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Convergence versus meshing granularity

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I have noticed interesting behaviour in my models.

If the model will solve using a coarser mesh, it may not solve well using a finer mesh.

And although use of the boundary layer tool can reduce the wall lift-off, it does not necessarily reduce convergence time, and often can lead to convergence instabilities. Sometimes I can only add 2 boundary layers.

Also, if I use an initial field from a coarsely meshed solution, the convergence behaviour for the finer mesh will only begin to act erratically earlier.

I have noticed these effects using the k-epsilon and k-omega turbulence models.

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