High-order entropy-stable isothermal wall boundary condition for the compressible Navier--Stokes equations

Abstract

Entropy-stable boundary conditions are critical for ensuring that the corresponding numerical scheme satisfies the discrete entropy inequality, thus mimicking the second law of thermodynamics for the compressible Navier-Stokes equations discretely. Provably entropy-stable adiabatic wall boundary conditions were introduced in Parsani 2014~\cite{parsani2014entropy}. These discrete wall boundary conditions were further generalized in Dalcin 2019~\cite{dalcin2019conservative} to a moving adiabatic solid wall or a wall with a prescribed heat flux for the compressible Navier--Stokes equations discretized by using summation-by-parts (SBP) and simultaneous-approximation-term (SAT) operators. Currently, no entropy-stable formulation using SBP-SAT operators for isothermal wall boundary conditions is available in the literature. This paper presents new isothermal no-slip wall boundary conditions that: 1) enforce the $T{|}_{y=0}=T^{wall}$ and $\vec{v}{|}_{y=0}={\vec{v}}^{wall}$ conditions with the design order of accuracy while maintaining stability, 2) provide the correct sign of entropy production at the wall, and 3) mimic the entropy balance of the Navier-Stokes equations at the discrete level. The structure-preserving and design-order properties of the proposed methodology are demonstrated and verified on standard benchmark problems for compressible flows.