Bifurcations in penetrative Rayleigh-Be′nard convection in a cylindrical container

The bifurcations of penetrative Rayleigh-B′enard convection in cylindrical containers are studied by the linear stability analysis(LSA) combined with the direct numerical simulation(DNS) method. The working ?uid is cold water near 4?C, where the Prandtl number P r is 11.57, and the aspect ratio(radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θ_m. The relationship between the normalized critical Rayleigh number(Rac(θ_m)/Rac(0)) and θ_m is formulated, which is in good agreement with the stability results within a large range of θ_m. The aspect ratio has a minor effect on Rac(θ_m)/Rac(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system.Moreover, two kinds of qualitatively different steady axisymmetric solutions are identi?ed.