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May 29, 2003, 13:32 
singularity problem in cylindrical coordinate

#1 
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Hi all,
I've been working on a secondorder 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 NavierStokes 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. 

May 29, 2003, 13:52 
Re: singularity problem in cylindrical coordinate

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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.


May 29, 2003, 14:03 
Re: singularity problem in cylindrical coordinate

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For Euler and NavierStokes 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.


May 29, 2003, 14:12 
Re: singularity problem in cylindrical coordinate

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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? 

May 29, 2003, 16:03 
Re: singularity problem in cylindrical coordinate

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In Euler & NS 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 axisymmetrical 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",SpringerVerlag, Berlin, 1986, but in reality the idea is all here.


June 2, 2003, 07:46 
Re: singularity problem in cylindrical coordinate

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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 finitedifference scheme for c threedimensional incompressible flows in cylindrical coordinates", c J. Comput. Phys. 123, pp402414, 1996. 

June 3, 2003, 11:47 
Re: singularity problem in cylindrical coordinate

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


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