Scalings for unsteady natural convection boundary layers on an evenly heated plate with time-dependent heating flux

It is of fundamental significance, especially with regard to application, to fully understand the flow behavior of unsteady natural convection boundary layers on a vertical plate heated by a time-dependent heat flux. Such an understanding is currently scarce. In this paper, the scaling analysis by Lin et al. [Phys. Rev. E 79, 066313 (2009)] using a simple three-region structure for the unsteady natural convection boundary layer of a homogeneous Newtonian fluid with Pr > 1 under isothermal heating was substantially extended for the case when the heating is due to a time-varying sinusoidal heat flux. A series of scalings was developed for the thermal boundary thickness, the plate temperature, the viscous boundary thicknesses, and the maximum vertical velocity within the boundary layer, which are the major parameters representing the flow behavior, in terms of the governing parameters of the flow, i.e., the Rayleigh number Ra, the Prandtl number Pr, and the dimensionless natural frequency f(n) of the time-varying sinusoidal heat flux, at the start-up stage, at the transition time scale which represents the ending of the start-up stage and the beginning of the transitional stage of the boundary-layer development, and at the quasi-steady stage. These scalings were validated by comparison to 10 full numerical solutions of the governing equations with Ra, Pr, and f(n) in the ranges 10⁶ <= Ra <= 10⁹, 3 <= Pr <= 100, and 0.01 <= f(n) <= 0.1 and were shown in general to provide an accurate description of the flow at different development stages, except for high-Pr runs in which a further, although weak, Pr dependence is present, which cannot be accurately predicted by the current scaling analysis using the simple three-region structure, attributed to the non-boundary-layer nature of the velocity field with high-Pr fluids. Some scalings at the transition time scale and at the quasi-steady stage also produce noticeable deviations from the numerical results when f(n) is reduced, indicating that there may be a further f(n) dependence of the scalings which also cannot be accurately predicted by the current scaling analysis