A numerical study of wave interaction with porous media

A NUMERICAL STUDY OF WAVE INTERACTION
WITH POROUS MEDIA












S. A. SUVINI ANUJA KARUNARATHNA

NATIONAL UNIVERSITY OF SINGAPORE

2005



A NUMERICAL STUDY OF WAVE INTERACTION WITH
POROUS MEDIA








S. A. SUVINI ANUJA KARUNARATHNA
(B.Sc.Eng. (Hons.), University of Peradeniya, Sri Lanka)










A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY

DEPARTMENT OF CIVIL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2005









Dedicated with love
to
my parents
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor, Assoc. Prof. Lin
Pengzhi for assisting me in my Ph.D. study at NUS. His valuable advice, constructive
feedback, suggestions and guidance was very much helpful for my research work. The
financial support during the final semester of my graduate study is also greatly
acknowledged. Without his assistance completion of this thesis would not have been
possible.
I would also like to thank the chairman of my thesis advisory committee, Prof.
Cheong Hin Fatt for valuable advice and suggestions on my research and thesis. I am
also thankful for the professors who examined my thesis. Their supportive comments
and suggestions were very much useful for the further improvement of the thesis.
Support from my former co-supervisor, Dr. Guo Junke John is also appreciated.
Thanks are also due to all the teaching faculty members from whom I have taken
courses during the graduate study.
The financial assistance from the National University of Singapore through the

NUS research scholarship is deeply appreciated. The support provided by Mr. Krishna
Sanmugam and Mrs. Norela Bte Buang in the use of computer facilities is also
appreciated.
Thanks are given to all the friends from the hydraulics group: Dr. Liu, Dr. Gu,
ChuanJiang, DongChao, Zhang Dan, XioHui, Didi, DongMing, WenYu, HaoLiang,
QuangHong and others for the support and friendship. Thanks are also extended to
Dulakshi, XinYing, Doan and Atique from the hydro-informatics group for sharing
enjoyable times. Warm appreciation goes to Pradeep and Sita, who were my constant

i
companions throughout my university life for the friendship, endless support and
encouragement. I also thank Dammi, Dumindu, Buddi, OG, and other Sri Lankan
friends for the support. The friendship shared with all these friends made my stay in
Singapore a pleasant period of time.
Special thanks are due to my friends in Sri Lanka, Dil and Prigie for loving
friendship, encouragement and endless support and care extended to my family during
my stay in Singapore.
I would like to express my deepest gratitude and love to my parents, Mr. S.A.
Karunarathna and Mrs. R.A.S. Ranasinghe for love, affection, endless support in my
life and continuous encouragement for my studies. I also like to thank my sister,
Neranja and brother-in-law, Ajith for love and support. I would like to remind my
grandmother, who always wished my success, with gratitude and love. I also like to
thank my mother-in-law, Mrs. E.A. Seelawathie for care and affection. Finally, I
would like to thank my best friend and husband, Dammika for love, care,
understanding and constant encouragement for my studies.

ii
TABLE OF CONTENTS
Page
Acknowledgements i

Table of contents iii
Summary viii
List of tables x
List of figures xi
Nomenclature xv

Chapter 1 Introduction 1
1.1 Wave interaction with porous media 1
1.2 Wave damping over porous seabeds 2
1.3 Wave transformation by porous breakwaters 4
1.4 Modeling of wave and porous media interaction 5
1.5 Objectives of the study 7
1.6 Outline of the thesis 11

Chapter 2 Literature review 13
2.1 Fundamental governing equations 13
2.2 Turbulent boundary layer 15
2.2.1 Turbulent boundary layer over an impermeable wall 15
2.2.2 Influence of injection and suction on turbulent
boundary layer 21
2.2.3 Effects of transpiration on velocity profiles of
turbulent boundary layers 24
2.3 Porous flow models 29
2.3.1 Stationary flows 30
2.3.2 Non-stationary flows 34

iii
Page
2.4 Wave damping over porous seabeds 38
2.4.1 Wave damping over rigid porous seabeds 38

2.4.2 Wave damping over non-rigid porous seabeds 39
2.5 Wave interaction with porous breakwaters 41
2.5.1 Interaction of periodic waves with porous breakwaters 41
2.5.2 Interaction of solitary waves with porous breakwaters 42
2.6 Conclusions 43

