On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case

Three-dimensional steady flows are simulated in a circular cylindrical cavity of aspect ratio A=H/D, where H is the height and D the diameter of the cavity. The cavity is heated from below and its sidewalls are considered to be adiabatic. The effect of the geometry of the cavity on the onset of convection and on the structure and symmetries of the flow is analyzed. The nonlinear evolution of the convection beyond its onset is presented through bifurcation diagrams for two typical aspect ratios A=0.5 and A=1. Axisymmetric (m=0) and asymmetric (m=1 and m=2) azimuthal modes [exp (imφ)] are observed. For A=0.5, the axisymmetric solution loses its stability to a three-dimensional solution at a secondary bifurcation point. Better understanding of the mechanisms leading to this instability is obtained by analyzing the energy transfer between the basic state and the critical mode. To study the influence of the Prandtl number on the flow pattern and on the secondary bifurcation, three values of the Prandtl number are investigated: Pr=0.02 (liquid metal), Pr=1 (transparent liquids), and Pr=6.7 (water).