Generalized formulations for the Rhie-Chow interpolation

In this paper, generalized formulations for the Rhie-Chow interpolation for co-located-grid discretizations are derived. These generalized formulations eliminate the major known defects in the standard Rhie-Chow interpolation, including the following: dependence of the converged solution on the value of the under-relaxation factor, saw-tooth pressure oscillations in transient problems with small time steps, and incorrect or non-converged solutions for problems with discontinuities. The generalized formulations are also shown to be applicable to a wider range of flow conditions than the standard Rhie-Chow interpolation. The derivation of the Rhie-Chow interpolation is first recalled and its numerical errors are analyzed. Then, the generalized formulations are presented and explained, and the way in which they eliminate or counter some of the known defects of the standard Rhie-Chow interpolation are outlined. The generalized formulations are then verified and validated with numerical experiments, including experiments with flow in porous media and in a packed bed. It is then concluded that the generalized formulations presented in this work represent an advance over the standard Rhie-Chow interpolation with a negligible increase in computational cost.