Geoscience Reference
In-Depth Information
literature. For example, Wu and Vieira (2002) simplified Eq. (5.2) to the following
form, by ignoring the momentum contribution from side flows and dividing by the
flow area
A
:
Q
A
β
Q
2
2
A
2
∂
∂
+
∂
∂
g
∂
z
s
+
x
+
gS
f
=
0
(5.54)
t
x
∂
β
where the correction factor
is not exactly the same as the momentum correction
β
factor
β
in Eq. (5.2). By comparing Eqs. (5.40) and (5.54), it is recognized that
is
similar in these two equations and thus can be determined using Eq. (5.41).
The numerical solution of the St. Venant equations is a classic problem in com-
putational river dynamics. Mahmood and Yevjevich (1975) and Cunge
et al
. (1980)
described some of the historical developments. One of the most widely used schemes is
the Preissmann (1961) scheme, based on two computational points in two time levels,
as presented in Eqs. (4.34)-(4.36). Its application to Eqs. (5.1) and (5.54) yields
ψ
1
−
ψ
A
i
)
+
θ
A
n
+
1
i
A
i
+
1
)
+
A
n
+
1
i
Q
n
+
1
i
Q
n
+
1
i
t
(
−
t
(
−
x
(
−
)
+
1
+
1
1
−
θ
Q
i
+
1
−
Q
i
)
−
θ
[
ψ
q
n
+
1
l
,
i
q
n
+
1
l
,
i
+
x
(
1
+
(
−
ψ)
]
1
+
q
l
,
i
+
1
+
(
q
l
,
i
]=
−
(
1
−
θ)
[
ψ
1
−
ψ)
0
(5.55)
Q
n
+
1
i
Q
n
+
1
i
A
n
+
1
i
Q
i
+
1
A
i
+
1
Q
i
A
i
ψ
1
−
ψ
1
A
n
+
1
i
+
1
−
+
−
t
t
+
⎡
⎣
β
n
+
1
2
⎤
⎦
Q
n
+
1
i
Q
n
+
1
i
A
n
+
1
i
2
−
β
n
+
1
+
θ
i
+
1
1
A
n
+
1
i
+
i
x
2
2
+
1
⎡
⎣
β
n
2
⎤
⎦
+
θ
Q
i
+
1
A
i
+
1
2
Q
i
A
i
−
β
n
1
−
θ
g
i
+
1
i
2
z
n
+
1
s
,
i
z
n
+
1
s
,
i
+
x
(
−
)
+
1
x
2
+
(
1
−
θ)
g
z
s
,
i
+
1
−
z
s
,
i
)
+
θ
R
S
n
+
1
f
,
i
S
n
+
1
f
,
i
(
g
[
ψ
+
(
1
−
ψ
)
]
R
+
1
x
R
S
f
,
i
+
1
+
(
S
f
,
i
]=
+
(
1
−
θ)
g
[
ψ
1
−
ψ
)
0
(5.56)
R
where
are the temporal and spatial weighting factors in the Preissmann scheme,
as shown in Fig. 4.5; and
θ
and
ψ
R
is the spatial weighting factor for friction slope in the
case of low flow depth, as described in Section 5.2.2.6.
ψ
5.2.2.2 Local linearization of discretized unsteady flow
equations
Eqs. (5.55) and (5.56) constitute a nonlinear system that needs to be solved iteratively.
During the iteration process, the following relations are expected:
