Note on Differencing the Viscous Dissipation Terms for a Newtonian Fluid

摘要

Most of the research effort on finite differencing the transient equations of fluid dynamics has been put into the convection terms and time marching schemes. As important as that effort has been, we cannot neglect the rest of the terms in the equations. In this report, we focus attention on the finite difference approximation used for the viscous dissipation term in the energy equation. As a concrete example, we use the COYOTE computational fluid dynamics (CFD) code (1) to show how the choice of difference methods can influence accuracy in subtle but significant ways. Terms with the same mathematical form as the viscous dissipation appear in various turbulence models, and this discussion applies to them as well. In Section 2, we present the basic governing equations. Section 3 summarizes some relevant equations from standard turbulence models. Section 4 expands these equations into components in Cartesian and cylindrical coordinates, a necessary step in creating the finite difference equations. Section 5 discusses some considerations on how to difference the viscous terms, and Section 6 gives numerical examples. We conclude with the summary in Section 7.