Pattern selection for finite-amplitude convection states in boxes of porous media

Abstract

When a box of fluid-saturated porous material is heated from below, it is known that either a two- or three-dimensional convection pattern can occur depending on the initial configuration. By means of an analytic eigenfunction-expansion technique and a study of the phase-space dynamics of finite-amplitude disturbances we obtain (i) the regions within the space of initial conditions which lead to one or other of these competing states, and thereby (ii) the probability that a certain pattern will be realized, as well as (iii) the explicit form of the heat transferred by the patterns as it depends on aspect ratios. Cubic and nearly cubic boxes are considered, and the analysis applies for values of Rayleigh parameter from convection onset to 1.5 times critical. Our results correct several details appearing in the literature and explain observations made in previous numerical studies.