Discrete regularization

Abstract

In this paper we discuss discrete regularization, more specifically about how to add finite dissipation to the discretized Euler equations so as to ensure the stability and convergence of numerical solutions of high Reynolds number flows. We will briefly review regularization strategies widely used in Lagrangian shockwave simulations (artificial viscosity), in Eulerian nonoscillatory finite volume simulations, and in Eulerian simulations of turbulent flow (explicit and implicit large eddy simulations). We will describe an alternate strategy for regularization in which we introduce a finite length scale into the discrete model by volume averaging the equations over a computational cell. The new equations, which we term Finite Scale Navier–Stokes, contain explicit (inviscid) dissipation in a uniquely specified form and obey an entropy principle. We will describe features of the new equations including control of the small scales of motion by the larger resolved scales, a principle concerning the partition of total flux of conserved quantities into advective and diffusive components, and a physical basis for the inviscid dissipation.