SIMPLE Method on Non-staggered Grids - Department of ...

4. Momentum Interpolation
4.1. Momentum interpolation for steady-state problem
BP = bP + ⎡⎣ 1−αu αu⎤⎦ aPuP
[Eq.(13)]. For the
n−1
Note that for the steady problems ( )
velocity component u at nodes P and E, Eq. (12) can be written as
u
u
P
E
( )
( a )
P
P
( )
( a )
αu ∑
i
au
i i
+ Bp α
P uΔy pe − pw
P
= − (14)
( )
( a )
P
E
P
P
( )
( a )
αu ∑
i
au
i i
+ Bp α
E uΔy pe − pw
E
= − (15)
Mimicking the formulation of u
E
interface velocity at the cell face e.
u
e
( )
( a )
P
e
P
E
and u
P
, we can obtain the following expression for the
( )
( a )
αu ∑
i
au
i i
+ Bp α
e uΔy pE − pP
= − (16)
P
e
where the terms on the right-hand side with subscript e should be interpolated in an
appropriate manner. The interface velocity at cell faces w, n, and s can be obtained
similarly.
In Rhie and Chow’s momentum interpolation, the first term and 1 ( aP ) in second
e
term of the Eq.(16) are linearly interpolated from their counterparts in Eqs.(14) and (15):
⎛∑ au + B ⎞ ⎛∑ au + B ⎞ ⎛∑ au + B ⎞
⎜ ⎟ ⎜ ⎟ ( 1 fe
) ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
i i i p + i i i p + i i i p
= fe
+ −
aP a
e P
a
E P P
(17)
1 + 1 + 1
= f + − (18)
a a a
e
( ) ( )
( 1 fe
) ( )
P e P E P P
where f + e
is a linear interpolation factor defines as
+ Δx
P
f
e
= (19)
2δ x
e
In order to have a better understanding of Eq. (16), substituting ( au B a )
Eq. (17) and ( ∑ au + B a ) , ( au B a )
i i i p P P
o o
(16) and omitting the term au , we obtain
P
P
i i i p P E
∑ + from
i i i p P e
∑ + from Eqs. (14) and (15) into Eq.
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