Geoscience Reference
In-Depth Information
At the channel bottom, the wall-function approach is applied. Its implementation in
the finite volume method has been described in Section 6.1.2. In the finite difference
model, Wu (1992) used the logarithmic law at the bottom point
P
:
U
∗
κ
Ez
P
)
U
s
,
P
=
ln
(
(6.113)
where
E
is the roughness parameter, defined in Eq. (6.13);
U
is the bed shear velocity,
related to the cross-section-averaged velocity
U
by
U
∗
=
√
g
U
∗
C
h
; and
z
P
U
∗
z
P
/ν
/
=
,
with
z
P
being the height above the bed.
At the outlet, a time series of water stage, a stage-discharge rating curve, or a non-
reflective wave condition is applied. The
U
x
-velocity at the outlet points is usually
extrapolated or copied from adjacent internal points. Vasiliev (2002) suggested that
the vertical velocity component changes linearly from
U
z
=
U
x
∂
z
b
/∂
x
at the bottom
to
U
z
=
∂
z
s
/∂
t
at the surface:
=
∂
z
s
∂
z
−
z
b
U
x
∂
z
b
z
s
−
z
U
z
+
(6.114)
t
h
∂
x
h
6.4.1.3 Numerical solutions
Hydrostatic pressure model
For gradually varied open-channel flows, the hydrostatic pressure assumption is often
adopted (Blumberg, 1977; Wu, 1992; Edinger
et al
., 1994; Li
et al
., 1994; Vasiliev,
2002). Under this assumption, the
z
-momentum equation (6.105) is simplified to
Eq. (2.64), and the
x
-momentum equation (6.104) takes the following form:
b
b
bU
x
)
∂
∂(
bU
x
)
+
∂(
+
∂(
bU
z
U
x
)
gb
∂
z
s
x
+
∂
t
∂
U
x
∂
+
∂
∂
t
∂
U
x
∂
=−
ν
ν
∂
t
x
∂
z
∂
∂
x
x
z
z
1
ρ
(
−
m
1
τ
x
1
+
m
2
τ
x
2
)
(6.115)
To overcome the difficulty of handling the free surface, the stretching coordinate
transformation (4.91) is applied, under which Eqs. (6.103) and (6.115) are converted
to (Wu, 1992)
b
∂
bhU
bU
+
ζ
∂
b
∂ζ
∂τ
+
∂(
h
ξ
)
h
∂(
ζ
)
+
=
0
(6.116)
∂ξ
∂ζ
b
∂(
bU
x
)
+
U
ξ
∂(
bU
x
)
+
U
ζ
∂(
bU
x
)
gb
∂
∂ξ
+
∂
z
s
t
∂
U
x
∂ξ
=−
ν
∂τ
∂ξ
∂ζ
∂ξ
b
H
2
h
2
∂
t
∂
U
x
∂ζ
1
ρ
(
+
ν
−
m
1
τ
+
m
2
τ
)
x
1
x
2
∂ζ
(6.117)
