The onset of natural convection in a horizontal fluid layer heated isothermally from below

摘要

The onset of buoyancy-driven convection in an initially quiescent fluid layer confined between two horizontal plates is analyzed theoretically. In case of isothermal heating it is well known that convective motion sets in when the Rayleigh number Ra exceeds 1708. For Ra > 1708, there are three characteristic times t_c, t_D and t_u which represent respectively, the critical time to mark the onset of intrinsic instability, the detection time of motion, and the undershoot time in a plot of the heating rate versus time. These characteristic times are analyzed by employing the numerical method under the single mode of instabilities and fitting some experimental t_u-values. The new measures to represent t_c and t_D are suggested, based on the growth rates of fluctuations. It is interesting that t_c is the invariant but the predicted t_D-and t_u-values are dependent upon the magnitude of initial conditions forced. It is shown that for the isothermally heated system of a large Prandtl number the relation of t_u≌7t_c agrees well with the available experimental t_u-values for Ra > 10~5 and t_D is located between t_c and t_u. This paper removes the confusion among the characteristic times, t_c, t_D and t_u in the literature on stability. Also the boundary-layer instability model is discussed in order to analyze turbulent thermal convection heat transfer characteristics in the fully developed state, based on the present numerical predictions.