A finite element method for compressible and incompressible flows

Abstract

In this study, we present a unified formulation of compressible and incompressible Navier–Stokes equations in the quasi-linear form for primitive variables. In this formulation, two thermodynamic parameters, coefficient of isothermal compressibility and coefficient of thermal expansion, are highlighted. The incompressible limit is obtained when the coefficients of isothermal compressibility and of thermal expansion are taken equal to zero and when the density is supposed constant. For the simulation of advection-dominated flows, a stabilized finite element method based on the Petrov–Galerkin formulation is proposed. The proposed unified formulation is tested and validated for different numerical simulations. Different test cases are processed, from simplified models to more elaborate models. Finally, we present the conclusions inspired by this work, as well as the perspectives envisaged.