momentum under-relaxation for compressible flow with SIMPLE
I'm writing a program for solving the 2D Euler equations with some source terms (it is in fact a simplification of Navier-Stokes equations for very particular conditions, when viscous effects can be expressed as source terms, a well accepted and used model, no problem). Equations (continuity, two momentum and energy when compressibility is important) are discretized on a triangular grid using colocated control volumes. The solution is then obtained using the SIMPLE algorithm for dealing with the pressure-velocity (and eventualy density) coupling. The approach is largely borrowed from Ferziger and Peric's book (Computational Methods for Fluid Dynamics, Springer, 1996).
The undesirable dependence on the momentum under-relaxation factor is eliminated by using Majumdar's approach ("Role of Underrelaxation in Momentum Interpolation for Calculation of Flow with Nonstaggered Grid", in Numerical Heat Transfer, vol. 113, pp. 125-132, 1988) for interpolating the normal cell face velocities. It works very well for incompressible flows but for compressible ones, I canot obtain a converged result (although divergence of the SILMPLE algorithm is not occuring and the residues are "stable"). If I switch to the classical momentum interpolation scheme of the cell face normal velocity, convergence is easely obtained but the results depend on the under-relaxation factor used for momentyum equations (they are not incorrect but hardly acceptable). The lack of convergence of the compressible SIMPLE algorithm (collocated control volumes) is then due to Majumdar's special interpolation scheme for normal cell face velocities?? I've never seen this problem discussed or cited and I used Majumdar's scheme exactly as for the incompressible flow (where it worked very well).
That is all my experience I hope my message was clear enough. Again, it is a problem that I have with collocated control volumes on triangular grids and for compressible flow. I will highly appreciate any comments, hints, references, discussions and sharing experience about this problem. Sincerely,
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