


Presented here is an example of a complete solution of a subsonic internal channel flow over a 10% bump.
This test case, a commonly used reference case, is run in 2DE by specifying an inflow total pressure of 1e5 N/m^2,
an inflow total Temperature of 300K, an outflow static pressure of 8e4 N/m^2 and a pressure ratio across the channel of 1.
Using a single block, structured euler grid of 96x32 cells this case serves to demonstrate the convergence improvements to be
achieved with the use of multigrid. Initially a single grid run is performed, which exhibits an extremely slow rate of convergence.
This problem is solved though the use of multigrid, as the results for a run using various 2, 3 and 4grid cycles demonstrate.
For all three computations the solver run is done using local timestepping, a CFL number of 7.5, implicit residual smoothing with variable
coefficients scaled with an epsilon of 0.8, and 2nd and 4th order artificial viscosity terms scaled with factors of 0.5 and 1/64 respectively. For the multigird cases
the coarse grid corrections are smoothed using a constant coefficient of 0.01 on all grid levels.

Comparative plot of the density residuals and the mass flow ratio across the channel for the various solver runs.
The singlegrid computations show a very poor rate of convergence in terms of the density residual and still
exhibits fluctuations in the mass flow ratio after 3000 cycles. A significant improvement can already be achieved
by using a 2vmultigrid cycle, which leads to a bottoming out of the density residual after 3000 cycles.
Using a 3vmultigrid cycle results in a fully converged solution after 1600 cycles, while machine accuracy is
achieved after 1400 cycles for both a 3w and a 4vcycle. For the 4wcycle a fully converged solution can be
reached after 1300 cycles, which is the lowest number of iterations of all of the multigrid cycles tested.

The benefit of running this case with multigrid also becomes evident when looking at the evolution of the
mass flowratio across the channel during the first 500 cycles of the various computations. The single grid run
shows strong oscillations which can be damped out much earlier using multigrid acceleration. Much like the
situation for the overall rate of convergence described above, using more grid levels as well as using a w
instead of a vcycle tends to speed up the damping out of oscillations in the mass flowratio, with the 4wmultigrid
cycle leading to a constant value after only about 150 cycles.

The 2Dcontour plot of the mach number distribution shows completely subsonic flow through the channel, with only a slight acceleration
on the top of the bump. The top image shows isolines of pressure superimposed on the grid of the finest level
used in this study.

Case

Cycles (to 1e13)

Mass flow [kg/s]

Mass flow Ratio

Runtime [s]

Ave. Time per Cycle [s]

Time to Convergence [s]

SG 
3000 () 
190.8 
1 
449 
0.149667 
 
2V 
3000 (2896) 
190.9 
1 
630 
0.21 
608.16 
3V 
2000 (1578) 
190.9 
1 
461 
0.2305 
363.729 
3W 
2000 (1362) 
190.9 
1 
566 
0.283 
385.446 
4V 
1500 (1352) 
190.9 
1 
346 
0.230667 
311.861784 
4W 
1500 (1253) 
190.9 
1 
364 
0.242667 
304.061751 
This table lists details pertaining to the convergence and run times for the various cases. The value in
parentheses in the cycle column is the iteration at which the density residual drops below a value of 1e13
for the first time. This convergence criteria is somewhat arbitrary but does allow for a fairly accurate
comparison of the timetoconvergence as the final density residual fluctuates around a value of 9e14 for
all cases. For all cases the physically relevant properties mass flow and mass flow ratio have reached identical
values at convergence. Of interest is the time to convergence, which is a product of the average runtime per
cycle and the iteration number at which the density residual reaches 1e13. It shows  with one notable exception 
that the improved convergence rate of wcycles over vcycles and of increasing the number of multigrid levels
also translates to an overall increase in efficiency, i.e. runtime to convergence. However the 4wcycle does
seem to be close to the optimum in terms of efficiency, as the improvement over the 4vcycle is relatively small.
The notable exception mentioned is the 3wcycle, which is slightly more expensive than the 3vcycle in terms
of compute time. A possible explanation for this is that here three rather expensive RungeKutta cycles on the
2nd finest grid level need to be performed per cycle, while only two are required for the 3vcycle. The extra
RKcycle on the 3rd finest grid level for the 4wcycle versus the 4vcycle on the other hand seems to be less
costly and more effective in improving convergence, making the time to convergence for the 4wcycle the quickest
of the cases tested here.



