A combined finite volumes ‐ finite elements method for a low‐Mach model

Abstract

Summary In this paper, we develop a finite volumes ‐ finite elements method based on a time splitting to simulate some low‐Mach flows. The mass conservation equation is solved by a vertex‐based finite volume scheme using a τ‐limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier‐Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

“…Tackling this problem amounts to introduce some upwinding in the scheme, and, to this purpose, many solutions have been explored in the context of the finite volume; finite‐volume convection operators respecting both some monotonicity and $$$ {L}^2 $$$‐stability properties (including, for the latter item, a local discrete entropy or, in the world of fluid flow, a kinetic energy balance) have been obtained in this way. Several authors have thus proposed discretizations combining finite elements and finite volumes, to take benefit of the best of both worlds, see for instance 1–6 and references therein. These works may address convection‐diffusion or Navier–Stokes equations, using preferably finite elements approximations of accuracy compatible with finite volumes, that is, low‐order elements.…”

A MUSCL-like finite volumes approximation of the momentum convection operator for low-order nonconforming face-centred discretizations

“…Tackling this problem amounts to introduce some upwinding in the scheme, and, to this purpose, many solutions have been explored in the context of the finite volume; finite-volume convection operators respecting both some monotonicity and L 2 -stability properties (including, for the latter item, a local discrete entropy or, in the world of fluid flow, a kinetic energy balance) have been obtained in this way. Several authors have thus proposed discretizations combining finite elements and finite 2 volumes, to take benefit of the best of both worlds, see for instance [1,9,[15][16][17]30] and references therein. These works may address convection-diffusion or Navier-Stokes equations, using preferably finite elements approximations of accuracy compatible with finite volumes, i.e.…”