Entropy stable far-field boundary conditions for the compressible Navier-Stokes equations

Abstract

We consider external compressible-flow problems with open boundaries. We begin by presenting the basic idea, namely the introduction of an infinite buffer zone around the computational domain, for the advection-diffusion equation. We discretise the problem with a summation-by-parts finite-volume scheme and prove convergence; a key step is the renormalisation of the solution. Next, we make the analogous rescaling of the entropy function for the Navier-Stokes equations and prove that a finite-volume scheme is entropy stable.Finally, we present numerical computations with a vortex governed by the Navier-Stokes equations interacting with the buffer zone. As proven, the scheme is stable, and with a sufficient number of grid points and with some numerical diffusion in the buffer zone, the reflections are modest.