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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 =−
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 (
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)
 
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