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Natural convection in two-dimensional irregular cavities Fournier, Martin

Abstract

Natural convection in two-dimensional irregular cavities was simulated by numerically solving the steady-state conservation equations written in terms of stream function, vorticity and temperature dependent variables and for a general orthogonal coordinate system. It was assumed that the Boussinesq approximations were valid, that the fluid was Newtonian and that the properties other than density were constant. The use of orthogonal coordinates and the above set of dependent variables was found to have several advantages over the use of Cartesian or non-orthogonal systems and the set of primitive dependent variables (velocities, pressure and temperature). The body-fitted orthogonal coordinate system was numerically generated by means of the weak constraint method of Ryskin and Leal [26], Special forms of the Wood and second-order vorticity boundary conditions were derived for a general two-dimensional body-fitted orthogonal coordinate system. Finite difference techniques were used to solve the resulting set of differential equations. The effects of the mapping characteristics, the vorticity boundary conditions and the finite difference grid size on the accuracy of the natural convection solution were investigated first. For the cavity geometries studied,, it was observed that, except for grid boundary conditions which led to undesirable grids, most combinations of grid and vorticity boundary conditions gave results of acceptable accuracy (relative error less than one percent) as long as a sufficiently fine grid size (28x28 or finer) was employed. The effects of the cavity geometry and the Rayleigh number on natural convection were investigated in Part II. It was found that increasing the Rayleigh number always acted to enhance both the natural convection circulation and the heat transfer rate, a result which was easily explained by examining the source term of the momentum equation. The effect of the cavity geometry was more complex but these results could also be interpreted by examining the influence of the cavity shape in impeding or enhancing fluid circulation and the opposing effects of the distance between isothermal walls on conductive and convective heat transfer. The possibility of using a similar numerical procedure to simulate a melting or a freezing process was investigated in Part III. Numerical predictions of the circulating flow in the liquid phase of an ice forming process were obtained by digitizing the photographic image of a real ice interface and using the true non-linear relationship between density and temperature for water at low temperature. The numerical results were in reasonable agreement with the flow visualization experiments carried out by Eckert [42].

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