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

Abstract

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 ≅ 7 t 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.