A finite element method for compressible and incompressible flows

摘要

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