Geoscience Reference
In-Depth Information
the discharge increment at the upstream point is
Q
n
+
1
Q
∗
δ
Q
up
=
−
(5.114)
Eq. (5.104) and one of Eqs. (5.107), (5.110), (5.112), and (5.114) are used to
determine the flow at a hydraulic structure. Eq. (5.104) can be written in the form
of Eq. (5.60), with the coefficients being
a
i
=
0,
b
i
=
1,
c
i
=
0,
d
i
=−
1, and
Q
up
. Eqs. (5.107), (5.110), (5.112), and (5.114) can be written as
Eq. (5.68). The coefficients are:
a
i
Q
down
−
p
i
=
z
s
,
up
,
b
i
1,
c
i
0,
d
i
=−
∂
/∂
=
=
=
f
0, and
p
i
f
∗
−
Q
∗
for Eq. (5.107);
a
i
1,
b
i
0,
c
i
1,
d
i
0, and
p
i
z
s
,
up
=
=
=
=−
=
=−
+
z
s
,
down
+
K
L
Q
∗
|
Q
∗
|
/(
2
gA
∗
2
for Eq. (5.110);
a
i
1,
b
i
0,
c
i
0,
d
i
)
=
=
=
=
0, and
p
i
=
Q
∗
for Eq. (5.114). Thus, Eqs. (5.104), (5.107), (5.110), (5.112), and (5.114) can be
intrinsically incorporated into the solution algorithm.
It should be noted that described above are the general methods for considering
hydraulic structures in a 1-D channel network model. For specific hydraulic structures,
empirical stage-discharge relations may be used (see Wu and Vieira, 2002).
z
n
+
1
z
s
,
up
for Eq. (5.112); and
a
i
=
0,
b
i
=
1,
c
i
=
0,
d
i
=
0, and
p
i
=
Q
n
+
1
−
−
s
,
up
5.2.2.5 Stability of Preissmann scheme for unsteady flow
equations
The numerical stability of the Preissmann scheme for the St. Venant equations was
studied by Lyn andGoodwin (1987) and Venutelli (2002). Lyn andGoodwin's findings
are introduced below.
Eqs. (5.1) and (5.2) are written as
∂
F
M
∂
F
+
x
=
b
(5.115)
∂
t
∂
where
F
=
(
u
,
h
)
,
M
is the coefficient matrix, and
b
is the vector of inhomoge-
neous terms.
Appling the Preissmann scheme (4.34)-(4.36) to discretize Eq. (5.115) and lineariz-
ing the discretized equation locally yields
F
n
+
1
i
F
i
+
1
)
+
(
F
n
+
1
i
F
i
)
+
F
n
+
1
i
F
n
+
1
i
ψ(
−
1
−
ψ)(
−
rM
0
[
θ(
−
)
+
1
+
1
F
i
+
1
−
F
i
)
]=
+
(
1
−
θ)(
b
t
(5.116)
where
r
x
, and
M
0
is the coefficient matrix of
M
at locally uniform state.
The Fourier component,
=
t
/
δ
=
δ
∗
e
−
i
(ω
t
−
σ
x
)
, corresponding to
F
is governed by
n
+
1
n
i
n
+
1
n
i
n
+
1
n
+
1
ψ(δ
−
δ
)
+
(
1
−
ψ)(δ
−
δ
)
+
rM
0
[
θ(δ
−
δ
)
+
1
i
+
1
i
i
+
1
i
n
i
n
i
+
(
1
−
θ)(δ
−
δ
)
]=
0
(5.117)
+
1
