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– 4.50 –<br />
iteration). The inner-iteration seeks the solution for each dependent variable, �� across the<br />
computational domain for given coefficient and source terms, A and B, which are<br />
constants. The solution for every dependent variable, � (� = u,v,w,p,k,�), is sought one<br />
after another. In the outer-iteration, solution is obtained for every dependent variable that<br />
all together satisfies the governing equations. In each outer-iteration, the coefficient and<br />
source terms are adjusted, where as in the inner-iteration, they are kept constant.<br />
When performing the inner-iteration, it is important to decide when to quit the solver.<br />
Since the solution of Eq. 4.97 for a particular variable, u for example, at a particular<br />
iteration level, �� �, does not necessarily satisfy all governing equations for u,v,w,p,k,�, it is<br />
inefficient to carry out a rigorous iteration at this stage. A restricted number of iterations<br />
and a moderate convergence criterion will do. A single or at most a two inner-iteration is<br />
generally sufficient to solve the momentum equation for u, v, and w since their equations<br />
are of convection types. The pressure correction however requires a number of sweeps<br />
over the entire domain to have a solution error within a sufficiently small allowable limit.<br />
The convergence of the k and � equations, being of convection-diffusion types, may also<br />
be slow. This is due to the need of a small relaxation factor in the iteration process to<br />
avoid oscillations. The residue of the solution of Eq. 4.97 is used as the basis to detect the<br />
solution error; it is defined as:<br />
�� � � b � a P�P �� anb�nb (4.98)<br />
all cells<br />
The calculation in the inner-iteration is stopped when either one of the following criteria<br />
is satisfied:<br />
mi<br />
�� � �1 (4.99a)<br />
mi mi�1<br />
�� �� � �2 (4.99b)<br />
mi � NIT �<br />
(4.99c)<br />
mi th<br />
in which �� is the sum of absolute residues over all cells after mi iterations for any<br />
variable �; �� and �2 are prescribed convergence criteria; and NIT is the maximum<br />
number of inner-iterations. Table 4.6 gives the default values of these criteria for each<br />
variable �.<br />
Table 4.6 Criteria to stop the inner iteration
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