Chapter 3 Turbulent boundary layer flows above a
porous surface subject to flow injection 45
3.1 Introduction 45
3.2 Derivation of analytical expressions 46
3.2.1 Mathematical formulation 46
3.2.1.1 Governing equations 46
3.2.1.2 Mathematical simplification 47
3.2.1.3 Analytical solution 49
3.2.2 Determination of and 54
+
b
u
b
y
3.2.3 Interpretation of analytical solution 56
3.2.3.1 Final solution 56
3.2.3.2 Asymptotic solution for low injection rate 58
3.2.4 Comparison with experimental data 60
3.2.4.1 Review on experiments 60
3.2.4.2 Methodology 63
3.2.4.3 Velocity profiles 64
3.2.4.4 Bed shear stress 64
3.3 Conclusions 65

iv
Page
Chapter 4 Description of the numerical model 74
4.1 Introduction 74
4.2 Governing equations 75
4.2.1 Flow motion outside of porous media 75
4.2.2 Flow motion inside of porous media 77
4.2.2.1 Spatially averaging of Navier-Stokes equations 77
4.2.2.2 Derivation of forces 79
4.2.2.3 Final governing equations 82
4.3 Initial and boundary conditions 83
4.3.1 Boundary conditions on the free surface 84
4.3.2 Boundary conditions on the rigid boundary 85
4.3.3 Boundary conditions on the permeable boundary 86
4.4 Numerical computation 88
4.4.1 Computational domain 88
4.4.2 Two-step projection method 90
4.4.3 Spatial discretization 91
4.4.4 Discretization of
ε
−
k
equations 94
4.4.5 Volume of fluid method 97
4.4.6 Computational cycle 99
4.4.7 Stability and error analysis 100
4.5 Model calibration 101
4.6 Conclusions 102

Chapter 5 Wave damping over rigid porous seabeds 103

5.1 Literature review 103
5.2 Motivation 107

v
Page
5.3 Model comparison with theories 109
5.3.1 Liu and Dalrymple’s theory (1984) 110
5.3.2 Gu and Wang’s theory (1991) 111
5.3.3 Numerical simulations 113
5.3.3.1 Setup of numerical experiments 113
5.3.3.2 Wave generation 115
5.3.3.3 Boundary conditions 116
5.3.3.4 Results and discussion 117
5.4 Model verification by experimental data 120
5.4.1 Savage’s (1953) experimental data 120
5.4.2 Sawaragi and Deguchi’s (1992) experimental data 126
5.5 Further discussion of other wave and porous bed parameters 128
5.5.1 Influence of the wavelength on wave damping 129
5.5.2 Influence of seabed thickness on wave damping 131
5.6 Conclusions 132

Chapter 6 Solitary wave interaction with porous breakwaters 134
6.1 Literature review 134
6.1.1 Interaction of periodic waves with porous breakwaters 134
6.1.2 Interaction of solitary waves with impermeable
structures 139
6.1.3 Interaction of solitary waves with porous breakwaters 141
6.2 Motivation 143
6.3 Model testing for solitary wave propagation on constant
water depth 145

6.4 Model validation 147
6.4.1 Model validation against various theories 148

vi
Page
6.4.1.1 Madsen’s theory (1974) 148
6.4.1.2 Vidal et al.’s theory (1988) 151
6.4.2 Model validation against experiments 155
6.4.2.1 Vidal et al.’s (1988) experimental data 155
6.4.2.2 Lynett et al.’s (2000) experimental data 157
6.5 Numerical results and discussions 159
6.5.1 Setup of numerical experiments 160
6.5.2 Tabulated results of RTD coefficients for different
combinations of
ha
b
and
hd
50
ratios 161
6.5.3 Analysis of solitary wave transformation 166
6.5.3.1 Wave reflection and transmission 166
6.5.3.2 Wave energy dissipation 169
6.5.4 Effects of depth of submergence 171
6.5.5 Effects of porosity 172
6.5.6 Scale effects 174
6.6 Conclusions 177

