the pole singulatity of cylindrical coordinate
 

Hello, everyone I am solving the Euler equations in cylindrical coordinate. The initial problem is the an imploding shock wave heading for the origin. However,when the shock arrive the origin,the computation can continue due to the singularity of pole. I have looked up some references, but I have not get a good answer, especially for the strong shock waves. I wish that someone could give me some proposals. thanks a lot!

Re: the pole singulatity of cylindrical coordinate
 

Check out these references:

# Numerical treatment of polar coordinate singlularities. J Comput. Phys. V.157, 789-795

# Numercial treatment of cylindrical coordinate centerline singularities, IJCFD, 2001, Vol 15, 251-263

Re: the pole singulatity of cylindrical coordinate
 

thanks a lot. In fact, I had checked out the two references listed above and other ones. But I still can't get the solution.

Re: the pole singulatity of cylindrical coordinate
 

Method 1: for ideal gas, use Guderlay (see Landau & Lifshitz classic Fluid Mechanics) self-similarity solution as asympotic solutin on 3-5 grid points near the pole . Method 2: (finitie volume layout). The first grid point is half-grid size away from the pole. Use reflection symmetry condition whenever needed.

Re: the pole singulatity of cylindrical coordinate
 

Do you (anyone) have a reference for the 2nd technique...that for Finite Volume?

Thank you.

Re: the pole singulatity of cylindrical coordinate
 

thanks! I would like to to adapt the first method, for my numerical method is based on the difference method.

In my pesent method, I used a symmetrical boundary condtition for pressure, density and other scalar variables, and antisymmetric boundary condition for velocity accross the pole. And the first grid point was set the half step distance from the pole. This method is feasible the weak imploding shock waves or not very large contact discontinuities, but failed the strong ones. So I want to find a better solution method for the pole singularities.

Re: the pole singulatity of cylindrical coordinate
 

During the 90's, I used (for different tasks) CFX-4, Fluent, and Flow-3d. The general problem of the polar singularity was treated the same by each - at least in the training sessions.

The recommendation was to abandon the cylindrical system and use block-structured boundary-fitted meshes in the Cartesian formulation. For Flow-3d, the recommendation was to use the FAVOR feature. Either of these approaches removes the singularity completely.

The reason for choosing a cylindrical system is usually because one or more of the boundaries is circular. Using the recommended approach, the boundary is accomodated without using a cylindrical coordinate system.

I don't know what the trainers for those three codes would recommend today. And I don't know what advanced features the codes may have today. But the marketing guys for the codes could certainly tell you!