Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations

TL;DR

This paper develops entropy stable wall boundary conditions for 3D compressible Navier-Stokes equations, ensuring non-linear stability and compatibility with various numerical methods, validated through subsonic and supersonic flow simulations.

Contribution

It introduces a novel entropy stable boundary condition framework compatible with multiple discretization schemes for 3D compressible flows.

Findings

Boundary conditions ensure entropy stability in 3D Navier-Stokes simulations.

Numerical tests confirm stability and accuracy for subsonic and supersonic flows.

Framework is adaptable to various spatial discretization methods.

Abstract

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.