High wavenumber convection in tall porous containers heated from below

In this paper we consider convection in a high-aspect-ratio porous container which is heated from below and whose sidewalls are insulated. This paper presents asymptotic analyses of weakly nonlinear and highly nonlinear convection in the limit of large wavenumber, a. When a is O(1), strongly nonlinear convection must be analysed using fully numerical methods, though, when a is large, some progress can be made using asymptotic methods without recourse to a direct simulation of the full governing equations.

The onset of convection is given by the usual linear stability analysis for the Darcy-Bénard problem, and the critical Rayleigh number is given by R_c ~ a² at leading order. As the Rayleigh number increases there are three nonlinear asymptotic regimes which may be considered, the first of which yields the familiar weakly nonlinear flow. In this regime, for which R — R_c is asymptotically small, the weakly nonlinear evolution is given by the solution of a cubic Landau equation for the amplitude. When R — R_c = O(1), the second regime, the flow is characterized by the fact that the mean correction to the temperature profile and the first harmonic appear at the same order in the asymptotic expansion. The amplitude of convection is now found to be given by an integro-differential equation and numerical solutions indicate that when R — R_c is large boundary layers develop at the top and the bottom of the container.

The final regime emerges when R is so large that it is of size R + O(a). The flow then separates into three distinct domains: two boundary layers form, one at either end of the container, and a core flow exists in the interior. These flows are given by a singular perturbation analysis and the equations describing the boundary layer flow constitute a nonlinear eigenvalue problem which requires numerical solution.