Chapter 7 Conclusions and recommendations 180
7.1 Conclusions 180

7.2 Recommendations for future research 184
References 186
Appendix A Calibration of the roughness function 202
Appendix B Velocity profile in a turbulent boundary layer
above a porous surface subject to flow suction 204
Appendix C Integral energy equation 211
List of publications 215


vii
SUMMARY
This thesis presents a study of wave interaction with porous seabeds and porous
breakwaters by employing a two-dimensional numerical model extended from an
earlier developed model named COBRAS (Liu et al., 2000). In the present numerical
model the flow outside the porous media is described by the Reynolds Averaged
Navier-Stokes (RANS) equations in which the turbulence is modeled by the
ε
−
k

model. The flow inside the porous media is described by the spatially averaged Navier-
Stokes equations while the effect of porous media is treated by additional resistance
forces.
In this study, a theoretical expression to describe the turbulent boundary layers
subject to flow injection is derived by solving momentum equations and turbulent
kinetic energy equation. The advection of turbulent kinetic energy is retained during
the derivation whereas the earlier studies have neglected it. The new solution reduces
to the universal logarithmic law in case of no flow injection. The leading order terms
in the series expansion of this solution for the small injection are equivalent to the
modified log law. The new solution can provide more accurate prediction of bed shear

stress for a wide range of flow injection rate, fluid type (e.g., from air to water) and
surface roughness.
A new porous flow model is derived for the flow motion inside the porous
media by spatially averaging Navier-Stokes equations while the effect of the porous
media is treated by additional inertial and drag forces. These forces are modeled by
assuming uniform equivalent spherical particles within the porous media. Unlike the
earlier porous flow models, Reynolds number (
Re
) is explicitly formulated in this
porous flow model and the model is found to be valid for wide range of porous flows.

viii
The above mentioned theoretical models are incorporated into the COBRAS
numerical model. The theoretical expression for velocity profile subject to flow
injection is used to calculate shear velocities at permeable boundaries to describe
boundary conditions for turbulent kinetic energy and its dissipation rate. The porous
flow model is used to describe the flow through porous media. The revised numerical
model is first validated for the study of wave damping over porous seabeds. The
effects of wave nonlinearity, wavelength and seabed thickness on wave damping are
studied by using the numerical model. It is found that in shallow water depths, the
wave nonlinearity has a distinct effect on wave damping whereas in deep water depths
wave damping is less affected by wave nonlinearity. The results also reveal that both
short waves and long waves have smaller wave damping when compared to damping
of waves in intermediate water depths.
The present numerical model is validated for both long wave and solitary wave
interaction with porous breakwaters. The model is employed to study solitary wave
interaction with fully emerged rectangular porous breakwaters. The reflection,
transmission and energy dissipation (RTD) coefficients when solitary waves interact
with porous breakwaters that cover the full range of breakwater lengths and porous
particle diameters are calculated and tabulated in convenient reference for engineering

design of porous breakwaters. The effects of depth of submergence and porosity on
RTD coefficients are discussed. The scale effects associated with small scale and large
scale model tests are also studied. The results reveal that when both Froude number
and Reynolds number criteria are satisfied the small scale models are able to predict
reflection and transmission coefficients accurately.


ix
LIST OF TABLES

Page
Table 3.1 Demonstration of the significance of higher order terms by
comparing the predicted values of velocities by using
universal log law, modified log law and new theory
59
Table 3.2 Experimental flow conditions of McQuaid (1967) and
comparison of the experimental shear velocities with the
predicted values of shear velocities by using the new theory
and the modified logarithmic law
70
Table 3.3 Experimental flow conditions of Simpson (1967) and
comparison of the experimental shear velocities with the
predicted values of shear velocities by using the new theory
and the modified logarithmic law
71
Table 3.4 Experimental flow conditions of Cheng and Chiew (1998b)
and comparison of the experimental shear velocities with the
predicted values of shear velocities by using the new theory
and the modified logarithmic law
72

Table 3.5 Comparison of the experimental shear velocities with the
predicted values of shear velocities by using the new theory
and the modified logarithmic law for an independent
experiment conducted by Julien et al. (1971)
72
Table 3.6 Comparison of percentage of sum of square errors between the
predicted results from the theories and McQuaid’s (1967)
experimental data in Fig. 3.5
73
Table 3.7 Comparison of percentage of sum of square errors between the
predicted results from the theories and Simpson’s (1967)
experimental data in Fig. 3.6
73
Table 3.8
Comparison of percentage sum of square errors between the
predicted results from the theories and Cheng and Chiew’s
(1998b) experimental data in Fig. 3.7
73
Table 5.1
Comparison of the numerical wave damping results to the
experimental and theoretical wave damping results for waves
with different wave heights
124
Table 6.1
Reflection coefficient for different combinations of
R
K
ha
b

