210
7. Solution of the Navier-Stokes Equations
solution, an error estimate is obtained. The errors are plotted against grid size
in Fig. 7.11. For both quantities on both uniform and non-uniform grids, the
error reduction expected of a second order method is obtained. The errors are
lower on non-uniform grids, especially for $ J ,
,
,
;
since the secondary vortex is
confined to a corner, the non-uniform grid, which is much finer there, yields
higher accuracy.
Calculations on uniform colocated grids were also performed using three
other discretization schemes: first order UDS, and cubic (fourth order) interpolation of the convective fluxes and midpoint rule integration, and fourth
order Simpson's rule integration and cubic polynomial interpolation. Comparison of the profiles of the horizontal velocity component in the vertical
symmetry plane, obtained with the four discretizations on grids ranging from
10 x 10 CV to 160 x 160 CV, are shown in Fig. 7.12. The first order scheme
is so inaccurate that the solution on the finest grid is still too far from the
grid-independent solution. The second order CDS shows monotone convergence; the ratio of errors on consecutive grids is a factor of four. Interpolation
by a cubic polynomial increases the accuracy from the third grid onwards;
on the first two grids, there are oscillations in the solution. Midpoint rule
integration is used in both cases (i.e., the integrals are approximated by the
product of the variable value at the cell face center and the cell face area).
Finally, the use of Simpson's rule integration and cubic interpolation (a true
fourth order method) gives the highest accuracy, except on the first two grids;
the velocity profiles can hardly be distinguished from each other for grids 40
x 40 and finer.
The quantitative analysis of the accuracy of different schemes is presented
in Fig. 7.13: it shows the convergence of the strength of the primary eddy towards the grid-independent solution and the estimated discretization errors.
Second and fourth order schemes converge rapidly, while the first order UDS
has a large error even on the finest grid (14.4%). Second order CDS and the
mixed scheme produce the same estimate of the grid-independent value when
Richardson extrapolation is applied (to within the convergence accuracy of
the iterative solution method, which was between 0.01 and 0.1%). The plot
of discretization error versus mesh spacing in Fig. 7.13 shows that the mixed
scheme and the standard CDS closely follow the theoretical error reduction
path for second order schemes. The UDS approaches the theoretical convergence rate as the grid is refined, but is more accurate than expected on
coarse grids. Finally, fourth order CDS shows the expected error reduction
from second t o third grid; the discretization error then becomes comparable
to the iteration convergence error, so that on finer grids the error is reduced
less than expected. In order to obtain the theoretical error reduction rate on
all grids, the iterative convergence error should have been a t least an order
of magnitude lower; on the finest grid this would mean iterating practically
to the round-off level, which was not done because it would increase the
computing time considerably.
7. Solution of the Navier-Stokes Equations
solution, an error estimate is obtained. The errors are plotted against grid size
in Fig. 7.11. For both quantities on both uniform and non-uniform grids, the
error reduction expected of a second order method is obtained. The errors are
lower on non-uniform grids, especially for $ J ,
,
,
;
since the secondary vortex is
confined to a corner, the non-uniform grid, which is much finer there, yields
higher accuracy.
Calculations on uniform colocated grids were also performed using three
other discretization schemes: first order UDS, and cubic (fourth order) interpolation of the convective fluxes and midpoint rule integration, and fourth
order Simpson's rule integration and cubic polynomial interpolation. Comparison of the profiles of the horizontal velocity component in the vertical
symmetry plane, obtained with the four discretizations on grids ranging from
10 x 10 CV to 160 x 160 CV, are shown in Fig. 7.12. The first order scheme
is so inaccurate that the solution on the finest grid is still too far from the
grid-independent solution. The second order CDS shows monotone convergence; the ratio of errors on consecutive grids is a factor of four. Interpolation
by a cubic polynomial increases the accuracy from the third grid onwards;
on the first two grids, there are oscillations in the solution. Midpoint rule
integration is used in both cases (i.e., the integrals are approximated by the
product of the variable value at the cell face center and the cell face area).
Finally, the use of Simpson's rule integration and cubic interpolation (a true
fourth order method) gives the highest accuracy, except on the first two grids;
the velocity profiles can hardly be distinguished from each other for grids 40
x 40 and finer.
The quantitative analysis of the accuracy of different schemes is presented
in Fig. 7.13: it shows the convergence of the strength of the primary eddy towards the grid-independent solution and the estimated discretization errors.
Second and fourth order schemes converge rapidly, while the first order UDS
has a large error even on the finest grid (14.4%). Second order CDS and the
mixed scheme produce the same estimate of the grid-independent value when
Richardson extrapolation is applied (to within the convergence accuracy of
the iterative solution method, which was between 0.01 and 0.1%). The plot
of discretization error versus mesh spacing in Fig. 7.13 shows that the mixed
scheme and the standard CDS closely follow the theoretical error reduction
path for second order schemes. The UDS approaches the theoretical convergence rate as the grid is refined, but is more accurate than expected on
coarse grids. Finally, fourth order CDS shows the expected error reduction
from second t o third grid; the discretization error then becomes comparable
to the iteration convergence error, so that on finer grids the error is reduced
less than expected. In order to obtain the theoretical error reduction rate on
all grids, the iterative convergence error should have been a t least an order
of magnitude lower; on the finest grid this would mean iterating practically
to the round-off level, which was not done because it would increase the
computing time considerably.