Convergence for axisymmetric flow
 

Hi everyone,

Question about axisymmetric flow: I'm using a 2-D planar flow solver that I want to modify to handle axisymmetric geomerty.

I've derived the NS equations in cylindrical coordinates, and coded the source terms (implicitly), but the solver does not converge anymore. Running 1st or 2nd order, inviscid or viscous, nothing can make it to converge.

Actually it does converge for a static-to-stagnation pressure ratio of .35, which is not very interesting as the flow separate right after the throat of the nozzle. No run with a lower pressure ratio converged.

Any solution to improve convergence ?

bl201

ps: I'm quite a beginner in CFD and on this forum; thanks for your help...

Re: Convergence for axisymmetric flow
 

Assuming that you have no bugs in your code, a likely reason for encountering problems with axisymmetric geometries as compared to their equivalent 2-d counterparts is that the flow variations are much larger in former as compared to latter so the inherent dissipation in your numerical scheme will trigger separation even in an inviscid flow which is physically incorrect. You need to increase grid points downstream of the throat substantially in both directions to avoid this numerical artifact and hopefully your code will converge. Keep increasing number of points until your inviscid solver does not yield a separated flow solution then add more points near the wall for capturing boundary layer for viscous computations in a given geometry.

Re: Convergence for axisymmetric flow
 

Check carefully the treatment of your BC, mainly for r=0. If you use FDM, this may cause difficulty, and special techniques were developed to overcome it. In the FVM this is not an issue, since the flux there is multiplied by a zero area.

Re: Convergence for axisymmetric flow
 

Thanks for your ideas.

Could tell me a bit more about these special techniques for r=0 BC ?

Re: Convergence for axisymmetric flow
 

A recent paper on this subject is

G. S. Constantinescu and S. K. Lele, A Highly Accurate Technique for the Treatment of Flow Equations at the Polar Axis in Cylindrical Coordinates Using Series Expansions, Journal of Computational Physics 183, 165–186 (2002).

I prefer, however, the weak forms (i.e., FVM and FEM), where - as mentioned in my former posting - this is not an issue at all.

Re: Convergence for axisymmetric flow
 

Actually, I'm using Finite-Volume method...

I can run 1st order, results are good (for 1st order, that is), but 2nd order blows up