Shallow cavity flow in a porous medium driven by differential heating

Abstract

This paper describes steady flow through a porous medium in a shallow two-dimensional cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. The main emphasis is on the situation where the Darcy–Rayleigh number $R$ is large and the aspect ratio of the cavity $L$ (length/depth) is of order $R^{1/2}$. For a monotonic temperature distribution at the upper surface, the leading-order problem consists of an interaction involving the horizontal boundary-layer equations, which govern the flow throughout most of the cavity, and the vertical boundary-layer equations which govern the flow near the colder sidewall. This problem is solved using numerical and asymptotic methods. The limiting cases where $L\,{\gg}\,R^{1/2}$ and $L\,{\ll}\,R^{1/2}$ are also discussed.