[GregCFD]

Hi Everybody,

I've done some searching through academic papers, the general internet and here and I can't find a clear answer to how to implement boundary conditions in a cell-centered (non-staggered) finite volume code. In particular I'm trying to adapt Chorin's artificial compressiblity method to a simple FV method using forwards Euler and central differences. The issue is that to remain stable I need a ridiculously tiny time-step, and I figure it might be because I haven't implemented the BC's correctly.

The question is then:
In FVM can I implement the wall boundary conditions directly without the use of ghost cells. Consider the northern wall of a system:

_N_

.C

Then for a generic internal cell we can write the U,V momentum as
U = -0.25*dx*(U(i,j) + U(i,nN))*(V(i,j) + V(i,nN)) + v*dx*(U(i,nN) - U(i,j))/dy
V = -0.25*dx*(V(i,j) + V(i,nN))*(V(i,j) + V(i,nN)) - dx*0.5*(P(i,j) + P(i,nN)) + v*(V(i,nN) - V(i,j))/dy;
and the continuity equation as:
C: (V(i,j) + V(i,nN))*dx*0.5;

As we need the flux at the cell interface can be directly implement it using the known values at the boundary?
U = -dx*UN*VN + v*dx*(UN - U(i,j))/(dy/2);
V = -dx*VN*VN - dx*P(i,j) + v*(VN - V(i,j))/(dy/2);
C = VN*dx;

Where I've assumed that dP/dN = 0 at the boundary.

[FMDenaro]

Your problem is not caused by the BCs but by the FTCS method your are using. There is a strong numerical stability issue in this method, you can get a stable solution only if the cell Reynolds number is equal to 1. But the accuracy is very low.


Have a look at this discussion https://www.researchgate.net/post/Is...fF2-yQfeirSdg4

[GregCFD]

Originally Posted by FMDenaro

Your problem is not caused by the BCs but by the FTCS method your are using. There is a strong numerical stability issue in this method, you can get a stable solution only if the cell Reynolds number is equal to 1. But the accuracy is very low.


Have a look at this discussion https://www.researchgate.net/post/Is...fF2-yQfeirSdg4

Thanks for that, I was worried it would be Re (or CFL-like) condition but I was hoping it would be the BC's.

Is the BC implementation actually correct though?, I've seen quite a few posts where people are using the ghost-cell method where as applying the velocities as a Dirichlet method seems so much easier. As well as some disagreement about how the pressure B.C. is implemented, it might be missing a velocity gradient term?

[FMDenaro]

Originally Posted by GregCFD

Thanks for that, I was worried it would be Re (or CFL-like) condition but I was hoping it would be the BC's.

Is the BC implementation actually correct though?, I've seen quite a few posts where people are using the ghost-cell method where as applying the velocities as a Dirichlet method seems so much easier. As well as some disagreement about how the pressure B.C. is implemented, it might be missing a velocity gradient term?

The Chorin projection method is tricky on a non-staggered grid. Depending on the pressure discretization you can get either an exact projection that has troubles in the spurious solutions or an approximate projection that does not satisfy exactly the divergence-free constraint. That is not only depending on the BCs but also on the stencil of the pressure equation