Question about the large expansion ratio influence to the numerical accuracy
Hi,
I'm wondering how's the large expansion ratio influence the numerical accuracy. In ICEM, expansion ratio is not a criteria for the grid quality test. But in the manual of autogrid5, it tells the expansion ratio need to be lower than 1.5. Sometimes, if we use small wall cell width, the expansion ratio has to be large. But the large wall width can't resolve all the fluid behavior near the wall. May I ask how to balance these two factors? 
Hi,
I haven't done excessive tests, but I found that for a lowRe turbulence model the difference in layer height should not be much more than a ratio of 1.2  even 1.5 as you suggest gives wrong values for the skinfriction and consequently a wrong drag coefficient. If you are in the loglayer region of your boundary layer a higher stretching ratio might be acceptable. Maybe one of the turbulence modelling experts could shed some more light on this. Of course it also depends on your discretisation, but I assume that most of today's finite volume CFD codes are fairly similar. It might be a good idea to do some tests and come up with an optimal stretching ratio for the different regions of a boundary layer. Cheers, Oliver 
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There is no way around a high cell count with a small expansion ratio if you want to accurately simulate the flow field near a solid boundary. This is one of the reasons why CPUs are never fast enough. 
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Do you know of any best practice guide for the stretching in the different regions? Cheers, Oliver 
To be honest, I have only little experience on flow simulation outside the academic world, but I can share a few thoughts.
Let us limit the discussion to RANSsimulations of turbulent boundary layers. The expansion ratio within the boundary layer should not be higher than 1.2 in wallnormal direction, I thing we agree on that. This ratio is constrained by numerical accuracy. Now in the transition between the boundary layer and the far field is constrained by a different consideration. Even if the gradients are small enough to be wellresolved by a mesh with high expansion ratio, you should not exaggerate here. The key word is volume jump. High volume jumps degrade the convergence behaviour of the simulation. So the smaller mesh size that could be achieved with a high expansion ratio between the far field and the boundary layer leads to a higher number of iterations necessary to converge the computation. For this reason, even in the far field, I would not go beyond 1.5 for the expansion ratio. 
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We go from highly anisotropic cells (ratio 5000) to isotropic cells in the far field. With the standard 1.2 ratio we end up with appr. 48 layers in the boundary layer. If we could use a ratio of 1.5 starting from y+ = 30, the number of layers would reduce to 33. This could be a huge reduction in the overall mesh size ... could be tested with the standard flat plate case. Cheers, Oliver 
using a stretching law, transition from a cell to the adjacent one must be smoot and regular, so that the derivatives of the stretching law are O(1).
You can also see paragraph 3.3.4 in the book of Ferziger and Peric 
Good morning,
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I had a quick look at the chapter you mentioned. What Peric does is trying to "save" the second order accuracy for stretched grids, because formally the discretisation is only first order accurate. He even gives the example of singularities in the stretching law. Peric argues that a nonequidistant grid will be 2nd order accurate if you refine the grid by dividing all cells in a half. The argument is that the region with formally 1st order become smaller and consequently the global truncation error will decrease like for a 2nd order scheme. A similar argument holds if we have one or two singularities in our stretching law for the boundary layer grid. The discussion is somewhat academical, however. In practice we are not going to refine the grid which means that we have to live with the truncation error we have. The singularity in the stretching law is no problem: The truncation error for the second derivative at any point i is O(dx_i+i  dx_i), no matter what the stretching of the neighbouring steps is. I still think it would be a worthwhile activity to look at the boundary layer grid and try to find a minimal point distribution which still delivers the correct skin friction. For a steady RANS simulation with a lowRe model the boundary layer can easily be 50% or more of the total cell count. Any saving here would be good! Cheers, Oliver 
But what about the influence of large expansion ratio at the other places of the fluid domain instead of close to the wall? Will that largely influence the numerical accuracy?

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