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

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.