Stability of multiple steady states of convection

A parametric study of multiple steady states, their stability, onset of oscillatory instabi-lity, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this conguration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two xed values of the Prandtl number: Pr = 0 and Pr = 0:015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0:015 6 Pr 6 0:03) and xed 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 reproduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in good agreement with the experimental results if the width ratio (width/height) of the experimental container is suciently large. The study is carried out numerically with the use of two independent numerical approaches based on the global Galerkin and nite-volume methods. 1