208
7. Solution of the Navier-Stokes Equations
A non-uniform grid and the streamlines for the Reynolds number, based on
cavity height H and lid velocity UL, Re = ULH/v = 1000 are shown in Fig.
7.9.
We first look at the estimation of iteration convergence errors. Several
methods were presented in Sect. 5.7.
First, an accurate solution was obtained by iterating until the residual
norm became negligibly small (of the order of the round-off error in double
precision). Then the calculation was repeated and the convergence error was
computed as the difference between the converged solution obtained earlier
and the intermediate solution.
Figure 7.10 shows the norm of the convergence error, the estimate obtained from Eqs. (5.86) or (5.93), the difference between two iterates, and
the residual on a 32 x 32 CV grid with under-relaxation factors 0.7 for velocity and 0.3 for pressure. Since the algorithm needs many iterations to
converge, the eigenvalues required by the error estimator were averaged over
the latest 50 iterations. The fields were initiated by interpolating the solution
on the next coarser grid, which is why the initial error is relatively low.
0
100
200
3 0 0
400
5 0 0
on a 32 x 3
2
~
~
grid, with CDS
Iter.
discretization
b
:
Q
.00001
.OOOOOI
.OOOOOOI
.IE-07
This figure shows that the error estimation technique gives good results
for the non-linear flow problem. The estimate is not good a t the beginning of
the solution process, where the error is large. Using the absolute level of either
the difference between two iterates or the residuals is not a reliable measure
of the convergence error. These quantities do decrease at the same rate as the
error, but need to be normalized properly to represent the convergence error
quantitatively. Also, they fall very rapidly initially, while the error reduction
is much slower. However, after a while all curves become nearly parallel and
-I\
-.
\
--. -..
- : \ \ \
-. --.
: \
.
-.
.
--.
.
--.
.
..
.
- :
. . . . .
Error, exac?..
-F
Error, estim. .'- - - - - - - - - -
. .
Difference
- ------ Residual
-
l
i
'
"
'
"
'
"
"
"
l
'
l
Fig. 7.10. Comparison of the
norms of the exact and estimated convergence error for the
SIMPLE method, the difference
between two iterates and the
residual for the solution of liddriven cavity flow at Re = lo3