Finite-amplitude cellular convection in a fluid-saturated porous layer

Abstract

The local nonlinear stability of thermal convection in fluid-saturated porous media, subjected to an adverse temperature gradient, is investigated. The critical Rayleigh number at the onset of convection and the corresponding heat transfer are determined. An approximate analytical method is presented to determine the form and amplitude of convection. To facilitate the determination of the physically preferred cell pattern, a detailed study of both two- and three-dimensional motions is made and a very good agreement with available experimental data is found. The finite-amplitude effects on the horizontal wavenumber, and the effect of the Prandtl number on the motion are discussed in detail. We find that, when the Rayleigh number is just greater than the critical value, two-dimensional motion is more likely than three-dimensional motion, and the heat transport is shown to have two regions for n = 1. In particular, it is shown that optimum heat transport occurs for a mixed horizontal plan-form formed by the linear combination of general rectangular and square cells. Since an infinite number of steady-state finite-amplitude solutions exist for Rayleigh numbers greater than the critical number λ^∗_υ, a relative stability criterion is discussed that selects the realized solution as that having the maximum mean-square temperature gradient.