Page 136 - Introduction to Computational Fluid Dynamics

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5.2 SIMPLE – COLLOCATED GRIDS
convergence. We, therefore, write R uf1,e in Equation 5.31, for example, as 12:28 115

l
l

l
AP u f1 u − A k u f1,k − D l ∂p
f1
= u 1 . (5.36)
R u f1,e +
V ∂x 1 e

e
This equation is the same as Equation 5.26 written for location e, but the net
momentum transfer terms are again multidimensionally averaged. This averag-
ing is done because, when computing on collocated grids, one does not have the
6
cell-face coefficients A k . Now, again using Equation 5.26, we get
l
AP u f1 u − A k u l − D l l
f1 f1,k u 1 ∂p
− . (5.37)
= R u f1,e
V ∂x 1
e e
Thus, effectively,

l
l
∂p
∂p
− + . (5.38)
R u f1,e = R u f1,e

∂x 1 ∂x 1 e
e
will
6. Now, R u f1,e is again evaluated in the manner of Equation 5.35. Thus, R u f1,e
contain residuals only at nodal locations P, E, N, S, NE, and SE. These residuals
will of course vanish at full convergence because momentum equations are
= 0 and
being solved at the nodal positions. Therefore, R u f1,e

l l
∂p ∂p
= − . (5.39)
R u f1,e

∂x 1 e ∂x 1
e
The practice followed here is same as that followed on staggered grids (see
item 3).
7. Now, to evaluate the multidimensionally averaged pressure-gradient in Equation
5.39, we write
l l
l
l
l 1 1 ∂p ∂p x 2,n ∂p /∂x 1 se + x 2,s ∂p /∂x 1 ne
∂p
= + +

∂x 1 2 2 ∂x 1 P ∂x 1 E x 2,n + x 2,s
e
l
1 p − p l W p l EE − p l P
E
= +
4 x 1,e + x 1,w x 1,e + x 1,w
l
l
1 x 2,s p + p l − p − p l
+ E NE P N
4 x 2,n + x 2,s x 1,e
l
l
1 x 2,n p + p l SE − p − p l S
P
E
+ . (5.40)
4 x 2,n + x 2,s x 1,e
6 Note that, in principle, evaluation of these coefficients can be carried out. However, the com-
putational effort involved will be prohibitively expensive in multidimensions. For example, in a
three-dimensional calculation, one will need to evaluate eighteen extra coefficients at the cell faces
in addition to the six coefficients evaluated at the nodal locations.