = 0.3 (figure 3.2(a)), the wave does not

break and the surface is smooth all the way during the simulation. For the case of

(ak)0 = 0.35 (figure 3.2(b)), the wave crest overturns a little but no plunging jet is

formed. It is a spilling breaking. For non-breaking and spilling breaking waves, no

air entrainment is observed and the wave crest moves forward with a constant wave

speed. For cases with (ak)0 = 0.4, 0.44, 0.55 (figure 3.2(c,d,e)), plunging jets form

61


X

Y+

t

0 0.5 1 1.5 2

1

2

3

4

5

6

7

X

Y+

t

0 0.5 1 1.5 2

1

2

3

4

5

6

7

X

Y+

t

0 0.5 1 1.5 2

1

2

3

4

5

6

7

(a) (b) (c)

Figure 3.2: See next page for caption.

62


X

Y+

t

0 0.5 1 1.5 2

1

2

3

4

5

6

7

X

Y+

t

0 0.5 1 1.5 2

1

2

3

4

5

6

7

(d) (e)

Figure 3.2: Free surface profiles for waves with different initial steepness (a) ak = 0.3;(b) ak = 0.35; (c) ak = 0.4; (d) ak = 0.44; (e) ak = 0.55.

63


ak non-breaking spilling plunging

0.3 X

0.35 X

0.4 X

0.44 X

0.55 X

Table 3.1: Breaking wave types for different initial wave slopes.

at the wave crests and then hit the front wave face. Air bubbles are entrained by

the plunging jets. Splashing-up is generated and entrains more air. Water droplets

are also generated. For the plunging breaking, the wave crest splits because of the

plunging and splash-up. The air bubbles entrained by the plunging jet move in a

low speed and form trajectories bifurcated to the left of the trajectory of the crest.

The jet tip moves faster than the crest and forms a trajectory bifurcated rightwards.

After all the bubbles burst out of the surface and all the water droplets fall into the

water, the bifurcated trajectories disappear.

3.2.2 Spectra of the free surface

As breaking approaches, wave energy is redistributed among different wave com-

ponents by a strong nonlinear wave–wave interaction process. In figure 3.3, the wave

surface spectra are plotted for different cases. For non-breaking waves (figure 3.3(a)),

64


the spectral energy of low wave number components does not have obvious change

and the spectra oscillate only in a small range. For spilling breaker(figure 3.3(b)),

the high wave number components increase until the incipient breaking when the

spectrum collapses to k−3. For plunging breaker (figure 3.3(c,d,e)), the saturated

k−3 spectra at incipient breaking are also observed. The saturated k−3 spectrum is

analytically obtained by Thornton [80] through dimensional analysis. Although dif-

ferent frequency spectra have been found in the field, the k−3 wavenumber spectrum

is found to be universal [81].

After spilling breaking (the dotted line in figure 3.3(b)), the low wave number

components lose a large amount of energy but the high wave number components do

not have much change. After plunging breaking, low wave number components lose

energy and high wave number components obtain energy. The difference for the low

wave number components comes from that disturbance of spilling breaker is confined

around the crest but the plunging breaker disturbs a large surface area.

It needs to be pointed out that second order accuracy is achieved in current code.

The grid space used in current simulation is around 0.004 and the numeric error

for the surface elevation is of order 10−5, thus the error of the spectra is of order

10−10. So the high frequency oscillation at large wavenumbers in figure 3.3 may be

contaminated by numerical error and can not be used for explaining physics.

65


k/k0

S(k

)

100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

t=0.025Tt=Tt=2Tt=4T

k/k0

S(k

)

100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

t=0.025Tt=Tincipient breakingt=5T

k-3

(a) (b)

k/k0

S(k

)

100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

t=0.025Tincipient breakingt=5T

k-3

k/k0

S(k

)

100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2


k-3

(c) (d)

k/k0

S(k

)

100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2


k-3

(e)

Figure 3.3: Surface spectra of wave surfaces for cases with: (a) (ak)0 = 0.3; (b)(ak)0 = 0.35; (c) (ak)0 = 0.4; (d) (ak)0 = 0.44; and (e) (ak)0 = 0.55.

66


X

η

0 0.2 0.4 0.6 0.8 1-0.1

-0.05

0

0.05

0.1

a1

a2

b1b2

Figure 3.4: Schematic of a nonlinear wave and the quantities used for definition ofskewness and asymmetry.

3.2.3 Steepness, skewness and asymmetry

As waves evolve to break, using ak to describe the steepness of wave becomes

confusing because of the asymmetry both horizontally and vertically. We use the

wave steepness definition [82]

Hk/2 =a1 + a2

2k. (3.6)

Here H is the wave height; a1 is the crest amplitude and a2 is the trough amplitude

as in figure 3.4; and k is the wavenumber.

The skewness

Sk = a1/a2 − 1, (3.7)

and the asymmetry

As = b1/b2 − 1 (3.8)

67


are introduced to describe the geometric deviation from sinusoidal wave [82]. Here

b1 and b2 are horizontal distances from the crest to the two zero crossing points as

in figure 3.4. The skewness describes the vertical asymmetry of wave. Zero skewness

indicates that the wave is symmetric vertically. Positive skewness indicates that the

wave crest is steeper than the trough, and vice versa. The asymmetry describes the

horizontal asymmetry of the wave. Zero asymmetry indicates that wave is horizontally

symmetric. Negative asymmetry indicates that the wave is lean forward, which is

usually observed before wave breaks.

For sinusoidal wave, both skewness and asymmetry are zero. In Stokes wave, the

initial asymmetry is zero and the skewness is positive. In figure 3.5, the steepness,

skewness, and asymmetry are plotted for the non-breaking wave with (ak)0 = 0.3. In

figure 3.5 (a), the wave steepness decreases with time because of the viscous dissipa-

tion. Oscillation with period 0.5T (T is the primary wave period) is observed, which

is caused by the standing wave with wave length the same as the horizontal domain

size [75].

In figure 3.5 (b,c), the skewness and asymmetry demonstrate a periodic behavior

with period approximately 2T . This kind of subharmonic is caused by the Benjamin-

Feir instability [72, 82]. The time series of the surface elevation at x = 0 is plotted

in figure 3.6. The local maximum of the surface elevation are connected with a

dashed line and this line oscillates with period 2T , which confirms the existence of

the subharmonic observed in figure 3.5 (b,c).

68


t/T

Hk/

2

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

(a)

t/T

Sk

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

(b)

t/T

As

0 1 2 3 4 5 6 7 8-0.3

-0.2

-0.1

0

0.1

0.2

0.3

(c)

Figure 3.5: Steepness, skewness and asymmetry evolution with time for steep non-breaking waves with (ak)0 = 0.3.

69


t/T

η

0 2 4 6 8-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 3.6: Surface elevation evolution with time at x = 0 for steep non-breakingwaves with (ak)0 = 0.3. The dash line enclosing the wave shows sub-harmonic withperiod two times the primary wave period.

The asymmetry changes its sign alternatively, which indicates that the wave is

leaning forward and backward periodically. In figure 3.7, the wave profile at time

t/T = 3.08, which corresponds to a negative asymmetry, deviates to the right of

the sinusoidal wave profile. The wave profile at t/T = 4.25, which corresponds to a

positive asymmetry, deviates to the left.

In figure 3.8, the steepness, skewness, and asymmetry for the spilling breaker

with (ak)0 = 0.35 are plotted. The steepness increases until breaking if we do not

consider the oscillation caused by standing wave. The skewness increases to one at

around 1.2T and then remains almost constant until the overturning happens. The

asymmetry decreases from zero to −0.45 before breaking occurs.

The steepness, skewness, and asymmetry for plunging breakers with ((ak)0 =

70


X

Y

0 0.2 0.4 0.6 0.8 10.46

0.48

0.5

0.52

0.54

0.56t/T=3.08t/T=4.25sine

Figure 3.7: Instantaneous wave profiles at time with opposite asymmetries (horizon-tally shifted to have two zero crossing points symmetric about x=0.5).

0.4, 0.44, 0.55) are plotted in figure 3.9. Because the breaking happens very fast and

there is standing wave, the general trend for the steepness is not very clear. In the

case with (ak)0 = 0.4, if we compare the two adjacent maximum, we still observe

an increase of the steepness. The skewness maxima are located between 0.8 and 1,

which indicates that there could be a limit for the wave skewness. The asymmetries

for all the three cases decrease from zero to −0.7, which indicates a profile similarity

for waves near breaking as stated by Caulliez [4]).

3.3 Velocity field under breaking waves

When waves break, part of the water conquers the restoring force of wave and

obtains speed larger than the wave phase speed. The horizontal velocity maxima

71


t/T

Hk/

2

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

(a)

t/T

Sk

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

(b)

t/T

As

0 0.5 1 1.5 2-1

-0.8

-0.6

-0.4

-0.2

0

(c)

Figure 3.8: Steepness, skewness and asymmetry evolution with time for spillingbreaker with (ak)0 = 0.35.

72


t/T

Hk/

2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

t/T

Hk/

2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

t/T

Hk/

2

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

(a) (b) (c)

t/T

Sk

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

t/T

Sk

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

t/T

Sk

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

(d) (e) (f)

t/T

As

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

t/T

As

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

t/T

As

0 0.1 0.2 0.3 0.4 0.5-1

-0.8

-0.6

-0.4

-0.2

0

(g) (h) (i)

Figure 3.9: Steepness, skewness and asymmetry versus time for plunging breaker with(ak)0 = 0.4 (a,d,g); (ak)0 = 0.44 (b,e,h); and (ak)0 = 0.55 (c,f,i).

73


t/T

u max

/c

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3(ak)0=0.3(ak)0=035(ak)0=0.4(ak)0=0.44(ak)0=0.55

Figure 3.10: Evolution of the maximum velocity with time for cases with (a) (ak)0 =0.35; (b) (ak)0 = 0.4; (c) (ak)0 = 0.44; and (d) (ak)0 = 0.55.

versus time for breaking wave with different intensities are plotted in figure 3.10. For

non-breaking wave with (ak)0 = 0.3, the maximum horizontal velocity is less than

the wave phase speed all the time and its variation is small. For breaking waves, the

maximum horizontal velocities are larger than the wave phase speed, which satisfy

the kinematic breaking criterion. The maximum horizontal velocity can reach as high

as three times the wave phase speed for the case of (ak)0 = 0.55.

The contours of the horizontal velocity when the maximum is achieved are plotted

in figure 3.11. The wave and the breaker can be divided by the dashed line with

u = c. The wave parts are similar for all the cases, which also confirms the profile

similarity [4]. The maximum horizontal velocity always happens at the front tip of

the breaker.

74


(a) (b)

(c) (d)

Figure 3.11: Velocity contours of cases with (a) (ak)0 = 0.35; (b) (ak)0 = 0.4; (c)(ak)0 = 0.44; and (d) (ak)0 = 0.55 when the maximum velocity is achieved.

75


t

Eto

tal

0 5 10 15 200

0.001

0.002

0.003

0.004

0.005

0.006

t

Eto

t/Eto

t_0

0 5 10 15 200

0.2

0.4

0.6

0.8

1

(a) (b)

Figure 3.12: Evolution of (a) the total mechanical wave energy and (b) the normalizedtotal mechanical wave energy with time for cases with different initial steepness: ————, (ak)0 = 0.3; – – – – , (ak)0 = 0.35; – · – · – , (ak)0 = 0.4; · · · · · · · ,(ak)0 = 0.44; and −−− −−− , (ak)0 = 0.55.

3.4 Energy dissipation by wave breaking

Wave breaking is the major wave energy sink in the ocean. The understanding

of the energy dissipation process is critical for wave modeling. In figure 3.12(a), the

time evolution of the total mechanical wave energy

Etotal = Ek + Ep =

∫ρu2 + v2

2dxdy +

∫ρgydxdy (3.9)

of different cases is plotted. Here Ek is the kinetic energy and Ep is the potential

energy.

For breaking waves in figure 3.12, three decaying regimes can be identified: the

pre-breaking regime when breaking has not occurred yet and the energy decays with

the same rate as non-breaking waves; breaking regime when breaking occurs and

76


energy decays very fast; post-breaking regime when breaking has almost finished and

the energy decay rate returns to that of non-breaking wave. The duration of the

pre-breaking slow decay decreases as the wave steepness increases. The strong decay

lasts for approximately 2 wave periods. The total energy loss increases with wave

steepness.

The total energy normalized by its initial value is plotted in figure 3.12(b). For

strong plunging breaker, the remaining energy percentage is almost independent of

the wave steepness.

3.5 Modeling of wave breaking

To fully resolve the details of wave breaking in large scale phase resolved wave

field simulation is still impossible because of the wide scale range involved. Tian

et al. [2, 62, 63] incorporated an eddy viscosity model into the high order spectral

(HOS) method to model the effect of wave breaking. In their simulation, second

order diffusion terms with eddy viscosity

νeddy = αhblbr/τb (3.10)

are added to both the kinematic and dynamic boundary conditions. Here, τb is the

breaking time scale defined as the duration from surface overturning to the diminish-

ing of surface disturbance front; lbr is the breaking length scale defined as the distance

from the incipient breaking to the diminishing point of the surface disturbance; hb

77


S

τ b/T

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

Current Simulationτb/T=3.58S+0.13Tian et al (2010)

Figure 3.13: Breaking time scale versus wave steepness S and comparison with Tian’s[2] model (dashed line) and data.

denotes the falling crest height; and α is a constant coefficient chosen as 0.02. This

breaking model is activated when local surface slope exceeds a criterion Sc = 0.95.

In their experiment, they found promising correlations between those breaking scales

and the wave steepness.

As pointed out by Tian et al. , their definition of scales does not follow quantitative

threshold and is subjective. In our simulation, all the flow details are known and we

can have more objective definitions. Here, the time scale of breaking is defined as

the time period when the maximum horizontal velocity (as in figure 3.10) is larger

than the wave phase speed. In figure 3.13, the time scale lbr is plotted versus wave

steepness S together with Tian’s fitting. The breaking length scale is defined as the

distance from the position of the crest overturning to the position where the last

air bubble bursts out of the surface. The length scale obtained from our simulation

versus wave steepness S (S corresponds to (ak)0 in our simulation) is plotted in figure

78


S

l br/L

0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

Current simulationlbr/L=3.86S-0.14Tian et al (2010)

Figure 3.14: Breaking length scale versus wave steepness S and comparison withTian’s [2] model (dashed line) and data.

