A DIV-CURL-GRAD FORMULATION FOR COMPRESSIBLE BUOYANT FLOWS SOLVED BY THE LEAST-SQUARES FINITE-ELEMENT METHOD

The present paper reports the development of the least-squares finite- element method for simulating compressible buoyant flows at low Mach n umbers. We propose a div-curl-grad formulation with unknowns including vorticity, velocity, heat fluxes, temperature and pressure variation. The formulation is proved to be elliptic such that permissible bounda ry conditions become self-evident for a well posed flow problem. In co ntrast to conventional approaches. the present method evades the predi cament of the 'singularity' problem of low-speed flows and no special treatment or artificial boundary condition is needed. Moreover, the as sembled coefficient matrix is symmetric and positive-definite: its inv ersion is implemented by an element-by-element jacobi conjugate gradie nt method. As a numerical example, we calculate two-dimensional compre ssible buoyant flows inside a square enclosure at various Rayleigh num bers. For Rayleigh number one million, four secondary vortices were fo und embedded in the primary vortex. Due to significant temperature var iations, the fluid flows are highly compressible in the interior. Alon g the walls, however, the flows are incompressible. The Nusselt number -Rayleigh number correlation deduced from the numerical result compare d favorably with previously reported data.