I'm trying to solve the problem of an homogeneous propellant burning into a rocket motor ( 2D ). The formulation is based on the conservation equations of mass, momentum, energy and species concentration, with consideration of finite-rate chemical reactions.

I'm using ALE (Lagrangian-Eulerian) finite volume formulation on a structured (finite difference) mesh with ICE (Implicit Continuous-Fluid Eulerian) method. Mesh rezoning is implemented in order to resolve the high gradients in the flame zone and solid degradation layer. (Solid cells may be converted into gas cells during the calculation.)

Here are my problems. When dealing with very different time scales and Mach numbers ranging from 0.001 to almost 1, I found a lot of problems with convergence and stability of the ICE-ALE algorithm. In adition, the geometry involves large aspect ratio cells (1:1000). I always like finite volumes because physics is clear for each term in the equations, but I'm afraid that they are not very efficient in order to perform all mach number calculation or even to deal with large aspect ratio cells. Is that true? shoul I migrate to another method? Can anybody give me a hint ?.

Thanks in advance Andres

Here is a brief description of my adapted ALE - ICE algorithm:

1) Initialize thermodynamics, transport and chemical kinetics data.

2) Load initial mesh, boundary and initial conditions.

3) Determine time step.

4) Integrate chemical reaction terms to update e and Y.

5) Integrate the heat diffusion term (ADI) to update e.

6) Integrate mechanical diffusion equation to update Y and e (Fick's law is used).

7) If neccesary, add artificial pressure

8) Advance (u,v) explicitly with momentum equation ( this would be the initial guess for step 10). Apply boundary conditions.

9) Evaluate relaxation factor on each cell (derivative of the function f = p - EOS, respect to p)

10) Obtain time advanced pressure implicitly (in a Lagrangian way) with a Newton-Raphson like solver applied to the momentum, energy and equation of state equations. (Maybe this step should require further explanation.)

11) Iterate step 9) sweeping through the whole mesh until no relevevant pressure corrections is found.

12) Obtain new (u,v) with the latest pressure field using the momentum equation, then apply boundary conditions again.

13) Complete the Lagrangian step updating the internal energy with the field pressure, and then move the mesh vertices with the fluid.

14) Mesh Rezoning : Vertices are moved with arbitrary velocities in order to: i) prevent the mesh from undergoing large distortions, ii) be able to resolve the different length scales involved int the integration domain, (donor cell scheme used) iii) and to change boundary conditions if any solid cell converts into gas cell (due to combustion).

15) Update thermodynamics, transport and chemical kinetics.

16) Swap vectors and repeat time marching ( steps 3 to 15 ) until solution is completed, that is when the propellant finishes.

where: (u,v) = vertex velocity p = cell pressure e = cell internal energy E.O.S. = equation of state

There are two things in your problem that are challenging. First, the low Mach number requires setting very tight limits on the pressure iteration. A good rule of thumb is that the relative error in the pressure field, abs(dP/P), must be less than the square of the Mach number. So for Mach 1.e-3, the pressure must be iterated to one part in a million. A safety factor of 0.1 is not a bad idea, just to make sure. Note that this requires that you run with 8 bytes in a floating point number (double precision on most computers) since 4 bytes gives you only about 6 significant figures. Also, ICE can have convergence problems for CFL numbers greater than about 10 for the SOR implementation. Second, the 1:1000 aspect ratio could be causing a lot of your trouble. I have experienced this, but I don't have a fix for it other than to use a less extreme grid. You might want to run some numerical experiments with ratios down around 10 or less to see if that helps. Hope this helps.

Thanks a lot for your response. The rule [abs(dp/p)< M**2] was of great help for me. Respect to the CFL number, you were absolutely rigth: I could reach a CFL number of around 10, and when I try to use greater CFL values then convergence begins to fail. I'll follow your advice, and run numerical experiments with lower cell's aspect ratio.

best regards, Andres