Enclosed Buoyant Convection With Internal Heat Generation Under Oscillating Sidewall Temperature

Abstract

Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division April 3, 2001; revision received September 6, 2001. Associate Editor: M. Faghri. Buoyant convection in an enclosure induced by uniform internal heat generation has been extensively studied (Acharya and Goldstein 1, Bergholz 3, Fusegi et al. 4, Kulacki and Goldstein 6, May 10). One canonical layout is a rectangular cavity with two vertical sidewalls at constant temperatures and two insulated horizontal endwalls, with large Rayleigh number Ra >105, Prandtl number O(1) and cavity aspect ratio O(1). In particular, when both vertical sidewalls are maintained at the same temperature T0, the overall flow is expectedly symmetric about the vertical centerline. In each half-cavity, the fluid rises (sinks) near the centerline (near the vertical sidewall) region, which forms a single circulation cell. Recently, buoyant flows induced by time-periodic boundary conditions, with no internal heat generation, emerge to be a subject of considerable interest (Antohe and Lage 2, Kwak and Hyun 7, Kwak et al. 8, Lage and Bejan 9). The overriding concern here is the existence of resonance, which was first observed in the numerical studies of Lage and Bejan 9 and was corroborated later by Antohe and Lage 2. It was illustrated that, when the frequency of the time-periodic boundary condition matches the proper resonance frequency, convective activities in the cavity are invigorated. This gives rise to a substantial increase in the amplitude of the fluctuating heat transfer coefficient in the cavity interior. The analytical endeavor of Lage and Bejan 9, based on the concept of a fluid wheel inside the cavity, suggested ways to make an order-of-magnitude estimate of the resonance frequency. In a related effort, Kwak and Hyun 7 and Kwak et al. 8 asserted that resonance is anticipated when the basic mode of the system eigenfrequencies is excited. They further argued that the system eigenfrequencies are characterized by the internal gravity oscillations in the interior, which are supported by the prevailing stratification. This paper describes the time-dependent buoyant convection in an enclosure, with the presence of internal heat generation, under a time-periodic thermal boundary condition. Estimations of the resonance frequencies will be made, and physical explanations will be offered. This configuration simulates simplified models of post-accident heat removal in nuclear reactors and geophysical problems. Also, natural convection is important in fluids undergoing electrolytic processes or exothermic chemical reactions. As emphasized earlier, utilization of time-periodic thermal boundary conditions is explored as a possible means to control these processes or reactions. Convection in an air-conditioned room, subject to the daily-varying environmental temperature, provides an easy example of technological applications of the flow configuration under present study. Consider a square cavity, filled with an incompressible Newtonian fluid with spatially-uniform internal heat generation of constant strength q0‴‴‴, as sketched in Fig. 1. The left vertical sidewall is maintained at constant temperature T0. The temperature Tr at the right vertical sidewall varies about T0 sinusoidally with time, Tr=T0+ΔT′ sinft, in which ΔT′ and f denote, respectively, the amplitude and frequency of oscillation. Flow is governed by the time-dependent Navier-Stokes equations, with the invocation of the Boussinesq-fluid relationship ρ=ρ0[1−βT−T0], which, after non-dimensionalization, read (1)∂U∂X+∂V∂Y=0,(2)∂U∂t+U ∂U∂X+V ∂U∂Y=−∂P∂X+PrRaI1/2∂2U∂X2+∂2U∂Y2,(3)∂V∂t+U ∂V∂X+V ∂V∂Y=−∂P∂Y+PrRaI1/2∂2V∂X2+∂2V∂Y2+θ,(4)∂θ∂t+U ∂θ∂X+V ∂θ∂Y=1RaIPr1/2∂2θ∂X2+∂2θ∂Y2+1RaIPr1/2.The associated boundary conditions can be expressed as (5)U=V=∂θ∂Y=0,atY=0,1;(6)U=V=θ=0,atX=0;(7)U=V=0,θ=ε sinωτ,atX=1.In the above, dimensionless quantities are defined as τ=tRaIPr1/2 κH2;X,Y=x,yH;(8)U,V=u,vRaIPr−1/2 Hκ;θ=kT−T0H2q0‴‴‴;P=p+ρ0gyH2ρ0κ2RaIPr;Pr=ν/κ;RaI=gβq0‴‴‴H5κνk,where u,v indicate dimensional velocity components in the horizontal (x) and vertical (y) directions, and ρ0 the reference density evaluated at the cold-wall temperature T0, and ν and κ denote the kinematic viscosity and thermal diffusivity, respectively, and β the isobaric coefficient of volumetric thermal expansion. The strength of internal heat generation is represented by the internal Rayleigh number, RaI. All fluid properties are assumed to be constant. Time is nondimensionalized by using the system Brunt-Va¨isa¨la frequency N, i.e., (9)N≡gβq0‴‴‴H2/kH1/2=RaIPr1/2 κH2.The nondimensional amplitude and frequency of the right-sidewall temperature fluctuation are (10)ε≡kΔT′q0‴‴‴H2;ω≡fN.In the above development, the system-wide temperature scale is derived from the internal heat generation, i.e., q0‴‴‴H2/k, and this scale is employed consistently, e.g., in Eqs. (8), (9), and (10a). Numerical solutions are acquired by using the well-documented finite-volume computational procedure (e.g., Hayase et al. 5, Patankar 12). For all the computations, the Prandtl number was set Pr=7.0 to simulate water. In the computations, the value of ε was set ε/θm=0.3. This choice of ε was based on the considerations explained below. First, ε should be small enough not to seriously distort the prevailing steady-state flow. In a series of detailed calculations, Kwak et al. 8 demonstrated that, when ε is small, i.e., ε⩽0.5, the amplitude of fluctuating Nusselt number in the interior is approximately proportional to ε. Secondly, the presence of multiple resonance was observed by Antohe and Lage 2 when the nondimensional amplitude of wall heat flux oscillation was larger than 0.1. The