Multiple states, stability and bifurcations of natural convection in a rectangular cavity with partially heated vertical walls

• Published in: Journal of fluid mechanics 2003-10, Vol.492, p.63-89, Article S0022112003005469
• Main Authors: ERENBURG, V., GELFGAT, A. YU, KIT, E., BAR-YOSEPH, P. Z., SOLAN, A.
• Format: Article
• Language: English
• Subjects: Buoyancy-driven instability; Convection; Convection and heat transfer; Exact sciences and technology; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Heat conductivity; Heat transfer; Heating; Hydrodynamic stability; Nonlinearity (including bifurcation theory); Pattern selection; pattern formation; Physics; Stability analysis; Studies; Turbulent flows, convection, and heat transfer
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112003005469

The multiplicity, stability and bifurcations of low-Prandtl-number steady natural convection in a two-dimensional rectangular cavity with partially and symmetrically heated vertical walls are studied numerically. The problem represents a simple model of a set-up in which the height of the heating element is less than the height of the molten zone. The calculations are carried out by the global spectral Galerkin method. Linear stability analysis with respect to two-dimensional perturbations, a weakly nonlinear approximation of slightly supercritical states and the arclength path-continuation technique are implemented. The symmetry-breaking and Hopf bifurcations of the flow are studied for aspect ratio (height/length) varying from 1 to 6. It is found that, with increasing Grashof number, the flow undergoes a series of turning-point bifurcations. Folding of the solution branches leads to a multiplicity of steady (and, possibly, oscillatory) states that sometimes reaches more than a dozen distinct steady solutions. The stability of each branch is studied separately. Stability and bifurcation diagrams, patterns of steady and oscillatory flows, and patterns of the most dangerous perturbations are reported. Separated stable steady-state branches are found at certain values of the governing parameters. The appearance of the complicated multiplicity is explained by the development of the stably and unstably stratified regions, where the damping and the Rayleigh–Bénard instability mechanisms compete with the primary buoyancy force localized near the heated parts of the vertical boundaries. The study is carried out for a low-Prandtl-number fluid with $\hbox{\it Pr}\,{=}\,0.021$. It is shown that the observed phenomena also occur at larger Prandtl numbers, which is illustrated for $\hbox{\it Pr}\,{=}\,10$. Similar three-dimensional instabilities that occur in a cylinder with a partially heated sidewall are discussed.