Geoscience Reference
In-Depth Information
As discussed in Section 4.4.4, the non-staggered grid is more convenient than
the staggered grid in the 3-D model for flows in channels with complex geometry.
The SIMPLE algorithm on the non-staggered grid used by Peric (1985) and Majum-
dar
et al
. (1992) for general flows is herein applied to acquire the pressure-velocity
coupling in the solution of open-channel flows (Wu
et al
., 2000a). The 2-D version of
this SIMPLE algorithm on fixed grid has been described in Section 4.4.4, and its 3-D
extension on moving grid is described below.
The discretized equations are obtained by integrating (4.152) over the control
volume shown in Fig. 4.22. The resulting algebraic equations for velocities
u
n
+
1
i
,
P
(
i
=
1,
2, 3) are
⎛
⎞
1
a
P
u
n
+
1
i
,
P
⎝
a
l
u
n
+
1
⎠
+
D
i
(
p
n
+
1
w
p
n
+
1
e
=
+
S
ui
−
)
i
,
l
l
=
W
,
E
,
S
,
N
,
B
,
T
p
n
+
1
s
p
n
+
1
n
p
n
+
1
b
p
n
+
1
t
D
i
(
D
i
(
+
−
)
+
−
)
(7.8)
where
D
i
a
P
.
The pressure correction is defined in Eq. (4.191). In analogy to Eq. (4.190), the
velocity correction at cell center
P
in the 3-D case is:
i
a
P
,
D
i
i
a
P
, and
D
i
i
=
(
J
α
ηζ)
P
/
=
(
J
α
ξζ)
P
/
=
(
J
α
ξη)
P
/
u
n
+
1
i
,
P
u
i
,
P
+
α
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
+
D
i
(
p
b
−
p
t
)
]
=
[
(7.9)
u
Using Rhie and Chow's (1983) momentum interpolation technique and the flux
definition in Eq. (4.144) yields the following relations for flux corrections at cell faces:
a
p
W
(
F
w
+
p
W
−
p
P
)
F
w
=
(7.10)
a
p
S
F
s
p
S
−
p
P
)
F
s
=
+
(
(7.11)
a
p
B
F
b
+
p
B
−
p
P
)
F
b
=
(
(7.12)
where
F
w
,
F
s
, and
F
b
are the fluxes in terms of the approximate velocities
u
i
,
w
,
u
i
,
s
,
and
u
i
,
b
;
a
p
W
=
α
w
Q
i
,
w
,
a
S
=
α
s
Q
i
,
s
, and
a
B
=
α
n
+
1
i
n
+
1
i
ρ
(
J
α
ηζ)
ρ
(
J
α
ξζ)
u
u
u
w
s
1
b
(
n
+
3
i
ξη)
b
Q
i
,
b
with
Q
i
,
w
=[
(
a
PW
+
a
P
]
(
1
i
w
,
Q
i
,
s
=[
(
ρ
J
α
1
−
f
x
,
P
)/
f
x
,
P
/
J
α
ηζ)
1
−
a
PS
+
a
P
]
(
2
i
s
, and
Q
i
,
b
=[
(
a
PB
+
a
P
]
(
3
i
f
y
,
P
ξη)
b
.
Integrating the 3-D continuity equation over the control volume and discretizing the
time-derivative term by the backward difference scheme yields
)/
f
y
,
P
/
J
α
ξζ)
1
−
f
z
,
P
)/
f
z
,
P
/
J
α
n
+
1
V
n
+
1
P
n
P
V
P
ρ
−
ρ
P
+
F
e
−
F
w
+
F
n
−
F
s
+
F
t
−
F
b
=
0
(7.13)
τ
where
is the time step in the moving, curvilinear grid system.
Note that the variation of flow density is considered in Eq. (7.13) for more general
applications where its effect needs to be considered (see Section 12.1).
τ
