Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below

Abstract

This paper outlines a combined theoretical and numerical study of the mass transfer effected by high Rayleigh number Bénard convection in a two-dimensional saturated porous layer heated from below. The focus of this study is on the Darcy flow, heat transfer and mass transfer scales of a single cell (roll) that exists in the steady two-dimensional convection regime. The numerical solutions are based on the complete governing equations for two-dimensional flow, and cover the Rayleigh number range 50–2000. The numerical results compare favorably with the theoretical conclusions of a scale analysis that is based on the recognition of 1. (i) two temperature difference scales in the cell, 2. (ii) a flow field without horizontal boundary layers, and 3. (iii) thermal top and bottom end-regions that are not slender enough to be boundary layers. Writing Le for the Lewis number, the overall mass transfer rate or Sherwood number is shown to scale as Le 1 2 Ra 7 8 if Le > Ra 1 4 , as Le 2 Ra 1 2 if Ra − 1 4 < Le < Ra 1 4 , and as O(1) if Le < Ra − 1 4 . The transition from the Darcy flow to the inertia-dominated Forschheimer flow and the scales of the Forschheimer regime are discussed in the closing section.