Study and implement a numerical code for the study of compressible flows

This report presents the study of the conservation properties for the Euler and Navier-Stokes equations ofcompressible flow, using Galerkin finite element methods. Most simulation methods for compressible flow attainnumerical stability at the cost of swamping the fine turbulent flow structures by artificial (numerical) diffusion.This study demonstrates that numerical stability can also be attained by preserving conservation laws at thediscrete level.First of all, in order to understand the Galerkin finite element methods a steady convection-diffusion trans-port problem is studied through the implementation of a Computational Fluid Dynamics (CFD) code, developedfrom scratch. Then, the numerical results of the simulations are analyzed, demonstrating that the standardGalerkin methods need stabilization techniques to avoid numerical wiggles and improve the results.Secondly, the conservation properties of Galerkin methods for the incompressible Navier-Stokes equationsare studied. A discrete formulation recently developed which conserves the energy, momentum and angularmomentum, without utilizing a strong enforcement of the divergence constraint, is shown.Finally, the compressible Euler equations and the compressible Navier-Stokes equations are studied,through the development of a second CFD code from scratch. Then, a novel discrete formulation conserv-ing energy, momentum and angular momentum is constructed for the compressible Navier-Stokes equations.Several numerical experiments are performed, which verify the theory and test the new formulation in the con-text of finite element.