Modal Selection for Inclined Darcy-Bénard Convection in a Rectangular Cavity

Abstract

Nonlinear free convection in an inclined rectangular porous cavity heated from below has been studied using a two-dimensional spectral decomposition. The code uses pseudo-arclength continuation to follow solution curves around fold bifurcations. The evolution with inclination of the pattern of convection is complicated and it relies strongly on both the Darcy–Rayleigh number and the aspect ratio of the cavity. When the inclination is large it is generally true that only one cell appears, and that it has a circulation that is consistent with the direction of the buoyancy forces along the heated and cooled boundaries. However, as the inclination decreases back towards the horizontal, this unicellular pattern evolves, sometimes initially via fold bifurcations, into patterns with different numbers of cells. Such evolutions always conserve the parity of the number of cells (such as one cell becoming three and then five, or two cells becoming four), but bifurcations also arise between patterns with different parities. These phenomena are illustrated using a suitable selection of solution curves that show the dependence of the Nusselt number on the inclination.