The `DFGBenchmark 1995/6':


Computational details


Short description and remarks


Aim of these simulations


Quantitative comparisons I


Quantitative comparisons II


Quantitative comparisons III


Conclusion


Visualization



Short description and remarks
Flow around a circle in a channel for a (maximum) Reynolds
number Re=100 (see the
Computational details).
This simulation is almost identical with the set of `DFG
benchmarks 1995/6' which can be
found
here .
In this simulation (until T=5), the inflow is nonsteady in
time, until it reaches its `final' profile at T=1.
The aim of the following simulations is to demonstrate
`graphically' the described results and problems via performing
such types of CFD simulations as explained in the cited paper.




Aim of these simulations
We aim to show how hard it is  even for this chosen small
Reynolds number  to calculate the
`grid independent' solution by simply refining the mesh in
space and time only if we ask for a
quantitatively exact representation of the drag and lift
values!
Additionally, we demonstrate how difficult it is to detect
whether the spatial or temporal mesh has to be refined.
In contrast, we give examples how quantitatively 'wrong'
results can be used to produce  online and in real time! 
qualitatively `good' flow patterns and movies which have nothing
at all in common with the `real' solution,
besides the fact that the produced graphical output looks
like `vortex shedding'!




Quantitative comparisons I
The following diagram shows the resulting lift and drag coefficients vs. the time. Here, we applied an adaptive time step control with very small error tolerances so that the shown results are `exact in time', that means that only the spatial error is visible!
For the mesh levels 5  7 (see the
Computational details), the shown lift values
have (almost) the same frequency and amplitude. The only
visible differences are shifts due to the more
or less accurate initial phase. However, being in the
periodically oscillating state, level 5 is already
sufficient.
In contrast, level 4 shows a slightly shifted frequency and
visible damping effects. Finally, on level 2 and 3, the
corresponding solution is (almost) steady!
Similar results are valid for the drag values, again the
results on level 6 and 7 are more or less identical,
as on level 5, too; while the solution on level 4 still might
be accepted. The results on level 2 (drag value about 4)
and 3 (drag value about 3.4) are not acceptable!




Quantitative comparisons II
The following diagram shows the resulting lift and drag
coefficients vs. the time on level 6. Here, we applied
fixed time stepping, with time step sizes
equal to or smaller than those for the previous
`reference calculations'.
`IE' means `Implicit Euler' as time stepping, with the same
higherorder upwinding in space
as before (time step 0.001111), while `UPW' stands for the
highorder Fractional Step scheme in time, but now
with the first order upwinding (time step again 0.001111).
It is remarkable that  even on this fine mesh  the results
look similar to the previously shown results for
level 4, due to the worse time stepping (IE) and the
worse spatial discretization (UPW)!




Quantitative comparisons III
The final diagram shows the resulting lift and drag
coefficients vs. the time, if the applied time step is too large!
The resulting flows typically show amplitudes much too large,
compared with the 'correct' time step sizes,
while often the corresponding frequency is quite good!
Additionally, even for very coarse meshes (level 3 and even
level 2, but with a different time scale for level 2!)
the resulting flow shows the typical nonsteady behaviour
which however vanishes for smaller time step
sizes (in fact, the flow gets steady!!!). While the
quantitative comparison fails completely with respect to the
reference solution, the flow might be used to produce 'nice'
movies, and this even in real time due to the
small requirements in CPU because of the large time steps and
small number of grid points!!!
However, do not forget that the corresponding results are
wrong and obtained by a lucky chance!!! (And many developers
use such mechanisms to demonstrate the superior behaviour of
their CFD software...)




Conclusion
Some conclusions for this kind of flow simulations in this Reynolds number regime are valid:
For the mathematical background concerning these observations
and with respect to general CFD for incompressible
flow problems at all, look at our paper archive, and there
especially at
http://www.featflow.de/ture/paper/habil.ps.gz
and
Stefan Turek's CFDbook, Springer.




Visualization




Please send any comments and suggestions to: featflow@featflow.de 