A Div-Curl-Grad Formulation for Compressible Buoyant Flows Solved by the Least-Squares Finite-Element Method

摘要

The present paper reports the development of the least-squares finite-element method for simulating compressible buoyant flows at low Mach numbers. For such flows, we propose a div-curl-grad formulation with unknowns including vorticity, velocity, heat fluxes,t emperature, and pressure variation. The formulation is then proved to be elliptic. As such, permissible boundary conditions become self-evident for a well posed flow problem. In sharp contrast to conventional approaches, the present method evades the awkward predicament of the "singularity" problem of low-speed flows and no special treatment is needed. Morevoer,t he 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 enclusure at various Rayleigh numbers. For Rayleigh number one million, four-secondary vortices are found embedded in the primary vortex. The duced from the numerical result compared favorably with previously reported data.