106 Computational Modelling in Hydraulic and Coastal Engineering
5.3.1 Governing equations
The mass conservation and momentum balance equations for a twodimensional system (Figure 5.11) can be written as follows:
∂
∂
+
∂
∂
+
∂
∂
=
ζ
t
hu
x
hv
y
( )
( ) 0
(5.38)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
+ +
∂
∂
+
∂
∂





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
sx
bx
h
h
(5.39)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
− +
∂
∂
+
∂
∂





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
sy
by
h
h
(5.40)
In these equations the water depth y(x,y,t) was replaced by the undisturbed
water depth h(x,y) under the assumption that ζ << h. Due to the complexity of Equations 5.38 to 5.40, the system can only be solved numerically.
w x
Mean water level
w y
v
u
ζ
h
z
y
x
Sea bed
Figure 5.11 Two-dimensional solution domain.