The `DFG-Benchmark 1995/6':
Channel flow around a circle at Re=100

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.

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'!

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!

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 higher-order upwinding in space as before (time step 0.001111), while `UPW' stands for the high-order 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)!

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...)

Some conclusions for this kind of flow simulations in this Reynolds number regime are valid:

• Level 4 (about 8,000 cells) is sufficient for a `good' simulation with respect to qualitative ratings, while at least level 5 (about 30,000) or even level 6 (more than 100,000 cells) is necessary for an `exact' quantitative representation of such drag and lift values. This may be improved by 'better' grids (by hand or by a rigorous error control!), but for larger Reynolds numbers, the number of mesh points and time steps will increase further!
• `Bad' time stepping (Implicit Euler) or `bad' discretization techniques (First Order Upwind) may completely destroy all high-accuray results and let the solution look like `coarser grid' solutions! Consequently, the question arises how to differ between spatial and temporal error!
• Projection-like techniques (as in Van Kan or Chorin schemes, many Pressure Correction or Fractional Step schemes) overshoot the solutions if the used time step is too large! Since the arising numerical instabilities look by a lucky chance as `vortex shedding', this `technique' may be used to calculate in real time even some very complex flow patterns. But, the corresponding numerical solution is wrong in a quantitative sense, compared with the exact solution of the underlying partial differential equations!!!

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 CFD-book, Springer.

• Distribution of temperature/concentration via Boussinesq model
  

  Visualization via temperature distribution from the heated cylinder only. The first row always shows the results 'exact in time', that means calculated with the described adaptive time step control, for the levels 7, 6 and level 5 which all look like very similar. The second row, with the same time stepping, contains the results for level 4 (left), and for levels 3 and 2 which give (more or less) wrong steady results!

  Additionally, the third row gives the results on level 6, but with Implicit Euler (left) or First Order Upwinding (right) which leads to huge damping of the flow structure!

  Finally, the fourth row gives results for `large' time steps, on level 3 and 2, which results again in `vortex shedding', but do not forget to compare the resulting quantitative comparisons for lift and drag values! All movies occupy about 2 MB at the most.
  

  
  

  
  

  
  

  
  
  

• Pressure
  
  

  Corresponding visualization via shaded pressure plots (about 1 - 2 MB).
  

  
  

  
  

  
  

  
  
  

• Streamfunction
  
  

  Corresponding visualization via shaded streamline plots (about 1 - 2 MB).
  

  
  

  
  

  
  

  
  
  

• Velocity
  
  

  Some representative vector plots (about 18 and 9 MB).