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 and Weber
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
vii

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
viii

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
ix

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
x

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 method
and coupled level-set/volume-of-fluid method for the stretching fluid
disk problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Profiling results of the coupled level-set/volume-of-fluid method code
using Craypat on Cray-XT5 supercomputer of the High Performance
Computing 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 steep
waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Friction velocity and drag coefficient for case II-1 during the breaking
process at different time. . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1 Values of α, θ, β, and γ for c/u∗ = 2 at different ak. Values of γ based
on the parameterization of Ref. [1], γ = 0.17(Uλ/2/c−1)2
(ωρa/ρw), are
listed 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. . . . . . . . . . . . . 7
2.2 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 with
ϕ = 0. The contour interval is 0.2. . . . . . . . . . . . . . . . . . . . . 16
2.3 Isosurface of ϕ = 0.1 of the level set function of a three dimensional
ellipsoid: (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 thick
black line is the interface. The shadowed area is the fluid area enclosed
by 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 and m3∆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 by
horizontal 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. . . . . . . 26
2.9 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 the
interface segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Instantaneous interfaces of two dimensional fluid disk in a vortical flow
field 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
2.11 Sketch of the coupled air–water simulation. . . . . . . . . . . . . . . . 30
2.12 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 shear
stress in ghost fluid method. . . . . . . . . . . . . . . . . . . . . . . . 37
2.14 Seven points stencil of the discretization of pressure poisson equation. 38
2.15 Illustration of the domain decomposition in current code. . . . . . . . 42
2.16 Result of speedup test. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.17 Instantaneous interfaces of Zaleski problem calculated with (a) split-
ting scheme and (b) ENO scheme after one rotation. The dashed lines
are the theoretical results. . . . . . . . . . . . . . . . . . . . . . . . . 45
2.18 Schematic of the static air bubble simulated. . . . . . . . . . . . . . . 46
2.19 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 pressure
distribution along X = 2 in (a,b,c) at the middle plane of bubble. . . 47
2.20 Schematic of the two-layer Couette flow. . . . . . . . . . . . . . . . . 48
2.21 Velocity profiles and error percentages of the two layer Couette flow
simulated with (a) Continuous surface force method and (b) Ghost
fluid method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.22 The air–water interface of a two dimensional air bubble with radius
1/3cm 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 bubble
bursting 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 = 0.1
and its comparison with linear theory. . . . . . . . . . . . . . . . . . . 53
2.25 Amplitude evolution of a gravity wave with initial wave slope ak = 0.1
with 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. . . . . . . . . 59
3.2 See next page for caption. . . . . . . . . . . . . . . . . . . . . . . . . 62
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
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
3.4 Schematic of a nonlinear wave and the quantities used for definition of
skewness and asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 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-breaking
waves 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 spilling
breaker with (ak)0 = 0.35. . . . . . . . . . . . . . . . . . . . . . . . . 72
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
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 is
achieved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 different
initial 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 with
Tian’s [2] model (dashed line) and data. . . . . . . . . . . . . . . . . 78
3.14 Breaking length scale versus wave steepness S and comparison with
Tian’s [2] model (dashed line) and data. . . . . . . . . . . . . . . . . 79
3.15 Normalized breaking length scale versus normalized breaking time scale. 80
3.16 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. . . . . . . . 85
4.2 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. . . . . . . . 89
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. . . . . . . . . . . . . . . . . . . . . . . 91
4.4 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) for
cases 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 prescribed
waves 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
xv

LIST OF FIGURES
4.8 Spanwise averaged profiles of wind wave around breaking for case II-1:
————, near breaking; – – – – , incipient breaking; ◦, [4]. The error
bar 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 side
for 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-1
during 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. . 106
4.14 The spanwise-averaged pressure, streamlines and vorticity at t = 1.1T
of case II-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.15 Sketch for pressure distribution over water wave before (a) and after
(b) breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.16 The form drag evolution with time for wind over breaking waves in
case II-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.17 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 with
horizontal velocity c. Here c is the phase speed of wave. . . . . . . . . 113
4.18 Instantaneous streamwise velocity u normalized by wave phase speed
c on a vertical cut for case II-1 with (ak)0 = 0.35. The time step is
0.145T. T is the linear wave period. . . . . . . . . . . . . . . . . . . . 114
4.19 Spanwise averaged underwater velocity vectors for breaking wave case
III-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 case
II-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: weak
turbulence regime; region 1: surface tension dominated regime; region
2: very strong turbulence regime; region 3: gravity dominated regime.
The region between the two dash lines represents the marginal breaking
region obtained by Brochini & Peregrine (2001). [5] . . . . . . . . . . 128
xvi

