A LEAST-SQUARES FINITE ELEMENT METHOD FOR DOUBLY-DIFFUSIVE CONVECTION

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Abstract
A least-squares finite element method (LSFEM) has been developed to investigate the phenomenon of natural convection caused by temperature and concentration buoyancy effects in rectangular enclosures with different geometric aspect ratios. The time dependent Navier-Stokes equations, the energy and the mass balance equations for an incompressible, constant property fluid in the Boussinesq approximation are reduced into a first-order velocity-pressure-vorticity-temperature-heat flux-concentration-mass flux (u-p-ω-T-q-C-J) formulation. The coupled system is discrelized by backward differencing in time. For the case of heat and mass transfer from side walls, results for both augmenting and opposing flows are obtained with Prandtl number Pr = 0.7, Schmidt number Sc = 0.6 and 0.7 (Le = 1), Grashof number Gr up to 106, buoyancy ratio χ = − 0.2 to − 5.0 and geometric aspect ratio of 1. For the case of heating from below, we test the LSFEM with two Rayleigh-Bénard convection problems. Then the LSFEM algorithm is used to solve the doubly-diffusive convection in a horizontal rectangular cavity heated from below with Grashof number 5.5 x 105 and geometric aspect ratio 7.