Natural Convection in a Periodically Heated Slot

Heat transfer resulting from the natural convection in a fluid layer contained in an infinite horizontal slot bounded by solid walls and subject to a spatially periodic heating at the lower wall has been investigated. The heating produces sinusoidal temperature variations along one horizontal direction characterized by the wave number α with the amplitude expressed in terms of a suitably defined Rayleigh number Rap. The maximum heat transfer takes place for the heating with the wave numbers α=0(1) as this leads to the most intense convection. The strength of convection decreases proportionally to α for small heating wave numbers, resulting in the heat transfer being dominated by periodic conduction with the Nusselt number decreasing proportionally to α. When α becomes large, the convection is confined to a thin layer adjacent to the lower wall with its strength decreasing proportionally to α, with the temperature field above the convection layer loosing dependence on the horizontal direction. Introduction Fluid systems exposed to spatially distributed heating patterns are ubiquitous in nature. Illustrative examples include convection in atmospheric boundary layer as surfaces of different colours heat up at different rates, e.g., patterns of forests/lakes in rural environments and patterns of roofs/streets in urban environments, convection induced by patterns of heat sources, e.g., systems of localized fires, computer chips, thermal patterning in microfluidic devices for biological applications, and many others. The characteristic property of such systems is the existence of horizontal gradients of buoyancy force which drive convection regardless of the intensity of the heating. This is in contrast with the well know Rayleigh-Bénard (RB) convection [1,2] where the heating must meet certain critical conditions in order to set the fluid into a motion. In spite of the obvious relevance, convection resulting from spatially distributed heating has not attracted much attention. The existing results are limited to simple sinusoidal temperature distributions. Analysis of arbitrary heating patterns is yet to be attempted. There have been a number of studies dealing with convection in slots formed by isothermal walls fitted with spatially-periodic grooves [3-5]. To our best knowledge, Kelly and Pal [6] were the first to study convection resulting from the sinusoidal heating using asymptotic methods in the long wavelength limit. Yoo and Kim [7] used Direct Numerical Simulation (DNS) to investigate convection with the heating wavelength comparable to the channel height and demonstrated existence of a steady convection replaced by a sequence of bifurcations as the heating intensity increased. As evidenced by the above discussion, the knowledge of even the fundamental features of convection driven by a periodic heating is very limited. The objective of this analysis is therefore to determine the basic characteristics of the heat transfer process in a simple reference system consisting of an infinite horizontal layer subject to heating from below. We shall focus attention on the simplest heating pattern represented by one Fourier mode and investigate system response in the complete range of the heating wave numbers α. Problem formulation Consider fluid contained in a slot between two parallel plane plates extending to ±∞ in the x-direction and placed at a distance 2h apart each other with the gravitational acceleration g acting in the negative y-direction, as shown in Fig.1. The upper plate is kept isothermal while the lower plate is subject to a periodic heating resulting in the temperatures of the lower (θL) and upper (θU) walls in the form x) cos( 1/2 (x) θL α = , 0 (x) θU = (1) where λ=2π/α is the wavelength of the heating, θ denotes the relative temperature scaled with the amplitude of the peak to peak temperature variations along the wall Td, i.e., θ = (T TU)/Td, T denotes the absolute temperature and TU denotes the temperature of the upper wall. The fluid is incompressible, Newtonian, with thermal conductivity k, specific heat c, thermal diffusivity κ=k/ρc, kinematic viscosity ν, dynamic viscosity μ, thermal expansion coefficient Γ and variations of the density ρ that follow the Boussinesq approximation. The temperature field is represented as a sum of the conductive field Θ and the convective field θ. Two temperature scales are used, i.e., Td defined above as the conductive temperature scale and Tv= Tdν/κ as the convective temperature scale. Tv/Td=Pr where Pr=ν/κ stands for the Prandtl number. Half of distance between the plates h is used as the length scale, ν/h Uv = is used as the (convective) velocity scale and v v ρU P = is used as the (dynamic) pressure scale. The complete dimensionless temperature θtot is scaled with the convective scale and takes the form ( ) ( ) ( ) y x, θ y x, Θ Pr y x, θ 1 tot + = − (2) where the conductive temperature Θ is determined analytically, has the form