Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

Abstract

SummaryIn this paper, discontinuous Galerkin (DG) discretization schemes for Navier‐Stokes equations of a nonconservative form was proposed, in which the constitutional equation for energy conservation is written in terms of pressure. The primary focus is to address the treatment of nonconservative products in the pressure equation, for which we formulate four different scheme variants by using the path‐conservative scheme or solely regarding the nonconservative products as source terms. In addition to the complete description of the discretization formulations, we highlight the positivity‐preserving properties of the proposed nonconservative DG schemes. The performance of the four schemes are thoroughly assessed and tested with the consideration of a series of classical flow configurations that involve steady/unsteady, inviscid/viscous, and laminar/turbulence flow physics. It is found that two out of the four schemes are able to provide the optimal convergence and similar accuracies as the fully conservative formulation. The nonconservative formulations provide an alternative way to model fluid flows with substantial thermodynamic and compositional complexities. The present work aims to lay the theoretical foundation for further algorithmic extensions to simulations of such fluid flows of practical relevance.