3.14 together with Tian’s result. With the time scale and length scale obtained, the

breaking velocity is calculated through

ubr = lbr/τb. (3.11)

The results are plotted in figure 3.15 together with Tian’s results. For (ak)0 = 0.55,

the breaking velocity is larger than the wave phase speed which deviates from Tian’s

measurement. This is because periodic boundary condition is used and multiple peri-

odic breakers are involved in our simulation. The breaker is caught up by the following

breakers before all the air bubbles bursts out, which makes the length scale large as in

figure 3.14. This indicates that the model for single breaking and continuous breaking

should be considered separately.

In phase-averaged wave field simulation, the wave breaking is usually modeled by

79


τb/T

l br/L

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5current simulationubr=cubr=0.75cubr=0.863c

Figure 3.15: Normalized breaking length scale versus normalized breaking time scale.

an energy dissipation rate [65]

dE

dt= b

ρc5

gΛ(c)dc (3.12)

Here b is the breaking parameter and Λ(c)dc is the probability of breaking front with

speed in the range [c, c+ dc]. In current simulation setup,

Λ(c)dc = 1/λ. (3.13)

The energy dissipation then becomes

dE

dt= bρc5/gλ =

bρwc5

gλ. (3.14)

Thus we have the breaking parameter

b =gλ

ρwc5dE

dt(3.15)

In figure 3.16, the breaking parameter b obtained in current simulation is plotted

together with Drazen’s [3] model. The parameter b of plunging breakers matches the

80


S

b

0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2Current resultb=0.25S5/2

b=0.31S2.77

Plunging breaker (Drazen et al)Spilling breaker (Drazen et al)

Figure 3.16: Dissipation parameter b versus wave steepness S and comparison withDrazen’s [3] model and data.

model very well. For spilling breaker, it deviates from the model. The deviation

is caused by the different mechanism of energy dissipation for plunging and spilling

breaking. In plunging breaker, significant amount of energy is consumed to do the

work against buoyancy of air entrained [76,83,84]. In spilling breaker, air entrainment

is not significant (figure 3.2(b)), so the parameter b is smaller than the model proposed

for plunging breaker. This is also confirmed by the experimental data compiled by

Drazen et al. [3].

81

Chapter 4

Numerical Study of High Wind

Over Steep/Breaking Water

Surface Waves

Wind forcing and wave breaking are the two most important factors in wave field

prediction. When wind speed is high, the two exist at the same time and interact

with each other. Heuristic coupled models are developed for the wave field with both

wind and wave breaking. Kukulka & Hara [85,86] added wave breaking effect into the

model of Hara & Belcher [87] and applied it in both mature and growing seas. The

total momentum budget is modified to add stress induced by wave breaking. Suzuki

et al. [88] performed large eddy simulation of turbulent airflow over young sea. In

the simulation, flat water surface is used and random wave breaking is modeled by

82

CHAPTER 4. NUMERICAL STUDY OF HIGH WIND OVERSTEEP/BREAKING WATER SURFACE WAVES

applying stress at the bottom. Liu et al. [64] coupled wind LES and high-order

spectral wave simulation. The wave breaking effect is represented by a numerical

dissipation. For further improvement of these wave models, more detailed study of

wind interaction with steep/breaking waves is expected.

Here we perform large eddy simulation of wind over steep/breaking water waves to

study the physics involved during the interaction between high wind and steep/breaking

waves.

4.1 Simulation setup and turbulence mod-

eling

4.1.1 Problem setup and parameters

Current simulations are performed in a three dimensional rectangular domain with

x, y, z representing the streamwise, spanwise, and vertical coordinates respectively.

The domain size is Lx × Ly × Lz = 2λ× 2λ× 1.5λ as shown in figure 4.1. Here λ is

the dominant wave length. Periodic boundary conditions are used in the horizontal

directions. No-slip boundary condition is used on the bottom boundary. On the top

boundary, a uniform velocity Ud is applied and it generates mean shear and injects

energy into the wind turbulence. The mean water level is located at the middle of

the domain and the water depth h is one wave length λ. So we are looking at the

83


deep water wave breaking and bottom effect is negligible.

Initially (i.e. t = 0), the wave profiles are given as third order Stokes waves with

surface elevation

η(x) = a cos(k(x− ct)) +1

2a2k cos(2k(x− ct)) +

3

8a3k2 cos(3k(x− ct)). (4.1)

Here, a is the wave amplitude; k = 2π/λ is the wave number; c is the wave phase

speed and c = ω/k; ω is the angular frequency. Nonlinear dispersion relationship ω =√gk(1 + (ak)2) is used here and g is the gravity acceleration. The initial underwater

velocities in x, y, z directions are given as

u = F31 cosh(k(h+ z)) cos(k(x− ct)) + F32 cosh(2k(h+ z)) cos(2k(x− ct))

+F33 cosh(3k(h+ z)) cos(3k(x− ct)),

v = 0,

w = F31 sinh(k(h+ z)) sin(k(x− ct)) + F32 sinh(2k(h+ z)) sin(2k(x− ct))

+F33 sinh(3k(h+ z)) sin(3k(x− ct)),

(4.2)

respectively. Here h is the water depth and coefficients

F31 = aσ/ sinh(kh)− a2kσ cosh2(kh)(1 + 5 cosh2(kh))/8/ sinh5(kh),

F32 = 34a2kσ/ sinh4(kh),

F33 = 364a3k2σ(11− 2 cosh(2kh))/ sinh7(kh).

(4.3)

For the wind side, a turbulent Couette flow is given initially. At the beginning,

the water side is moving according to the above analytic solution and the air flow

can develop freely. As time evolves, the wind field gradually achieves steady state

84


Figure 4.1: Sketch of the setup for the wind-wave breaking problem.

statistically. Numerical tests show that the statistic of steady state does not depend

on the initial condition.

Three different wave slopes (ak = 0.1, 0.35, 0.55) are considered in our simulation.

For ak = 0.1 and ak = 0.35, wave length is 0.262m. For ak = 0.55, wave length is 20m.

For each wave slope, different wind speeds are considered. In table 4.1, simulation

parameters of all the cases are listed.

After the wind side becomes statistically steady, we allow the wave to develop

freely under the action of the wind. For the cases with ak = 0.35 and ak = 0.55, the

waves are steep and will break because of their inherent instability.

4.1.2 Numerical method

Numerical method introduced in chapter 2 is used for the coupled air–water sys-

tem. The grid resolution used in our simulation is Nx ×Ny ×Nz = 256× 128× 192.

85


Case (ak)0 λ (m) U(m/s) Re Fr2 Ud/U u∗(m/s) z0(m) U10(m/s) Cd c/u∗

I-1 0.1 0.262 3.5 2.92e5 14.99 1.35 0.176 0.00137 3.91 0.002023 3.63

I-2 0.1 0.262 6.0 5.00e5 44.05 1.35 0.296 0.00187 6.35 0.002170 2.16

I-3 0.1 0.262 7.9 6.59e5 76.36 1.35 0.456 0.00197 9.72 0.002199 1.40

I-4 0.1 0.262 10.0 8.34e5 122.36 1.35 0.640 0.00206 13.58 0.002220 1.00

II-1 0.35 0.262 3.5 2.92e5 14.99 1.35 0.288 0.0089 5.06 0.003242 2.22

II-2 0.35 0.262 6.0 5.00e5 44.05 1.35 0.640 0.0098 11.08 0.003334 1.00

II-3 0.35 0.262 7.9 6.59e5 76.36 1.35 0.880 0.0111 14.97 0.003458 0.73

II-4 0.35 0.262 10.0 8.34e5 122.36 1.35 1.164 0.0115 19.70 0.003491 0.55

III-1 0.55 20 20 1.27e8 6.41 2.0 4.0 1.105 22.03 0.03298 1.60

III-2 0.55 20 30 1.91e8 14.43 2.0 5.6 1.154 30.24 0.03430 1.14

III-3 0.55 20 40 2.55e8 25.65 2.0 7.0 1.187 37.30 0.03523 0.91

III-4 0.55 20 50 3.18e8 40.07 2.0 8.0 1.221 42.05 0.03619 0.80

III-5 0.55 20 70 4.46e8 78.54 2.0 11.2 1.331 56.47 0.03933 0.57

Table 4.1: Simulation parameters for different cases of wind over initially steep waves.

The grids are equally spaced in horizontal directions. Vertically the grids are al-

gebraically stretched and are clustered both at the top boundary and around the

air-water interface.

4.1.3 Turbulence modeling

Direct numerical simulation of the Navier-Stokes (NS) equations is not feasible in

current simulation because the high Reynolds number requires very high resolution

to resolve the smallest vortical structure. Large eddy simulation is used to address

the high Reynolds number flow with only large scale motion resolved. The large scale

86


flow field f can be obtained by filtering the subgrid-scale motions f ′ from f through

f(x, t) ≡∫

G(x− x′)f(x′, t)dx′. (4.4)

Here, G(x) is the spatial grid filter. Applying the filter to single fluid NS equations,

we have the following filtered NS equations

∂u

∂t+∇ · (uu) = −∇p+∇ · (τ0 − τ) + g. (4.5)

∇ · u = 0 (4.6)

Here u is the resolved grid-scale velocity vector; p is the resolved pressure; τ0 is the

resolved viscous stress tensor; g is the gravity acceleration; and τ is the subgrid-scale

stress tensor coming from the filtering of nonlinear convection term

τij = uiuj − uiuj. (4.7)

The SGS stress tensor τ is modeled by an eddy viscosity model

τij −1

3δijτkk = −2νsgsSij. (4.8)

The eddy viscosity is calculated through the model based on renormalization group

analysis [89,90] as

νt = ν0

[1 +H

(c2s∆

4νtν30

(2S : S)− C

)]1/3(4.9)

Here the ramp function H(x) = x for x > 0 and 0 otherwise; ∆ =√∆x∆y∆z is

the equivalent filter scale; cs = 0.0062 is the Smagorinsky constant; C = 75 is the

87


model constant; and the total equivalent kinematic viscosity νt = ν0 + νsgs; ν0 is the

material viscosity. When the strain rate is small as that in low Reynolds number

flow, νt becomes ν0 automatically and no model effect is included. When strain rate

is large, the above equation can be transformed into a cubic equation to solve.

4.2 Wind over prescribed steep waves

Before waves break, they experience a stage during which the slope of the wave

increases from a small regular value to as large as O(1). This crest sharpening process

is usually caused by energy input from wind in deep ocean and topological change of

the bottom boundary in shallow water. The wind flow dynamics over the wave during

this stage is also important for the understanding of the entire interaction process of

wind and breaking wave. According to the field measurement of Babanin et al. [91],

significant wind pressure increase is detected during this stage. In this section, the

results of wind flow over prescribed waves with different steepness and wind speeds

are discussed to help us understand the flow mechanism during this stage. Because

waves are prescribed and wind flow achieves statistical steady state, transitional effect

is excluded.

88


(a) (b)

(c) (d)

Figure 4.2: Phase averaged horizontal wind velocity vector field over prescribed waterwaves of case: (a) II-1; (b) II-2; (c) II-3; (d) II-4.

4.2.1 Wind field above prescribed waves

Near the air–water interface, the airflow is affected by the surface undulation and

orbital velocity of the waves, and exhibits dependence on the wave phase. In current

wind-wave simulation in the prescribed stage, phase averaging technique is used to

obtain the mean phase-dependent flow characteristics. In figure 4.2, the normalized

phase averaged wind velocity vectors for cases with wave slope ak = 0.35 at four

different wind speeds are plotted. As wind speed increases (from case II-1 to II-4),

the normalized horizontal velocity above the wave trough decreases faster toward the

surface, which indicates that a universal law like that of flow over flat surface does not

exist when waves exist. The wave condition such as wave age needs to be included

in the velocity profile near the surface. This can also be explained by replacing the

89


wave with an oscillating flat plate. When normalized by the wind speed and the

same length scale, high wind speed cases have lower oscillating frequency and the

corresponding Stokes layer is thicker.

In figure 4.3, the streamlines of the phase averaged flow field in the reference frame

moving with wave phase speed are plotted. The wave surface is a streamline in this

moving frame. This moving reference frame allows the streamline and particle path

to coincide [92]. Critical height where the wind speed is equal to the wave phase

speed is very small because the wave is very slow compared with the wind flow above.

The “cat’s eye” above the wave trough is close to wave surface but does not extend

to the surface. As wind speed increases, the top boundary of the “cat’s eye” moves

down and the “cat’s eye” becomes flatter, which also indicates the influence of wave

to the wind flow above becomes smaller.

Away from surface, influence of wave is small and velocity profiles tend to be

identical for different wave phases. Plane averaged streamwise velocity profile in this

region can be described by the universal log law

U(z)

u∗ =1

κln(

z

z0). (4.10)

Here u∗ =√

τa/ρa is the friction velocity; τa is the wind shear stress on the water

surface; ρa is the air density; κ is the von Karman constant and we use 0.4 here;

z0 is the roughness length scale determined by the wave field. In figure 4.4, the

velocity profiles U(z) from slightly above the wave crest to the middle of the air

domain are plotted and the region where log law holds can be identified. As wind

90


Figure 4.3: Streamline pattern of wind flow over prescribed waves for case (a) II-1;(b) II-2; (c) II-3; and (d) II-4.

speed increases, the profile becomes steeper which indicates larger u∗. By fitting the

profiles with equation 4.10, the friction velocity u∗ and roughness length scale z0 are

obtained and listed in table 4.1. For cases with the same wave slope, both friction

velocity and roughness length scale increase as wind speed increases.

The wave speed relative to wind can be represented by wave age c/u∗ which is

also calculated and listed in table 4.1. For all the cases simulated here, the wave age

c/u∗ < 5 and waves are under high wind conditions.

The 10 meter height mean wind velocity U10 can be obtained by extrapolation

according to equation 4.10. The drag coefficient

Cd =τa

ρaU210

= (u∗

U10

)2 (4.11)

was calculated and listed in table 4.1. For cases with the same wave slope, the

91


U (m/s)

z(m

)

0 2 4 6

0.05

0.1

(a)

U (m/s)

z(m

)

0 2 4 6

0.05

0.1

(b)

U (m/s)

z(m

)

0 20 40 60 802

4

6

8

10

(c)

Figure 4.4: Mean horizontal velocity above the water surface (a) for cases I-1∼I-4 withwavelength 0.262m and wave slope ak = 0.1; (b) for cases II-1∼II-4 with wavelength0.262m and wave slope ak = 0.35; (c) for cases III-1∼III-5 with wavelength 20m andwave slope ak = 0.55.

