Instabilities in a cylindrical cavity heated from below with a free surface. I. Effect of Biot and Marangoni numbers

Convective instabilities in a cylindrical cavity heated from below, with a free surface at the top, are numerically investigated using a spectral-element code. Both buoyancy and surface tension forces are taken into account, and heat exchange is considered at the upper surface. This configuration corresponds to the Bénard-Marangoni situation. The primary thresholds associated with azimuthal eigenmodes and corresponding to the onset of convection are first given as a function of the aspect ratio of the cavity A (radius/height), the Biot number Bi, and the Marangoni number Ma. Particular attention is paid to the influence of the Biot and Marangoni numbers: a stabilizing surface tension effect (Ma>0) induces an increase of the primary thresholds, which is magnified for small values of Bi, but may also change the flow structure by creating counter-rotating rolls near the free surface. The nonlinear evolution of the convection beyond its onset is given through bifurcation diagrams for A=1.5. Two different branches of axisymmetric solutions, either with upflow or downflow at the center, emerge at the onset. The destabilization of these solutions and the further dynamical evolution of the flow has been highlighted for widely varying Biot numbers.