Geoscience Reference
In-Depth Information
scheme yields
n
+
1
n
P
ρ
−
ρ
P
A
P
+
F
e
−
F
w
+
F
n
−
F
s
=
0
(4.198)
τ
Substituting Eqs. (4.196) and (4.197) and two similar equations for the fluxes at
faces
e
and
n
into Eq. (4.198) leads to the equation for pressure correction:
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
(4.199)
where
a
P
=
a
p
W
+
a
E
+
a
S
+
a
N
, and
S
p
=−
(ρ
n
+
1
P
F
e
−
F
w
+
F
n
−
F
s
)
.
The computation procedure of the SIMPLE algorithm on the non-staggered grid is
similar to that on the staggered grid, as introduced in Section 4.4.3.
−
ρ
)
A
P
/τ
−
(
P
SIMPLEC algorithm
Following Van Doormaal and Raithby (1984), the term
a
k
u
i
,
k
is kept in the
derivation of Eq. (4.190), thus yielding
u
l
a
P
u
i
,
P
=
α
a
l
u
i
,
l
+
α
u
a
P
[
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
]
(4.200)
=
E
,
W
,
N
,
S
u
a
l
u
i
,
P
is then subtracted from both sides of Eq. (4.200), yielding
The term
α
⎛
⎝
a
P
−
α
⎞
⎠
u
i
,
P
=
α
u
l
u
l
a
l
a
l
(
u
i
,
l
−
u
i
,
P
)
=
E
,
W
,
N
,
S
=
E
,
W
,
N
,
S
+
α
u
a
P
[
D
i
(
p
w
−
p
e
)
+
D
i
(
p
s
−
p
n
)
]
(4.201)
Assuming that all
u
i
,
k
are of about the same order as
u
i
,
P
and neglecting the first
term on the right-hand side of Eq. (4.201) leads to
[
D
i
(
D
i
(
u
n
+
1
i
,
P
u
i
,
P
+
α
p
w
−
p
e
)
+
p
s
−
p
n
)
]
=
(4.202)
u
u
l
=
E
,
W
,
N
,
S
a
l
/
D
i
D
i
/(
a
P
)
where
1, 2.
Using the momentum interpolation technique introduced above,
=
1
−
α
,
m
=
the velocity
corrections at cell faces
w
and
s
are derived as
u
Q
i
,
w
(
u
n
+
1
i
,
w
u
i
,
w
+
α
p
W
−
p
P
)
=
(4.203)
u
Q
i
,
s
(
u
n
+
1
i
,
s
u
i
,
s
+
α
p
S
−
p
P
)
=
(4.204)