Geoscience Reference
In-Depth Information
7.2.4 SIMPLE algorithm
Eqs. (7.18)-(7.20) can be written as Eq. (4.152) and discretized using the finite volume
method presented in Section 7.1.3.2. Discretizing the momentum equations (7.19) and
(7.20) leads to the following equation for horizontal velocities
u
n
+
1
i
,
P
(
i
=
1, 2
)
:
⎛
⎞
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
,
B
,
T
(7.39)
where
D
i
1
i
a
P
,
D
i
2
i
a
P
, and
p
gz
s
.
In analogy to Eq. (7.9), the velocity correction at cell center
P
is as follows:
=
(
J
α
ηζ)
/
=
(
J
α
ξζ)
/
=
ρ
P
P
u
n
+
1
i
,
P
u
i
,
P
+
α
u
[
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
]
=
(7.40)
-grid lines are along the vertical direction and cell faces
w
and
s
are on vertical planes. Using Rhie and Chow's momentum interpolation technique
and the flux definition in Eq. (4.144) yields the horizontal flux corrections at cell faces
w
and
s
, as described in Eqs. (7.10) and (7.11).
Integrating the 3-D continuity equation over each control volume leads to Eq. (7.13).
Summing it over all control volumes along a vertical grid line and using the boundary
conditions at the water surface and channel bed yields
Suppose that the
ζ
K
K
K
K
p
n
+
1
P
p
P
−
A
P
+
F
e
,
l
−
F
w
,
l
+
F
n
,
l
−
F
s
,
l
=
0
(7.41)
g
τ
l
=
1
l
=
1
l
=
1
l
=
1
where
K
is the total number of control volumes at the vertical grid line, the subscript
l
is the control volume index in the vertical direction, and
A
P
is the area of the
2-D control volume that is obtained by projecting the 3-D control volume onto the
horizontal plane.
Substituting Eqs. (7.10) and (7.11) into Eq. (7.41) leads to the following equation
for pressure correction:
b
p
b
p
b
p
b
p
b
p
P
p
P
=
W
p
W
+
E
p
E
+
S
p
S
+
N
p
N
+
S
p
(7.42)
=
l
=
1
a
p
=
l
=
1
a
p
=
l
=
1
a
p
=
l
=
1
a
p
where
b
p
W
W
,
l
,
b
p
E
,
l
,
b
p
S
,
l
,
b
p
N
,
l
,
b
p
=
E
S
N
P
b
p
W
+
b
E
+
b
S
+
b
E
+
A
P
/(
τ)
g
, and
⎛
⎞
K
K
K
K
p
P
−
p
P
)
⎝
F
e
,
l
−
F
w
,
l
+
F
n
,
l
−
F
s
,
l
⎠
.
S
p
=−
(
A
P
/(
g
τ)
−
l
=
1
l
=
1
l
=
1
l
=
1
Once the pressure correction is calculated using Eq. (7.42), the horizontal fluxes at
cell faces
w
and
s
are corrected using Eqs. (7.10) and (7.11). Because the vertical flux
