Hi all,

I've been working on a second-order differential equation using FDM. It is ok when working in Cartesian coordinate. However, in cylindrical system, there is a term '1/r' being introduced, which is singular at r=0. I dont know how to work around it. I know there is some paper talking about Navier-Stokes equations in cylindrical coordinate. But my problem is different in this case.Could anyone suggest some reference book or paper on a general description of solving differetial equations in cylindrical system using FDM? I appreciate that.

See <a href=http://www.flatface.net/~praveen/refs.html>this</a> page under the heading "Polar/Cylindrical Coordinates" for some references which I got on this forum a few days back.

For Euler and Navier-Stokes equations in cylindrical case we have source term which is proportional to v/r. At r=0 this is avoidable singularity point 0/0 (v=0 at r=0). One widely used approach is to replace v/r at r=0 by dv/dr.

Peter,

I dont quite understand, why v=o at r=0? if the cylinder is symmetric, dv/dr = 0 makes sense to me. can you give me some reference?

In Euler & N-S case after change of coordinate system we have source term which is like nu*V*RO/r, where nu=0 for plane case and nu=1 for axi-symmetrical case (V is Y velocity and ro - density). In plane case nu=0 and we have no singularity. For axisymmetrical case by symmetry condition V=0 for r=0. So we can replace v/r = dv/dr / dr/dr = dv/dr. I am not sure where it was published, but when I was busy with CFD, it was usual approach. Probably you can find something in Pirumov, UG, Roslyakov, GS, "Gas Flow in Nozzles",Springer-Verlag, Berlin, 1986, but in reality the idea is all here.

Verzicco & Orlandi have implemented a excellent finite difference method using staggered mesh and q1=V_t, q2=r*V_r, q3=V_z to handle the singularity of r=0. The recent book of Orlandi has the source code, and the reference J.C.P. paper is

c 1. R. Verzicco and P. Orlandi, "A finite-difference scheme for c three-dimensional incompressible flows in cylindrical coordinates", c J. Comput. Phys. 123, pp402-414, 1996.

Look for a recent paper by LELE S.K. in Journal of Computational physics