[Anna Tian]

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?

[ogloth]

Hi,

I haven't done excessive tests, but I found that for a low-Re 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 skin-friction and consequently a wrong drag coefficient. If you are in the log-layer 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

[flotus1]

Originally Posted by Anna Tian

Hi,

In ICEM, expansion ratio is not a criteria for the grid quality test.

This is why you should not rely on a single "quality" estimator for the grid.

Sometimes, if we use small wall cell width, the expansion ratio has to be large

I would not subscribe to this statement. If the width of the first cell has to be small due to Yplus-constraints, the adjacent cells have to be of similar size in order to capture the large gradients of the flow near the wall.
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.

[ogloth]

Originally Posted by flotus1

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.

That is certainly true! Any layer you can save, however, saves a fair amount of computational time. Different regions of the flow field have different constraints on the stretching ratio. For a hybrid grid (prism + tets/polys) the ratio in the laminar sub-layer is very important. However, the ratio at the transition point between boundary layer mesh and far-field is not as important. Of course you still have some constraints and cannot just jump by a factor of 100 ... ;-)

Do you know of any best practice guide for the stretching in the different regions?

Cheers,
Oliver

[flotus1]

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 RANS-simulations of turbulent boundary layers.

The expansion ratio within the boundary layer should not be higher than 1.2 in wall-normal 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 well-resolved 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.

[ogloth]

Originally Posted by flotus1

The expansion ratio within the boundary layer should not be higher than 1.2 in wall-normal direction, I thing we agree on that. This ratio is constrained by numerical accuracy.

Is this true for the whole boundary layer, or only for the viscous sub-layer? In the log-layer the gradients are not as steep. Assume the following:

We go from highly an-isotropic cells (ratio 5000) to iso-tropic 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

[FMDenaro]

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

[ogloth]

Good morning,

Originally Posted by FMDenaro

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

How do you derive this rule and what does it guarantee if you fulfil this requirement?

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 non-equidistant 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 low-Re model the boundary layer can easily be 50% or more of the total cell count. Any saving here would be good!

Cheers,
Oliver

[Anna Tian]

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?