A Numerical Study of Two–Dimensional Parallel-Plate Convection

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Abstract
Convection between horizontal plates maintained at two different temperatures is studied numerically for a Rayleigh number of 6.75×105 and a Prandtl number of 0.71. The two-dimensional and Boussinesq approximations are applied. The computed motion eventually takes the form of large-scale and nearly steady-state vortices which induce plumes of warm (cool) air to impinge strongly upon the cool (warm) plate. When the width is restricted by lateral, insulated walls and is equal to the height, eddies which appear in corners suppress the single large-scale vortex which forms and cause the heat transport to be about 20 per cent smaller than is observed experimentally for a region of width much greater than height. With a width twice the height, a large-scale vortex pair develops. The effect of corner eddies is then diminished, and the heat flux becomes 18 per cent larger than the experimental value. When the lateral walls at this width are replaced by cyclic boundaries, no small-scale eddies appear, and the heat flux is 34 per cent too large. The tendency for air leaving the vicinity of either plate to be concentrated into narrow plumes, with temperature anomaly diminishing with distance from the plate, allows the convective heat flux to occur in the central region in the absence of an unstable lapse rate. The absence of aperiodic, turbulent motions in the two-dimensional model of limited width is corroborated by measurements within a “two-dimensional” convection chamber of width equal to height. A quasi-linear model of thermal convection is found to yield a heat transport at least 55 per cent too large, and a kinetic energy 270 per cent too large, in comparison with the non-linear model. The effect of replacing the rigid-surface boundary conditions by free-surface conditions is to increase the heat transport by 190 per cent.

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