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 elevation
for the weak turbulence case of (Fr2
= 0.8, We = 40). The solid line
denotes the dispersion relationship (equation 5.5). The dash-dot line
denotes the characteristic frequency of each wavenumber component
(equation 5.6). The dashed line denotes the characteristic frequency
obtained 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 = 32
but different Weber numbers: · · · · · · We = ∞; – · – · – We = 40;
————We = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.10 Vertical velocity fluctuations of cases with (a) the same Weber number
We = ∞ but different Froude numbers, (b) the same Froude number
Fr2
= 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 cases
with (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 cases
with (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 of
vertical turbulent normal stress by the vertical turbulent velocity. . . 144
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
plunges downward 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 vertical cut is extracted for analysis in figure
5.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.16 Energy dissipation and turbulent Reynolds stress transport associated
with 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 and
pressure (normalized by ρau2
∗) distribution on the surface of broadband
waves (cm/u∗ = 12.3). The air domain is lifted up for better visualization.152
6.2 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 ( ) of
Ref. [1]; (b) pressure profiles over waves with different steepnesses; (c)
pressure profiles over waves with different wave ages. The wind and
wave are from left to right. The wave phase is shown in the sketch at
the 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 in
Ref. 7; , numerical results of Refs. 9 and 11; ▹, numerical results of
Ref. 13; ×, current results for monochromatic waves. The lines are the
current 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 flow
condition 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 wave
crest, and (b) when a wave trough arrives at the front surface of the
object. the inflow is in the x-direction. the vertical planes show the
streamwise velocity contours. the velocity field inside the small black
window is enlarged and shown in figure 6.9. the pressure on the object
surface and the wave surface are shown. vortices are plotted with grey
color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.9 Enlarged streamwise velocity contours from figure 6.8: (a) above wave
crest when a crest arrives at the object; (b) above wave trough when
a trough arrives at the object. . . . . . . . . . . . . . . . . . . . . . . 167
7.1 Wave field generated by the turbulent wind with 10 meter hight speed
30m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.2 Evolution of the root-mean-square surface elevation with time. . . . . 174
7.3 One dimensional surface spectra of the wave field generated by wind. 174
7.4 Illustration of the coupled LS/SPH simulation . . . . . . . . . . . . . 176
7.5 Zero-energy mode of SPH simulation with cubic spline kernel. . . . . 178
7.6 Coupled LS/SPH simulation of a two dimensional linear wave with ini-
tial wave 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
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 FLOW
SIMULATION
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

SIMULATION
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

SIMULATION
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

SIMULATION
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 =
ϕn
i,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)
ˆϕi,j,k =
˜ϕn
i,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 =
ˆϕn
i,j,k + (∆t/∆z)( ˆGi,j,k−1/2 − ˆGi,j,k+1/2)
1 − (∆t/∆z)(wi,j,k+1/2 − wi,j,k−1/2)
. (2.10)
ϕn+1
i,j,k = ¯ϕ − ∆t
( ˜ϕi,j,k
∆x
(ui+1/2,j,k − ui−1/2,j,k)
+
ˆϕi,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 = ϕn
i,j,k +
∆x
2
(
1 − ui+1/2,j,k
∆t
∆x
)
ϕn
i+1,j,k − ϕn
i−1,j,k
∆x
, (2.12)
else
ϕi+1/2,j,k = ϕn
i+1,j,k −
∆x
2
(
1 + ui+1/2,j,k
∆t
∆x
)
ϕn
i+2,j,k − ϕn
i,j,k
∆x
. (2.13)
9

SIMULATION
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 = (ϕ2
xϕyy − 2ϕxϕyϕxy + ϕ2
yϕxx + ϕ2
xϕzz − 2ϕxϕzϕxz + ϕ2
zϕxx
+ϕ2
yϕzz − 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

SIMULATION
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

SIMULATION
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
√
ϕ2
0 + 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

SIMULATION
A global area-preserving reinitialization is proposed by Chang et al. [32]. A per-
turbed Hamilton-Jacobi equation
∂
∂t
ϕ + (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