and
hd
50
for
1.0=hH
and
5.0
=
n

163
Table 6.2
Transmission coefficient for different combinations of
T
K
ha
b
and
hd
50
for
1.0=hH
and
5.0
=
n

164
Table 6.3
Dissipation coefficient for different combinations of

D
K
ha
b

and
hd
50
for
1.0=hH
and
5.0
=
n

165


x
LIST OF FIGURES

Page
Fig. 2.1 Semi-logarithmic and linear plots of mean velocity distribution
across a turbulent boundary layer with zero pressure gradient –
the figure is from Cebeci and Smith (1974)
16
Fig. 2.2 Roughness function for impermeable walls where data is from
Nikuradse (1933)
20
Fig. 3.1 Illustration of flow in the inner region of a turbulent boundary

layer above a horizontal permeable bed subject to injection
47
Fig. 3.2 Graphical representation of four general explicit solutions for
horizontal velocity when , and
1=
+
s
v 5=
+
b
u 1.0=
+
b
y
53
Fig. 3.3 Graphical representation of four specific explicit solutions for
horizontal velocity after defining the deflection point as the
reference point when and
1=
+
s
v 15.0=
+
b
y
55
Fig. 3.4 Velocity profiles under zero (conventional logarithmic law),
low, critical and high injection velocity
58
Fig. 3.5 Comparison of the velocity profiles with McQuaid’s (1967)

experimental measurements
67
Fig. 3.6 Comparison of the velocity profiles with Simpson’s (1967)
experimental measurements: Date, Traverse Index - (a) 122766,
M=1 (b) 122366, M=1 (c) 122066, M=4 and (d) 121966, M=1
68
Fig. 3.7 Comparisons of the velocity profiles with Cheng and Chiew’s
(1998b) experimental measurements: Run (a) M8-104 (b) M23-
400 (c) M22-600 and (d) E3-100
69
Fig. 4.1 The sketch of a rectangular control volume composed of
uniform spherical particles
78
Fig. 4.2
Drag force coefficient for a single smooth sphere as a
function of Reynolds number
Ds
C
Re
81
Fig. 4.3 Staggered grid system in the numerical model 89
Fig. 5.1 Definition sketch of wave damping over a porous seabed 109
Fig. 5.2
(a) The mesh arrangement from
mx 200
=
to
mx 240
=
within the

computational domain (b) The setup of numerical experiments
114
Fig. 5.3 Illustration of the application of universal logarithmic law to
determine shear velocity at the rigid boundary
116
Fig. 5.4 Illustration of the application of the derived new theory for
turbulent boundary layers subject to flow injection to determine
shear velocity at the permeable boundary
117

xi

Page
Fig. 5.5 Grid independence test: dotted line – coarse mesh, solid line –
fine mesh
118
Fig. 5.6 Comparison of the wave damping rates predicted from the
numerical model, Liu and Dalrymple’s theory (1984) and Gu
and Wang’s theory (1991)
120
Fig. 5.7
Simulation results of free surface profile at
Tt 25
=
for the Run
No. 1 of Savage’s (1953) experiments
122
Fig. 5.8 Comparison of numerical results for wave damping along the
porous bed with the corresponding experimental results (Run
No. 1 of Savage, 1953) and theoretical results (Liu and

Dalrymple, 1984; Gu and Wang, 1991)
122
Fig. 5.9 Variation of wave damping rates with wave nonlinearity for
waves with in different water depths
sT 27.1=
125
Fig. 5.10 Comparison of numerical results for the wave damping along
the porous seabed to the experimental (Sawaragi and Deguchi,
1992), theoretical (Liu and Dalrymple, 1984; Gu and Wang,
1991) and numerical (Chang, 2004) results for case J-2
127
Fig. 5.11 Comparison of numerical results for the wave damping along
the porous seabed to the experimental (Sawaragi and Deguchi,
1992) and theoretical (Liu and Dalrymple, 1984; Gu and Wang,
1991) results for case J-6
128
Fig. 5.12 Variation of (a) dimensional and (b) non-dimensional wave
damping rate with for different porous beds
hk
w
130
Fig. 5.13 Variation of wave damping rate with the seabed thickness for
different porous beds
131
Fig. 6.1 Definition sketch of solitary wave interaction with a porous
breakwater
144
Fig. 6.2 Comparison of numerical and analytical free surface profiles for
solitary wave propagation in a constant water depth at different
times

()
21
hgt
=13.3 (A), 44.6 (B), 76.0 (C): solid line –
numerical free surface profile, dotted line – analytical free
surface profile
147
Fig. 6.3 Standing wave envelope for long wave interaction with the
porous breakwater with
06.0
50
=
hd

