Conservative Properties of Finite Difference Schemes for Incompressible Flow. 2nd Report. Schemes in Staggered and Collocated Grid Systems.

Abstract

In this report, finite difference schemes in staggered and collocated grid systems are considered in light of the analytical requirements that were discussed in the first report. The standard second-order accurate finite difference scheme with divergence form in a staggered grid system is proper. Proper second-order accurate advective and skew-symmetric forms have been proposed in the staggered grid system. Existing fourth-order accurate convective schemes in a staggered grid system are not proper. Proper higher-order accurate finite difference schemes in a staggered grid system are proposed. This is novel because higher-order staggered grid schemes that conserve momentum and kinetic energy simultaneously do not previously appear in the literature. The pressure term is not conservative in the kinetic energy equation in the collocated grid system, although we can construct proper-second and fourth-order accurate convective schemes in this system.