Free surface flows 109
and the three components of the radiation stresses are approximated by the
expressions
S
T
hu
g
h d
dt
dt
xx
t
t T
ρ
ζ
ζ
=
+
−
+
∫
1
2
3
2
2
2
(5.47)
S
T
hv
g
h d
dt
dt
yy
t
t T
ρ
ζ
ζ
=
+
−
+
∫
1
2
3
2
2
2
(5.48)
S
S
T
huv
xy
yx
t
t T
ρ
ρ
=
=
+
∫
1
dt
(5.49)
where T is the wave period.
The wave-generated currents terms appear on the right-hand side of the
momentum equilibrium equations in the form
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
+ +
∂
∂
+
∂
∂
u
t
u
u
x
v
u
y
g x
fv
u
x
u
y
h
ζ
ε
2
2
2
2
+
−
−
∂
∂
+
∂
∂
τ
ρ
τ
ρ
ρ
sx
bx
xx
xy
h
h
h
S
x
S
y
1
(5.50)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
− +
∂
∂
+
∂
∂
v
t
u
v
x
v
v
y
g y
fu
v
x
v
y
h
ζ
ε
2
2
2
2
+
−
−
∂
∂
+
∂
∂
τ
ρ
τ
ρ
ρ
sy
by
yx
yy
h
h
h
S
x
S
y
1
(5.51)
5.3.4 Numerical solution scheme
The numerical solution of the governing equations is based on a simple
explicit and stable scheme, with minimal numerical errors, and easily programmable. The flow domain is discretized by means of a staggered grid
and the three components of the radiation stresses are approximated by the
expressions
S
T
hu
g
h d
dt
dt
xx
t
t T
ρ
ζ
ζ
=
+
−
+
∫
1
2
3
2
2
2
(5.47)
S
T
hv
g
h d
dt
dt
yy
t
t T
ρ
ζ
ζ
=
+
−
+
∫
1
2
3
2
2
2
(5.48)
S
S
T
huv
xy
yx
t
t T
ρ
ρ
=
=
+
∫
1
dt
(5.49)
where T is the wave period.
The wave-generated currents terms appear on the right-hand side of the
momentum equilibrium equations in the form
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
+ +
∂
∂
+
∂
∂
u
t
u
u
x
v
u
y
g x
fv
u
x
u
y
h
ζ
ε
2
2
2
2
+
−
−
∂
∂
+
∂
∂
τ
ρ
τ
ρ
ρ
sx
bx
xx
xy
h
h
h
S
x
S
y
1
(5.50)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
− +
∂
∂
+
∂
∂
v
t
u
v
x
v
v
y
g y
fu
v
x
v
y
h
ζ
ε
2
2
2
2
+
−
−
∂
∂
+
∂
∂
τ
ρ
τ
ρ
ρ
sy
by
yx
yy
h
h
h
S
x
S
y
1
(5.51)
5.3.4 Numerical solution scheme
The numerical solution of the governing equations is based on a simple
explicit and stable scheme, with minimal numerical errors, and easily programmable. The flow domain is discretized by means of a staggered grid