A complete flux scheme for Boussinesq model with applications to convection problems

A complete-flux approximation scheme for a system of thermally-coupled Navier–Stokes equations has been proposed which has applications in convection problems. The governing equations have been discretized on a 2D staggered grid in space by finite volume method. The convective and viscous momentum fluxes are approximated by solving appropriate local nonlinear boundary value problems (BVPs). This numerical-flux approximation scheme is second order accurate and strongly depends on the transverse flux gradient, pressure gradient, and thermal buoyancy force. Similarly, the heat conduction and diffusion fluxes are approximated by solving local BVPs which have a significant influence of the thermal cross flux. The numerical validation of the scheme has been done for natural convection in a square cavity with heated bottom wall for various Prandtl and Rayleigh numbers. Furthermore, the effect of the domain’s aspect ratio on heat flux at the bottom wall has been studied for several combinations of parameters. The increase of the thermal buoyancy force leads the formation of a multicellular cat’s-eyed circulation pattern and a centrally located sharp thermal plume in the temperature field.