Bifurcation analysis of multiple steady flow patterns for Rayleigh-Bénard convection in a cubical cavity atPr=130

Abstract

The bifurcation diagram of steady convective flow patterns inside a cubical cavity with adiabatic lateral walls heated from below and filled with silicone oil (Pr = 130) was determined for values of the Rayleigh number (Ra) up to 1.5 x 10(5). A continuation procedure based on the Galerkin spectral method was used to determine the steady convective solutions as a function of Ra. Bifurcations leading to either new steady or time-dependent solutions were identified and new steady solution branches were also continued. A total of fifteen steady solutions were tracked and the stability analysis predicted that six flow patterns were stable and that two, three, or even four of these patterns coexisted over certain ranges of Ra in the studied domain. Predicted flow patterns and transitions are in agreement with flow visualizations previously reported in the literature. The variation of the Nusselt number (Nu) as a function of Pr was investigated for three of the stable flow patterns identified: a x or y roll, a diagonal oriented roll and a pattern formed by four connected half rolls. It was found that whereas the Nusselt changes within the region 0.71 < or = Pr < or = 10 it tends to an asymptotic value with increasing Pr.