On oscillatory instability of convective flows at low Prandtl number

Numerical investigation of the oscillatory instability of convective flows in laterally heated rectangular cavities is presented. Cavities with no-slip isothermal vertical boundaries, no-slip adiabatic lower boundary, and stress-free adiabatic upper boundary are considered. Dependence of the critical Grashof number and the critical frequency of oscillations on the aspect ratio (A = length/height) of the cavity are investigated The stability diagrams were obtained for the whole interval of the aspect ratio 1 less than or equal to A less than or equal to 10. The study was carried out for two values of the Prandtl number, Pr = 0 and 0.015. It was shown that the oscillatory instability sets in as a result of the Hopf bifurcation. It was found that at two different values of the Prandtl number considered the instability is caused by different infinitely small dominant perturbations, which means that the convective heat transfer strongly affects stability of the flow even for cases having small Prandtl number. No asymptotic behavior large aspect ratios was found up to A = 10. Slightly supercritical oscillatory flows were approximated asymptotically by means of the weakly nonlinear analysis of the calculated bifurcation.