momentum under-relaxation in colocated variable arrangement
 

Hy,

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 problem is that the solution shows an undesirable (and largely unacceptable) dependence on the momentum under-relaxation factor. The bug comes from the momentum interpolation used for the normal cell face velocity (formulations of Rhie or Peric for this interpolation seem very close). In Peric's book it was mentioned as beeing of less concern. Somewhere else (Majumdar, S., "Role of Underrelaxation in Momentum Interpolation for Calculation of Flow with Nonstaggered Grid", in Numerical Heat Transfer, vol. 113, pp. 125-132, 1988) it was stressed that the role of underrelaxatation can be important and a modified (not radically) cell face velocity interpolation was proposed. It was not very clear what was the effect of underrelaxtion or if it was eliminated. Finally Majumdar underlined that the effect is problem dependent and is eliminated only when pressure is a purely linear function of space variables. It is exactly what I'm obtaining, excepting perhaps that I have a stronger influence on the underrelaxtaion parameter (it might be normal for the particular problem I'm dealing with).

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 (for staggered grid on rectangles the problem is not addressed, there is no need for a special interpolation of the celle face normal velocity, everything works well). I will highly appreciate any comments, hints, references, discussions and sharing experience about this problem. Sincerely,

Mihai ARGHIR

Re: momentum under-relaxation in colocated variable arrangement
 

(1). Are the equations you are trying to solve Inviscid equations? or Viscous equations? (2). Are those compressible equations? or Incompressible equations? (3). Are those equations steady state equations? or Transient equations? (4).In these equations, what are the primary dependent variables? (5).What is the actual problem you are trying to solve? Flow in a cavity? Flow over a wing? or Flow through a pipe? (6). Have you tried to solve a test case with known solution? (7). I would strongly suggest that a test case with known analytical solution be selected and used to check out your program. (8). If you want to develope methods, the best place to start is : someone's PhD dissertation, then a good report from a government laboratory, and a technical paper presented at a meeting. (9). By the time a technical paper is published , most important materials would have been gone. And by the time a book is published, it is most likely that it is just for information only. (10). Anyway, were you able to solve your Euler equation with viscous source term? Were you able to solve your physical problem, which is still unknown to me?

Re: momentum under-relaxation in colocated variable arrangement
 

I am not sure what your question is but if you are having problems with large source terms in the momentum equations which result in unrealistic pressure fields then I may be able to help.

Let me know,

Jeff

Re: momentum under-relaxation in colocated variable arrangement
 

Hi,

I've seen this one before: the problem is that in the interpolated velocity onto the face you've got the "under-relaxed" terms and the balance between the "real" and "additional" terms messes up your solution. For example, your velocity is

a_P U_P = \sum_N a_N U_N + (1 - alpha)/alpha U_old

and then you interpolate this lot onto the face. The last term depends on the under-relaxation factor and therefore introduces dependence on the convergence history (read: alpha). The solution is: when you interpolate the U_P from the above equation to the face, take out the interpolated equivalent of the last term and use the face flux instead - that one does NOT depend on convergence history.

P.S. It works!