projection method(3D)
 

Hi

I want to simulation the flow in helical pipe.
I have one Continuity equation and three momentum equations in s=>(u),r=>(v),teta=>(w).
I used staggered mesh and descritized all of terms in central(finite difference).
I want to use projection method for solution these navier-stokes equations.

first problem:
What is stability condition for helical coordinates (three dimensional)in projection method?because I don’t know which number of interval in s,r,teta causes convergence!

second problem :
In descritized scheme we have i+2,i-2,j+2,j-2,k+2,k-2 indexes for u,v and w.
what do I have to do for the nodes of near the boundary? It means that two nodes produced out of domain but I don’t know value of them and they are unknowns.

Third problem:
We have to apply boundary conditions.in s and teta I have to apply periodic
Conditions.what are periodic boundary conditions for u,v and w in s and teta?

Is there any paper or books(references) for programming of projection method(three dimensional)?

Thanks for your guidance and aid

One possible way to resolve the second problem is to define so-called fictitious or ghost cells on each side of the mesh - two layers in your case. You will solve the equations in real cells, say up to index IMAX in the x-direction, and the ghost cells will store variable values in cells IMAX+1 and IMAX+2, so that you can reach out to use them without worrying about exceeding array dimensions.

The ghost cells may also help in imposing periodic boundary conditions.