Evaluation of "diffusive flux" of Compressible Navier-Stoke in the corner
 

Dear all,
I have a question about the evaluation “diffusive flux” at the corner A like this figure.

http://img96.imageshack.us/img96/1622/68nm.png

In my code, I’m developping Navier-Stokes compressible equation 2D (written in DENSITY BASE form) with collocated grid and Explicit Scheme Euler 1st order with Upwind TVD 2nd order and Splitting Flux with ROE approximation scheme used. For the boudary condition, I use two layers of dummy cells because my stencil of solver has second order in space.
For the “diffusive flux”, I can calculate easily for all points in my domain except the point in the corner for exemple A (where two boundary conditions are intersected). To estimate the “diffusive flux”, I used the gradient of the variables at the faces of cells. But in the corner for instance A, I can not evaluate the gradient du/dy at the face F because I don’t have information of the point B. I tried to find the information on Internet but I have not yet found the solution.
I hope you could give me some helpful advices.
Many thanks and best regards.

what kind of boundary you have at flux-section F?

Dear FMDenaro,
The boundary condition at face F either outlet condition or wall condition but I would like to find a method which is able to apply for any boundary condition. I wonder if it's possible. Anyway, I need an advice.
Thank to you.

well, a method can not be really general as the type of boundary addresses different choices.
For example, if F is a wall, you have u=v=0 but also du/dy=0. For an outflow you must assume some physical beahviour of the flow.

I supppose you are using linear extrapolation to prescribe values in the ghost points. Therefore, to set a value in B you could work along the same idea. That is, assuming the point A (i,j), B (i+1,j+1), consider a bi-linear law involving points (i,j), (i+1,j), (i,j+1) and (i+1,j+1). Once the first three values are prescribed, you need a condition to compute the extrapolation.
You have to use some physical information about the boundary

Dear FMdenaro,
You can explain your method of bi-lineatmr more clearly. We know the value of three points but the other information is not available. It's enough to find the value of B. :(
Besides, in the case of no-slip wall, we have du/dy=0 but for the symetry condition the gradient du/dy is still curious. You have some ideas for the determination of this gradient?
I think the problem of corner is still difficult for the numerical method and there are not many people who focus in this problem. If you have read the documents concerning this problem. Thank to share it with me.
Best regard and thanks very much.

Dear Tranhophu,

Have you found a better solution for the boundary condition at the corner? I simulate flow with multi species. Somehow I get fluctuations near the wall corner.

Best.