Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

Abstract

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.