Page 137 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
May 20, 2005
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2D CONVECTION – CARTESIAN GRIDS
To simplify the evaluation, we introduce the following definitions: 12:28
x 1,w p E + x 1,e p W
p = , (5.41)
x 1 ,P
x 1,w + x 1,e
x 2,s p N + x 2,n p S
p x 2 ,P = , (5.42)
x 2,s + x 2,n
1
p = (p x 1 ,P + p x 2 ,P ), (5.43)
P
2
x 1,e p EE + x 1,ee p P
p x 1 ,E = , (5.44)
x 1,e + x 1,ee
x 2,s p NE + x 2,n p SE
p x 2 ,E = , (5.45)
x 2,s + x 2,n
1
p = (p x 1 ,E + p x 2 ,E ). (5.46)
E
2
Substituting these definitions in Equation 5.40 and replacing p EE and p W in
favour of p E and p P , we can show that

1 p − p P p − p P 1 ∂(p + p )
l l l l l l l
E
E
∂p
= + = , (5.47)
∂x 1 2 x 1,e x 1,e 2 ∂x 1 e
e
and, therefore, from Equation 5.39
l l
1 ∂(p − p ) ∂p
= = sm , (5.48)
R u f1,e
2 ∂x 1 ∂x 1
e e
where
1
l
l
p sm = (p − p ). (5.49)
2
The suffix sm here stands for smoothing pressure correction.
8. Repeating items 4, 5, 6, and 7 at other cell faces, we obtain

∂p ∂p ∂p
= sm , = sm , = sm . (5.50)
R u f1,w R u f2,n R u f2,s
∂x 1 ∂x 2 ∂x 2
w n s
Thus, substituting these equations in Equation 5.31, it follows that

∂p sm ∂p sm

˙ m R = AE x 1 − AW x 1

∂x 1 e ∂x 1 w
∂p ∂p
+ AN sm x 2 − AS sm x 2 . (5.51)

∂x 2 ∂x 2
n s
9. In evaluating coefficients AE, AW, AN, and AS, we need AP coefficients
at the cell faces (see Equation 5.29). However, these can be evaluated by