Page 132 - Introduction to Computational Fluid Dynamics

P1: IWV
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0521853265c05
5.2 SIMPLE – COLLOCATED GRIDS
where the convective coefficients are given by May 20, 2005 12:28 111
C e = ρ e r e u f1,e x 2 , C w = ρ w r w u f1,w x 2 ,
C n = ρ n r n u f2,n x 1 , C s = ρ s r s u f2,s x 1 , (5.9)
and the diffusion coefficients are

eff,e r e x 2
eff,w r w x 2
d e = , d w = ,
x 1e x 1w

eff,n r n x 1
eff,s r s x 1
d n = , d s = . (5.10)
x 2n x 2s
Now, in terms of the notation just introduced, the discretised mass conservation
equation (5.6) (with = 1) can be written as
V

o
ρ P − ρ P + C e − C w + C n − C s = 0. (5.11)
t
Further, the expressions for C at the cell faces can be generalised to account
for any of the convection schemes introduced in Chapter 3. When this is done and
Equation 5.11 is employed, it can be shown that Equation 5.8 reduces to

AP P = AE E + AW W + AN N + AS S + D, (5.12)

where
AE = d e [A + max(−P ce , 0)] , P ce = C e /d e , (5.13)

AW = d w [A + max(P cw , 0)] , P cw = C w /d w , (5.14)
AN = d n [A + max(−P cn , 0)] , P cn = C n /d n , (5.15)

AS = d s [A + max(P cs , 0)] , P cs = C s /d s , (5.16)
o
ρ V
p
AP = AE + AW + AN + AS + , (5.17)
t
o
ρ V
o
P
D = S V + . (5.18)
P
t
In these equations
⎧
1 (UDS)
⎪
⎪
⎪
⎪
max(0, 1 − 0.5 |P c |) (HDS)
⎪
⎨
A =
5
⎪max 0, (1 − 0.1 |P c |) (Power)
⎪
⎪
⎪
⎪
⎩
1 − 0.5 |P c | (CDS). (5.19)
From the point of view of computer coding, the utility of this generalised rep-
resentation for all variables (scalars as well as vectors) is obvious.