A stream function finite element solution for two‐dimensional natural convection with accurate representation of nusselt number variations near a corner

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Abstract
A finite element stream function formulation is presented for the solution to the two‐dimensional double‐glazing problem. Laminar flow with constant properties is considered and the Boussinesq approximation used. A restricted variational principle is used, in conjunction with a triangular finite element of C1 continuity, to discretize the two coupled governing partial differential equations (4th order in stream function and second order in temperature). The resulting non‐linear system of equations is solved in a segregated (decoupled) manner by the Newton‐Raphson linearizing technique.Results are produced for the standard test case of an upright square cavity. These are for Rayleigh numbers in the range 103−105, with a Prandtl number of 0.71. Comparisons are made with benchmark results presented at the 1981 International Comparison study in Venice. In the discussion of results, emphasis is placed on the variation of local Nusselt number along the isothermal walls, particularly near the corner. This reveals a noticeable source of error in the evaluation of the maximum Nusselt number by lower order discretization methods.