92


drag coefficient increases slightly with wind speed. Powell et al. [93] analyzed data

measured in tropical cyclones and found that drag coefficient tends to decrease at

very high wind speeds. We find for prescribed waves the drag coefficient is more

sensitive to the wave slope and height instead of wind speed. Since wave breaking

constrains the wave slope and height, it should be a major factor that causes the drag

saturation.

4.2.2 Wind forcing over prescribed waves

The wind forcing applied on the wave surface is critical for the understanding of

the momentum and energy transfer between wind and wave. When waves present, the

total air-sea momentum flux can be expressed as the sum of three parts: turbulent

stress contributed by turbulence fluctuation, wave-coherent stress contributed by the

wave motion, and viscous stress contributed by the mean flow. In the measurements

of Veron et al. [94], the wave-coherent stress is the dominant one. The wave-coherent

stress can be further divided into form drag and wave-coherent tangential stress.

Critical layer theory propose by Miles [95] can not apply to very slow waves because

the critical layer is too close to the surface. Direct measurement of force on the

wind side is difficult because of the motion of wave and contamination of water vapor

and droplets to the probes. Banner & Peirson [96] measured the tangential stress

beneath wind-driven air-water interfaces through the PIV technology. The velocity

field in the viscous sublayer is obtained to calculate velocity gradient and tangential

93


Figure 4.5: Phase averaged dynamic pressure field of wind flow over prescribed wavesfor case (a) II-1; (b) II-2; (c) II-3; and (d) II-4.

stress. The stress in the breaking and trough region where direct measurement is not

feasible is obtained by interpolation or setting to zero. They found that, in the stage

before waves are developed, the total wind stress is mainly composed of tangential

stress. After waves are developed, the form drag becomes the dominant part of the

total stress. The wave coherent tangential stress only accounts for 5 percent of the

momentum flux. The tangential stress is insensitive to wave conditions for given wind

speed. They also pointed out that the airflow over water wave is only transitional

rough over a wide range of sea states.

In figure 4.5, the phase averaged dynamic pressure fields of case II-1∼II-4 normal-

ized by the air density ρa and wave phase speed c are plotted. For all these four cases,

pressure maximum is located on the windward wave surface and pressure minimum is

94


located on the leeward. The normalized pressure magnitudes increase as wind speeds

increase, which implies the increased wind forcing associating with the increased wind

speeds. As the wind speed increases, the pressure maximum moves toward the trough

and the minimum moves toward the crest. The low pressure region becomes flatter

as wind speed increases.

In figure 4.6, phase averaged pressure on the water surface for all cases are plotted.

The pressure distribution is very different for cases with different wave slope ak. As

ak increases from 0.1 to 0.35, the pressure maxima moves downwind and the pressure

profile becomes flat in the trough. For the cases with ak = 0.55, the pressure maxima

moves to the position close to wave crest.

4.3 Wind over breaking steep waves

Starting from the fully developed wind turbulence over prescribed steep waves,

the numerical simulation started with the wave evolving freely under the wind effect,

which is an analogy of wind over mechanically generated waves. For cases II-1∼II-4

and III-1∼III-5, the steep waves will evolve to break because of the inherent instabil-

ity.

The wave breaking process can be divided into different stages by their visual and

dynamic characteristics. Liu & Babanin [97] divided wave breaking process in the

field into four stages: incipient breaking; developing breaking; subsiding breaking;

95


kx

p d/ρ ac

2 ,kη

0 2 4 6 8 10 12

-2

-1

0

1

2surfaceI1I2I3I4

(a)

kx

p d/ρ ac

2 ,kη

0 2 4 6 8 10 12

-2

-1

0

1

2surfaceII1II2II3II4

(b)

kx

p d/ρ ac

2 ,kη

0 2 4 6 8 10 12

-2

-1

0

1

2surfaceIII1III2III3III4III5

(c)

Figure 4.6: Phase averaged dynamic pressure field over the wave surface of case (a)I-1∼I-4, (b) II-1∼II-4, and (c) III-1∼III-5.

96


Figure 4.7: The instantaneous breaking water surface and streamwise velocity con-tours on two vertical planes for case II-1 at (a) t=0.29T, (b) 0.44T, (c) 0.58T, (d)0.87T, (e) 1.16T, (f) 1.45T, (g) 1.74T, (h) 2.03T

97


x/λ

η/a c

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5t=0.44Tt=0.73TCaulliez 2002

Figure 4.8: Spanwise averaged profiles of wind wave around breaking for case II-1:————, near breaking; – – – – , incipient breaking; , [4]. The error bar representsthe standard deviation of the experimental results of [4].

and residual breaking. Tulin & Waseda [98] divided wave breaking process into four

phases: steepening-crest phase; plunging-jet phase; splashing-ploughing phase; and

decaying-scar phase according to their surface geometry. The corresponding stages

with Tulin’s classification in our simulation of case II-1 are plotted in figure 4.7.

Instantaneous horizontal velocity contours on two vertical cuts and the water surface

are plotted. In the steepening-crest phase (figure 4.7(a, b)), front face of wave becomes

more and more steep and concave, eventually becomes vertical. In the plunging-jet

phase (figure 4.7(c, d)), a small jet is formed and plunges toward front surface. In the

splashing-ploughing phase (figure 4.7(e-g)), the jet plunges into water and generates

splash-up, then it ploughs on water surface for several times. A highly turbulent

surface region is generated in this phase. In the decaying-scar phase (figure 4.7(h)),

ploughing motion stops and underwater turbulence interacts with free surface and

dissipates its energy.

The asymmetric wave profiles near breaking are found to be scale independent.

98


Caulliez [4] performed experimental study on breaking of both wind generated short

gravity waves and mechanically generated waves with wind blowing above. They

found that the geometric properties of the waves near breaking such as the crest-

slope distribution are scale invariant. In figure 4.8, spanwise-averaged wave profiles

of case II-1 at the time near and shortly after breaking are plotted together with

the experimental results of Caulliez [4]. The profile near breaking collapses very well

with the experimental result. The asymmetrical wave crest has steep wave front with

incline angle larger than 30o and a mild rear face with incline angle around 15o. The

profile shortly after breaking also shows good match with the near breaking profile

of Caulliez [4] on the windward.

4.3.1 Wind field above the breaking waves

Wave breaking process could significantly affect the wind field above. In figure

4.9 and figure 4.11, the spanwise-averaged and plane-averaged streamwise velocity

fields of case II-1 at different time are plotted. At the early stage of breaking, there

exists a strong negative velocity region on the front face of the wave crest. As the

wave breaks, the breaking jet sweeps over the surface and the negative velocity region

becomes smaller and eventually disappeared.

Wave breaking also affects the turbulence intensity above the wave surface. The

spanwise-averaged streamwise turbulence intensity u′ is plotted in figure 4.10. After

breaking, the turbulence fluctuation above the water surface is enhanced. This is

99


(a)

(b)

(c)

Figure 4.9: The spanwise-averaged streamwise velocity on the air side for case II-1.(a) t=0.29T; (b) t=1.16T; (c) 2.03T.

100


(a)

(b)

(c)

Figure 4.10: The spanwise-averaged streamwise velocity fluctuation on the air sidefor case II-1. (a) t=0.29T; (b) t=1.16T; (c) 2.03T.

caused by the energy transfer from the wave to wind through the sweeping jet and

the newly generated rough surface also brings more disturbance to the wind.

4.3.2 Shear stress, drag coefficient, and roughness

Wave breaking plays important roles in the momentum transfer between wind

and waves. The plane averaged streamwise velocity profiles at different time instants

during the breaking process are plotted in figure 4.11. The lower parts of the profiles

shift right to larger value as time evolves, which implies a decrease of the friction

velocity and shear stress according to equation 4.10. The fitted shear velocities are

101


U/U10

z(m

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.02

0.04

0.06

0.08

0.1t=0.29Tt=0.44Tt=0.58Tt=0.87Tt=1.16Tt=1.45Tt=1.74Tt=2.03T

Figure 4.11: The mean streamwise velocity above the water surface for case II-1during the breaking process at different time.

t=0.29T t=0.87T t=1.45T t=1.74T t=2.03T

u∗ (m/s) 0.284 0.272 0.256 0.240 0.208

Cd (×103) 3.182 3.112 2.988 2.805 2.515

Table 4.2: Friction velocity and drag coefficient for case II-1 during the breakingprocess at different time.

listed in table 4.2.

The relation between the shear stress and the wind speed is important for the mod-

eling of the atmospheric boundary layer over the ocean. According to the laboratory

studies in the wind–wave tank, Plant & Wright [99] gives

u∗ ∼ 0.022(±10%)U1.5∞ . (4.12)

Here, U∞ is the free stream wind velocity in his wind tunnel.

In figure 4.12(a), the friction velocity u∗ obtained in our simulation at the pre-

scribed stage and those for breaking case II-1 listed in table 4.2 ares plotted together

with the experimental results of other researchers. Since the wind and wave condi-

102


U10

u*

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2ak=0.1 Prescribeak=0.35 Prescribeak=0.35 BreakingReul 2008Banner 1990Plant & Wright 1977Large & Pond 1981

(a)

U10

Cd

0 5 10 15 200

0.002

0.004

0.006

0.008

0.01ak=0.1 Prescribeak=0.35 Prescribeak=0.35 BreakingReul 2008Banner 1990Plant & Wright 1977Large & Pond 1981

(b)

Figure 4.12: Friction velocity u∗ and drag coefficient Cd obtained in current simulationand presented in other literatures.

103


tions are different for those data sets, direct comparison is not feasible. In current

simulation, we have a continuous periodic wave train. In the experiment, only single

breaker is generated. So the water surface in current simulation is rougher than that

of the experiment and the simulated friction velocity is also larger than the experi-

ment. From figure 4.12(a), we still can make the following arguments: the friction

velocity increases nonlinearly with the increase of wind speed; wave slope significantly

affects the friction velocity; and the friction velocity has large variability during the

wave breaking process.

Drag coefficient is the dimensionless parameter that reflects the relationship be-

tween the friction velocity and the wind speed (equation 4.11). The drag coefficients

corresponding to the friction velocities in figure 4.12(a) are plotted in figure 4.12(b).

The formula given by Plant & Wright [99] is plotted as the solid line. Banner [100]

measured the friction velocity over the incipient and the almost continuous breaking

waves. His results are plotted in figure 4.12 as the downward triangles. By compar-

ing the stress over incipient and continuous breaking waves, he concluded that wave

breaking enhances the wind stress. Large & Pond [101] studied the measurement

data from a weathership and obtained drag coefficient

Cd =

1.205× 10−3, 4 ≤ U10 < 11m/s,

0.49 + 0.065U10, 11 ≤ U10 ≤ 25m/s.

(4.13)

They observed that drag coefficient measured is smaller during rising winds than

during falling winds or after wind direction change. Their result is plotted in figure

104


4.12 as the dashed line.

Savelyev et al. [102] experimentally studied momentum flux from strong wind to

the water waves with maximum 10 meter height wind speed as high as 26.9m/s and

wave age U10/Cp = 16.6. They found that their drag coefficient responsible for the

form drag is approximately

Cd = 0.146(ak)2, (4.14)

which indicates strong dependence of drag coefficients on the wave slope. The growth

rate γ and sheltering coefficient G used by Donelan et al. [1] are also examined.

Maat & Makin [103] performed numerical simulation of airflow over breaking

waves on a boundary fitted grid. The wave form is prescribed and the breaking is

represented by a wave height increase and a sudden roughness jump. They found

both mechanism influence the form drag and total wind stress. The increase of wave

height moves the pressure maximum downwind. The roughness increase shifts the

pressure minimum downwind. According to equation 4.10, roughness length scale has

the one-on-one relation with the drag coefficient as

Cd =

(κ

log(10z0)

)2

. (4.15)

Roughness length scales obtained in current simulations normalized by wave height

are plotted in figure 4.13 together with Jones & Toba’s data and the conservative

expression

gz0u2∗= 0.02

(σpu∗

g

). (4.16)

105


2πωu*/g

z 0/H

10-2 10-1 100 10110-4

10-3

10-2

10-1

100

Jones & Toba (1995)Conservative expressionak=0.1ak=0.35ak=0.55(ak)0=0.35, breaking

Figure 4.13: Roughness length scale normalized by wave height versus wave age.

Our wave age is at the edge of the data set. If we extend Jones & Toba’s data

further into the younger wave age, they will be in the same range as that of current

simulation.

4.3.3 Wind pressure above breaking waves

Wave breaking comes with strong and fast topological and kinematic changes of

the water surface. These changes modify wind flow above and affect wind pressure

applied on the water surface. Span-wise averaged dynamical wind pressure contours

after breaking at t = 1.1T are plotted in figure 4.14(a). The pressure maximum goes

across wave crest and is located just above the breaking jet. This kind of pressure

106


(a)

(b)

(c)

Figure 4.14: The spanwise-averaged pressure, streamlines and vorticity at t = 1.1Tof case II-1.

107


(a)

(b)

Figure 4.15: Sketch for pressure distribution over water wave before (a) and after (b)breaking.

distribution is very different from the pressure distribution we have observed over

non-breaking waves. To understand the reason of this distribution, let us neglect the

viscous effect and consider potential flow over breaking and non-breaking waves. In

a reference frame moving with wave phase speed c, the wave surface is a streamline

as shown in figure 4.15 and we can apply the Bernoulli equation over it. According

to the Bernoulli equation, for the non-breaking wave

(ut − c)2

2+

ptρa

+ gzt =(uc − c)2

2+

pcρa

+ gzc. (4.17)

Here, ut and uc are the velocities at trough and crest respectively; zt and zc are

corresponding vertical coordinates. The difference between the non-breaking and

breaking waves is mainly the velocity of the breaking front. The flow above the wave

108


trough almost stay the same. For breaking waves, applying Bernoulli equation we

have

(ut − c)2

2+

ptρa

+ gzt =(ub − c)2

2+

pbρa

+ gzb. (4.18)

Here, ub and zb are the velocity at the breaking crest and the corresponding vertical

coordinate. According to the equations above, we have

(uc − c)2

2+

pcρa

+ gzc =(ub − c)2

2+

pbρa

+ gzb. (4.19)

For the spilling breaker, |ub − c| < |uc − c|, so the dynamic pressure increases over

the breaking front.

The streamlines over the wave are plotted in figure 4.14(b). The “cat’s eye” is

pushed downwind by the sweeping jet and the tip of the jet forms the critical point

for air flow separation. The spanwise vorticity ωy is plotted in figure 4.14(c). Strong

counter-clockwise vorticity is observed just above the sweeping jet, which is another

evidence of the flow separation. This could also be the reason of the drag decrease

after breaking.

