Penetrative convection in porous medium bounded by a horizontal wall with hot and cold spots

This paper reports a series of numerical simulations and a scale analysis of the penetrative convection occurring along the unevenly heated horizontal wall of a semi-infinite porous medium. Modeling the problem as two-dimensional, and assuming that the horizontal wall temperature varies between alternating hot and cold spots, it is found that the natural circulation consists of a row of counter-rotating cells situated near the wall. Each cell penetrates vertically into the porous medium to a distance approximately equal to λ Ra 1 2 λ , where λ is the distance between a hot spot and the adjacent cold spot, and Ra gl is the Darcymodified Rayleigh number based on λ and on the temperature difference between each spot and the porous medium situated far enough from the wall. The ability of each cell to convect heat between two adjacent spots increases with the Rayleigh number. The results of numerical simulations in the Ra λ range 1–100 are found to support a number of scaling laws derived based on pure scaling arguments.