I’m sure plenty of people will have views on this, it would be great to hear what others have to say.
I see there are really 3 questions here:
- How do I adapt the resolution for different orders?
- How do I know when and how much anti-aliasing to use?
- Should I curve the boundaries?
The first question is probably one of the oldest questions in CFD. A reasonable zeroth-order metric for the resolution of the method might be the points per wavelength. I looked at this for FR in a 2018 AIAA-J paper; however, this is not the whole story. The amount of resolution required will depend on the solution, which will depend on the case. It will also depend on how accurate you want the result to be. If you are doing DNS, then you can look at the time average and instantaneous Kolmogorov scales and make a judgement based on that. (But with DNS the more the merrier). If you are doing ILES, then a mesh convergence study will be what you’ll want to do. You’ll make a few meshes of different resolutions and look at the convergence of one or more functionals. Lots of journals would expect this if presenting LES results. With experience, you will pick up rules of thumb for mesh generation, but this will be problem dependent. For FV, Paul Tucker has a whole chapter in his book about mesh generation more focused on turbomachinery.
On the topic of anti-aliasing, as you say, this will be dependent on the mesh resolution. Lower resolution, will lead to high order modes in your solution being more excited, that coupled to a non-linear flux function will cause more aliasing. Aliasing will redistribute the unresolved modal energy to other modes. The upshot is that although there are plenty of papers on how much AA you should have for Navier–Stokes for example, really it depends on how much energy isn’t getting resolved. So if you have a course mesh then you can try adding some in a similar way to the mesh convergence study. Modal filtering is also an option.
On the topic of mesh curving, and you might spot a theme here, you should use as much or as little as you need. In that Bassi and Rebay paper, if they had discretided the wall with a trillion points they clearly wouldn’t need to curve the mesh. But if you are doing ILES, then you probably will need some curvature. However, for a square cylinder, you probably shouldn’t curve the walls as it makes the mesh a less accurate representation of the geometry, not more. I can come up with counterexamples to this though, for example, the Mach 3 forwards facing step, when solved with Euler, there should be a singularity at the corner. To get an accurate result it is best to round the corner just a little. Curving the mesh can also give rise to aliasing, and non-conservation of linear invariants (although that is a topic for another time). So if you curve the mesh you may need to keep track of aliasing.
That was a long one but I hope it helps. I would be interested to hear from the community if they have anything more to add for specific geometries and cases. E.g. laminar vs. turbulent boundary layers, sharp features in cases like delta wings, and how order plays with cell aspect ratio.
@tdzanic you have great experience in meshing, any wisdom from you?