150
Fig. 6.4 Comparison of the reflection and transmission coefficients for
linear long wave interaction with rectangular porous
breakwaters with different of
hd
50
between the present
numerical results and the results from Madsen’s theory (1974)
151





xii

Page
Fig. 6.5 Time histories of free surface profiles recorded in gauge 1 and
gauge 2 for solitary wave interaction with porous breakwater
with
0.2=ha
b
and
05.0
50
=
hd

153
Fig. 6.6 Comparison of the variation of reflection and transmission
coefficients ( and ) with the increase of between the
present numerical results and the theoretical results (Vidal et.
al., 1988); (a) small porous size and (b) large porous size:
dashed dotted line – numerical , dotted line – numerical ,
solid line – theoretical , dashed line – theoretical
R
K
T
K
ak
w
R
K
T
K
R

K
T
K
154
Fig. 6.7
Comparison of reflection and transmission coefficients ( and
) of solitary wave interaction with porous breakwaters:
diamonds – experimental , circles - experimental , solid
lines - present numerical results, dashed dotted lines –
theoretical results from Silva et al. (2000), dashed lines -
numerical results from Liu and Wen (1997), and dotted lines –
theoretical results from Vidal et al. (1988)
R
K
T
K
R
K
T
K
157
Fig. 6.8
Comparison of reflection and transmission coefficients ( and
) of solitary wave interaction with porous breakwaters:
diamonds – experimental
R
, circles – experimental
T
, solid
lines – present numerical results, and dashed lines – numerical

results from Lynett et al. (2000)
R
K
T
K
K K
159
Fig. 6.9 Setup of the numerical experiments 161
Fig. 6.10
Variation of reflection coefficient (a) as the function of
ha
b

for different
hd
50
and (b) as the function of
hd
50
for different
ha
b

168
Fig. 6.11
Variation of transmission coefficient (a) as the function of
ha
b

for different

hd
50
and (b) as the function of
hd
50
for different
ha
b

169
Fig. 6.12
Variation of dissipation coefficient (a) as the function of
ha
b

for different
hd
50
and (b) as the function of
hd
50
for different
ha
b

170
Fig. 6.13 Effects of submergence on reflection, transmission and energy
dissipation coefficients for
0.2
=

ha
b
and
02.0
50
=
hd

172
Fig. 6.14 Effects of porosity on reflection, transmission and energy
dissipation coefficients for
0.2
=
ha
b
and
02.0
50
=
hd

174
Fig. 6.15 Comparison of RTD coefficients for different scale factors for
fixed
0.2=ha
b

177

xiii

Page
Fig. A.1 Least square fit of (3.23) with the data determined from the
experiments
203
Fig. B.1 Illustration of velocity profile of a turbulent boundary layer
above a horizontal permeable bed subject to flow suction
205
Fig. B.2 Graphical representation of four explicit solutions for horizontal
velocity
208


xiv
NOMENCLATURE
A
area of source region
a
shear coefficient
0
a
a coefficient
1
a
an integration constant
b
a
length of the breakwater
p
a

conductivity or the coefficient in the linear friction term
w
a
wave amplitude
wi
a
wave amplitude in th wave mode
i
w
a
averaged wave amplitude
B
roughness function
1
B
independent component in the roughness function
+
s
v
2
B
independent component in the roughness function
+
s
k
s
B
roughness function under injection
sc
B

roughness function under seepage condition (Cheng and Chiew, 1998b)
0
b
a coefficient
b
b
height of the breakwater
p
b
coefficient in the nonlinear friction term

xv
C
wave celerity
1
C
, , empirical coefficients
2
C
3
C
ε
1
C
, coefficients
ε
2
C
D
C

drag force coefficient for porous media
Ds
C
drag force coefficient for a single sphere
b
C
a parameter
d
C
empirical coefficient
f
C
a parameter
,
1
c
, , coefficients
2
c
3
c
4
c
A
c
added mass term
,
1D
c
coefficients for linear and nonlinear resistances

