Multiple buoyancy-driven flows in a vertical cylinder heated from below

Abstract

Abstract The structure of axisymmetric buoyancy-driven convection in a vertical cylinder heated from below is probed by finite-element solution of the Boussinesq equations coupled with computer-implemented perturbation techniques for detecting and tracking multiple flows and for determining flow stability. Results are reported for fluids with a Prandtl number of one and for cylinders with aspect ratio Λ (defined as the ratio of height to radius of the cylinder) between 0.5 and 2.25. Extensive calculations of the neutral stability curves for the static solution and of the nonlinear motions along the bifurcating flow families show a continuous change of the primary cellular motion from a single toroidal cell to two and three cells nested radially in the cylinder, instead of the sharp transitions found for a cylinder with shear-free sidewalls. The smooth transitions in flow structure with Rayleigh number and Λ are explained by nonlinear connectivity between the first two bifurcating flow families formed either by a secondary bifurcation point for Λ ⩽ Λ* ≃ 0.80 or by a limit point for Λ ⩽ Λ*. The transition between the two modes may be described by the theory of multiple limit point bifurcation.