10 - The Covolume Approach to Computing Incompressible Flows

Summary

Introduction

This article contains a summary account of covolume methods for incompressible flows. Covolume methods are a recently developed way to solve both compressible and incompressible flow problems on unstructured meshes. The general idea is to use complementary pairs of control volumes to discretize flux, circulation and other expressions which occur in the governing equations. These complementary volumes (covolumes for short) are related by an orthogonality property which is a basic feature of the covolume approach. One of the simplest mesh configurations which is suitable is the Delaunay- Voronoi mesh pair. This is introduced in the next section. After that we proceed through div-curl systems to the stationary Stokes equations and the Navier-Stokes equations. We will show that for uniform meshes the covolume equations for the stationary Stokes equations specialize to the MAC (staggered mesh) scheme, and that the MAC scheme itself is actually equivalent to a velocity-vorticity scheme. Some numerical results are presented in the last section.

Since this article is intended only as an overview, we will present most of the results in a two dimensional setting. Almost all of the ideas and techniques do generalize nicely to three dimensions but are harder to visualize than in two dimensions. Given our limited aims it would be inappropriate to present proofs of most of the mathematical results. We will refer to the original sources for these and other details.

One of the reasons for introducing covolume methods is to find lower order methods for viscous flows which are free of “spurious mode” problems.