Poly-region boundary element method for two-dimensional Boussinesq flows

A sparse boundary element method (BEM) for steady non-uniformly heated viscous fluid flow is presented. The new boundary integral formulation with a poly-region approach involves the use of region-based convective velocity along with free-space fundamental solutions with slight compressibility in order to eliminate pressure from the governing equations. The boundary element formulation developed previously by the authors for incompressible isothermal viscous fluid flows is extended for non-isothermal flows governed by Boussinesq equations. The new BEM is then applied to the problem of natural convection in a square enclosure, to the classical Rayleigh-Benard problem and to a stratified flow over a backward-facing step. In all three cases, the boundary element results are in good agreement with published finite difference and finite element solutions. However, in some instances, the boundary element solutions are more accurate, particularly in terms of resolving surface tractions and heat flux. Consequently, results are presented in detail to provide data for future comparison.