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

摘要

It is of fundamental significance, especially with regard to application, to fully understand the flow behaviornof unsteady natural convection boundary layers on a vertical plate heated by a time-dependent heat flux. Suchnan understanding is currently scarce. In this paper, the scaling analysis by Lin et al. [Phys. Rev. E 79, 066313n(2009)] using a simple three-region structure for the unsteady natural convection boundary layer of a homogeneousnNewtonian fluid with Pr > 1 under isothermal heating was substantially extended for the case when the heating isndue to a time-varying sinusoidal heat flux. A series of scalings was developed for the thermal boundary thickness,nthe plate temperature, the viscous boundary thicknesses, and the maximum vertical velocity within the boundarynlayer, which are the major parameters representing the flow behavior, in terms of the governing parametersnof the flow, i.e., the Rayleigh number Ra, the Prandtl number Pr, and the dimensionless natural frequency fnnof the time-varying sinusoidal heat flux, at the start-up stage, at the transition time scale which represents thenending of the start-up stage and the beginning of the transitional stage of the boundary-layer development, andnat the quasi-steady stage. These scalings were validated by comparison to 10 full numerical solutions of thengoverning equations with Ra, Pr, and fn in the ranges 106 u0002 Ra u0002 109, 3 u0002 Pr u0002 100, and 0.01 u0002 fn u0002 0.1 andnwere shown in general to provide an accurate description of the flow at different development stages, except fornhigh-Pr runs in which a further, although weak, Pr dependence is present, which cannot be accurately predictednby the current scaling analysis using the simple three-region structure, attributed to the non-boundary-layer naturenof the velocity field with high-Pr fluids. Some scalings at the transition time scale and at the quasi-steady stagenalso produce noticeable deviations from the numerical results when fn is reduced, indicating that there may be anfurther fn dependence of the scalings which also cannot be accurately predicted by the current scaling analysis.