This paper is concerned with a comparative study of the stream function-vorticity formulation and penalty function formulation of the two-dimensional equations governing natural connection in enclosures. The penalty function formulation presented herein is the only correct way of describing it for the problem at hand. The penalty-finite element model developed herein is novel with this work, and involves two velocities, temperature, and stream function as degrees of freedom at each node. The model includes, as a special case, the penalty-finite element model of natural convection in enclosures reported in the literature. Numerical results obtained using the two formulations are compared for several geometries, and boundary conditions, and the effects of Rayleigh number and Prandtl number on the flow and heat transfer are studied.

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