Comparison of low Mach number models for natural convection problems

We investigate in this paper two numerical methods for solving low Mach num ber compressible flows and their application to single-phase natural convec tion flow problems. The first method is based on an asymptotic model of the Navier-Stokes equations valid for small Mach numbers, whereas the second i s based on the full compressible Navier-Stokes equations with particular ca re given to the discretization at low Mach numbers. These models are more g eneral than the Boussinesq incompressible flow model, in the sense that the y are valid even for cases in which the fluid is subjected to large tempera ture differences, that is when the compressibility of the fluid manifests i tself through low Mach number effects. Numerical solutions are computed for a series of test problems with fixed Rayleigh number and increasing temper ature differences, as well as for varying Rayleigh number for a given tempe rature difference. Numerical difficulties associated with low Mach number e ffects are discussed, as well as the accuracy of the approximations.