How Rhie and Chow relation eliminates pressure oscillaiton in collocated grid? thank you

I have same problem as you in Collocated grid (cartesian velocity at cell center and contravariant velocity at cell surface). I employ fractional method (Mac-type). I remember that Rhie adn Chow used SIMPLE method. Does pressure oscillation occur even with SIMPLE method?

I think two characteristics of collocate grid cause this problem. (1) contravariant velocity satisfies mass conservation while cartesian velocity does not. (2) contravariant velocity is corrected by pressure like staggered grid while cartesian velocity is corrected by pressure with central difference scheme.

Before Rhie and Chow, Maliska scheme, where basis variable is cartesian velocity with staggered grid, was proposed. How about Maliska scheme ?

thank you for your reply. I dont know Maliska scheme. yes pressure oscillation occurs in SIMPLE scheme because of 2delta difference pressure is used. Could you please explain briefly about cartesian and contravarient velocities. what is the definition and basic difference of these two velocities. thank you very much

My program is based on Zang scheme ( Journal of computational physics 114, 118-33 (1994) ). This is also collocated grid but Mac-type schme. contravariant velocity is defined at cell surface, and nromal to cell surface. In Zang scheme,

1, convection term and viscous term is calculated by cartesian velocity at cell center explicitly.

2, After that, contravariant velocity is calculated by interpolation of cartesian velocity.

3 poisson equation is calculated using contravariant velocity.

4, Cartesian velocity and contravariant velocity is corrected by pressure correction.

In detail, please refer to Zang paper.

Maliska paper is as follows.

International journal for numerical methods in fluids, vol. 4, 519-537, 1984

Zang paper also refers to Maliska paper.

From your brief explanation should we conclude that , the cell center velocities ( main grid points) are cartesian velcities and face velocites (flux point) velocities are contraverient velocities , but could you please then tell me what is the difference between covarient and contraverient velocities thanks.

for a good explanation of Rhie and Chow see the book of Peric: "Computational Mehtods for fluid dynamics". You can also download the example code from Peric "caffa.f".

- Check-out the following paper: Frederic N Felten and Thomas S Lund, "Kinetic Energy Conservation issues associated with the collocated mesh scheme for incompressible flow", Journal of Computational Physics, Vol 215, 2006, pp 465-484

- It will help you understand the purpose of the Rhie-Chow interpolation and some of the issues associated with using collocated grids.