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and Chow's (1983) momentum interpolation technique on the non-staggered grid is
adopted. An upwind scheme is often used to discretize the convection terms. When the
central difference scheme is used, artificial dissipations or TVD limiters are often used
to suppress potential numerical oscillations. Some of these methods are used to simu-
late dam-break and overtopping flows, as discussed in Section 9.1. Described in this
subsection are the SIMPLE(C) algorithm, the projection method, and the vorticity-
based method, which are widely used in the simulation of common open-channel
flows.
6.1.3.1 SIMPLE(C) algorithm
Discretization of governing equations
In the curvilinear coordinate system, Eqs. (6.1)-(6.3), (6.10), and (6.11) can be written
in the following tensor notation form:
ρ
∂
∂
φ)
+
∂
∂ξ
j
∂φ
∂ξ
m
j
m
t
(ρ
hJ
hJ
u
m
ˆ
φ
−
h
φ
J
α
α
=
hJS
φ
(6.21)
m
m
where
φ
stands for 1,
U
x
,
U
y
,
k
, and
ε
, depending on the equation considered;
φ
=
ρ(ν
+
ν
t
/σ
φ
)
φ
is the diffusivity of the quantity
;
S
φ
is the source term in the
equation of
, including the cross-derivative diffusion terms;
J
is the Jacobian of
the transformation between the Cartesian coordinate system
x
i
φ
(
x
1
=
x
,
x
2
=
y
)
and the
i
U
i
; and
i
curvilinear coordinate system
x
i
.
As described in Section 4.4, the primary variables can be arranged in a staggered
or non-staggered (collocated) pattern. The staggered grid approach for the depth-
averaged 2-D model can be found in Lu and Zhang (1993) and Kim
et al
. (2003),
whereas the non-staggered grid approach is applied here.
Eq. (6.21) is integrated over the control volume shown in Fig. 4.21. The convection
terms can be discretized using one of the following schemes: hybrid, exponential,
QUICK, HLPA or SOUCUP, presented in Section 4.3.1.1. The normal-derivative
diffusion terms are usually discretized using the central difference scheme. The time-
derivative term is discretized using the first-order backward scheme (4.23) or the
three-level implicit scheme (4.43) and treated in analogy to Eq. (4.126). The discretized
momentum equations give velocities
U
n
+
1
i
,
P
ξ
(ξ
=
ξ
,
ξ
=
η)
;
u
m
ˆ
=
α
α
=
∂ξ
/∂
m
1
2
m
(
i
=
1, 2
)
at cell center
P
as
⎛
⎝
⎞
⎠
+
1
a
P
U
n
+
1
i
,
P
a
l
U
n
+
1
D
i
(
p
n
+
1
w
p
n
+
1
e
D
i
(
p
n
+
1
s
p
n
+
1
n
=
+
S
ui
−
)
+
−
)
i
,
l
l
=
W
,
E
,
S
,
N
(6.22)
h
n
+
1
P
h
n
+
1
P
where
D
i
1
i
a
P
,
D
i
2
i
a
P
, and
p
is the pressure defined
=
(
J
α
η)
/
=
(
J
α
ξ)
/
P
P
=
ρ
as
p
gz
s
.
The relations of the velocity and pressure corrections in the depth-averaged 2-D
model are similar to Eqs. (4.190), (4.191), (4.194), (4.195), and (4.202)-(4.204).
Thus, they are not repeated here.
