7.8 Examples
213
all equations three orders of magnitude using various combinations of underrelaxation parameters and a 32 x 32 CV uniform grid.
Fig. 7.14. Numbers of outer iterations required to reduce the residual level in all
equations three orders of magnitude using various combinations of under-relaxation
parameters and a 32 x 32 CV uniform grid with staggered (left) and colocated
(right) arrangement of variables (lid-driven cavity flow at Re = 1000)
This figure shows that the dependence on the under-relaxation parameter
a, is almost the same for the two types of grids; the range of good values
is somewhat wider for the colocated grid, but overall the behavior of both
methods is similar. When the velocity is more strongly under-relaxed, we can
use any value of a, between 0.1 and 1.0, but the method converges slowly.
For larger values of a,, the convergence is faster but the useful range of a,
is restricted.
Patankar (1980) suggested using a, = 0.5 and a, = 0.8 in the SIMPLE
method. We see from Fig. 7.14 that this is not optimum. The value of a,
suggested by Eq. 7.50 is nearly optimum; a, = 1.1 - a, gives the best results for this flow and yields an improvement by about a factor of five over
Patankar's recommendation.
In Fig. 7.15 we show the effect of the under-relaxation factor for velocity,
a,, on convergence rate for both staggered and colocated arrangements when
optimum value of a, is used, for two grids. We see that the method behaves
in the same way for both grids. The dependence on a, is stronger on the
refined grid.
We next investigate the influence of the under-relaxation parameter for
velocity on the solution for colocated grids. We choose a, = 0.9 for one case
and a, = 0.5 for the other. The difference in the solutions (after residual
levels were reduced five orders of magnitude, to exclude convergence errors)
213
all equations three orders of magnitude using various combinations of underrelaxation parameters and a 32 x 32 CV uniform grid.
Fig. 7.14. Numbers of outer iterations required to reduce the residual level in all
equations three orders of magnitude using various combinations of under-relaxation
parameters and a 32 x 32 CV uniform grid with staggered (left) and colocated
(right) arrangement of variables (lid-driven cavity flow at Re = 1000)
This figure shows that the dependence on the under-relaxation parameter
a, is almost the same for the two types of grids; the range of good values
is somewhat wider for the colocated grid, but overall the behavior of both
methods is similar. When the velocity is more strongly under-relaxed, we can
use any value of a, between 0.1 and 1.0, but the method converges slowly.
For larger values of a,, the convergence is faster but the useful range of a,
is restricted.
Patankar (1980) suggested using a, = 0.5 and a, = 0.8 in the SIMPLE
method. We see from Fig. 7.14 that this is not optimum. The value of a,
suggested by Eq. 7.50 is nearly optimum; a, = 1.1 - a, gives the best results for this flow and yields an improvement by about a factor of five over
Patankar's recommendation.
In Fig. 7.15 we show the effect of the under-relaxation factor for velocity,
a,, on convergence rate for both staggered and colocated arrangements when
optimum value of a, is used, for two grids. We see that the method behaves
in the same way for both grids. The dependence on a, is stronger on the
refined grid.
We next investigate the influence of the under-relaxation parameter for
velocity on the solution for colocated grids. We choose a, = 0.9 for one case
and a, = 0.5 for the other. The difference in the solutions (after residual
levels were reduced five orders of magnitude, to exclude convergence errors)