Analytical and Numerical Study of Natural Convection in a Stably Stratified Fluid along Vertical Plates and Cylinders with Temporally-periodic Surface Temperature Variations

This paper describes one-dimensional (parallel) laminar and transitional regimes of natural convection in a viscous stably stratified fluid due to temporally-periodic variations in the surface temperature of infinite vertical plates and cylinders. Analytical solutions are obtained for the periodic laminar regime for arbitrary values of stratification, Prandtl number and forcing frequency. The solutions for plates and cylinders are qualitatively similar and show that (i) the flows are composed of two waves that decay exponentially with distance from the surface; a fast long wave and a slow short wave, (ii) for forcing frequencies less than the natural frequency, both waves propagate away from the surface, while (iii) for forcing frequencies less than this natural frequency, the short wave propagates away from the surface while the long wave propagates toward the surface. The analytical results are complemented, for the plate problem, with three-dimensional numerical simulations of flows that start from rest and are suddenly subjected to a periodic thermal forcing. The numerical results depict the transient (start-up) stage of the laminar flow and the approach to the periodicity, and confirm that the analytical solutions provide the appropriate description of the periodic regime for the laminar convection case. Preliminary numerical data are presented for transition from the laminar to turbulent convection. NOMENCLATURE k – nondimensional wavenumber; R – nondimensional radial coordinate; ' T –perturbation temperature; T∞ – ambient temperature; W – nondimensional velocity along the plate; β – buoyancy parameter; θ – nondimensional temperature perturbation; κ – molecular thermal diffusivity; ξ – nondimensional plate-normal coordinate; τ – nondimensional time. INTRODUCTION Unsteady natural convection flows abound in nature and technology. Such flows are notoriously difficult to analyze theoretically because of the intrinsic coupling between the temperature and velocity fields. A notable exception is the classical problem of unsteady laminar one-dimensional natural convection along an infinite vertical plate, a class of flows for which the Boussinesq equations of motion and thermodynamic energy reduce to a set of linear partial differential equations that may be solved analytically in a number of circumstances [1]. These solutions are among the few known exact solutions of unsteady natural convection. Such solutions are prized for the insights they provide into flow behavior and for their utility as benchmark solutions for verification of computational fluid dynamics algorithms. In the 1950's and 1960's, analytical solutions for unsteady one-dimensional natural convection along an infinite vertical plate in an unstratified fluid were obtained for a variety of surface forcings [1]. These solutions described resting fluids that were abruptly set into motion through the agency of a surface heat flux or temperature perturbation. The 4 International Conference on Computational Heat and Mass Transfer flows were characterized by a sudden burst of convection along the plate followed by an inexorable outward growth of the boundary layer. The extension of the one-dimensional convection framework to include ambient stratification is a relatively recent development [2-5]. Shapiro and Fedorovich [4] considered convection in a stratified flow adjacent to a single infinite plate. Analytical solutions were obtained for a Prandtl number of unity for the cases of impulsive (step) change in plate perturbation temperature, sudden application of a plate heat flux, and for arbitrary temporal variations in plate perturbation temperature or plate heat flux. Vertical motion in a stably stratified fluid was associated with a simple negative feedback mechanism: rising warm fluid cooled relative to the environment, whereas subsiding cool fluid warmed relative to the environment. Because of this feedback, convective flows in stably stratified fluid adjacent to a double-infinite plate eventually approached a steady state, whereas the corresponding flows in an unstratified fluid did not. In a companion paper, Shapiro and Fedorovich [5] explored the Prandtl number dependence of convection of a stably stratified fluid along a single vertical plate both numerically and analytically. The developing boundary layers were thicker, more vigorous, and more sensitive to the Prandtl number at smaller Prandtl numbers (<1) than at larger Prandtl numbers (>1). The gross temporal behavior of the flow after the onset of convection was of oscillatorydecay type for Prandtl numbers near unity, and of non-oscillatory-decay type for large Prandtl numbers. In the present paper, analytical solutions are obtained for laminar natural convection flows of linearly-stratified fluid along infinite vertical plates and cylinders with temporally-periodic surface temperature variations. The plate and cylinder solutions, which are rather similar in the periodic regime, are valid for arbitrary ambient stratification, forcing frequency and Prandtl number. However, these solutions only apply to the purely periodic regime, and do not describe the transient (start-up) stage of a flow started from rest. In order to study this start-up flow stage and verify that the analytical solutions provide the appropriate description of the terminal state of the laminar convective flow started from rest, a direct numerical simulation is invoked. Numerical results are presented also for the plate flow undergoing a transition from the laminar to turbulent regime. ANALYTICAL SOLUTIONS The governing equations for one-dimensional (parallel) laminar natural convection in the Boussinesq approximation are discussed in detail in Shapiro and Fedorovich [4]. Under the onedimensional restriction, the Boussinesq form of mass conservation is satisfied identically, while the horizontal equations of motion reduce to statements that the horizontal components of the pressure gradient are zero. The dimensionless vertical equation of motion and thermodynamic energy equation for a linearly-stratified fluid reduce to 2 2 W W θ τ ξ ∂ ∂ = + ∂ ∂ , 2 2 1 Pr W θ θ τ ξ ∂ ∂ = − + ∂ ∂ , (1) where ξ is the horizontal (plate-normal) coordinate, τ is time, θ is perturbation temperature, W is vertical velocity, and Pr=ν/κ is the Prandtl number, where the kinematic viscosity ν and thermal diffusivity κ are considered to be constant. These nondimensional variables (ξ, τ, θ, W) are related to their dimensional counterparts (x, t, , w) by , , ' T 1/ 2 1/ 4 ( ) x ξ ν γβ − ≡ 1/ 2 ( ) t τ γβ ≡ 0 '/ ' T T θ ≡ , , where is the surface temperature perturbation, γ ≡ 1/ 2 1/ 2 0 / ' W w T γ β − ≡ 0 ' T / / p dT dz g c ∞ + is the stratification parameter, z is height, and is a linearly-varying ambient temperature. ( ) T z ∞ The corresponding inviscid system [dimensional version of (1) with ν=κ=0] admits traveling wave solutions of the form sin( ) w kx βγτ ∝ − and ' cos( ) T kx βγτ ∝ − , where / r g T β ≡ is the buoyancy parameter ( is the reference temperature); and en masse temporal oscillations of the form r T sin( w ) βγτ ∝ and ' cos( T ) βγτ ∝ . These inviscid solutions have a frequency equal to the Brunt-Väisälä (buoyancy) frequency βγ . The nondimensional value of this frequency is unity. The plate is located at ξ=0, and fluid fills the semiinfinite domain ξ>0. The no-slip condition is imposed at the surface, W(0, τ) = 0. The perturbation temperature at the surface is a temporal oscillation with circular frequency ω and an amplitude of unity, θ(0, τ) = cosωτ. This fixed amplitude is a consequence of the nondimensionalization, and does not represent a loss of generality. Since the thermal forcing originates at the plate, and the medium is viscous, the disturbance is considered to vanish far from the plate, except for the case of resonance (imposed frequency equal to the natural frequency of 4 International Conference on Computational Heat and Mass Transfer the inviscid system, ω = 1), where an extension of the disturbance to infinity is found to be unavoidable. Solution for the plate flow: We seek solutions of (1) in the form of simple harmonic oscillations: [ ( )exp( )] A i θ ξ ωτ =R − [ ( )exp( )] i , W B ξ ωτ =R − , (2) where, A and B are complex-valued functions, and, without loss of generality, ω is considered to be positive. Application of (2) in (1) yields 2 2 1 Pr d A i A B d ω ξ − = − + , 2 2 d B i B A d ω ξ − = + . (3) The solution of (3) is 1 (1 )exp( )cos( ) 2 1 (1 )exp( )cos( ) 2 F k k