Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier–Stokes equations

Abstract

Non-linear entropy stability and a summation-by-parts (SBP) framework are used to derive entropy stable interior interface coupling for the semi-discretized three-dimensional (3D) compressible Navier–Stokes equations. A complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order [1,2] with an entropy stable coupling condition for the inviscid terms, and a local discontinuous Galerkin (LDG) approach with an interior penalty (IP) procedure for the viscous terms. The viscous penalty contributions scale with the inverse of the Reynolds number (Re) so that for Re → ∞ their contributions vanish and only the entropy stable inviscid interface penalty term is recovered. This paper extends the interface couplings presented [1,2] and provides a simple and automatic way to compute the magnitude of the viscous IP term. The approach presented herein is compatible with any diagonal norm summation-by-parts (SBP) spatial operator, including finite element, finite volume, finite difference schemes and the class of high-order accurate methods which include the large family of discontinuous Galerkin discretizations and flux reconstruction schemes. This note relies on the formalism introduced in [1,3] and complements the new class of interior entropy stable SBP operators of any order for the 3D compressible Navier–Stokes equations on unstructured grids that was proposed in [1,2]. To keep the notation as simple as possible, a uniform Cartesian grid is considered in the derivation. However, the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi-discrete geometric conservation [4]. The proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two-dimensional (2D) and 3D viscous subsonic and supersonic flows presented in [1,5,6].