Geoscience Reference
In-Depth Information
water level:
2
z
s
2
z
s
∂
∂
+
∂
S
z
g
=
(6.47)
∂
x
2
y
2
where
U
x
∂
2
∂
2
2
U
x
∂
2
U
y
y
∂
U
y
∂
U
y
∂
+
∂
U
x
∂
2
∂
U
x
∂
∂
S
z
=−
−
x
−
−
x
y
x
2
∂
x
∂
y
U
y
2
U
x
2
U
y
∂
2
T
xx
∂
2
T
xy
2
T
yy
∂
y
+
∂
2
∂
y
+
∂
∂
1
ρ
∂
−
+
+
∂
x
∂
y
2
x
2
∂
x
∂
y
2
x
τ
bx
h
y
τ
by
h
1
ρ
∂
∂
1
ρ
∂
∂
−
−
The stream function, vorticity, and water level equations (6.45)-(6.47) are appar-
ently more complicated than Eqs. (6.40)-(6.42), but they are still typical partial dif-
ferential equations and easy to solve. In addition, the mass is conserved automatically,
and the difficulty in solving Eq. (6.40) is avoided.
Note that because the definition of stream function in Eq. (6.43) is not valid for
Eq. (6.1), the aforementioned stream function and vorticity method cannot be used
for the depth-averaged 2-D simulation of unsteady flows. However, it can be used
in steady and quasi-steady cases. In particular, it can be used in the stepwise quasi-
steady model for the long-term simulation of flow and sediment transport to reduce
computational effort (Wu
et al
., 1995).
Boundary conditions for stream function and vorticity
For the river reach shown in Fig. 6.1, the stream function is set to zero on the left bank
and
Q
on the right bank. Here,
Q
is the total flow discharge. The stream function
along the island should have a constant value. The gradient of stream function along
the flow direction at the outlet is set to zero. Corresponding to Eq. (6.17), the stream
function at the inlet can be determined by
B
y
Q
y
0
Uhdy
=
h
1
+
r
dy
h
1
+
r
dy
=
(6.48)
0
0
where the transverse coordinate
y
starts from the left bank.
The water level is given at the outlet, as usual in the case of subcritical flow. The
water level at the inlet should be extrapolated from the values at adjacent internal
points. The water level gradient in the direction normal to rigid wall boundaries, such
as banks and islands, can be set to zero.
The vorticity at the boundaries can be determined using Eq. (6.44), with the velocity
derivatives calculated using the following one-sided schemes:
