4.7 Examples
87
100
1000
10000
100000
No. of CV
Fig. 4.6. Convergence of total flux of 4 through the west wall (left) and the error
in computed flux as a function of grid spacing, for r = 0.001
scribed above by adjusting the boundary conditions (prescribed values of 4 at
west and south boundaries, outflow conditions at north and east boundaries).
We show below the results obtained using the UDS and CDS discretizations.
Since diffusion is not present in this case, the equation to be solved is (in
differential form) :
For this case, the UDS on a uniform grid in both directions gives the very
simple equation:
which is readily solved in a sequential manner without iteration. On the other
hand, CDS gives a zero value for the coefficient on the main diagonal, Ap,
making solution difficult. Most iterative solvers would fail to converge for
this problem; however, by using the deferred correction approach described
above, it is possible to obtain the solution.
If the flow is parallel to x-coordinate, both schemes give the correct result:
the profile is simply convected downstream. When the flow is oblique to grid
lines, UDS produces a smeared step profile at any downstream cross-section,
while CDS produces oscillations. In Fig. 4.8 we show the profile of 4 at
x = 0.45 for the the case when the flow is at 45' to the grid (u, = u,),
obtained on a uniform 10 x 10 CV grid. The same figure shows the profile
at x = 0.475 for the same case obtained on a uniform 20 x 20 CV grid.
The effect of numerical diffusion is clearly seen in the UDS solution; little
87
100
1000
10000
100000
No. of CV
Fig. 4.6. Convergence of total flux of 4 through the west wall (left) and the error
in computed flux as a function of grid spacing, for r = 0.001
scribed above by adjusting the boundary conditions (prescribed values of 4 at
west and south boundaries, outflow conditions at north and east boundaries).
We show below the results obtained using the UDS and CDS discretizations.
Since diffusion is not present in this case, the equation to be solved is (in
differential form) :
For this case, the UDS on a uniform grid in both directions gives the very
simple equation:
which is readily solved in a sequential manner without iteration. On the other
hand, CDS gives a zero value for the coefficient on the main diagonal, Ap,
making solution difficult. Most iterative solvers would fail to converge for
this problem; however, by using the deferred correction approach described
above, it is possible to obtain the solution.
If the flow is parallel to x-coordinate, both schemes give the correct result:
the profile is simply convected downstream. When the flow is oblique to grid
lines, UDS produces a smeared step profile at any downstream cross-section,
while CDS produces oscillations. In Fig. 4.8 we show the profile of 4 at
x = 0.45 for the the case when the flow is at 45' to the grid (u, = u,),
obtained on a uniform 10 x 10 CV grid. The same figure shows the profile
at x = 0.475 for the same case obtained on a uniform 20 x 20 CV grid.
The effect of numerical diffusion is clearly seen in the UDS solution; little