2D
c
n
c
empirical coefficient
p
c
coefficient for acceleration term
∞
c
a coefficient associated with Vidal et al.’s theory
D
energy dissipation
DNS
direct numerical simulation
D
ˆ
characteristic length
d
seabed thickness
dx
length of the control volume for the porous media

xvi
dy
height of the control volume for the porous media
dz
width of the control volume for the porous media
15
d

diameter of the particles that exceed of the sample
%15
50
d
mean particle diameter
90
d
diameter of the particles that exceed of the sample
%90
e
d
diameter of a sphere
p
d
particle diameter
s
E
roughness coefficient
I
EF
total incident wave energy flux
R
EF
total reflected wave energy flux
T
EF
total transmitted wave energy flux
kin
E
kinetic energy

pot
E
potential energy
F
volume of fluid function
E
F
energy flux
Fr
Froude number
Di
F
drag resistance within the fluid volume in
−
i
direction
Dsi
F
drag resistance for a single sphere in
−
i
direction
Ii
F
inertia resistance within the fluid volume in
−
i
direction

xvii

FkYFkX,
advection of turbulent kinetic energy
YFXF
ε
ε
,
advection of turbulent kinetic energy dissipation rate
f
linearized friction coefficient
0
f
linearized resistance coefficient
Di
f
drag resistance force in
−
i
direction
Ii
f
inertia resistance force in
−
i
direction
M
f
linearized friction coefficient in Madsen’s (1974) theory
i
f
resistance force in

−
i
direction
l
f
a complex variable
p
f
friction coefficient related to Reynolds number
Di
f
spatially averaged drag resistance force in
−
i
direction
Ii
f
spatially averaged inertia resistance force in
−
i
direction
i
f
spatially averaged resistance force in
−
i
direction
g
gravitational acceleration
i

g
gravitational acceleration in
−
i
direction
H
wave height
R
H
reflected wave height
T
H
transmitted wave height
max
H
maximum wave height

xviii
min
H
minimum wave height
x
H
wave height at a distance
x

h
water depth
0
h

height over a selected datum
I
hydraulic gradient
K
specific permeability
KC
Keulegan-Carpenter number
D
K
dissipation coefficient
R
K
reflection coefficient
T
K
transmission coefficient
k
kinetic energy
i
k
imaginary part of complex wave number
im
k
measured wave damping rate
p
k
permeability
r
k
real part of complex wave number

s
k
surface roughness
+
s
k
dimensionless surface roughness
⎟
⎠
⎞
⎜
⎝
⎛
=
ν
s
ku
*

w
k
wave number
L
Prandtl’s mixing length

xix
l
length scale
MAC
marker and cell method

m
a parameter
N
number of spheres
n
porosity
n
~
outward normal direction
P
production of turbulent kinetic energy
p
pressure
D
p
dynamic pressure
S
p
static pressure
*
p
instantaneous pressure
p
′
fluctuation of pressure
0
p
effective pressure
0
p

averaged effective pressure
0
p
′′
fluctuation of effective pressure
R
permeability parameter
RANS
Reynolds Averaged Navier Stokes equation
RTD
Reflection, transmission, energy dissipation coefficients
w
R
wave variable

xx
Re
Reynolds number
h
R
hydraulic radius
w
S
wet surface area
s
mass source function
T
wave/oscillation period
TD
total energy dissipation

eff
T
effective wave period
t
time
t
~
tangential direction
i
U
, velocity within the porous media in
j
U
−
i
,
−
j
direction
max
U
maximum particle velocity
max
U

maximum filter velocity
i
U
′′
, fluctuation of velocity in

j
U
′′
−
i
,
−
j
direction
U
discharge velocity in horizontal direction
c
U
characteristic velocity
i
U
,
j
U
spatially averaged velocity in
−
i
,
−
j
direction
U
ˆ
characteristic velocity
U

~
velocity defined on the rigid surface
u
velocity in the horizontal direction

xxi
+
u
dimensionless horizontal velocity
⎟
⎠
⎞
⎜
⎝
⎛
=
*
u
u

1
u
free stream velocity
a
u
horizontal velocity at a distance
a
y
b
u

horizontal velocity at the reference point B
,
i
u
mean velocity in
j
u
−
i
,
−
j
directions
s
u
slip velocity
+
a
u
dimensionless horizontal velocity at a distance
a
y
⎟
⎠
⎞
⎜
⎝
⎛
=
*

u
u
a

+
b
u
dimensionless horizontal velocity at B
⎟
⎠
⎞
⎜
⎝
⎛
=
*
u
u
b

,
i
u
′
fluctuation of velocity in
j
u
′
−
i

,
−
j
directions
,
*
i
u
instantaneous velocity in
*
j
u
−
i
,
−
j
directions
*
u
shear velocity
i
u
~
intermediate velocity
V
control volume
VOF
volume of fluid method
V

discharge velocity in vertical direction
f
V
volume of fluid
a
V
volume of air
VISk
viscous diffusion of turbulent kinetic energy

xxii