Babanin et al. [82] studied the role played by wave breaking in the wind–wave

interaction process. The wind energy flux is enhanced by a factor of 2 through wave

breaking. They also parameterized a modified exponential growth parameter

γ = γ0(1 + bT ). (4.20)

Here, γ0 is the wave growth rate without wave breaking and bT is the breaking prob-

ability. In figure 4.16, the form drag of wind over breaking waves

109


t/T

Fp/

ρu*2

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 4.16: The form drag evolution with time for wind over breaking waves in caseII-1.

Fp =

∫ λ

0

p dS/λ (4.21)

is plotted versus time. Here S is the surface segment. The form drag increases at

the early stage and reaches the maximum at around 1.1 wave period. After that, it

decreases. The mechanism that induced the increase and decrease will be discussed

in the following sections.

4.3.4 Airflow separation

As pointed out by Banner & Melville [104] and Gent & Taylor [92], when waves

start breaking, the water particle velocity at the wave crest becomes equal or larger

than the wave phase speed, which forms a critical point for flow separation to hap-

pen. This velocity criterion is the only criterion for separation occurrence and the

110


separation is only meaningful in the reference frame moving with the wave phase

speed. Negative stress is not directly related to separation in the wind-wave environ-

ment. Wind flowing over breaking waves satisfies the criterion completely. For large

non-breaking waves with strong wind drift, above criterion can also be satisfied.

Air flow separation over water waves has long been considered the key mecha-

nism of the momentum transfer enhancement and drag saturation during high wind.

Longuet-Higgins [105] developed a model of flow separation over spilling breaker

by representing the turbulence effect with a constant eddy viscosity and expressing

the tangential stress across the laminar/turbulent interface using a drag coefficient.

Mueller & Veron [106] proposed a surface stress model which incorporates the air-flow

separation effect and is able to predict the drag coefficient saturation. Mizutani &

Hashimoto [107] experimentally studied the air flow characteristics over wind waves

with particle image velocimetry(PIV). The wind speed is around 10 m/s. They ob-

tained airflow separation with inverse flow in 6% of their cases. In 60% of their cases

they did not observe flow separation. The wind speed at the windward wave surface

is found comparable to the mean wind speed at the lowest point from conventional

point sensor.

In figure 4.17, a vertical cut from the simulation domain of case II-1 is extracted

and the velocity vector, streamwise velocity contour, and the streamlines are plotted

on it. Above the wave trough, there is a low speed but highly turbulent zone riding

over the water surface. A shear layer starting from the overturning tip separates the

111


trapped flow zone and the outer flow. Strong back flow velocity as large as 20 percent

of the phase speed is observed in the trapped region. The low pressure associated

with the separation is the major reason of the form drag increase at the early stage.

In the later stage, the wave is destroyed gradually. The decreasing wave height makes

the form drag decrease.

4.3.5 Turbulence and coherent structures gener-

ated by breaking

The vorticity field obtained by Perlin el al. [69] shows that the flow is irrotational

and potential theory holds until the breaking happens. After waves break, they gen-

erate turbulence in the water which accelerates energy dissipation. The instantaneous

horizontal velocity field on a vertical cut of the breaking process of case II-1 are plot-

ted in figure 4.18. In the steepening-crest and plunging-jet phase, the flow field in the

water is still laminar. During the splashing-ploughing phase, a strong surface shear

layer is formed. The surface undulation induces flow separation and vertex shed-

ding. The flow becomes turbulent. The turbulence generated also interacts with the

free surface and makes it rougher. Part of the turbulence is transported downward

and makes the turbulent zone thicker. In this case, a large scale mean vortex is not

observed.

In figure 4.19, the spanwise averaged velocity field under water is plotted. At

112


(a)

(b)

(c)

Figure 4.17: Instantaneous flow field cut of case II-1 in the free developing stage:(a)velocity vector; (b) horizontal velocity contour; (c) surface streamlines. The ve-locities are plotted in a moving reference frame with horizontal velocity c. Here c isthe phase speed of wave.

113


Figure 4.18: Instantaneous streamwise velocity u normalized by wave phase speed con a vertical cut for case II-1 with (ak)0 = 0.35. The time step is 0.145T . T is thelinear wave period.

114


t = 1.33T , no strong vortical motion is observed. At t = 1.78T , strong clockwise

rotated vortical structures are formed and some are going to be formed. At t = 2.22T ,

small co-rotating vortices with strength weaker than that at t = 1.78T are observed

and their rotating centers are deeper. At t = 2.67T , co-rotating vortices under each

wave coalesce into one large vortex. This large vortex is consistent with that observed

by Melville et al. [71].

4.3.6 Surface current generated by breaking

Wave breaking is an important source of surface current. In figure 4.20 and

figure 4.21, the spanwise-averaged and plane-averaged streamwise velocity contours

on the water side are plotted. In the steepening-crest and plunging-jet phase (figure

4.20(a)), the current generation is small. During the splashing-ploughing phase (figure

4.20(b,c)), the strong shear on the surface starts generating current. The current

increases significantly from the middle of the splashing-ploughing phase until the

decaying-scar phase. The current generated is larger than that of the experiment

without wind by Rapp & Meville (1990). Two factors cause this difference: the first

one is that we have consecutive breakers in our simulation since periodic boundary

condition is used; the second one is the wind shear. For single breaker, the current

is generated in a local area and can be weakened by the environmental flow without

current. With consecutive breakers, the current generated does not need to consume

energy to push the fluid around. It is obvious that wind shear will generate current

115


x/λ

z/λ

0 0.5 1 1.5 2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(a)

x/λ

z/λ

0 0.5 1 1.5 2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(b)

x/λ

y/λ

0 0.5 1 1.5 2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(c)

x/λ

z/λ

0 0.5 1 1.5 2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(d)

Figure 4.19: Spanwise averaged underwater velocity vectors for breaking wave caseIII-1 with (ak)0 = 0.55. (a) t=1.33T; (b) t=1.78T; (c) t=2.22T; (d) t=2.67T. HereT is the wave period.

116


(a)

(b)

(c)

(d)

Figure 4.20: Spanwise-averaged horizontal velocity on water side of case II-1: (a)t = 0.29T ; (b) t = 1.16T ; (c) t = 2.03T ; (d) t = 2.90T .

U (m/s)

z(m

)

0 0.01 0.02 0.03 0.04 0.05-0.1

-0.08

-0.06

-0.04

-0.02

0

t=0.29Tt=0.44Tt=0.58Tt=0.87Tt=1.16Tt=1.45Tt=1.74Tt=2.03T

Figure 4.21: Horizontal plane-averaged streamwise velocity on water side of case II-1at different time.

117


since in the field the mean current is found to be 3% of the wind speed [108], which

is consistent with our simulation result.

4.3.7 Effect of wind speed on wave breaking

Wind also affects the wave breaking process. Chambarel et al. [109] studied the

wind effect on the wave breaking generated by dispersive focusing. Boundary integral

method is used and Jeffrey’s sheltering formula is used to model the wind effect. They

found increased amplitude and lifetime of highest wave associated with wind and the

area where it is formed is shifted downwind. They also found that deep water waves

are more affected by wind than shallow water waves.

In figure 4.22, the instantaneous streamwise velocity fields and the free surfaces

of case II-1∼II-4 at time t ≈ T when waves are breaking are plotted. For case II-1,

the wave profile is approximately two dimensional and the three dimensional surface

structures happen only at the tip of the breaking jet. For case II-2, both the wave

crest and breaking jet become three dimensional. For case II-3, the jet almost sweeps

the entire surface and the entire surface becomes three dimensional. Water droplets

can be observed in this case. For case II-4, the surface is even rougher than case

II-3 and fine scale water droplets are observed. The overturning of the free surface

for high wind speed happens a little bit earlier than that for low wind speed. The

strong three dimensional surface structures for high wind cases also indicate that

wind could be one of the major sources of three dimensional instability of waves.

118


(a) (b)

(c) (d)

Figure 4.22: Instantaneous flow field and the free surface at t = 0.417s ≈ 1T of case(a) II-1; (b) II-2; (c) II-3; (d) II-4.

119


At very high wind (60 ∼ 80 m/s), the droplets torn down by the wind shear can

form a “slippery surface” which can further affect the drag coefficients as pointed

out by Soloviev & Lukas [110]. Compared to another major source of water droplets

generated by bursting bubbles, the spume droplets generated by wind shear plays

more significant roles on the turbulence mixing around the surface as pointed out

by Kudryavtsev [111]. In figure 4.22, the turbulence intensity over the wave crest is

stronger for high wind cases than those in the low wind cases, which indicates the

wind enhancement of turbulence mixing through the spray generation. Since large

eddy simulation is used, we do not expect to resolve all the droplets generated which is

not affordable nowadays. Further discussion about spray droplets is out of the scope

of current thesis and it will rely more on the experiments and field measurements.

4.4 Conclusions

Strong wind turbulence flowing over steep and breaking waves is simulated using

large eddy simulation with the RNG subgrid-scale model. The air–water interface is

captured using the coupled level set/VOF method with improved mass conservation.

Cases with different wind speeds and wave slopes are considered.

Wind flow near the wave surface is modified by the wave motion and geometry.

Velocity profiles normalized by wind speed near wave surface are dependent on wave

age. Away from the wave surface, the log law of turbulent boundary layer holds.

120


Wind pressure distribution over waves is dependent on both wind speeds and wave

slopes. High wind speed is associated with high pressure maximum and flatter low

pressure region. As wave slope increases, the pressure maximum moves toward the

wave crest. The drag coefficient is found very sensitive to wave slope.

The asymmetric wave profile near breaking is scale independent and collapses very

well with the experiment. Wind field above the wave is significantly modified by the

breaking process. The lower part of the streamwise velocity profile is shifted towards

larger value and reduced drag coefficient is observed. When breaking happens, airflow

separation happens and comes with an increase in form drag. After breaking, the high

pressure region originally located on the back of the wave moves downwind with the

sweeping jet. Strong counter-clockwise vorticity is found above the jet tip in the

air. The form drag is found to increase to a maximum value and then decrease

as the jet sweeps forward and wave amplitude reduces. On the water side, large

coherent clockwise rotating vortices are generated by strong plunging breaker but

are not observed in the spilling breaker. Formation process of the large vortex from

multiple small vortices is illustrated. Strong current generation is observed by the

consecutive breaking and wind shear. The wind speed is also found to affect the wave

breaking. Waves under higher wind speed break slightly earlier and exhibit more

three dimensional structures.

In current simulation, steep waves is used as initial condition to generate breaking

waves. To be closer to real field condition, breaking evolved from mild broadband

121


wave field is more favorable and will be our future work.

122

Chapter 5

Numerical Simulation of Strong

Free-Surface Turbulence for

Mechanistic Study

Modeling of violent free-surface turbulent flows, which is of vital importance to

many naval applications, requires a deep understanding of the fundamental physics

of strong free surface turbulence (SFST). The interaction of turbulence with surface

waves is complex in many ways. For example, the turbulence can be substantially

distorted by the periodic orbital motion and the surface drift associated with the

waves. Wave breaking is an important source of turbulence in upper ocean and near

naval structures. On the other hand, the turbulence scatters and dissipates surface

waves. The turbulence pressure and shear stress may also amplify waves and trigger

123

CHAPTER 5. NUMERICAL SIMULATION OF STRONG FREE-SURFACETURBULENCE FOR MECHANISTIC STUDY

wave breaking [112].

The complex surface processes pose great challenges to the simulation of SFST.

For direct numerical simulation (DNS), the violent surface motion makes the kine-

matic and dynamic boundary conditions difficult to represent in numerical schemes

(cf. the discussion in Shen, 2007 [113]). For large-eddy simulation (LES), besides

the question of the applicability of subgrid-scale (SGS) models originally developed

for other flows [114], the multiphase nature of SFST also introduces issues in the

formulation of LES itself. For example, new filters based on component-weighted,

volume-averaging procedure are required; commutativity between filter and deriva-

tive needs to be accounted for; and additional interfacial SGS terms need to be

modeled [115–117].

Here we perform direct numerical simulation of SFST with the numerical method

introduced in chapter 2 to study the characteristics of the flow and surface and the

physics involved.

5.1 Problem setup and numerical approach

5.1.1 Setup of numerical simulation

We consider the simulation of a canonical free-surface and turbulence interaction

problem (figure 5.1) with DNS method. In this simulation, turbulence is generated

in the deep water and is then transported to the free surface to interact with the

124


surface. The air part is initially quiescent and its motion is driven by the water side.

Because of the very small air-to-water density ratio, the influence of air on the water

motion is relatively weak, and the motion of the free surface is mainly generated by

the turbulence in the water.

To obtain steady free-surface turbulence statistics, we choose a forced turbu-

lence field in the deep water as the turbulence source. We adopt a linear forcing

method [118,119] to generate quasi-steady isotropic homogeneous turbulence. In this

method, a body force proportional to the turbulent velocity is added to the momen-

tum equations:

∂u

∂t+ u · ∇u =

1

ρ(−∇p+∇ · τ + ρg + σκδ(xs) + c0F (zc)u

′). (5.1)

Here u is the velocity vector; u′ is the velocity fluctuation. Since there is no mean flow

in this problem, u′ = u. And ρ is the density; p is the pressure; τ = µ(∇u+∇uT ) is

the shear stress tensor; µ is the dynamic viscosity; g is the gravitational acceleration;

σ is the surface tension coefficient; κ is the surface curvature; xs denoted the surface

location; δ(x) is the Dirac delta function; zc is the vertical coordinate with its origin

located at the center of the water domain; F (zc) is the forcing distribution function;

c0 is the forcing coefficient. The function F (zc) has the form

F (zc) =

1, zc ≤ lb, bulk region,

12

(1− cos

[πld(zc − lb − ld)

]),

lb < zc ≤ lb + ld, damping region,

0, zc > lb + ld, free region,

(5.2)

125


which is symmetric about the origin zc = 0. The forcing coefficient c0 is prescribed

and it determines the strength of the turbulence generated.

With the linear forcing, steady isotropic and homogeneous turbulence is generated

at the center of the forcing region. The turbulence is then transported upward from

the deep water to the free surface to disturb the surface and generate waves, dimples,

scars, and even spays and bubbles.

air

water

free surface

Lz

ld

2lb

ld

bulkregion

dampingregion

freeregion

meanwater level

dampingregion

Lx

Ly

turbulence

x

zy

zc

0

Figure 5.1: Sketch of the multi-phase flow simulation setup of the free-surface turbu-lence problem.

