Geoscience Reference
In-Depth Information
Note that the use of primary variables
h
and
q
in Eq. (9.2) is restricted in rectangular
channels. However, this arrangement provides convenience for extension of a 1-D
algorithm to the 2-D model.
The 2-D shallow water equations (6.1)-(6.3) are written in the following conserva-
tive form:
∂
∂
+
∂
F
(
)
∂
+
∂
G
(
)
=
S
(
)
(9.5)
t
x
∂
y
where
,
F
(
)
,
G
(
)
, and
S
(
)
represent the vectors of unknown variables, fluxes,
and source terms:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
h
hU
x
hU
y
hU
x
hU
x
+
gh
2
=
,
F
(
)
=
/
,
2
hU
x
U
y
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
hU
y
hU
x
U
y
hU
y
+
0
G
(
)
=
,
S
(
)
=
gh
(
S
0
x
−
S
fx
)
(9.6)
gh
2
/
2
(
S
0
y
−
S
fy
)
gh
The stress effects are usually omitted in Eq. (9.5) for dam-break flow. The convection
terms can be rewritten as
∂
F
(
)
∂
A
∂
∂
∂
G
(
)
B
∂
∂
=
x
,
=
(9.7)
x
∂
y
y
where
A
and
B
are the Jacobian matrices:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
0
1
0
0
0
1
U
x
−
A
=
gh
−
2
U
x
0
,
B
=
U
x
U
y
U
y
U
x
(9.8)
U
y
0
U
y
gh
−
−
U
x
U
y
U
y
U
x
Many traditional numerical schemes, such as the Preissmann (1961) scheme,
designed for solving the 1-D shallow water equations in common flow situations,
are insufficient for dam-break flow simulation, producing non-physical oscillations.
Numerous numerical schemes based on finite volume, finite difference, and finite ele-
ment methods have been developed recently for simulation of dam-break flow. As an
example, a finite volume approach is introduced here, which uses the non-staggered
grid shown in Fig. 9.1 for the 1-D problem. The computational domain is divided
into
I
segments. Each segment is a control volume (cell) embraced by two faces. The
primary variables
h
and
q
are defined at cell centers and represent the average values
over each cell, while the fluxes are defined at cell faces.
Integrating Eq. (9.1) over the
i
th control volume yields
x
i
+
1
/
2
x
i
+
1
/
2
x
i
+
1
/
2
∂
∂
∂
F
(
)
∂
t
dx
+
dx
=
S
(
)
dx
(9.9)
x
x
i
−
1
/
2
x
i
−
1
/
2
x
i
−
1
/
2