SIMULATION
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+1
ijk =



ϕn
ijk − δt
δx
(
sign(ϕ0
ijk)|ϕn
ijk| − Dijk
)
if (i, j, k) ∈ Σ∆
ϕn
ijk − ∆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 =
ϕ0
ijk
max(
[
ϕ2
x + ϕ2
y + ϕ2
z
]1/2
ijk
, ϵ)
. (2.30)
Here ϵ is a small positive number to avoid singularity; and
Σ∆ = {(i, j, k): ϕ0
i,j,kϕ0
i−1,j,k < 0 or ϕ0
i,j,kϕ0
i+1,j,k < 0 or ϕ0
i,j,kϕ0
i,j−1,k < 0 or
ϕ0
i,j,kϕ0
i,j+1,k < 0 or ϕ0
i,j,kϕ0
i,j,k−1 < 0 or ϕ0
i,j,kϕ0
i,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

SIMULATION
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

SIMULATION
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

SIMULATION
X
-4
-2
0
2
4
Y
-4
-2
0
2
4
Z
-4
-2
0
2
4
Y
X
Z
X
-4
-2
0
2
4
Y
-4
-2
0
2
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 dimensional
ellipsoid: (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

SIMULATION
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

SIMULATION
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

SIMULATION
(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 is
the interface. The shadowed area is the fluid area enclosed by reconstructed line
segments.
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

SIMULATION
(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 and m3∆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

SIMULATION
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,
1
2 (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 = m2
1/ 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 = [m2
3(3m12 − m3) + m2
1(m1 − 3m3) + m2
2(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

SIMULATION
Here a = 1; p0 = c/3 − b2
/9; q0 = (bc − 3d)/6 − c3
/27; and θ = 1
3
arccos(q0/
√
−p3
0).
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 =
ϕn
i,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)
ˆFi,j,k =
˜Fn
i,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 =
ˆFn
i,j,k + (∆t/∆z)( ˆfi,j,k−1/2 − ˆfi,j,k+1/2)
1 − (∆t/∆z)(wi,j,k+1/2 − wi,j,k−1/2)
. (2.47)
Fn+1
i,j,k = ¯F − ∆t
( ˜Fi,j,k
∆x
(ui+1/2,j,k − ui−1/2,j,k)
+
ˆFi,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.
López 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

SIMULATION
(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−1/2
δ(ϕ)(ϕ − ai,j,k(x − xi) − bi,j,k(y − yj)
−ci,j,k(z − zk) − di,j,k)2
dxdy.
(2.50)
24

SIMULATION
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+1
j′=j−1
∑k′=k+1
k′=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

SIMULATION
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

SIMULATION
(a) (b) (c)
Figure 2.9: 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 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

SIMULATION
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

SIMULATION
X
Y
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
X
Y
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
(a) (b)
X
Y
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
X
Y
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
(c) (d)
Figure 2.10: Instantaneous interfaces of two dimensional fluid disk in a vortical flow
field 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

SIMULATION
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 and
coupled 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 as



ρ(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

SIMULATION
incompressible Navier-Stokes equations as
∂⃗uw
∂t
+ (⃗uw · ∇)⃗uw = −∇p
ρw
+ ∇2
⃗uw + ⃗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
+ ∇2
⃗ua + ⃗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
+ ∇ · (⃗u⃗u)) = −∇p + ∇ · (2µ(ϕ) ¯D) + ⃗g + σκδ(ϕ)⃗n.
∇ · ⃗u = 0
(2.59)
Here ¯D = 1
2
(∇⃗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 σ
ρ(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
= −∇ · (⃗u⃗u) − 1
ρ(ϕ)
∇p + 1
Re
1
ρ(ϕ)
∇ · (2µ(ϕ) ¯D)
+ 1
Fr2
⃗k + 1
We
1
ρ(ϕ)
κδ(ϕ)⃗n.
(2.60)
31

SIMULATION
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 =
ρwU2
L
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,
∇ ·
∇pp
ρ
= −
∇ · u∗
p
∆t
, (2.65)
up − u∗
p
∆t
= −
∇pp
ρ
; (2.66)
32

SIMULATION
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,
∇ ·
∇pc
ρ
= −
∇ · 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

SIMULATION
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

SIMULATION
counterparts
H(ϕ; ϵ) =
1
2
(1 +
ϕ
ϵ
+
1
π
sin(
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

SIMULATION
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 1
ρ
∂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

SIMULATION
(a) (b)
Figure 2.13: Schematics of the treatment of discontinuity for pressure and shear stress
in ghost fluid method.
With the location of the interface known, the pressure jump condition can be written
as
p+
0 − p−
0 = [p],
1
ρa
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 as
p−
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
∂2
p
∂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

