128 Computational Modelling in Hydraulic and Coastal Engineering
u
u
x
v
u
y
g
h h z
h
x
fv
u
u
u
u
u
b
o
u
h
∂
∂
+
∂
∂
= −
∂
+ +
−
∂
+
+
∂
[(
)
]
o
∆ρ
ε
2
u u
x
u
y
h
h
u
u
ix
o o
bx
u u
∂
+
∂
∂





 +
−
2
2
2
τ
ρ
τ
ρ
(5.74)
∂
∂
+
∂
∂
+
∂
∂
= −
∂
+ +
−
 
 
∂
−
v
t
u
v
x
v
v
y
g
h h z
h
y
u
u
u
u
u
o
u
b
o
(
) ∆ρ
f fu
v
x
v
y
h
h
u
h
u
u
iy
o o
by
u u
+
∂
∂
+
∂
∂





 +
−
ε
τ
ρ
τ
ρ
2
2
2
2
(5.75)
where Δρ is the relative density difference
∆ρ
ρ ρ
ρ
=
−
u
o
u
(5.76)
The numerical solution is accomplished similarly to the previous nearly
horizontal free surface flow models, by using a FTCS (forward in time, central in space) explicit scheme on a staggered computational grid, through
successive computations of the velocity and depth variables.
5.4.2 One-dimensional stratified system
For the case of a one-dimensional stratified system in a narrow basin confined by vertical shore boundaries (full reflection) the governing equations
(Equations 5.70 to 5.76) are reduced as follows:
∂
∂
+
∂
∂
=
h
t
h u
x
o
o o
(
) 0
(5.77)
∂
∂
+
∂
∂
= −
∂ + +
∂
+
∂
∂
+
u
t
u
u
x
g
h h z
x
u
x
h
o
o
o
o
u
b
h
o
s x
o o
(
) ε
τ
ρ
2
2
− −
τ
ρ
ix
o o
h
(5.78)
∂
∂
+
∂
∂
=
h
t
h u
x
u
u u
(
) 0
(5.79)