Free surface flows 111
u
u
t
u u
u
x
v u
i j
n
i j
n
i j
n
i j
n
i j
n
m
,
,
,
,
,
(
)
(
+
+
−
−
= −
−
−
1
1
1
2
∆
∆
i i j
n
i j
n
i j
n
i j
n
m
h
i
u
y
g
x
fv
u
,
,
,
,
)
+
−
−
+
−
−
−
+
+
1
1
1
1
2∆
∆
ζ ζ
ε
, ,
,
,
,
,
,
( )
j
n
i j
n
i j
n
i j
n
i j
n
i j
n
u
u
x
u
u
u
+
−
+
+
−
−
+
−
1
2
1
1
2
2
∆
( ( )
,
,
∆y
h
h
sxi j
mx
bxi j
mx
2





 +
−
τ
ρ
τ
ρ
(5.53)
v
v
t
u v
v
x
v v
i j
n
i j
n
m
i j
n
i j
n
i j
n
i
,
,
,
,
,
+
+
−
−
= −
−
(
) −
1
1
1
2
∆
∆
, ,
,
,
,
j
n
i j
n
i j
n
i j
n
m
h
i
v
y
g
y
fu
v
+
−
−
+
−
(
) −
−
−
+
1
1
1
1
2∆
∆
ζ ζ
ε
, ,
,
,
,
,
,
( )
j
n
i j
n
i j
n
i j
n
i j
n
i j
n
v
v
x
v
v
v
+
−
+
+
−
−
+
−
1
2
1
1
2
2
∆
( ( )
,
,
∆y
h
h
syi j
my
byi j
my
2





 +
−
τ
ρ
τ
ρ
(5.54)
where
u
u
u
u
u
m
i j
n
i j
n
i j
n
i j
n
=
+
+
+
+
+−
−
1
11
1
4
,
,
,
,
(5.55)
v
v
v
v
v
m
i j
n
i j
n
i j
n
i j
n
=
+
+
+
+
−+
−
,
,
,
,
1
11
1
4
(5.56)
h
h
h
mx
i j
n
i j
n
=
+ −
,
,
1
2
(5.57)
h
h
h
my
i j
n
i j
n
=
+ −
,
, 1
2
(5.58)
In the case of a wave-generated circulation, the components of the radiation stresses (S ij ) can be computed in the centre of the grid cells, using the
mean over-the-cell ‘i-j’ values:
u
u
i j
n
i j
n
,
,
+ +1
2
,
v
v
i j
n
i j
n
,
,
+ +1
2
, and ζ i j
n
, .
Example 5.4
This exercise investigates the flow pattern in a coastal region of constant depth, where a breakwater for a marina project has been built.
The solution domain is 350 m × 200 m discretized into a 70 × 40