5.1.2 Simulation parameters

The non-dimensionalized Navier-Stokes equations

∂u

∂t+ u · ∇u = −∇p

ρ+

1

Re

1

ρ∇ · τ +

1

Fr2k

+1

We

1

ρκδ(xs) + c0F (zc)u

′. (5.3)

126


are solved on a dimensionless domain of size Lx × Ly × Lz = 2π × 2π × 6.5π. The

mean water level is located at z = 5π.

In the free surface–turbulence interaction process, gravity and surface tension play

an important role in stabilizing the surface from the disturbance of the turbulence

beneath. Let q denote the turbulence velocity fluctuation magnitude and l the tur-

bulence integral scale. According to Brocchini & Peregrine (2001), the influence of

gravity and surface tension can be characterized by four flow regimes in the q − l

space as: weak turbulence regime, surface tension dominated regime, gravity domi-

nated regime, and very strong turbulence regime. After non-dimensionalization, the

effects of gravity and surface tension are represented by Fr and We, respectively. In

this study, Re = 1000 and c0 = 0.1 are fixed. We map the four flow regimes in the

Fr − We space as shown in figure 5.2. The marginal breaking region is marked by

two colored dash lines.

5.2 Characteristics of the free surface

5.2.1 Free surface disturbed by turbulence

We first discuss the geometrical characteristics of the free surface, which is dis-

turbed by the turbulence underneath. The aforementioned four flow regimes have

substantially different appearance in the instantaneous free surface.

127


10-2 10-1 100 101 102 10310-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Fr2

We-1

(a) region 0 (b) region 1

(d) region 2(c) region 3

Figure 5.2: Diagram of the flow regimes in the Fr − We space. Region 0: weakturbulence regime; region 1: surface tension dominated regime; region 2: very strongturbulence regime; region 3: gravity dominated regime. The region between the twodash lines represents the marginal breaking region obtained by Brochini & Peregrine(2001). [5]

Weak turbulence regime

In this regime (region 0 in figure 5.2a), turbulence disturbance to the free surface is

relatively weak because of the large gravity and surface tension stabilizing effects. The

free surface appears flat and smooth, as shown in figure 5.3(a) where instantaneous

elevation contours for the case of (Fr2 = 0.8,We = 40) are plotted. For flows in this

regime, linearized dynamic and kinematic free surface boundary conditions can be

utilized, and analysis of one phase fluid is often performed.

128


(a) (b)

(c) (d)

Figure 5.3: Instantaneous free surface elevation for the cases of: (a) (Fr2 = 0.8,We = 40) that is in the weak turbulence regime, (b) (Fr2 = 128, We = 40) thatis in the surface tension dominated regime, (c) (Fr2 = 4, We = ∞) that is in thegravity dominated regime, and (d) (Fr2 = 32, We = ∞) that is in the very strongturbulence regime.

Surface tension dominated regime

In this regime (region 1 in figure 5.2b), the surface tension effect is strong and

the gravity effect is weak. The cohesion of the water is maintained by the surface

129


tension, and the surface is smooth. Because the gravity effect is small, it cannot keep

the surface flat. As a result, the surface has a smooth round shape (figure 5.3b),

which is called “knobbly” by Brocchini & Peregrine (2001).

Gravity dominated regime

This regime (region 3 in figure 5.2c) is the most common one that is often observed

in oceans, lakes, and rivers. Due to the gravity effect, if the turbulence is not strong

enough, the surface cannot have very large deformation. But small scale surface

structures such as dimples, scars, and waves are often present. In figure 5.3(c), an

instantaneous surface for the case of (Fr2 = 4,We = ∞) is shown. A scar is observed

on the left corner, and some dimples also exist nearby.

Very strong turbulence regime

In this flow regime (region 2 in figure 5.2d), both the gravity and surface tension

effects are weak. The motion at the free surface is violent, and the surface cannot

keep flat or smooth. The turbulence can bring the water to a significant height and

make the surface break. Large amount of spays and air entrainments may occur. The

region near the surface becomes a air-water mixture. In figure 5.3(d), an instantaneous

surface for the case of (Fr2 = 32,We = 1) is shown. At this moment, the surface

geometry is quite complex. The surface elevation cannot be described by a single-value

function because the surface is multi-connected. Water jet shoots up and reenters the

130


water later, playing an important role in the atmosphere–ocean exchange of mass,

momentum, and energy.

5.2.2 Surface spectra

Having illustrating the instantaneous surface elevation, we next present its spectral

statistics, which is again quite different under different gravity and surface tension

effects. Surface tension has more influence on small scale surface structures because

of their relatively large curvature. The surface spectra of a gravity dominated case

and a surface tension dominated case are plotted in figure 5.4(a) and figure 5.4(b),

respectively. For the former, the surface spectrum has a slope of k−2.5. For the latter,

high wavenumber components are damped by the strong surface tension, and the

surface spectrum has a much steeper slope of k−5.5.

100

101

102

10−4

10−3

10−2

10−1

100

k

S(k

)/S

(k=

1)

Fr2=4, We=∞k−2.5

100

101

102

10−10

10−8

10−6

10−4

10−2

100

k

S(k

)/S

(k=

1)

Fr2=32, We=1

k−5.5

(a) (b)

Figure 5.4: Surface elevation spectra of (a) the gravity dominated case of (Fr2 = 4,We = ∞) and (b) the surface tension dominated case of (Fr2 = 32, We = 1).

131


5.2.3 Surface wave and roughness

When the free surface is disturbed by the turbulence from below, it can respond

passively and locally in the form of surface roughness. The energy received can also

propagate away in the form of surface waves.

Figure 5.5: Normalized frequency–wavenumber spectrum of the surface elevation forthe weak turbulence case of (Fr2 = 0.8,We = 40). The solid line denotes the dis-persion relationship (equation 5.5). The dash-dot line denotes the characteristic fre-quency of each wavenumber component (equation 5.6). The dashed line denotes thecharacteristic frequency obtained by linearized kinematic boundary condition (equa-tion 5.7).

The normalized frequency–wavenumber spectrum of the surface elevation [112]

ΦNη (|k|, ω) = 1

(2π)3(ηrms)2

·∫T

∫Sη(x, t)η(x+ r, t+ τ)e−i(k·r+ωτ)drdτ

(5.4)

is used to study the free surface waves and roughness. Here T is sampling duration; S

is the horizontal plane; ω is the temporal frequency; |k| is the module of the horizontal

wavenumber vector.

132


In figure 5.5, ΦNη (|k|, ω) for the weak turbulence case of (Fr2 = 0.8,We = 40) is

plotted. Two ridges are observed. One is represented by the solid line in the figure,

which corresponds to the dispersion relationship of capillary–gravity waves

ω =

√k

Fr2+

k3

We. (5.5)

The other denotes the characteristic frequency of the surface elevation at each wavenum-

ber (the dash-dot line in figure 5.5)

ω =

√Φη(k)

Φηt(k). (5.6)

Here Φη and Φηt are the one-dimensional spatial spectra of the surface elevation

and its time derivative, respectively. The lower ridge corresponds to the turbulence

induced roughness. It extends to high wavenumbers, indicating that the turbulence

roughness is dominant at small spatial scales. Its time scale is also much smaller than

the wave period.

If the linearized free-surface kinematic boundary condition (KBC) ηt = w is used,

equation 5.6 becomes the one used by Borue et al. (1995) [120]

ω =

√Φη(k)

Φw(k). (5.7)

It is plotted as the dashed line in figure 5.5. It deviates from the ridge, suggesting

that the nonlinearity plays an important role in the dynamics of surface roughness.

133


5.2.4 Intermittency

For very violent free surfaces with surface breaking such as the case of (Fr2 =

32,We = ∞), using surface elevation to describe the free surface is no longer appro-

priate because it is multiple-valued. The flow near the surface is often an air–water

mixture. The water phase is the focus of our study here. We define the phase indicator

as

I =

1 water,

0 air.

(5.8)

After averaging, the phase indicator becomes the intermittency factor

γ(z) =< I(x, y, z, t) > . (5.9)

It is also the averaged volume fraction of water, which is an important quantity in

the modeling of multi-phase turbulent flows.

In figure 5.6(a), we plot the intermittency factor with respect to the water depth

for cases with different Froude and Weber numbers. Among different cases, the

intermittency factors have different lengths of extension but a similar shape. It is

found that their shape can be fitted by the complementary error function erfc(z).

We define the intermittency layer thickness based on an analogy to the ηrms of

mild surface cases. Here ηrms denotes the root-mean-square of the surface elevation.

We plot the histogram of the surface elevation for the mild surface case of (Fr2 =

32,We = 1) in figure 5.7(a). It fits the Gaussian function well. Therefore, we express

134


γ

z

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

γ

z/η σ

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

(a) (b)

Figure 5.6: (a) Intermittency factors of the cases with violent free surfaces: · · ·· · · , (Fr2 = 32,We = ∞); – · · – · · – , (Fr2 = 32,We = 500); ————,(Fr2 = 32,We = 40); – · – · – , (Fr2 = 8,We = ∞); (b) intermittency factors withz normalized by the equivalent thickness ησ.

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25

30

35

η η/ηrms

-4 -3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

P

γ

γ=0.159

γ=0.841

(a) (b)

Figure 5.7: (a) Histogram of the surface elevation of the mild surface case of (Fr2 =32,We = 1) and the fitted Gaussian function. (b) Relationship between intermittencyfactors and the surface elevation probability density function.

135


the probability density function of η as

P (η = z) =1√

2πη2rms

e−z2/2η2rms . (5.10)

The intermittency factor is calculated as

γ(z) = 1−∫ z

−∞ P (z′)dz′

= 1−∫ 0

−∞ P (z′)dz′ −∫ z

0P (z′)dz′

= 0.5− 0.5erf(z/√2ηrms)

= 0.5erfc(z/√2ηrms).

(5.11)

We have γ(z = ηrms) ≈ 0.159 and γ(z = −ηrms) ≈ 0.841. These values are case

independent. We have

ηrms = (z|γ=0.159 − z|γ=0.841)/2. (5.12)

Analogously, we define the intermittency layer thickness for cases with violent surfaces

as

ησ = (z|γ=0.159 − z|γ=0.841)/2. (5.13)

After z is normalized by ησ, lines plotted in figure 5.6(a) almost become a single line

(figure 5.6(b)).

The intermittency layer thickness for cases with different Froude numbers but the

same Weber number We = ∞ is plotted in figure 5.8(a). A straight line through

the origin fits the data. The linear fitting can be explained by the balance between

the turbulent kinetic energy (TKE) and the gravity potential energy (surface tension

136


Fr2

η σ

0 5 10 15 20 25 30 350

0.05

0.1

0.15

Current Numerical SimulationLinear fitting

1/We

η σ

-0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.02

0.04

0.06

0.08

0.1

0.12

(a) (b)

Figure 5.8: Intermittency layer thickness for cases with: (a) the same Weber numberWe = ∞ but different Froude numbers; (b) the same Froude number Fr2 = 32 butdifferent Weber numbers.

energy is zero since We = ∞), which can be described as

1

2q2ησ ∼ η2σ

2Fr2. (5.14)

As a result, ησ/(2Fr2) is comparable to q2/2, which is about the same for all cases.

The intermittency layer thickness is also plotted in figure 5.8(b) with respect to

1/We for cases with the same Froude number Fr2 = 32. The larger the Weber

number, the thicker the intermittency layer. But a linear relationship between the

thickness and the Weber number does not exist, because the gravity effect still exists

in these cases. In other words, the energy balance is among TKE, gravity potential

energy, and surface tension energy. The ratio between the gravity potential energy

and the surface tension energy is not a constant, and the surface elevation is also

affected by the surface curvature 1/κ through the surface tension energy. Therefore,

137


simple relationship between ησ and 1/We does not exist in figure 5.8(b).

5.3 Turbulence statistics and structures

5.3.1 Turbulence statistics

5.3.1.1 Horizontal velocity fluctuation u′

In figure 5.9, the vertical profiles of the horizontal velocity fluctuations u′ for differ-

ent cases are plotted. The velocity fluctuations are normalized by the corresponding

value at the water depth z = −0.5.

u’/u’ z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

u’/u’ z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

(a) (b)

Figure 5.9: Horizontal velocity fluctuations of cases with (a) the same Weber numberWe = ∞ but different Froude numbers: · · · · · · Fr2 = 32; – – – – Fr2 = 4; – · · –· · – Fr2 = 1, (b) the same Froude number Fr = 32 but different Weber numbers: ·· · · · · We = ∞; – · – · – We = 40; ————We = 1.

In figure 5.9(a), the profiles of u′ for cases with the same Weber number We = ∞

138


but different Froude numbers are plotted. The u′ is smaller for cases with larger

Froude numbers. For the case of Fr2 = 1, u′ increases significantly when approaching

the mean water level. In figure 5.9(b), u′ for cases with the same Froude number

Fr2 = 32 but different Weber numbers is plotted. At the surface, u′ is smaller for

cases with larger Weber numbers. Similar to the case of (Fr2 = 1, We = ∞), u′ for

the case of (Fr2 = 32, We = 1) increases towards the free surface. For cases where

the surface elevation is small, the strong blockage effect turns the vertical motion into

horizontal directions. The smaller the Weber number and Froude number, the larger

the blockage effect. Therefore, the flow in region 0 (weak turbulence) has the largest

blockage effect and the flow in region 2 has the weakest blockage effect.

5.3.1.2 Vertical velocity fluctuation w′

The vertical velocity fluctuation w′ as a function of water depth is plotted in figure

5.10. In general, w′ shows the opposite trend as u′ does when the Froude and Weber

numbers change. This is because the vertical motion is blocked by the free surface

and its energy is transferred to the horizontal motion when the surface is approached.

It is also interesting that for the case of (Fr2 = 32,We = ∞), the vertical velocity

fluctuation even increases slightly as the water surface is approached from below,

because the blockage effect is countered by the turbulence generation due to strong

surface breaking (figure 5.3(d)).

139


w’/w’ z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

w’/w’ z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

(a) (b)

Figure 5.10: Vertical velocity fluctuations of cases with (a) the same Weber numberWe = ∞ but different Froude numbers, (b) the same Froude number Fr2 = 32 butdifferent Weber number. (See figure 5.9 for line legend.)

5.3.1.3 Phase averaged Reynolds stress

The phase averaged turbulent normal Reynolds stress < u′u′I > and < w′w′I >

are plotted in Figs.5.11 and 5.12, respectively. The phase averaged Reynolds stress is

determined by both the turbulence intensity and the intermittency. Above the mean

water level, the intermittency factor is small. The phase averaged Reynolds stress is

mainly determined by the intermittency factor, and they thus have the similar shape.

