Geoscience Reference
In-Depth Information
The depth-integrated continuity equation (6.1) is discretized as
g
t
p
n
+
1
P
p
P
−
=
A
P
(
F
e
−
F
w
+
F
n
−
F
s
)
(6.23)
where
A
P
is the area of the cell centered by
P
; and
F
e
,
F
w
,
F
n
, and
F
s
are the convection
fluxes across cell faces
e
,
w
,
n
, and
s
, defined as
n
+
1
i
w
U
n
+
1
i
,
w
F
w
=
(ρ
h
)
(
J
α
η)
(6.24)
w
n
+
1
2
i
s
U
n
+
1
F
s
=
(ρ
h
)
(
J
α
ξ)
(6.25)
s
i
,
s
It seems that the pressure (water level) can be calculated from the discretized
continuity equation (6.23), but in fact node-to-node oscillations may exist on
the non-staggered grid if the fluxes at the cell faces are linearly interpolated
from the quantities stored at the cell centers, as explained in Section 4.4. To
avoid this, Wenka (1992), Ye and McCorquodale (1997), and Minh Duc (1998)
applied Rhie and Chow's (1983) momentum interpolation technique to evaluate
the variable values at the cell faces from the quantities at the cell centers in the
depth-averaged simulation of open-channel flows. In the formulations of Ye and
McCorquodale (1997) and Minh Duc (1998), the pressure correction was defined
as
p
p
n
, which forms an explicit algorithm for pressure. To form a
semi-implicit algorithm, which allows for longer time steps, the pressure correc-
tion was defined as
p
p
n
+
1
=
−
p
n
+
1
p
∗
by Wu (2004). Wu's formulation is introduced
=
−
below.
Using Rhie and Chow's (1983) momentum interpolation procedure as described in
Section 4.4.4 yields the flux correction equations (4.196) and (4.197). For the depth-
averaged 2-D SIMPLE algorithm, the coefficients
a
p
W
and
a
S
in these equations are
derived as
a
p
W
n
+
1
1
1,
w
Q
1,
w
+
α
1
2,
w
Q
2,
w
)
=
α
(ρ
h
)
(
J
η)
(α
(6.26)
u
w
w
a
p
S
n
+
1
2
1,
s
Q
1,
s
+
α
2
2,
s
Q
2,
s
)
=
α
(ρ
h
)
(
J
ξ)
(α
(6.27)
u
s
s
where
Q
i
,
w
a
PW
+
a
P
]
h
n
+
1
w
1
i
w
;
Q
i
,
s
a
PS
=[
(
1
−
f
x
,
P
)/
f
x
,
P
/
(
J
α
η)
=[
(
1
−
f
y
,
P
)/
+
s
; and
a
PW
and
a
PS
stand for
a
P
when Eq. (6.22) is applied in
the control volumes centered by
W
and
S
, respectively.
For the depth-averaged 2-D SIMPLEC algorithm,
a
P
]
h
n
+
1
s
2
i
f
y
,
P
/
(
J
α
ξ)
the coefficients
a
p
W
and
a
p
S
replaced by
Q
i
,
w
and
are determined by Eqs. (6.26) and (6.27) with
Q
i
,
w
and
Q
i
,
s
Q
i
,
s
defined in Eqs. (4.203) and (4.204).
Inserting Eqs. (4.196) and (4.197), as well as two similar equations for
F
e
and
F
n
,
into Eq. (6.23) yields the pressure correction equation:
a
p
a
p
a
p
a
p
a
p
P
p
P
=
W
p
W
+
E
p
E
+
S
p
S
+
N
p
N
+
S
p
(6.28)
