[arkie87]

I've been studying Rhie Chow interpolation for fixing of checkerboard pressure profiles in collocated CFD simulations and have a question.

The way I understand it, Rhie Chow is necessary because if central differencing about the cells is used to obtain the pressure force in the momentum equations (with collocated variables), then in the continuity/pressure correction equation, a large stencil will result with 9 points (P, W, E, N, S, WW, EE, SS, NN). Not only is this hard to solve, but it can result in a checkboard pressure profile.

Conversely, if central differencing about the faces is used to make the pressure correction equation, a 5 point stencil (P, W, E, S, N) is obtained, which is easier to solve and is oscillation free, but is inconsistent with the way pressure is handled in the momentum equation and can lead to instability.

To fix this, the 9 point stencil resulting from central differencing of the cells is treated explicitly, and the 5 point stencil is added and subtracted from the continuity equation, with one term treated implicitly and one term treated explicitly. Thus, the stencil for solving the pressure correction equation has only 5 points, and an explicit term involving the difference between the 5 point and 9 point stencils is obtained.

This is essentially what Rhie Chow interpolation is, and in fact, it is very similar to a standard "deferred correction" approach, where a more favorable stencil is added and subtracted, with half being treated implicitly and half being treated explicitly, such that an convergence, the implicit and explicit terms cancel. It is typically done to arrive at the solution using a higher order discretization scheme; however, it does not remove the oscillations in the solution, but rather, just provides an convenient way to arrive at the solution.

TL;DR: thus, my question is: how does Rhie Chow interpolation remove the oscillations in the pressure field, given that it is essentially just a deferred correction of the pressure correction equation (and deferred correction approaches do not remove oscillations from high order discretization schemes)?

[Gerry Kan]

Dear Rapheal:

The Rhie and Chow interpolation prevents pressure checkboard in collocated grid solvers by allowing face pressure to be reconstructed using face and cell values. The interpolation is, if memory serves, fourth order accurate, so in theory it will not suppress numerical oscillations beyond this.

The momentum equation is only a function of pressure gradient. So in a collocated grid elliptic solver, the gradient will be reconstructed from the pressure on the cell faces using, say, central difference. Since the grid is collocated, the face values are constructed by taking the average between neighboring cells.

Going through this derivation, you will quickly realize that the pressure gradient depends only on the pressure of the neighboring cells, and not the cell point in question. That means there will be two possible pressure fields, for the alternating 'row' and 'column' sets, that will satisfy the same pressure gradient. This is the origin of the pressure checkerboard problem.

What the Rhie and Chow interpolation does, is to use the pressure correction equation (effectively mass conservation) to reconstruct the face pressures as functions of the cell point and its immediate neighbors, so that only one pressure field will be present instead of two alternating ones, thereby eliminating the checkerboard.

Gerry.

[FMDenaro]

Question about Rhie-Chow Interpolation

[arkie87]

Originally Posted by Gerry Kan

Dear Rapheal:

The Rhie and Chow interpolation prevents pressure checkboard in collocated grid solvers by allowing face pressure to be reconstructed using face and cell values. The interpolation is, if memory serves, fourth order accurate, so in theory it will not suppress numerical oscillations beyond this.

The momentum equation is only a function of pressure gradient. So in a collocated grid elliptic solver, the gradient will be reconstructed from the pressure on the cell faces using, say, central difference. Since the grid is collocated, the face values are constructed by taking the average between neighboring cells.

Going through this derivation, you will quickly realize that the pressure gradient depends only on the pressure of the neighboring cells, and not the cell point in question. That means there will be two possible pressure fields, for the alternating 'row' and 'column' sets, that will satisfy the same pressure gradient. This is the origin of the pressure checkerboard problem.

What the Rhie and Chow interpolation does, is to use the pressure correction equation (effectively mass conservation) to reconstruct the face pressures as functions of the cell point and its immediate neighbors, so that only one pressure field will be present instead of two alternating ones, thereby eliminating the checkerboard.

Gerry.

That is all well and good, but does not directly address my claims or question.

[arkie87]

Originally Posted by FMDenaro

Question about Rhie-Chow Interpolation

Are you saying the answer to this question is in that thread? If so, which comment?

[Gerry Kan]

Originally Posted by arkie87

That is all well and good, but does not directly address my claims or question.

You asked how the Rhie and Chow interpolation solves the pressure checkerboard. I explained to you how it works, i.e., making sure that all immediate terms are present, albeit without any math. Then you came around and told me that my explanation is no good. I don't know what other claims and questions I have missed.

[FMDenaro]

Originally Posted by arkie87

Are you saying the answer to this question is in that thread? If so, which comment?

If you read carefully the comments and the references you will find the answers

[arjun]

Originally Posted by arkie87

That is all well and good, but does not directly address my claims or question.

This is your claim or question:

Originally Posted by arkie87

thus, my question is: how does Rhie Chow interpolation remove the oscillations in the pressure field, given that it is essentially just a deferred correction of the pressure correction equation (and deferred correction approaches do not remove oscillations from high order discretization schemes)?

The basis of your question is wrong because Rhie and Chow correction is not deferred correction approach.

The deferred correction is this:

phi_face  =   phi_face_implicit +  ( phi_face_HigherOrder    - phi_face_lowerOrder)

Here phi_face_implicit is basically phi_face_lowerOrder treated implicitly.

This is not what Rhie and Chow is. Rhie and Chow is correction now and is defect correction scheme. If there is a defect then it is corrected now (not deferred correction).

The link posted above explains in detail.