The onset of buoyancy-driven convection in a horizontal fluid layer heated suddenly from below

Abstract

The onset of buoyancy-driven convection in an initially quiescent fluid layer confined between the two infinite horizontal plates is analyzed theoretically. In the case of isothermal heating it is well known that a convective motion sets in for the Rayleigh number Ra≥1708. By using the linear stability theory, for Ra⪢1708 the onset time of instantaneous instability tc is analyzed in the self-similar coordinate. In the self-similar coordinate, the propagation theory based on the quasi-steady state approximation (QSSA) represents the eigenanalysis and the initial value approach without QSSA and therefore it seems to be a quite reasonable. It is assumed that, besides the theoretical tc, there are two more characteristic times: (1) the theoretical time tm which marks the maximum disturbance or fluctuation growth rate and (2) the experimental time scale tD which marks the detection time of motion. By employing the numerical method under the single mode of instabilities, tc and tm are analyzed. It is interesting that tc is the invariant but the predicted tm-value is dependent upon the magnitude of initial conditions forced. It is shown for the isothermally heated system of a large Prandtl number, tm(≈4tc) obtained by fitting some experimental tD-value agrees well with the available experimental tD-values for Ra>105.