Scaling Analysis of Unsteady Natural Convection Boundary Layers on an Evenly Heated Plate with a Time-dependent Temperature

In this paper, a scaling analysis using a simple three-region structure was conducted for the unsteady natural convection boundary layer (NCBL) of a homogeneous Newtonian fluid with Pr > 1 adjacent to a vertical plate evenly heated with a time-dependent sinusoidal temperature. A series of scalings were developed for the thermal boundary thickness, the viscous boundary thicknesses, the maximum vertical velocity within the boundary layer, and the local and average Nusselt number across the plate, 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 fn of the time-dependent sinusoidal temperature, 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. Introduction As a classic fluid mechanics problem, NCBL flow has been widely studied. Earlier studies had focused on experimental and analytical investigations of the steady behavior of the flow, in particularly that on a heated semi-infinite vertical wall and in a rectangular cavity with differentially heated sidewalls. More recent studies have focused on the transient flow behavior. In particular, scaling analysis has proven to be a very effective tool to reveal the transient behavior of such a flow since Patterson and Imberger [1] made a pioneering scaling analysis of the transient NCBL flow in a two-dimensional rectangular cavity with differentially heated sidewalls. This study has inspired many subsequent studies to extend scaling analysis to many different aspects of transient NCBLs under various configurations and flow conditions. The readers are referred to our recent papers (e.g.[2]) for a more detailed review of some of these studies. Unsteady NCBLs on a vertical plate heated by a time-dependent heat flux or temperature are found in many applications, such as in the Trombe wall system of a passive solar house and in a solar chimney for electricity generation. In the Trombe wall case, the wall, which is usually painted in black or with a solar selective coating, absorbs solar radiation and converts it into heat which is then transported to the dwelling by the heated air via NCBL flow in the channel formed by the glazing and the wall. A solar chimney operates in a similar manner. For both cases, the time-dependent solar radiation, which varies sinusoidally under a clear sky condition (only in the first half of the sinusoidal cycle), serves as the heat flux for the NCBL flows. Although there have been numerous studies on NCBLs on a vertical plate heated by a heat flux, the majority of these studies have been on the cases where the applied heat flux is either uniformly constant or spatially varied but not time dependent. Lin and Armfield [3] recently carried out a scaling analysis to develop scalings for the unsteady NCBL of a homogeneous Newtonian fluid with Pr > 1 adjacent to a vertical plate evenly heated with a time-varying sinusoidal heat flux, which were validated and quantified by a series of direct numerical simulations. In the current study, this scaling analysis is extended to the unsteady NCBL of a homogeneous Newtonian fluid with Pr > 1 adjacent to a vertical plate evenly heated with a time-dependent sinusoidal temperature, which, to our best knowledge, has not been done so far, although its fundamental significance and practical application importance. Scaling Analysis Under consideration is the unsteady NCBL of a homogeneous Newtonian fluid with Pr > 1 adjacent to a vertical plate evenly heated with a time-dependent sinusoidal temperature in the form of Tw(t) = Ta +Tw,msin(2π f t) (1) where t is time, Ta is the initial fluid temperature at t = 0, Tw,m and f are the amplitude and the natural frequency of the timedependent sinusoidal temperature applied to the plate, respectively. The flow is assumed to be two-dimensional and the fluid is initially at rest. The plate lies at X = 0 with the origin at Y = 0 (X and Y are the horizontal and vertical coordinates, respectively), with the plate boundary conditions U =V = 0, Tw(t) = Ta +Tw,msin(2π f t) atx = 0 for Y > 0 where Tw,m and f are assumed to be constant for a specific timedependent temperature condition. The governing equations of motion are the Navier-Stokes equations with the Boussinesq approximation for buoyancy, which together with the temperature equation can be written in the following two-dimensional forms, ∂U ∂X + ∂V ∂Y = 0 (2) ∂U ∂t + ∂(UU) ∂X + ∂(VU) ∂Y =− 1 ρ ∂P ∂X +ν ( ∂2U ∂X2 + ∂2U ∂Y 2 )