SIMPLE and energy equation convergence
 

Hello everybody. Here is my question: I have small a CFD code (I wrote it myself) to solve Navier-Stokes-like 2D equations (actually, I have strong simplifications in one direction, but this is not important here). At the moment, I solve for continuity and momentum (coupling them with the SIMPLE algorithm) and energy. My domain is enclosed in two curved sides. I inject mass, momentum and energy from one of them (side -A-) and I extract them from the other (side -B-). At side -A- I have fixed input fluxes, at side -B- the fluxes are function of the flow variables (density, velocity and temperature).

I did a few tests: 1. Continuity and momentum only, fixed temperature: ok (very slow convergence, but I do not implement (at least not yet) any convergence acceleration method, so I am not terribly surprised).

2. Energy only, with zero velocity field: ok. Please note that in this case I had to change the boundary condition on the B side to fixed temperature, since a computation of the out-fluxes in this case would give me zero flux, because of the zero velocity field.

3. Energy + continuity-momentum: no. More precisely, a first test crashed, and a second one shows EXTREMELY slow convergece rate (more than 1 order of magnitude slower than the previous cases taken separately).

I tried to get an idea of what the reason could be, and I am coming up with two possibilities - The boundary conditions are probably not terribly nice. Specifying the flux at both inlet and outlet boundaries could also, in some cases lead to an ill-posed problem. So I could probably think of some new set of boundary conditions. - The coupling between the continuity-momentum and energy equations is not optimal. On this point, one more consideration: at present , my scheme runs like this: 1. Guess the velocity

2. Get the pressure and velocity corrections

3. Based on the new pressure value, update the density

4. Solve for the energy

5. Back to step 1

In points 2 and 3, whenever I have to link the pressure to density variations (my scheme is compressible) I use an iso-thermal law (perfect gas). I am now wondering if this is not perturbing the energy (I can turn the momentum equation in an equation for the kinetic energy, not for the total one), so slowing down the convergence. So are my questions:

1. Is there any experience around with similar problems? (I hope so)

2. Does people adopt other choices, different from iso-thermal, when using the compressible pressure-correction scheme and coupling with the energy equation? In particular, I am thinking trying with an adiabatic law, which seems to me more physically sound. However, any other experience report and/or literature on this point would be welcomed.

3. Any more general literature suggestion on partial or total coupling schemes (I would higly welcome some review literature on this point).

4. Any other suggestions? (They would be very welcomed too!).

Thank you to everybody for your time/suggestions

Fabio