Conservative properties of finite difference schemes for incompressible flow

1. Motivation and objectives The purpose of this research is to construct accurate finite difference schemes for incompressible unsteady flow simulations such as LES (large-eddy simulation) or DNS (direct numerical simulation). Experience has shown that kinetic energy conservation of the convective terms is required for stable incompressible unsteady flow simulations. Arakawa (1966) showed that a finite difference scheme that conserves the enstrophy in the absence of viscous dissipation is required for long-time integration in the two-dimensional vorticity-streamfunction formulation. The corresponding conserved variable is kinetic energy in velocity-pressure formulation, and some energy conservative finite difference schemes have been developed for the Navier-Stokes equations in three dimensions. Staggered grid systems are usually required to obtain physically correct pressure fields. The standard second order accurate finite difference scheme (Harlow &: Welch 1965) in a staggered grid system conserves kinetic energy and this scheme has proven useful for LES and DNS. However, the accuracy of the second order finite difference scheme is low and fine meshes are required (Ghosal 1995). Spectral methods (Canuto et al. 1988) offer supreme accuracy, but these methods are limited to simple flow geometries. Existing fourth order accurate convective schemes (A-Domis 1981, Kajishima 1994) for staggered grid systems do not conserve kinetic energy. Higher order staggered grid schemes that conserve kinetic energy have not been presented in the literature. The conservation of kinetic energy is a consequence of the Navier-Stokes equations for incompressible flow in the inviscid limit. In contrast, energy conservation in a discrete sense is not a consequence of momentum and mass conservation. It is possible to derive numerical schemes that conserve both mass and momentum but do not conserve kinetic energy. It is also possible to derive schemes that conserve kinetic energy even though mass or momentum conservation are violated. In this report, conservation properties of the continuity, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discretized equations. Existing finite difference schemes in staggered grid systems are checked for satisfaction of the requirements. Proper higher order accurate finite difference schemes in a staggered grid system are then proposed. Plane channel flow is simulated using the proposed fourth order accurate finite difference scheme and the results compared with those of the second order accurate Harlow and Welch (1965) algorithm.