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This paper describes the total energy formulation of the compressiblereacting Navier-Stokes equations which is solved numerically using a fullyconservative discontinuous Galerkin finite element method (DG). Previousapplications of DG to the compressible reacting Navier-Stokes equationsrequired nonconservative fluxes or stabilization methods in order to suppressunphysical oscillations in pressure that led to the failure of simple testcases. In this paper, we demonstrate that material interfaces with atemperature discontinuity result in numerical unphysical pressure oscillationsif the species internal energy is nonlinear with respect to temperature. Wedemonstrate that a temperature discontinuity is the only type of materialinterface that results in unphysical pressure oscillations for a conservativediscretization of the total energy formulation. Furthermore, we demonstratethat unphysical pressure oscillations will be generated at any materialinterface, including material interfaces with at which the temperature iscontinuous, if the thermodynamics are frozen during the temporal integration ofthe conserved state. Additionally, we demonstrate that the oscillations areamplified if the specific heat at constant pressure is incorrectly evaluateddirectly from the NASA polynomial expressions. Instead, the mean value, whichwe derive in this manuscript, should be used to compute the specific heat atconstant pressure. This can reduce the amplitude of, but not prevent,unphysical oscillations where the species concentrations numerically mix. Wethen present solutions to several test cases using the total energy formulationand demonstrate spurious pressure oscillations were not generated for materialinterfaces if the temperature is continuous and that it is better behaved thanfrozen thermodynamic formulations if the temperature is discontinuous.

A Conservative Discontinuous Galerkin Discretization for the Total Energy Formulation of the Reacting Navier Stokes Equations