High Rayleigh number thermal convection in a shallow laterally heated cavity

Abstract

The flow near the end of a shallow laterally heated cavity enters a nonlinear convective régime when the Rayleigh number R , based on cavity height, is of the same order of magnitude as the aspect ratio L (length/height). In the present work the asymptotic structure of the flow that develops in the limit as is R/L →∞ considered for the case where the horizontal surfaces of the cavity are thermally insulated. A model is discussed in which the formation of a vertical boundary layer on the end wall involves an unexpectedly large contribution to the local ambient temperature field. Expulsion of fluid from the base of the layer, and its subsequent return to the core through a horizontal boundary layer, maintains the necessary lateral heat transfer in the cavity. Implications of the model for the flow throughout the cavity are also discussed. The evolution of the end-zones leads to a change in the amplitude of the main Hadley circulation when R = O ( L 12/7 ). Various properties of the solution for this new régime are determined, including the Nusselt number for the lateral heat transfer, which is found to be proportional to L 3/7 . A comparison is made with both numerical and experimental results.