DIRECT SPECTRAL METHODS FOR THE LOW MACH NUMBER EQUATIONS

The low Mach number approximation of the Navier-Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space- and time-varying density that introduces a supplementary non-linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two-dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive-time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non-homogeneous flows.