Stability of multiple steady states of convection in laterally heated cavities

A parametric study of multiple steady states, their stability, onset of osc illatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is pre sented. Cavities with four no-slip boundaries, isothermal vertical and perf ectly insulated horizontal boundaries are considered. Four distinct branche s of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/hei ght) of the cavity varying continuously from 1 to 11 and for two fixed valu es of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represente d as stability diagrams showing the critical parameters (critical Grashof n umber and the frequency at the onset of the oscillatory instability) corres ponding to transitions from steady to oscillatory states, appearance of mul ti-roll states, merging of multiple states and backwards transitions from m ulti-roll to single-roll states. For better comparison with the existing ex perimental data, an additional stability study for varying Prandtl number ( 0.015 less than or equal to Pr less than or equal to 0.03) and fixed value of the aspect ratio A = 4 was carried out. It was shown that the dependence of the critical Grashof number on the aspect ratio and the Prandtl number is very complicated and a very detailed parametric study is required to rep roduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in g ood agreement with the experimental results if the width ratio (width/heigh t) of the experimental container is sufficiently large. The study is carrie d out numerically with the use of two independent numerical approaches base d on the global Galerkin and finite-volume methods.