SIMULATION
Figure 2.14: Seven points stencil of the discretization of pressure poisson equation.
(figure 2.13(b)). The stress jump condition now becomes
µa
ui+1 − u0
∆x+
− µw
u0 − ui
∆x−
= [µux], (2.80)
and the velocity on the interface u0 is obtained. The second order derivative becomes
∂2
u
∂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

SIMULATION
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 ˆp in Kˆp = pi
vi = Aˆp
αi = ρi/(vi, ¯r0)
s = ri − αivi
Kq = s ⇒ q
u = Aq
ωi = (u, s)/(u, u)
xi+1 = xi + αi ˆp + ωiq
39

SIMULATION
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 + AL
x + AU
x )D−1
(D + AL
y + AU
y )D−1
(D + AL
z + 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 As
i (can be AL
x , AU
x , AL
y ,
AU
y , AL
z , or AU
z )
As
i =









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
As
i is split into a tridiagonal matrix B and the dyadic of two vector u and v, and
(As
i )−1
= (B + uvT
)−1
= B−1
−
B−1
uvT
B−1
1 + vT B−1u
. (2.85)
40

SIMULATION
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 + bc
a1,1









(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
Tref Nref /TNP E
. (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

SIMULATION
Figure 2.15: Illustration of the domain decomposition in current code.
wall-clock simulation time, and TNP E
is 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

SIMULATION
NP
Speedup
10
0
10
1
10
2
10
3100
10
1
10
2
103
17M Grids
67M Grids
Ideal 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

SIMULATION
# 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 using
Craypat 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

SIMULATION
X
Y
0 20 40 60 80 100
0
20
40
60
80
100
X
Y
0 20 40 60 80 100
0
20
40
60
80
100
(a) (b)
Figure 2.17: Instantaneous interfaces of Zaleski problem calculated with (a) splitting
scheme and (b) ENO scheme after one rotation. The dashed lines are the theoretical
results.
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

SIMULATION
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

SIMULATION
(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 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 pressure distribution along X = 2 in
(a,b,c) at the middle plane of bubble.
47

SIMULATION
ρ2=1
µ2=0.1
ρ1
=1
µ1
=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

SIMULATION
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
CSF
Exact Solution
Error%
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
GFM
Exact Solution
Error%
(a) (b)
Figure 2.21: Velocity profiles and error percentages of the two layer Couette flow
simulated 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

SIMULATION
x (cm)
y(cm)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
x (cm)
y(cm)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
(a) (b)
x (cm)
y(cm)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
x (cm)
y(cm)
0 0.5 1 1.5 2
0
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 radius
1/3cm 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

SIMULATION
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

SIMULATION
(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

SIMULATION
t/T
a
0 5 10 15 20 25 30 35
0.02
0.04
0.06
0.08
0.1
Simulation
Theory
Figure 2.24: Amplitude evolution of a gravity wave with initial wave slope ak = 0.1
and its comparison with linear theory.
because of viscous dissipation. According to Lamb [60], the wave amplitude
a(t) = a(0) e− 1
Re
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

SIMULATION
t/T
a
0 1 2 3 4 5
0.02
0.04
0.06
0.08
0.1
64x64
128x128
256x256
Figure 2.25: Amplitude evolution of a gravity wave with initial wave slope ak = 0.1
with 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
ω We
ekz
cos(kx),
w(x) = ak3
ω We
ekz
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

SIMULATION
t/T
a
0 2 4 6 8
0.02
0.04
0.06
0.08
0.1
Figure 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

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 DIMENSIONAL
WAVE 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

WAVE BREAKING
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

WAVE BREAKING
X
Y
0 0.2 0.4 0.6 0.8 1
0
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
2
a2
k cos(2kx) +
3
8
a3
k2
cos(3kx). (3.1)
Here a is the wave amplitude and k is the wave number. The initial velocity compo-
59

WAVE BREAKING
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) − a2
kσ cosh2
(kh)(1 + 5 cosh2
(kh))/8/ sinh5
(kh),
F32 = 3
4
a2
kσ/ sinh4
(kh),
F33 = 3
64
a3
k2
σ(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

WAVE BREAKING
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

WAVE BREAKING
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

WAVE BREAKING
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

Numerical Study of Strong Free Surface Flow and Breaking Waves

Numerical Study of Strong Free Surface Flow and Breaking Waves

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