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

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.

Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

“…Within the time periodic flow regime, they showed that the interface, which was shown to temporally oscillate, could lead to crystal striations. Flow bifurcation in a vertical cylinder heated from below was also investigated by Yamaguchi et al [8] and Ma et al [9], who found multiple steady-state flows at different Rayleigh numbers and aspect ratios. A notable recent work is the study of 3-D axisymmetry-breaking instabilities in a destabilized Bridgman-like configuration using the linear stability analysis by Gelfgat et al [10].…”

“…Eventually, an infinite number of frequencies without any specific peak will be observed. At this moment, chaos was set in at a Rayleigh number that is larger than 1.3 Â 10 8 . The process that the solutions evolve firstly from the fixed point to the periodic attractor and then to the two-torus limit set constitutes the route to chaos.…”

Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

“…One approach is to solve the momentum and continuity equations directly in coupled form (e.g., Yamaguchi, Chang, and Brown [45] and Bathe and Dong [4]). This direct approach is general and robust; however, it can be inefficient and memory intensive for large, three-dimensional problems, in particular, for high-order methods.…”

Petrov‐Galerkin methods for natural convection in directional solidification of binary alloys

“…This qualitative change in interface shape was accompanied by achange in the sign of the determinant of the Jacobian matrix, which indicated the probable loss of temporal stability of the discrete equation set. 6 We believe that the oscillations in the interface are a numerical instability associated with the onset of morphological instability along the melt/solid interface, which cannot be adequately resolved by either our model for interface shape or by the finite element mesh. For the analysis of a planar interface, neglecting the correction to the melting temperature caused by interface curvature and the surface free energy results in the prediction of morphological instability for all spatial wavelengths, with the fastest growing wavelength being the smallest one.…”