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

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.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This article focuses on the compressible Navier-Stokes equations under the no-slip adiabatic wall boundary conditions. A low-order summation-by-parts scheme is proposed and its linear stability and entropy stability are proved through the complete linearisation procedure. A linearly stable third-order scheme with the same boundary conditions is also introduced, but its nonlinear stability is not provable. Simulations demonstrate the robustness of both boundary treatment schemes.