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December 13, 2007, 13:51 
Asymmetry induced by the mesh

#1 
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Dear all,
I have symmetric domain and symmetric boundary conditions, but I get asymmetric solutions. I am simulating the whole 360° physical domain. Can the asymmetry be due to the mesh? Is there any evidence of this? Can anyone briefly explain this to me, or give me the references to books or papers dealing with this subject? Thank you very much, Ale 

December 13, 2007, 16:05 
Re: Asymmetry induced by the mesh

#2 
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It depends on whether the steady symmetric solution is stable or not. If it is stable, then mesh asymmetry would induce a small amount of asymmetry in the solution. If the steady symmetric solution (with steady boundary conditions) is (physically) unstable, then the physical solution may be either a steady grossly asymmetric solution, or a timeperiodic/chaotic solution that is spatially asymmetric at almost every instant of time. Although the timeaveraged statistics of the latter could be spatially symmetric. If you are dealing with this situation, i.e., the steady symmetric solution exists but is unstable, then it would be very hard to capture numerically without exceedingly careful and precise treatment. In general, even if you began with symmetric initial conditions, the numerical solution would drift off to steady asymmetric or to a periodic/chaotic solution if it had sufficient numerical perturbations (such as algorithm asymmetry or mesh asymmetry). I think that you might be able to somewhat stabilize a physically unstable symmetric solution by the use of sufficient numerical viscosity (so that you get a smeared out but still symmetric numerical solution), but significant mesh asymmetry would be enough to completely destroy the symmetry.
As examples, you could investigate the 2D flow of a uniform stream past an infinite circular cylinder in crossflow. The inviscid flow displays not only upper/lower symmetry but also fore/aft symmetry of the pressure and the streamlines. The viscous flow exhibits stable upper/lower symmetry at lower Reynolds numbers, an unstable upper/lower symmetry at intermediate Re (which leads to a timeperiodic von Karman vortex street), and stable upper/lower symmetry of the timeaveraged solution at turbulent large Re. You can investigate for yourself the difficulties of numerically capturing a symmetric solution in all these regimes. This is not an area I have studied, so I cannot provide specific references offhand, but I am sure it has been much studied in the CFD literature. 

December 14, 2007, 10:19 
Re: Asymmetry induced by the mesh

#3 
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Except for the mesh, there could be other sources of asymmetry coming from an error in the code itself.
It is true that if the problem is symmetric, then the mesh should be chosen to match the symmetry of the problem. The errors (e.g. in finite differences) due to the accuracy of the method (say 2nd, 3rd,..nth order) and even the truncation of the maching (e.g. 1.e16 for single precision in FOTRAN) can add up over thousands of time step and create an asymmetry in the solution, especially if you don't have anything to damp it. A solver can easily introduce energy into a system and produce numerical instabilities. Also the wrong imposition of boundary conditions (e.g. when the BCs are imposed on the primitive variables rather than on the inflowing characteristics of the flow) can produce errors that propagate inwards at the sound speed and can poison your solution. These are just thoughts. Maybe if you explain what problem you are trying to solve and the solver you are using, etc... there might be more we might be able to do to help. For example if the steady solution you are looking for is from a delicate ballance between two very large forces, then even the smallest assymmetry between these forces (the way they are approximated and treated discretely/numerically) can produce 'explosive' results. 

December 14, 2007, 15:44 
Re: Asymmetry induced by the mesh

#4 
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If you are using upwinding schemes, these could introduce asymmetry (as opposed to a central difference scheme)


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