Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations

TL;DR

This paper extends entropy stable p-nonconforming discretizations with summation-by-parts property from Euler to Navier-Stokes equations, ensuring stability, conservation, and accuracy in complex flow simulations.

Contribution

It introduces a flexible coupling procedure with planar interpolation for nonconforming elements, preserving entropy stability and geometric conservation laws for viscous flows.

Findings

Achieves ~p+1 convergence in viscous shock simulations

Maintains entropy stability with viscous interface dissipation operators

Demonstrates comparable accuracy and stability to conforming schemes in turbulent flow tests

Abstract

The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators are developed that retain the entropy stability of the base scheme. The accuracy and stability properties of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme (achieving ~p+1 convergence) in the context of the viscous shock problem, the Taylor-Green vortex problem at a Reynolds number of Re=1,600, and a subsonic turbulent flow past a sphere at Re = 2,000.