Mesh Adaptation for Computational Fluid Dynamics, Volume 1

1CFD Numerical Models

This chapter defines two fluid mechanics models and two particular numerical approximations for their discretization, which will be used in the examples of mesh adaptation algorithms that constitute the main part of this book. We first consider compressible fluid flows and introduce an approximation method, referred in the sequel as mixed element volume (MEV), which relies mainly on a standard continuous P₁-Galerkin approximation. Its stabilization is obtained by introducing high-order Godunov upwinding. Second, we consider a multifluid model based on the incompressible Navier–Stokes equations and introduce an approximation method based on the continuous P₁-Galerkin approximation. Pressure stabilization is obtained by projection. Advective stabilization is obtained by introducing high-order upwinding.

1.1. Compressible flow

1.1.1. Introduction

The simulation of compressible flows experienced, in the 1990s, a small revolution with the development of new algorithms that are able to compute flows through (or around) any kind of shape. This was due to new numerical algorithms and mesh generation algorithms. For both types, the main innovation was related to unstructured meshes, and the way to do it was first to rely on tetrahedrizations. Unstructured mesh generation and in particular tetrahedrization has been the object of many research and advances, and we refer, for example, to previous studies (Frey and George 2008; Borouchaki and George 2017; George ...