CFD Online Discussion Forums - grid singularity

There are probably a few options you can consider (assuming you are dealing with the compressible Euler or NS equations):

1) Move to finite-volume discretization. In this case, your flow properties will be located at the cell center (instead of cell nodes), which makes it much easier to deal with grid singularities. On the other hand, implementation of boundary conditions is a little more complicated (you will need to create ghost cells, reflect the normal velocity component for a surface BC instead of just setting it equal to zero, etc.).

2) If the freestream flow is supersonic, you can probably get away without including the region before the discontinuity (for an example similar to a sharp cone, anyway). This is really only appropriate, though, if your singular point corresponds to the stagnation point, and you are sure that only oblique shock waves will result (i.e., a cone/wedge with a large surface angle will not meet this requirement).

3) Use a multi-block approach to divide your grid domain into regions that do not contain singularities. Can be a complicated process depending on the geometry, however, and you will still have to deal with transfer of flow properties at the boundaries of each block. Basically, you are adding complexity to eliminate the singular point.

Personally, I prefer option #1 as it is (in my opinion) the easiest and most robust method of dealing with complex geometry containing singularities. Option #2 is an easy method for some cases, but may only work for simple cases (e.g., cone/wedge/plate in supersonic flow). Option #3 is probably only worth considering if you already have a multi-block grid generator/solver available (if you are writing your own, you might be better off considering option #1).

Keep in mind that most of my work is in finite-volume compressible Euler/NS, so I may very well be missing some other possiblities in the above explanation...