Rayleigh-Bénard convection in a vertical annular container near the convection threshold.

The instabilities and transitions of flow in an annular container with a heated bottom, a cooled top, and insulated sidewalls are studied numerically. The instabilities of the static diffusive state and of axisymmetric flows are investigated by linear stability analysis. The onset of convection is independent of the Prandtl number but determined by the geometry of the annulus, i.e., the aspect ratio Gamma(outer radius to height) and radius ratio delta(inner radius to outer radius). The stability curves for onset of convection are presented for 0.001 <= delta <= 0.8 at six fixed aspect ratios: Gamma = 1, 1.2, 1.6, 1.75, 2.5, and 3.2. The instability of convective flow (secondary instability), which depends on both the annular geometry and the Prandtl number, is studied for axisymmetric convection. Two pairs of geometric control parameters are chosen to perform the secondary instability analysis-Gamma = 1.2, delta = 0.08 and Gamma = 1.6, delta = 0.2-and the Prandtl number ranges from 0.02 to 6.7. The secondary instability exhibits some similarities to that for convection in a cylinder. A hysteresis stability loop is found for Gamma = 1.2, delta = 0.08 and frequent changes of critical mode with Prandtl number are found for Gamma = 1.6, delta = 0.2. The three-dimensional flows beyond the axisymmetry-breaking bifurcations are obtained by direct numerical simulation for Gamma = 1.2, delta = 0.08.