A div-curl-grad formulation for compressible buoyant flows solved by the least-squares finite element method Original Research Article

The present paper reports the development of the least-squares finite-element method for simulating compressible buoyant flows at low Mach numbers. 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 boundary conditions become self-evident for a well posed flow problem. In contrast to conventional approaches, the present method evades the predicament of the ‘singularity’ problem of low-speed flows and no special treatment or artificial boundary condition is needed. Moreover, the assembled coefficient matrix is symmetric and positive-definite; its inversion is implemented by an element-by-element jacobi conjugate gradient method. As a numerical example, we calculate two-dimensional compressible buoyant flows inside a square enclosure at various Rayleigh numbers. For Rayleigh number one million, four secondary vortices were found embedded in the primary vortex. Due to significant temperature variations, the fluid flows are highly compressible in the interior. Along the walls, however, the flows are incompressible. The Nusselt number-Rayleigh number correlation deduced from the numerical result compared favorably with previously reported data.