Low Mach Navier-Stokes equations on unstructured meshes

Numerical methods for fluxes with strong density variations but with speeds much lower than the sound speed, known as low Mach flows, present some particularities with respect to incompressible formulations. Despite having similar application ranges, incompressible formulations using the Boussinesq approximation with constant fluid properties cannot correctly describe fluxes with high density variations. According to Gray and Giorgini [1], use of the Boussinesq approximation can be considered valid for variations of the density up to 10% with respect to the mean value, so when strong density variations are present, variable density formulations are required. In the low Mach number Navier-Stokes equations the velocity divergence is not zero and acoustic waves are not considered. Also, the pressure is split into a dynamic pressure and a thermodynamic part. The latter is used to evaluate the density, by means of the ideal gas state law. Here will be presented an extension of an incompressible pressure projection-type algorithm (fractional-step) to simulate low Mach fluxes, using a Runge-Kutta/Crank-Nicolson time integration scheme, similar to the one presented by Najm et al. [2] and Nicoud in [3]. The use of a predictor-corrector substeps is related to the instabilities introduced by the density time derivative into the constant coefficient Poisson equation, as reported in [2, 3]. The spatial discretisation is performed by means of an unstructured finite volume technique, using both the collocated formulation by Felten [4] and the staggered formulation by Perot [5]. Finally, the algorithm is tested against benchmark test cases, such as the differentially heated cavity with large temperature differences and cases with reactive fluxes.