At lower heights, the fluid is mainly the water phase, and as expected, the < u′u′I >

and < w′w′I > have the similar behavior as the turbulence velocity fluctuations

discussed earlier.

140


<u’ 2I>/<u’ 2I> z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

<u’ 2I>/<u’ 2I> z=-0.5

z

0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(a) (b)

Figure 5.11: Phase weighted horizontal turbulent normal stress < u′u′I > of caseswith (a) the same Weber number We = ∞ but different Froude numbers; (b) thesame Froude number Fr2 = 32 but different Weber numbers. (See figure 5.9 for linelegend.)

<w’ 2I>/<w’ 2I> z=-0.5

z

0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

<w’ 2I>/<w’ 2I> z=-0.5

z

0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(a) (b)

Figure 5.12: Phase weighted vertical turbulent normal stress < w′w′I > of caseswith (a) the same Weber number We = ∞ but different Froude numbers; (b) thesame Froude number Fr2 = 32 but different Weber numbers. (See figure 5.9 for linelegend.)

141


5.3.2 Flow structures

To further understand the turbulence statistics, we next investigate flow struc-

tures. It is found that events such as splat and surface breaking play an important

role in the turbulent energy transport and dissipation underwater.

5.3.2.1 Splat and antisplat

When an fluid element moves towards a surface, the surface blockage effect turns

the motion from the surface normal direction to the outward horizontal ones that

are radially along the surface, a process called splat. When the radial flows from

different splats encounter each other, the flow may be forced to return to the bulk

flow, a process called antisplat. For the free-surface problems, the surface blockage is

caused by the gravity and surface tension effects.

In figure 5.13(a), a horizontal slice near the free surface for the case of (Fr2 = 32,

We = 1) is plotted with horizontal velocity vectors and vertical velocity contours.

Three splats can be seen in the region with large positive vertical velocity and radial

horizontal velocity vectors. Antisplats are located at the edge of the splat with

negative vertical velocity. The antisplat regions are long and thin.

A vertical cut through a splat is plotted in figure 5.13(b), the dynamic pressure

contours show a high pressure region where the flow hits the surface. On the sides of

this splat, two counter rotating vortices are formed because of the radial flow motion.

The vortex structures presented in figure 5.13(c) by the iso-surface of the second

142


(a) (b)

(c)

Figure 5.13: Instantaneous flow structures of the case of (Fr2 = 32,We = 1): (a)horizontal slice close to the interface; (b) vertical slice through a splat; (c) free surfaceand vortex structures.

eigenvalue λ2 of the velocity gradient tensor show vortex tubes parallel to the free

surface. There are also airside vortex structures generated because of the radial

motion on the free surface.

For the strong turbulence cases, the surface is so violent that splat can be found

directly from the surface geometry. In figure 5.14(a), we plot an instantaneous free

surface and velocity vectors for the case of (Fr2 = 32, We = ∞). In this figure, a

large dome-like surface geometry is located at the center of the surface, which is the

143


result of a large splat there. We also plot the vertical velocity contours on a vertical

cut through the dome-like geometry. Strong upward motion exists in the dome (figure

5.14b), which eventually breaks the dome and brings the fluid to a significant height

(plotted in figure 5.3d).

(a) (b)

(c) (d)

Figure 5.14: Instantaneous flow structure for the case of (Fr2 = 32,We = ∞): (a)free surface and velocity vectors; and on a vertical cross-section, distributions of (b)vertical velocity; (c) transport of horizontal turbulent normal stress by the verticalturbulent velocity; (d) transport of vertical turbulent normal stress by the verticalturbulent velocity.

Splat also plays an important role in turbulence stress and energy transport. In

figure 5.14(c) and figure 5.14(d), the transport of horizontal and vertical normal stress

on the vertical cut through the splat are plotted. In the dome area, there is strong

144


positive vertical transport, meaning that velocity fluctuations associated with the

splat transports kinetic energy toward the surface.

5.3.2.2 Breaking surface

Strong splats bring water blobs to a significant height and generate surface break-

ing. When a breaking occurs, the surface sheet overturns, reenters the water, splashes,

and entrains a large amount of air. For very strong turbulence cases (large Froude

number and Weber number), surface breaking is a common phenomenon. In figure

5.15, a surface breaking process is presented.

(a) (b) (c)

Figure 5.15: A surface breaking process in the case of (Fr2 = 16,We = ∞). (a)A water sheet is brought up and begins to overturn. (b) The water sheet plungesdownward to the free surface. (c) The water sheet reenters and then splashes up.Surface elevation contours and the velocity vectors of water are plotted. A verticalcut is extracted for analysis in figure 5.16.

When the water sheet impinges on the water surface below, there exists strong

shear in the contacting region. The shear causes high dissipation rate of the kinetic

energy. In figure 5.16(a), the distribution of energy dissipation on a vertical cut

is presented. Large dissipation in the contacting region is clearly shown. Surface

145


breaking also enhances Reynolds stress transport. In figure 5.16(b), the horizontal

transport of the horizontal normal stress is plotted. When the water sheet reenters,

the strong shear brings a large amount of horizontal normal stress into the water.

(a) (b)

Figure 5.16: Energy dissipation and turbulent Reynolds stress transport associatedwith surface breaking: (a) viscous dissipation rate; (b) horizontal transport of thehorizontal turbulent normal stress.

5.4 Conclusions

In this study, we perform a systematic study on the canonical problem of ho-

mogeneous turbulence interacting with a free surface. With the recently developed

simulation capability that combines the strengths of several free-surface flow simula-

tion tools, we are able to obtain an accurate description of the free surface and the

turbulence flow field. Such information is important for the modeling of free-surface

turbulence.

Different flow regimes are demonstrated by the instantaneous surface geometries

of representative cases. The surface elevation spectrum also demonstrates large dif-

146


ference among flows under different gravity and surface tension effects. We identify

surface waves and turbulence induced surface roughness in the normalized frequency–

wavenumber spectrum. The turbulence roughness is dominant at small spatial scales.

The intermittency factors for violent surface cases are calculated. An equivalent inter-

mittency layer thickness is defined based on the intermittency factors, and is discussed

for different Froude and Weber numbers.

We also investigate the influence of the gravity and surface tension effects on

the turbulence statistics. The blockage effect of the free surface turns the vertical

motion into horizontal motion. It is strongly dependent on the Froude number and

the Weber number. The smaller the Froude number and the Weber number are, the

stronger the blockage effect is. The phase averaged Reynolds stress is discussed and it

is determined by both the intermittency factor and the turbulent velocity fluctuation

magnitude.

The two most important flow structures in SFST, namely splat and surface break-

ing, are also discussed. Splat is the major mechanism that turns vertical motion to

horizontal motion. Strong splats generate surface breaking, and surface breaking

enhances dissipation and turbulence transport significantly.

147

Chapter 6

Multi-Scale Numerical Simulation

of Wind-Wave-Structure

Interaction

Most of the existing numerical simulations concerning wind, wave, and structures

focus either on large-scale flow or on local-scale flow–structure interaction with simple

boundary conditions to approximate wind and wave environment. Simulation of

the interaction among wind, wave, and structure with realistic environment input is

challenging because of the scale difference.

In this chapter, we introduce a multi-scale modeling approach developed for the

simulation of wind and wave coupling dynamics and the simulation of wind and wave

past a surface-piercing object. Preliminary results of large-scale wind over broadband

148

CHAPTER 6. MULTI-SCALE NUMERICAL SIMULATION OFWIND-WAVE-STRUCTURE INTERACTION

waves [64] and local-scale wind–wave–structure interaction [121] are given, which show

the wave effect on the atmospheric boundary layer and the wind load on the structure.

6.1 Large scale wind-wave simulation

6.1.1 Numerical methods and simulation setup

To obtain a more direct description of the wavefield with finely-resolved spatial

and temporal details, it is desirable to resolve the wave phases in the simulation.

Such information will be valuable for the mechanistic study of wind-wave dynamics

which may eventually lead to improved modeling for the wave spectrum simulation.

Recent advancement in computing power and algorithm development has facilitated

the phase-resolved simulation of nonlinear wave interaction involving a large number

of wave modes (e.g., O(104) modes in each direction as shown by Wu [11]). Central to

the development is an efficacious high-order spectral (HOS) method that is capable

of directly capturing nonlinear wave interaction at a reasonable computational cost.

The HOS method directly simulates the evolution of surface elevation η and surface

potential Φs, which is defined as the surface value of the velocity potential Φ. With

a perturbation series of Φ with respect to the wave steepness to the order of M and

149


Taylor series expansion about the mean water level z = 0,

Φ(x, y, z, t) =∑M

m=1Φ(m)(x, y, z, t) ,

Φs(x, y, t) =∑M

m=1

∑M−ml=0

ηl

l!∂lΦ(m)

∂zl|z=0 ,

(6.1)

and an eigenfunction expansion of each Φ(m) with N modes,

Φ(m)(x, y, z, t) =∑N

n=1Φ(m)n (t)Ψn(x, y, z) , z ≤ 0 , (6.2)

the kinematic and dynamic free surface boundary conditions are written as [122]

∂η∂t = −∇hη · ∇hΦ

s + (1 + |∇hη|2)

×[∑M

m=1

∑M−mℓ=0

ηℓ

ℓ!

∑Nn=1Φ

(m)n (t) ∂ℓ+1Ψn(x,y,z)

∂zℓ+1 |z=0

],

∂Φs

∂t = −gη − 12 |∇hΦ

s|2 +DΦ − pa(x,y,t)ρw

+ 12(1 + |∇hη|2)

×[∑M

m=1

∑M−mℓ=0

ηℓ

ℓ!

∑Nn=1Φ

(m)n (t) ∂ℓ+1Ψn(x,y,z)

∂zℓ+1 |z=0

]2.

Here, ∇h ≡ (∂/∂x, ∂/∂y) is the horizontal gradient; DΦ is the wave dissipation; pa

is the atmospheric pressure at the wave surface; ρw is the density of water; and g is

the gravitational acceleration. By using a pseudo-spectral method, the HOS method

accounts for the nonlinear interactions among all the N wave modes up to the desired

perturbation order M in wave steepness, with a computational cost proportional to

MN lnN . Complete review of the HOS method is provided in Ch. 15 of Mei et al.

. [123]

While the HOS method provides a vehicle for the nonlinear wave simulation in

a phase-resolved framework, the wave breaking dissipation DΦ and the wind forcing

pa remain to be specified in a phase-resolved context. The model development for

150


these terms is still at an early stage, and in this section we investigates the charac-

teristics of pa. [1, 82, 124–128] For this purpose, we perform coupled simulations of

wind turbulence and wave evolution (figure 6.1). To our best knowledge, this is the

first simulation-based study that addresses the two-way dynamical coupling between

wind and narrowband/broadband waves with all of the essential nonlinear wave in-

teraction processes being resolved to high order. (We note that recently a powerful

numerical capability has been developed for the large-scale simulation of air–water

coupled flows including wind-driven waves on top-ranked supercomputers such as the

Earth Simulator. [129]) For wind over simple wave trains, we use direct numerical

simulation (DNS) for wind turbulence. The approach of DNS has been proven to

faithfully capture the pressure variation in wind over water waves. [18, 127, 128] For

broadband waves, in order to resolve the wind–wave interaction over a wide range of

wavenumbers, we use large-eddy simulation (LES).

For the wind simulation, we consider as a canonical problem Couette air flow over

water waves. We simulate the Navier–Stokes equations for the air motion on a time-

dependent boundary-fitted grid. For spatial discretization, we use a hybrid pseudo-

spectral and finite-difference scheme. Time integration of the momentum equation

is realized through a fractional step scheme. Details of the numerical scheme and

the validation using the data in the literature [1, 125–128] are provided by Refs. 13–

15 [18,130,131].

Different from Refs. [18] and [130] where the waves do not evolve dynamically, in

151


Figure 6.1: Illustration of wind turbulence and water wave coupled simulation. Plot-ted are streamwise velocity (normalized by Uλm/2) of the wind and pressure (normal-ized by ρau

2∗) distribution on the surface of broadband waves (cm/u∗ = 12.3). The

air domain is lifted up for better visualization.

this paper the wind turbulence simulation is dynamically coupled with the HOS wave

simulation through a fractional step method with two-way feedbacks. At the sub-

timesteps, the HOS simulation provides the wind simulation with the wave surface

geometry and the normal and tangential surface velocities as the Dirichlet boundary

conditions; the windfield evolves dynamically, subject to the wave form and friction

drags that are generated naturally from the retardance by the wave calculated by the

HOS simulation; the wind simulation provides pa(x, y, t) on the wave surface, which

the HOS simulation uses as the wind forcing in equation 6.3 to advance the wave in

time. After the above coupling, the entire wind-and-wave field advances to the next

timestep. The numerical details of the dynamic coupling are provided by Yang. [131]

152


t0 200 400 600 8000

0.2

0.4

atk

(b)0 200 400 600 8000

0.2

0.4

ack

(a)

Figure 6.2: Evolution of (a) ack and (b) atk: −−, c/u∗ = 2; − · −, c/u∗ = 2 (fromlinear wave simulation); · · ·, c/u∗ = 5; − · ·−, c/u∗ = 10; −−−, c/u∗ = 14. The timeis normalized by λ/Uλ/2.

6.1.2 Wind over monochromatic waves

We discuss simple wave trains first. Note that the wave is initially monochromatic.

Due to nonlinear wave interaction and the excitation by the turbulent eddies in the

wind, other wave components are generated. During the course of our simulation,

these wave modes have much less energy than the dominant wave, and this paper

focuses on the growth of the latter. We use k to denote wavenumber, λ for wavelength,

a for wave amplitude, ω for wave frequency, c for wave phase velocity, u∗ for wind

friction velocity, Uλ/2 for the mean wind velocity at a distance of λ/2 above the wave

surface, and c/u∗ (and also c/Uλ/2) for wave age. Various cases of different wave

ages are simulated, and the growth of the waves provides results for the different

evolution stages of ak (cf. Tables 6.1 and 6.2). Figure 6.2 shows the time evolution

of the wave steepness based on the wave crest and trough amplitude, ac and at,

153


respectively. The wave nonlinear effect is shown through the difference between ack

and atk (hereafter, we define a = (ac + at)/2) and the difference between linear and

nonlinear waves. Figure 6.2 shows that the wave grows fast for young waves with the

wave ages c/u∗ = 2 and 10. At c/u∗ = 10 and 14, the wave grows slowly.

