[Matheus.Costa]

Hello,

I'm doing my thesis right now, which consists on the discretization of flow on steady state through an orifice plate using the central finite difference of second order applied to the Navier-Stokes equations, a also using a Poisson to calculate de pressure. I'm using Matlab for codding by the way, and using a staggered grid (Cartesian coordinates), and the SCGS( Symmetric-Coupled Gauss-Seidel) as the solving method for the equations' system. I'm having some trouble implementing the method, specially on how to deal with the boundary conditions. I would be very thankful if any of you guys could shed some light on that matter. The graphic results that I want are the velocity gradient and the pressure associated to it. Thanks for the attention.

[Gerry Kan]

Dear Matheus:

If you are using finite difference method with staggered grid, your grid will have uniform spacing. How you incorporate boundary conditions into the grid will depend on the stagger arrangement. Typically (and I mean typically) velocity components will be face centered, while other scalars will be cell centered.

For the face centered variables they are relatively trivial; the boundary node are at the boundary so you simply have to write an equation there to reflect the boundary condition.

For cell centered variables, your boundary is staggered. You need an extra cell beyond each boundary, and effectively write your linearized equation so that the boundary conditions can be maintained.

Of course, I am assuming that your boundary conditions are first order accurate. You will need to pad more cells beyond the first and last cells if you need higher order interpolation.

I don't really have too much time to write out the equations, but I hope the descriptions would help you get started.

Sincerely, Gerry.

[Matheus.Costa]

Thank you, indeed the velocities are face centered and the pressure is cell centered, I forgot to mention it.
So I have to use an extra cell outside my control volume for the face centered variables on the boundary, that makes everything clearer!
Once more thanks for the information.
Best regards, Matheus.