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

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 tc, tD and tu 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 tu-values. The new measures to represent tc and tD are suggested, based on the growth rates of fluctuations. It is interesting that tc is the invariant but the predicted tD- and tu-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 tu ≅ 7tc agrees well with the available experimental tu-values for Ra > 105 and tD is located between tc and tu. This paper removes the confusion among the characteristic times, tc, tD and tu 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.