The wind turbulence simulation provides detailed information on the distribution

of pa along the waveform. Figure 6.3 shows the phase-averaged results of pa. In

figure 6.3(a), we validate our results via comparison with the corresponding case in

Ref. [1]. In figure 6.3(b), we illustrate the dependence of pa on the wave steepness.

For mild waves (from ak = 0.1 to 0.15), the magnitude of both the maximum and

minimum pressure increases with ak. When the wave becomes steeper (from ak =

0.15 to 0.20 to 0.25), the pressure maximum increases slightly and moves downstream,

while the lower part of the pressure profile changes from a flat shape to a trough.

Figure 6.3(c) shows the dependence of pa on the wave age. At c/u∗ = 2, the maximum

pressure is located downstream of the wave trough. As the wave age increases (c/u∗ =

5), it moves further downstream toward the wave crest. As the wave age increases

to c/u∗ = 10, the pressure profile moves upstream. For c/u∗ = 14, the maximum

pressure is located on the wave trough.

We next quantify the wave growth rate. Due to the orthogonality of different

trigonometric components, for a wave component with the surface elevation η(x, t) =

a cos(kx− ωt), we focus on the corresponding surface pressure component expressed

as pa(x, t) = αak cos(kx−ωt+θ). Here α denotes the amplitude ratio of the pressure

154


θηpa

p a/p a,

max

0 0.5 1 1.5 2-1

0

1(a)

p a/ρ au

*2

0 0.5 1 1.5 2-10

0

10(b)

x/λ

p a/ρ au

*2

0 0.5 1 1.5 2-10

0

10(c)

Figure 6.3: Surface pressure profiles over monochromatic waves: − · ·−, (c/u∗ = 2,ak = 0.05); −−−, (c/u∗ = 2, ak = 0.1); − · −, (c/u∗ = 2, ak = 0.15); · · ·, (c/u∗ = 2,ak = 0.2); −−, (c/u∗ = 2, ak = 0.25); −−, (c/u∗ = 5, ak = 0.1); −−, (c/u∗ = 10,ak = 0.1); −−, (c/u∗ = 14, ak = 0.1). (a) Comparison of simulation result withfield measurement data (N) of Ref. [1]; (b) pressure profiles over waves with differentsteepnesses; (c) pressure profiles over waves with different wave ages. The wind andwave are from left to right. The wave phase is shown in the sketch at the bottom.

to the wave steepness; and θ denotes the phase difference between the pressure and

the waveform (figure 6.3). As in the literature, the wave growth rate parameter β

and the fractional rate of energy input γ are defined respectively as

β = (α sin θ)/ρau2∗ , γ = ω(ρa/ρw)(u∗/c)

2β . (6.3)

Tables 6.1 and 6.2 show the dependence of α, θ, β, and γ on ak and c/u∗. The

comparison of the γ value with the parameterization by Ref. [1] shows good agreement.

The decrease of α as ak increases suggests that the pressure does not simply scale with

ak as was assumed in many of the literature. The dependence of β on c/u∗ and ak

can be understood through the variation of α and θ (equation 6.3). Table 6.1 shows

155


ak 0.05 0.10 0.15 0.20 0.25

α 40.2 36.3 35.1 30.4 26.3

θ 118 107 100 96 92

β 36.9 34.6 34.4 30.1 26.3

γ (×10−3) 2.22 2.08 2.07 1.81 1.58

γ (×10−3) [Ref. 10] 2.23 2.09 1.95 1.80 1.66

Table 6.1: Values of α, θ, β, and γ for c/u∗ = 2 at different ak. Values of γ basedon the parameterization of Ref. [1], γ = 0.17(Uλ/2/c − 1)2(ωρa/ρw), are listed forcomparison.

that when c/u∗ = 2 and ak increases, the variation of θ is small, and β is mainly

controlled by α, which decreases as ak increases; the value of γ decreases at the same

rate according to equation 6.3. Table 6.2 shows that when ak = 0.1, for different

c/u∗ cases, the change in β is affected by both α and θ, with the latter determining

the efficiency of wind input. As c/u∗ increases from 2 to 5, α increases (because the

magnitude of the minimum pressure increases as shown in figure 6.3), θ gets closer

to 90, and β becomes larger as a result. As c/u∗ further increases to 10 and 14, θ

approaches 180 and β decreases significantly. As c/u∗ increases, the variation of β

together with the proportionality to (u∗/c)2 (equation 6.3) makes γ decrease.

6.1.3 Wind over broadband waves

We next investigate waves with broadband spectra (figure 6.1). We construct a

wavefield based on the wave spectrum obtained during the Joint North Sea Wave Ob-

156


c/u∗ 2 5 10 14

c/Uλ/2 0.12 0.33 0.64 0.86

α 36.3 50.0 27.5 36.2

θ 107 79 152 177

β 34.6 49.0 17.9 1.62

γ (×10−3) 2.08 1.18 0.215 0.0139

Table 6.2: Values of α, θ, β, and γ at ak = 0.1 with different wave ages.

servation Project (JONSWAP) (details are provided in Ref. [131]). We consider three

cases with different wave ages based on the phase velocity at the peak (denoted by the

subscript “m”) wavenumber km, cm/u∗ = 5, 12.3, and 16 (the corresponding values of

cm/Uλm/2 obtained from the simulations are 0.27, 0.66, and 0.84, respectively), which

hereinafter are referred to as cases I, II, and III, respectively.

For the broadband wavefield, it is essential to quantify the wind input for different

wave components. At each k, we perform analysis in a way similar to that in the

monochromatic wave case (equation 6.3). The variation of β and γ with k is shown in

figure 6.4. To help understand their behavior, we also indicate the values of c/u∗ at

different k for the three cases in figure 6.4. Note that according to the wave dispersion

relationship, c/u∗ decreases as k increases and has different ranges for the three cases.

For case I, β reaches its peak around k = km and decreases as k further increases; this

is consistent with the monochromatic wave case, in which c/u∗ = 5 has a larger value

157


k/km

0 1 2 3 4 5 6 7 8-10

0

10

20

30

40

50

60

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

β γβ γβ γβ γβ γβ γβ γβ γβ γβ γβ γβ γβ γβ γβ γ

16.0 11.3 9.22 8.00 7.14 6.52 6.03 5.64

12.3 8.69 7.09 6.14 5.49 5.02 4.64 4.34

5.00 3.54 2.89 2.50 2.24 2.04 1.89 1.77c/u*

Figure 6.4: Values of β (lines) and γ (symbols) for broadband waves: −−− and ,cm/u∗ = 5 (case I); − · − and , cm/u∗ = 12.3 (case II); − − − and , cm/u∗ = 16(case III).

of β than c/u∗ = 2 does. For cases II and III, β first increases with k and reaches its

peak around k = 2km ∼ 4km (note that in the monochromatic wave case c/u∗ = 14

has smaller value of β than c/u∗ = 5 does), and then decreases as k further increases.

For cases II and III, the corresponding c/u∗ values of peak β deviate from the value of

c/u∗ = 5 shown in the monochromatic wave case, probably due to the sheltering effect

of dominant waves on relatively short and small waves. At large k, our simulation

does not show significant values of β. At these small scales, the pressure induced by

the short waves is relatively small compared with the pressure fluctuation in the wind

turbulence and is thus difficult to quantify. On the other hand, γ, which measures the

fractional rate of wave growth (equation 6.3), does not reduce as rapidly compared

158


××

×

×

××××

c/u*0 10 20

0

20

40

60

80

100

β

Figure 6.5: Wave growth rate parameter β: •, experimental results compiled in Ref. 7;, numerical results of Refs. 9 and 11; , numerical results of Ref. 13; ×, currentresults for monochromatic waves. The lines are the current broadband wave results(see the line legend in Fig. 6.4).

with β (γ even increases with k for case I). In reality, local small waves may grow

rapidly to break, serving as an important vehicle for atmosphere–ocean momentum

and energy transfer. [13,14]

Figure 6.5 summarizes our results of β and the comparison with other studies.

Our data of monochromatic waves agree with the previous numerical results and (to

a less extent) experimental data. As shown by Yang and Shen, [18] many factors,

including wave steepness and wind induced surface drift, can affect the value of β

(an example for the same c/u∗ = 2 but different wave steepness and surface drift

conditions is shown in figure 6.5 using the data of Ref. [18]).

Figure 6.5 shows that for the broadband waves, the long wave components (i.e.,

the right parts of each curve with relatively large λ and c and thus large c/u∗) have

β values close to those from the study of monochromatic waves. This result suggests

that for phase-resolved simulation of broadband wavefield using the HOS method,

159


the wind input model for large wave components can be developed based on the

analysis of monochromatic waves as a first step of the study. Therefore, it is valuable

to have precise laboratory measurement [132] of wind over waves and to perform

mechanistic study using numerical simulation. For short waves, as discussed earlier,

the mean value of β drops rapidly to have large deviation from the simple wave

results, while locally the wind input may have large fluctuations. Therefore, in the

HOS method, for wind input to small waves, stochastic modeling is called for. This

strategy of deterministic and stochastic wind input modeling for long and short waves

respectively is consistent with the philosophy of the HOS method. [11,131]

6.2 Local scale wind-wave-structure sim-

ulation

The local-scale wind–wave–structure simulation is implemented through a hybrid

interface capturing (discussed in chapter 2) and immersed boundary method. The

large-scale simulation data provides inflow condition to the local-scale simulation.

160


6.2.1 Immersed boundary method for flow-structure

interaction

Immersed boundary method (IBM) is first introduced by Peskin in 1970s [133,134]

to simulate the effect of heart muscle on the blood flow. In immersed boundary

method, the grid is not required to conform to the immersed boundary, which could

avoid time consuming mesh generation when complex geometries present.

Since the grid does not conform to the immersed boundary, the governing NS

equations or their discretization need to be modified at the grid points around the

boundary. In IBM, a forcing term fb is introduced into the NS equation to represent

the solid boundary effect. The governing NS equations then become

∂u

∂t= RHS + fb, (6.4)

∇ · u = 0. (6.5)

According to when the force is calculated, the immersed boundary method can be

classified into continuum force method and discrete force method. In Peskin’s seminal

work, the boundary is elastic heart muscle and continuum force method is used. The

elastic force applied on the boundary points was calculated according to the Hooke’s

law

Fib = κ(xib − x0) (6.6)

Here xib and x0 are the current and equilibrium positions of the boundary; κ is the

elastic coefficient. After the elastic force is obtained, it is smoothed onto the grid

161


Figure 6.6: Schematic of immersed boundary method (discrete force method).

points around the boundary as

fb =∑

Fibδ(xib − xb) (6.7)

Here δ(x) is the smoothed Dirac delta function and can have different forms [134].

For rigid boundary, the above continuum force method can be applied by assuming

the material very stiff and setting a very large κ. However, it could generate very

stiff matrix which slows the convergence of the code. Discrete force method which

calculates force after discretization of the governing equation is more suitable for rigid

body and is adopted in current model.

As shown in figure 6.6, the force is only applied on those forcing points which are

located immediately outside the solid boundary with at least one adjacent grid point

inside the solid body. The grid points inside the solid body do not need to be solved,

which could save computational resource especially in three dimensional problems.

162


In discrete force method, we advance the simulation without considering the

boundary first and have

u∗ − un

∆t= RHSn. (6.8)

The updated u∗ is then used together with the velocity of the boundary to interpolate

the velocity ub on those forcing points and the force is calculated as

fb = −RHSn +ub − un

∆t(6.9)

6.2.2 Inflow boundary condition

Uniform inflow boundary condition is widely used in existing simulations of flow–

structure interaction. For the environmental wind and wave inputs to be included,

large-scale and local-scale simulations are combined together for a multi-scale wind–

wave–structure simulation. A subdomain of the large-scale simulation data is ex-

tracted and used by the local-scale simulation as the inflow boundary condition. This

process is displayed in figure 6.7.

6.2.3 Outflow boundary condition

Radiative outflow boundary condition is used in current local scale simulation.

The velocity components on the outflow boundary satisfy

∂u

∂t+ u

∂u

∂x= 0. (6.10)

163


Figure 6.7: Illustration of multi-scale wind–wave–structure simulation. The flowcondition inside the small black window is provided to local scale wind–wave–structuresimulation as inflow condition.

∂v

∂x= 0, (6.11)

∂w

∂x= 0. (6.12)

The spatial derivative is calculated using one-sided finite difference scheme. The level

set function ϕ on the outflow boundary satisfies

∂ϕ

∂t+ u

∂ϕ

∂x= 0. (6.13)

If we calculate ∂ϕ∂x

using only the two grid points near the outflow boundary, the water

level there may deviates significantly from the inflow water level. A modification is

applied on the discretization as

∂ϕ

∂x= α

ϕo − ϕNX

xo − xNX

+ βϕo − ϕi

xo − xi

(6.14)

Here ϕi, ϕo, and ϕNX are the signed distance function values at inflow, outflow, and

the NXth grid point respectively; α and β are the weight and α + β = 1. We use

164


β = 0.1. With this modification, the water level of the inflow is incorporated into the

outflow boundary condition directly and the water level drop can be avoided.

The total volume conservation is enhanced by mandatorily setting the total out-

flow volume equal to the inflow volume as

∑uo =

∑ui. (6.15)

Here, ui and uo are the streamwise velocities at the inflow and outflow boundary

respectively. The summation is applied on both fluids. With this correction, the

convergence of the poisson equation can be guaranteed.

6.3 Multi-scale simulation of wind-wave

interaction with surface piercing body

In figure 6.7, the large-scale wind–wave simulation result of turbulent wind over

complex broadband wave field is presented on the left. The wave age c/u∗ defined

as the ratio between the phase speed of the dominant wave c and the turbulence

friction velocity u∗ is 2. The air domain with streamwise wind velocity contours is

lifted up for better visualization. The pressure field on the water surface shows that

the windward pressure is larger than the leeward pressure statistically. The pressure

difference provides forcing for the wave growth.

In figure 6.8, the local-scale wind–wave–structure simulation results at two differ-

165


(a) (b)

Figure 6.8: Wind and wave fields around a surface piecing body: (a) when a wavecrest, and (b) when a wave trough arrives at the front surface of the object. the inflowis in the x-direction. the vertical planes show the streamwise velocity contours. thevelocity field inside the small black window is enlarged and shown in figure 6.9. thepressure on the object surface and the wave surface are shown. vortices are plottedwith grey color.

ent wave phases are presented. Vertical streamwise velocity, surface elevation, and

pressure on the object surface are plotted. Complex vortical structures in the wake

of the surface piercing object are shown. Strong wave-coherent flow patterns near the

water surface are seen. In figure 6.9, the enlarged wind velocity contours above the

wave crest and trough are presented. Above the wave crest, the wind shear is strong

and the wind velocity is large. Above the wave trough, the wind velocity is small

because of the sheltering effect behind the wave crest. When a wave crest arrives at

the front surface of the object, the pressure on it is obviously larger than that when

a wave trough arrives. The wind drag coefficients are calculated and the difference is

found to be 24%. The phase dependence of wind load could severely affect the energy

166


(a) (c)

Figure 6.9: Enlarged streamwise velocity contours from figure 6.8: (a) above wavecrest when a crest arrives at the object; (b) above wave trough when a trough arrivesat the object.

input to wind turbines and could also incur structural vibration and damage.

