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1. Introduction
The compressible Navier-Stokes equations are commonly used to model flow problems when compressibility effects become relevant. Some areas that require a compressible flow description are the aerodynamics and aeroacoustics fields, in which applications range from turbo-machinery design to speech therapy. The compressible Navier-Stokes equations consists of the conservation of mass, momentum and energy equations. Thermodynamical properties and constitutive relations of the fluid close the mathematical description. We restrict our problem to perfect fluids in the gas state, which we model using Newton ' s law for fluids together with the caloric equation, the ideal gas law, and Fourier’s heat law. However, other compressible fluids can be modeled by the compressible Navier-Stokes equations using the appropriate constitutive definitions.
In this work we are interested in the finite element method (FEM) approximation of compressible flow problems. In particular, we solve this set of equations using the conservative variables formulation, that is, defining density, momentum, and total energy as the problem unknowns. When these equations are approximated by the classical Galerkin approach, numerical instabilities may appear due to the hyperbolic nature of the equations. The first attempt to deal with this instability was the introduction of a stabilizing term into the Galerkin finite element formulation. Within this concept, the Streamline Upwind Petrov Galerkin (SUPG) method by Tezduyar and Hughes (1983) was the first method adopted for solving the compressible Navier-Stokes equations. In that formulation, the authors applied the stabilization methods previously developed for the convection-diffusion equation. The main idea was to optimally introduce numerical diffusion along the streamlines using a stabilization term that contained a matrix of algorithmic parameters, a certain operator applied to the test function, and the residual of the differential equation (e.g. Brooks and Hughes, 1981). An important contribution of this pioneering work was the inclusion of the largest eigenvalue of the hyperbolic system into the matrix of algorithmic parameters. More recently, Polner (2005), Billaud et al. (2010), and Sevilla et al. (2013) applied the SUPG stabilization into their compressible flow formulations. Some later stabilizations were formulated based on the Galerkin Least Squares method. In all of these works (Shakib, 1989; Hauke and Hughes, 1998; Hauke et al., 2005), the authors transformed the conservative variables into entropy variables using Jacobian transformation matrices.