Scaling theory of melting with natural convection in an enclosure

This study identifies the most basic scales and regimes of the phenomenon of melting with natural convection in an enclosure heated from the side. In the first part of the study the method of scale analysis is used to show that the phenomenon consists of a sequence of four regimes: (a) the pure conduction regime, (b) the mixed regime in which the upper portion of the liquid gap is ruled by convection and the lower portion by conduction, (c) the convection regime and, finally, (d) the last or ‘shrinking solid’ regime. For the first three regimes the scaling theory predicts a Nusselt number vs time curve that has features similar to a van der Waals isotherm, in particular, a clear Nu minimum of order Ra 1 4 at a time SteFo of order Ra− 1 2 , where Ste is the liquid superheat Stefan number and Fo the Fourier number based on H. The corresponding average melting front position has an inflexion point at a time of order Ra− 1 2 . The theory shows further that during the fourth regime the solid disappears during a Ste Fo time interval of order Ra− 1 4 . The second part of the study consists of numerical experiments the purpose of which is to verify the correctness of the theory constructed in the first part. The numerical simulations are based on the quasi-stationary front approximation and the quasi-steady natural convection assumption. The parametric domain covered by these simulations is 0 ⩽ Ra ⩽ 108, 0 < SteFo < 0.2, Pr = 50 and H/L = 1, where L is the horizontal dimension of the enclosure and Ra the Rayleigh number based on H. Closed form correlations for both Nu and the melting front location time functions are developed by combining the theoretical and numerical conclusions of the study.