value of ε/θm=0.3 was selected to meet the aforementioned two dynamical issues. The nondimensional frequency of oscillation ω encompasses a wide range, 0.005⩽ω⩽O(1). In the analysis of the computed results, it is advantageous to introduce the following notations, i.e., (11a)NuX*τ≡NuXτ−NuXBNuX=1B,(11b)ANuX*≡Max{NuX*τ}−Min{NuX*τ}2,τ0⩽τ⩽τ0+2πω.In the above, NuX*τ indicates the difference between the instantaneous Nu value (at X=X and τ=τ) and the corresponding value NuXB of the case of non-oscillating sidewall temperature (ε=0). The amplitude of Nu-value fluctuations is expressed by ANuX*.As stressed by Lage and Bejan 9, the amplitude of Nu-fluctuation, ANu*, at the centerline X=0.5 is of particular interest. This is representative of the intensity of time-dependent heat transfer activities in the interior core. Figure 2 displays the collection of numerical data of ANuX* in the interior region versus ω for varying RaI. It is clear that the amplitude of Nu-fluctuation, ANuX*, peaks at certain particular frequencies; this was interpreted to manifest the existence of resonance in the context of enclosed buoyant convection (Lage and Bejan 9). Frames (a), (b), and (c) of Fig. 2 show that the frequency at which the primary peak occurs, ωr1, is largely unaffected by RaI. Furthermore, the secondary peak at ωr2 is visible for very high values of RaI, i.e., RaI⩾109.The prior expositions by Kwak et al. 78 ascertained that resonance is anticipated when the eigenfrequencies of the system are excited. Furthermore, these intrinsic eigenfrequencies are characterized by the internal gravity oscillations in the stably stratified interior core. Paolucci and Chenoweth 11 calculated the modes of these oscillations, which, for a square cavity, can be expressed, in the present nondimensional form, as (12)ωn≡fnN=∂θ/∂Y1/21+1/n21/2.In the above, ∂θ/∂Y denotes the average strength of stratification in the interior. In the practical calculation of ∂θ/∂Y, a linear curve-fitting was applied to the interior vertical temperature profile at a given location X. Here, n is the mode index, which is the ratio of the wave numbers in the horizontal and vertical directions. The primary mode is identified to be n=1, i.e., the scale of the whole cavity. The theoretical values of ω1 obtainable from Eq. (12) are 0.098, 0.079, and 0.062 for RaI=108,109, and 1010, respectively. The computed primary resonance frequencies ωr1, as illustrated in Fig. 2, are respectively 0.098, 0.080, and 0.065, which are in close agreement with the results of Eq. (12). This comparison gives credence to the assertion that the resonance frequencies are determined by the modes of internal gravity oscillations. The secondary peak in Fig. 2 is identified to be the consequence of the excitation of ω2 mode of Eq. (12). In this case, fluid motions based on n=2, which have the size of half-cavity, are under focus. Again, ω2, calculated by Eq. (12), are 0.124, 0.100, and 0.078, while the present computed data for ωr2 are 0.123, 0.103, and 0.082, respectively, for RaI=108,109 and 1010. As before, this favorable agreement reinforces the finding that the modes of internal gravity oscillations are the principal eigenfrequencies of the system. The time histories of NuX* are revealing. For the primary resonance mode ω≅ωr1, as demonstrated in Fig. 3(a), convective fluid motions are intensified throughout the entire interior region of the cavity. Because of the dominant convective activities, the amplitudes of Nu-oscillation are substantial. The invigorated convective flows are propagated to the bulk of the interior from the sidewall of oscillating temperature; much of Nu in the cavity interior core does not show appreciable phase lags. When the sidewall temperature oscillates at the secondary resonance frequency ω≅ωr2, as depicted in Fig. 3(b), considerable phase lags are discernible in the interior. The phase lag is π radian between X=0.25 and X=0.75. This implies that the convective motions at ωr2 take place with opposite phases between these two locations. At higher values of ω, the effect of sidewall temperature oscillation is confined to the region close to the wall, and the influence of sidewall temperature oscillation is meager at far-away locations. As a result, the amplitude of NuX* oscillation at X=0.75 is larger than at other locations. The behavior of temperature fluctuation in the interior at resonance is plotted in Fig. 4. The horizontal profiles of interior temperature oscillation at mid-height Y=0.5 clearly exhibit [see Fig. 4(a)] the periodic tilting with ωr1, which points to the source of internal gravity oscillations. This pattern is qualitatively similar to the observations of Kwak and Hyun 7 for a differentially-heated cavity with q0‴‴‴=0. The spatial behavior of thermal field fluctuations at the secondary resonance frequency ωr2 is slightly more complex. A closer inspection of Fig. 4(b) discloses that there exist two branches of tilting, with a phase difference of approximately π radian, in the left and right regions of the cavity. This points to the fact that the gradients of fluctuating temperature in both regions have opposite signs at each time instant. The primary-peak resonance at ωr1 is distinct. The secondary-peak resonance at ωr2 is detected for higher RaI. The instantaneous Nusselt number fluctuation at ωr1 indicates the presence of cavity—scale motions, which are qualitatively similar to those of a differentially—heated cavity. For the secondary resonance frequency ωr2, the tilting of the interior isotherms is of a concave/convex shape. The theoretical predictions of the resonance frequencies, based on the modes of internal gravity oscillations, are in accord with the present numerical data. This work was supported by the grants from NRL, South Korea.