Scalings for unsteady natural convection boundary layers on a vertical plate at time-dependent temperature

In this paper, a scaling analysis using a simple three-region structure was used to develop scalings for the unsteady natural convection boundary layer (NCBL) of a homogeneous Newtonian fluid with Pr>1 adjacent to a finite vertical plate evenly heated with a time-varying sinusoidal temperature, where Pr is the Prandtl number. Three distinct sub-stages are identified in the heating stage, i.e., a start-up stage, a transitional stage, and a quasi-steady stage. After the heating stage, there is a very short cooling stage, in which a double thermal boundary layer structure is developed, which leads to a new, thinner thermal boundary layer adjacent to the plate co-existing with the original thermal boundary layer. A series of scalings are developed for the thermal and viscous boundary thicknesses, the maximum vertical velocity within the boundary layer, and the Nusselt number which are the major parameters representing the flow behavior, in terms of the governing parameters of the flow, i.e., the Rayleigh number Ra, Pr, and the dimensionless natural frequency of the time-varying sinusoidal temperature at various flow development stages. These scalings are tested by full numerical solutions of the governing equations and are shown to, in general, provide an accurate description of the flow.