Page 134 - Introduction to Computational Fluid Dynamics

P1: IWV
0 521 85326 5
May 20, 2005
CB908/Date
0521853265c05
5.2 SIMPLE – COLLOCATED GRIDS
where, p is the mass-conserving pressure correction. Thus, Equation 5.23 will 12:28 113

m
read as 5
1 ∂ ρ l+1 r α V ∂p m
1 ∂ ρ l+1 r α V ∂p m

+
r ∂x 1 AP u f1 ∂x 1 r ∂x 2 AP u f2 ∂x 2
∂(ρ l+1 ) 1 ∂ 1 ∂
= + r ρ l+1 u l f1 + r ρ l+1 u l f2
∂t r ∂x 1 r ∂x 2
1 ∂ ρ r α V 1 ∂ ρ r α V
l+1
l+1
− R uf1 + R uf2 , (5.25)
r ∂x 1 AP u f1 r ∂x 2 AP u f2
where residuals per unit volume, R uf1 and R uf2 , are given by
l
AP u f1 u − A k u l f1,k − D l ∂p l
f1
= u 1 + , (5.26)
R u f1
V ∂x 1
l
AP u f2 u − A k u l − D l ∂p l
f2 f2,k u 2
= + . (5.27)
R u f2
V ∂x 2
The discretised version of the mass-conserving pressure-correction Equation
5.25 will read as

AP p m,P = AE p m,E + AW p m,W + AN p m,N + AS p m,S − ˙ m P + ˙ m R , (5.28)
where
ρ l+1 2 2 ρ l+1 2 2
r α x
r α x
AE = 2 , AW = 2
AP uf1 AP uf1
e w,
ρ l+1 2 2 ρ l+1 2 2
r α x
r α x
AN = 1 , AS = 1 .
AP uf2 AP uf2
n s
AP = AE + AW + AN + AS, (5.29)

l+1 l+1
˙ m P = ρ ru l − ρ ru l x 2
f1 e f1 w
V

l+1 l+1
l+1 o
+ ρ ru l − ρ ru l x 1 + ρ − ρ , (5.30)
f2 n f2 s P P
t
˙ m R = AE R uf1 x 1 | e − AW R uf1 x 1 | w + AN R uf2 x 2 | n − AS R uf2 x 2 | s .
(5.31)
A number of comments with respect to Equations 5.25–5.31 are now in order.
1. On both staggered and collocated grids, the pressure is stored at node P and the
mass conservation equation is solved over the control volume surrounding node
P. Therefore, Equation 5.25 is applicable to both types of grids.

5 In deriving Equation 5.25, it is assumed that A k u = A k u = 0. This is consistent
k f1,k k f2,k
with the SIMPLE-staggered grid practice [51].