6.4 Conclusions

In this study, we develop a multi-scale wind–wave–structure simulation approach.

This approach combines large-scale environmental wind–wave simulation and local-

scale wind–wave–structure simulation, and may provide data for offshore wind energy

applications.

Our simulation results show that the wind flow in the atmospherical boundary

layer is highly wave phase dependent, which makes the wind drag on surface-piercing

structure also wave phase dependent. For turbine efficiency and structure stability,

it may be necessary to consider the coupling dynamics of wind, wave, and structure.

167

Chapter 7

Summary and Future Work

7.1 Summary

Numerical tool that combines the strength of several simulation methods is de-

veloped. The interface is captured by the level set method, in which the air and

water are simulated together as a coherent flow system with varying physical prop-

erties such as density and viscosity. To improve the mass conservation of level-set

simulations, the volume-of-fluid method is coupled with the LS method. To avoid

the interface being artificially smoothed as in the continuous surface force method,

the ghost fluid method is incorporated to treat the interface in a sharp fashion. The

immersed boundary method is adopted to represent solid structures on the Cartesian

grid. The code is parallelized with message passing interface basing on the domain

decomposition technique to achieve high resolution simulation.

168

CHAPTER 7. SUMMARY AND FUTURE WORK

Direct numerical simulation of two dimensional wave breaking without wind effect

is performed. The relationship between the breaker type and the initial wave steep-

ness is investigated. The skewness, asymmetry and steepness evolution with time

before the breaking onset are examined. The energy loss and energy dissipation rate

are quantified. With the direct numerical simulation data, the empirical dissipation

models are validated and the model coefficients are quantified.

Large eddy simulation of turbulent wind interacting with steep breaking waves

is performed. In this study, the waves are prescribed at the beginning until the

wind turbulence is fully developed, then the waves are released and break under the

wind forcing. The wind profiles during the breaking process are studied. The wind

stress and drag coefficients are quantified. Surface current and underwater turbulence

generation are also quantified. The occurrence of airflow separation over a breaking

wave is identified and the flow field in the separation bubble is found to be highly

turbulent. The effect of separation on the wind input to wave growth is also quantified.

The data obtained from the simulation are valuable for the cross comparison with

measurement and will help establish a physical basis for mechanistic study and model

development.

A simulation based study of strong free surface turbulence (SFST) is performed.

Finely-resolved turbulence and wave fields under different gravity and surface tension

effects are obtained for systematic analysis. Different flow regimes are identified by

the surface geometries. Surface elevation spectra under different gravity and surface

169


tension effects are investigated. The thickness and distribution of the intermittency

layer is calculated for different Froude and Weber numbers. The influence of the grav-

ity and surface tension effects on the blockage effect of the free surface is investigated.

Splat is the major mechanism that turns vertical motion to horizontal motion. Strong

splats generate surface breaking, and surface breaking enhances dissipation and tur-

bulence. The results of this study may be useful for the development of improved

turbulence models for SFST and steep/breaking waves.

A multi-scale modeling approach is developed for the simulation of wind–wave–

structure interaction. In this approach, the large-scale simulation is performed through

large eddy simulation of wind on a boundary fitted grid over wave field simulated by

high order spectra method. The local-scale simulation is performed with the numeri-

cal tools discussed above with inflow condition from the large-scale simulation. With

this approach, realistic wind input is used in the local-scale simulation and makes the

simulation results more reliable. With the data obtained through this approach, the

wave phase dependence of the wind drag is investigated.

Through current study of the wind-wave-turbulence system, more elaborate and

solid understanding of the ocean surface wave field is obtained and can be useful for

improving the wave models currently in use.

170


7.2 Future Work

7.2.1 Wind wave generation and growth

Wind wave generation and its growth is important for the fully understanding of

the wind–wave system. Phillips [135] theoretically studied the wave generation by

turbulent wind. He divided the developing process of wave field before nonlinearity

is significant into two stages: the initial stage when the water surface is disturbed by

turbulence and the mean-square water surface displacement is bounded; the principle

stage when resonance between wind pressure and wave surface happens and waves

grow linearly with time. Caulliez [136] experimentally studied the generation of wind-

waves and identified three regions with different surface characteristics: the smooth

zone, the streak zone, and the uniformly rough zone. They also found an explosive

growth in surface elevation beyond a critical fetch in the streak zone. This exponential

growth is accompanied by the drop of the near surface velocity caused by a laminar-

turbulent transition. Veron et al. [137] studied the wind–wave generation process

experimentally. They divided the process into four stages: the uniform and monotonic

surface acceleration stage; the wind wave generation and growth stage; the Langmuir

circulation generation and evolution stage; and the fully developed turbulence stage.

The surface velocity is found to increase first and then decrease in the Langmuir

circulation stage to a relatively stable low value. They pointed out that during low

wind speed without wave breaking, Langmuir circulation is the dominant source of

171


surface turbulence. Lin et al. [138] studied the wind-wave generation at the very

initial stage with direct numerical simulation. Two stages of wave growth, the early

linear growth and the later exponential growth stages, are confirmed. Liberzon [139]

studied the initial stages of wind-waves experimentally. In this study, waves are

generated by wind from flat in a 5m wind wave flume. The wind field characteristics,

wave statistics, and the momentum transfer from wind to wave are investigated.

Numerical simulation of the wind-wave generation process is performed. The

simulation setup is similar to the cases in Chapter 4 except that the water surface is

flat initially. The domain size is L × W × H = 40m × 30m × 40m. Different wind

speed U10 are considered and the case with U10 = 30m/s is demonstrated here. In

figure 7.1, the wave surfaces generated by wind at different time are plotted.

The evolution of the root-mean-square (rms) surface elevation is plotted in figure

7.2. Significant wave height is four times the rms surface elevation. As shown in

figure 7.2, the wave growing process can be divided into three stages: the first stage

when wave grows linearly; the second stage when wave grows exponentially and the

fastest growth rate occurs in this stage; and the third stage when the growth slows

down and nonlinear effect becomes significant.

Surface spectra at different time are plotted in figure 7.3. At early stage, high

wave number components are dominant. As time evolves, the peak shifts towards

low wave number (downshifting), which is observed in the field. In the later stage,

downshifting slows down and energy of the peak keeps increasing. In the ocean, wave

172


(a)

(b)

(c)

Figure 7.1: Wave field generated by the turbulent wind with 10 meter hight speed30m/s.

173


t (s)

η rms

(m)

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

Figure 7.2: Evolution of the root-mean-square surface elevation with time.

k (m-1)

Sk

(m2 )

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.002

0.004

0.006

0.008

0.01

0.012

0.014t=26.5 st=53.1 st=79.6 st=106.1 s

Figure 7.3: One dimensional surface spectra of the wave field generated by wind.

174


field keeps developing as wind blows. The JONSWAP spectrum [140]

S(ω) =αg2

ω5exp

[−5

4

(ωp

ω

)4]γr (7.1)

is used to describe the developing sea. Here F is the fetch (distance from the coast);

r = exp[− (ω−ωp)2

2σ2ω2p

]; α = 0.076

(U210

Fg

)0.22; γ = 3.3; σ = 0.07 for ω ≤ ωp and 0.09 for

ω > ωp. The peak angular frequency

ωp = 22

(g2

U10F

)1/3

. (7.2)

Using linear dispersion relationship, the peak wave length becomes

λp = 6.066× 10−3(U10F )2/3. (7.3)

The spectrum is determined by the wind speed U10 and fetch F . In table 7.1, the

peak wave length and significant wave height obtained in the simulation are listed

together with that of JONSWAP spectrum.

Further analysis such as the nonlinear energy transfer and effect of Langmuir

circulation will be conducted in the future.

7.2.2 Coupled LS/SPH Method

Wave breaking is associated with large amount of water droplets and sprays whose

scale is much smaller than the scale of wave and the overturning jets. Lagrangian

particle method is good at simulating water droplets and sprays in small scale and

could complement the Eulerian simulation. Here smoothed particle hydrodynamics

175


t(s) 26.5 53.1 79.6 106.1

Fetch (m) 796 1592 2387 3183

Peak Wave Length (m)(JONSWAP) 5.03 7.99 10.46 12.67

Peak Wave Length (m) 4.0 5.00 6.67 10.0

Significant Wave Height (m) (JONSWAP) 0.38 0.56 0.69 0.81

Significant Wave Height 0.272 0.58 0.76 0.90

Table 7.1: Peak wave length and significant wave height from simulation and corre-sponding value from JONSWAP spectrum.

Figure 7.4: Illustration of the coupled LS/SPH simulation

(SPH) method is embedded in the Eulerian simulation to simulate the small scale

flow structures as in figure 7.4.

Smoothed particle hydrodynamics method is first introduced by Gingold & Mon-

aghan [141] and Lucy [142] for the simulation of astrophysical problems. It is based

on the fact that an arbitrary function f(x) can be written as∫f(x′)δ(x′−x)dx′. Here

δ(x) is the Dirac delta function. It can then be discretized as

f(xi) ≈N∑j=1

mj

ρjfjW (xi − xj, h) =

N∑j=1

mj

ρjfjWij (7.4)

176


Here W (r, h) is the smoothed Dirac delta function or kernel function. It satisfies∫Ω

W (x− x′, h)dx′ = 1, (7.5)

and

limh→0

W (x− x′, h) = δ(x− x′). (7.6)

The smoothing length h is usually chosen as 2∆p and ∆p is the diameter of the

particles. Gaussian kernel function

W (R) =1

πh2e−(ra−rb)

2/h2

(7.7)

satisfies equation 7.5 and 7.6, but it is not compact supported and each particle need

to know the information of all the other particles to update itself. Cubic spline kernel

W (R) =15

7πh2

23−R2 + 1

2R3 0 ≤ R < 1

16(2−R)3 1 ≤ R < 2

0 R ≥ 2

(7.8)

is widely used. This kernel can cause the zero-energy mode (hourglass arrangement

of particles) for some problems such as still water (figure 7.5). Wendland kernel

W (R) =

7

4πh2 (1− 0.5R)4(2R + 1) R < 2

0 R ≥ 2

(7.9)

is a fifth order kernel and can efficiently avoid the zero-energy mode. In our simula-

tion, the Wendland kernel function is adopted.

In SPH, the continuity equation and the momentum equation are discretized as

dρidt

=N∑j=1

mj(vi − vj) · ∇Wij =N∑j=1

mj ij · ∇Wij. (7.10)

177


X

Y

3.4 3.6 3.8 4 4.22.2

2.4

2.6

2.8

3

Paired particles

Hourglass Shape

Figure 7.5: Zero-energy mode of SPH simulation with cubic spline kernel.

dvαidt

= −∑j

mj

(piρ2i

+ fracpjρ2j

)∂Wij

∂xαi

+∑j

mj

(µiϵ

αβi

ρ2i+

µjϵαβj

ρ2j

)∂Wij

∂xβi

(7.11)

Here

ϵαβi =N∑j=1

mj

ρjvβji

∂Wij

∂xαi

+N∑j=1

mj

ρjvαji

∂Wij

∂xβi

−

(2

3

N∑j=1

mj

ρjvji · ∇iWijδ

αβ

)(7.12)

is the strain rate tensor. An artificial equation of state

p = B

[(ρ

ρ0

)γ

− 1

](7.13)

is used and no energy equation is needed. Here γ = 7 is used for water; B is a

constant that makes the flow weakly compressible and ca > 10cp; ca is the acoustic

wave speed; and cp is the wave phase speed.

In the overlap region of the Eulerian simulation and the SPH simulation, the

particles are moving with the local velocity of the Eulerian simulation. When a

particle moves out of the SPH domain, it is deleted. When a particle moves from the

178


area within ∆p distance from the boundary of SPH domain into the inside area, an

extra particle is added at the location xa = xi − v∆t. Here xi is the location of the

particle that moves in and v is its velocity.

A linear sinusoidal wave with initial steepness ak = 0.05 is simulated with the

coupled method. The simulation setup is sketched in figure 7.6(a). The simulation

results at different time are plotted in figures 7.6(b-e). The wave surfaces of the SPH

simulation match very well with those of the Eulerian simulation. The underwater

velocity field also matches very well.

For problems with wave breaking, wave slope at the overlap region is large and the

compressibility of SPH would incur hydraulic jump on the surface. Seamless matching

under this kind of condition is challenging. Further development of the coupling will

be our future work.

179


(a)

X

Y

2 3 4

2

3

4-0.035 -0.021 -0.007 0.007 0.021 0.035

u/c:

X

Y

2 3 4

2

3

4

X

Y

2 3 4

2

3

4-0.035 -0.021 -0.007 0.007 0.021 0.035

u/c:

X

Y

2 3 4

2

3

4

(b) (c)

X

Y

2 3 4

2

3

4-0.035 -0.021 -0.007 0.007 0.021 0.035

u/c:

X

Y

2 3 4

2

3

4

X

Y

2 3 4

2

3

4-0.035 -0.021 -0.007 0.007 0.021 0.035

u/c:

X

Y

2 3 4

2

3

4

(d) (e)

Figure 7.6: Coupled LS/SPH simulation of a two dimensional linear wave with initialwave slope ak = 0.05: (a) numerical setup; and horizontal velocity contours at (b)t = 4.25T , (c) t = 4.5T , (d) t = 4.75T , (e) t = 5T . Here T is the wave period.

180

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Vita

Yi Liu was born in Jingmen, Hubei, China in 1982.

He received B. S. degree in mechanical engineering and

B. Eng. degree in electronic engineering from Univer-

sity of Science and Technology of China in 2003. From

2003 to 2006, he was a graduated student in Beijing

Institute of Applied Physics and Computational Math-

ematics and earned M. S. degree in fluid mechanics. After that he enrolled in the

civil engineering Ph.D. program at Johns Hopkins University. In 2009, he earned M.

S. E. degree in mechanical engineering from Johns Hopkins University.

201

numerical study of strong free surface flow and breaking waves

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