Page 280 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: IWV/KCX
0 521 85326 5
CB908/Date
0521853265c09

9 Convergence Enhancement May 11, 2005 15:41

9.1 Convergence Rate

In all the preceding chapters it was shown that discretising the differential transport
equations results in a set of algebraic equations of the following form:

AP P = A k k + S, (9.1)

where suffix k refers to appropriate neighbouring nodes of node P. In pure conduc-
tion problems ( = T ), A k and S may be functions of T . In the general problem

of convective–diffusive transport, may stand for any transported variable and A k
and S may again be functions of the under consideration or any other relevant
to the system. In curvilinear grid generation, = x 1 , x 2 , and A k and S are again
functions of x 1 and x 2 . In all such cases, if there are N interior nodes, we need
to solve N equations for each variable in a prespecified sequence. An iterative
solution is particularly attractive when the algebraic equations for different s are
strongly coupled through coefficients and sources.
In an iterative procedure, convergence implies numerical satisfaction of Equa-
tion 9.1 at each interior node for each . This satisfaction is checked by the residual
in Equation 9.1 at each iteration level l (say). Thus
l l
R = AP − A k − S. (9.2)
P P k
The whole-field convergence is declared when

) * 0.5
2
R
all nodes P

R = < CC, (9.3)
R norm
where CC stands for the convergence criterion and R norm is a dimensionally correct
normalising quantity defined by the CFD analyst. For example, in a problem with
total inflow ˙ m in and average property in , R norm = ˙ m in × in (say). If no such
representative quantity is found then R norm = 1. Ideally, CC must be as small as

259