RESUMEN
The flow instabilities of Rayleigh-Bénard convection in a cylinder with effect of uniform internal heat source are investigated numerically. The instabilities of the static state and of axisymmetric flows are investigated by linear stability analysis. The convection threshold depends on the strength of internal heat source q and the aspect ratio of the cylinder Γ. The stability of axisymmetric flows is strongly affected by these two parameters, as well as the Prandtl number Pr. Depending on the value of q, three regimes are identified: weak internal heating, moderate internal heating, and strong internal heating regime. In a weak internal heating regime, the instability characteristics are similar to Rayleigh-Bénard convection. In a moderate internal heating regime, intense interaction of buoyancy instability and hydrodynamic instability result in complex instability curves. When q is large enough, the internal heating effect overwhelms the boundary heating effect. Specifically, the influence of Pr on instability is studied at a moderate internal heat strength q=6.4. An extremely multivalued stability curve is observed. At most five critical Rayleigh numbers can be determined for the axisymmetry-breaking instability at a certain Prandtl number. An axisymmetric unsteady instability mode is observed as well. By nonlinear simulation, the oscillatory flow patterns are obtained, and the axisymmetry-breaking bifurcation of the unsteady toroidal flow is studied.
RESUMEN
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 Γ (outer radius to height) and radius ratio δ (inner radius to outer radius). The stability curves for onset of convection are presented for 0.001≤δ≤0.8 at six fixed aspect ratios: Γ=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-Γ=1.2, δ=0.08 and Γ=1.6, δ=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 Γ=1.2, δ=0.08 and frequent changes of critical mode with Prandtl number are found for Γ=1.6, δ=0.2. The three-dimensional flows beyond the axisymmetry-breaking bifurcations are obtained by direct numerical simulation for Γ=1.2, δ=0.08.
RESUMEN
Three-dimensional steady Rayleigh-Bénard convection in a vertical cylinder is investigated by numerical simulation and bifurcation analysis. The complex pattern formation beyond the onset of the convection is presented by a bifurcation diagram. The coexistence of multiple stable states is observed near the threshold of the first bifurcation and two group symmetries are summarized for the corresponding primary branches. The first stable target pattern originates through a subcritical bifurcation. A multiplicity of flow states for the Rayleigh number of 14200 is validated numerically in comparison with the experiment, and a four-spoke pattern is observed.