Effects of gravity modulation on convection in a horizontal annulus

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Abstract

Convection in the annulus between two horizontal coaxial cylinders resulting from gravity modulation in a microgravity environment is investigated for the first time. The three-dimensional transient equations of fluid motion and heat transfer are solved to study the unsteady flow structures in a large-gap annulus. The gravity fluctuations are shown to induce recirculating flows that reverse direction of rotation in response to the gravitational reversals. Except for a short period of time following flow reversal, at low modulation frequencies the centers of rotation of these flows are below and above the horizontal angular plane when the acceleration acts in the upward and downward directions, respectively, whereas at high frequencies they are above and below this plane. The effects of gravity modulation on development of secondary flows that arise in narrow- and moderate-gap annuli owing to thermal instability are also investigated. It is found that supercritical transverse rolls repeatedly form, dissipate, and re-form in the upper and lower regions of a narrow-gap annulus as a result of the fluctuating gravity field. At the same frequency, the supercritical rolls in a moderate-gap annulus are much slower to develop between flow reversals. The results show that gravity modulation has a stabilizing effect on the secondary flows compared to the case of a constant gravity force, and this effect diminishes with reduction in either frequency or annulus radius ratio R. The effects of frequency on heat transfer in large-, moderate-, and narrow-gap annuli are also studied. It is found that the time-averaged Nusselt number approaches that of pure conduction at high frequencies and increases toward that for terrestrial natural convection as frequency is reduced. As R is decreased, the increase in time-averaged Nusselt number brought about by reducing frequency becomes a smaller percentage of the value for terrestrial natural convection. The results for large-, moderate-, and narrow-gap annuli subjected to gravity modulation under microgravity are compared to results for steady natural convection under terrestrial conditions to clarify differences in flow behavior. The present work provides the first description of convection in a cylindrical annulus under microgravity, and practical information on the influence of gravity fluctuations on heat transfer in a space environment.

Introduction

Buoyancy-driven convection in microgravity resulting from gravity fluctuations has gained considerable attention owing to the possibility of conducting research in the low-gravity environment of space, and because of interest in the fundamental effects of gravity modulation on fluid systems. On spacecraft, fluctuating accelerations can originate from a variety of sources such as crew activities, vibrations from onboard equipment, and structural oscillations of the spacecraft. Typical peak g levels, although small relative to terrestrial gravity [1], are at least 100 times larger than the steady g levels in space. The fluctuating accelerations act on density gradients in the fluid caused by heat and/or mass transfer between the fluid and boundaries, producing convective motions. These motions may increase heat transfer significantly beyond that of pure conduction, and can strongly affect certain processes such as alloy solidification.

Previous studies of buoyancy-induced fluid motion and heat transfer resulting from gravity modulation under microgravity have focused on a few basic fluid systems, as well as some specific applications. Kamotani et al. [2] conducted a linearized numerical analysis for a two-dimensional rectangular enclosure with two opposing walls at unequal temperatures. They concluded that modulation normal to the direction of the temperature gradient is most critical. Amin [3] analytically studied heat transfer from a sphere. It was concluded that for high modulation frequencies heat transfer is negligible, while for low frequencies heat transfer will be non-trivial for fluids of small viscosity and sufficiently large Prandtl number. The effects of non-linearity of the governing equations for the rectangular enclosure problem were investigated by Biringen and Danabasoglu [4]. They established some limits on applicability of the linear models in the excitable frequency range. Biringen and Peltier [5] considered the effects of three-dimensionality as well as non-linearity of the governing equations for both sinusoidal and random modulations. They found that sinusoidal modulations are more stabilizing than random modulations, and the similarity between terrestrial systems and systems at low-gravity increases with increasing modulation amplitude. Cyr et al. [6] also conducted a numerical study of convection in a two-dimensional rectangular enclosure. Their results showed that the flow transitions from stable, to periodic, to non-periodic behavior as the frequency of the modulation is varied. The average heat transfer rate was found to increase significantly as the frequency was decreased to low values. Other investigations have been conducted to determine the effects of gravity modulation on melt processing systems in microgravity [7], [8], [9].

A problem that has been widely studied owing to its many practical applications is natural convection in the annulus between horizontal concentric cylinders. Depending on the outer to inner cylinder radius ratio R and the Rayleigh number, various types of laminar flow structures can arise in a sufficiently long horizontal annulus. Based on their experimental results and those of previous investigators, Powe et al. [10] classified various types of laminar natural convective regimes as (i) a unicellular steady regime for small Rayleigh numbers at any value of R, (ii) a multicellular regime for higher Rayleigh numbers and R < 1.24 (narrow-gap annulus), (iii) a spiral flow regime for higher Rayleigh numbers and R between 1.24 and 1.71 (moderate-gap annulus), and (iv) an oscillating regime for higher Rayleigh numbers and R > 1.71 (large-gap annulus). Most of the prior studies of buoyancy-induced convection in horizontal cylindrical annuli have dealt with two-dimensional flow occurring in annuli with large length to gap-thickness ratios. There are fewer studies of three-dimensional flow, which occurs owing to the presence of the endwalls or the onset of instabilities at higher Rayleigh numbers. Previous numerical studies of three-dimensional natural convection in large- and moderate-gap annuli have focused on classification of different convective regimes [11], the effects of annulus inclination [12], flow patterns in a short annulus [13], temporal development of the flow and temperature fields [14], turbulent flow [15], and high Rayleigh number laminar flow [16]. Three-dimensional buoyancy-driven convection in a narrow-gap annulus has been investigated by Dyko and Vafai [17]. The results of this study showed the existence of four different supercritical states characterized by the orientations and directions of rotation of counter-rotating convective rolls that form in the upper part of the annulus owing to thermal instability. A fifth supercritical state was studied by Dyko and Vafai [18].

In the present work, we extend the problem of buoyancy-induced convection in horizontal annuli to the case of gravity modulation in a microgravity environment, which has application to systems such as materials processing furnaces. In this three-dimensional numerical investigation, the governing equations are formulated in terms of vorticity and vector potential. The parabolic equations are solved by a three-dimensional three-level time-splitting ADI method, and the elliptic equations are solved by the extrapolated Jacobi method. The flow structures that arise in large-, moderate-, and narrow-gap annuli under microgravity as a result of periodic gravity modulation, and the influence of modulation frequency on the flow and heat transfer are presented.

Section snippets

Governing equations and solution

The inner and outer horizontal concentric cylinders have radii of ri and ro, respectively, and are of length l. The temperature of the inner cylinder Ti is greater than that of the outer cylinder To, and the two axial endwalls are impermeable and adiabatic. The dimensionless length of the annulus is defined as L = l/ri. The annulus geometry, which is characterized by the radius ratio R = ro/ri and gap aspect ratio A = l/(ro  ri), is shown in Fig. 1. In all of the numerical simulations, the

Results

The unsteady flow and temperature fields in horizontal cylindrical annuli resulting from periodic modulation of the gravity field in a microgravity environment are presented. The effects of modulation frequency on the flow patterns, temperature distributions, and heat transfer are investigated for a large-gap annulus. The development of secondary flows that arise in moderate- and narrow-gap annuli owing to the onset of thermal instabilities, and the effects of frequency on heat transfer in

Conclusions

An investigation of buoyancy-driven convection in a cylindrical annulus subjected to gravity modulation in a microgravity environment has been conducted for the first time. Simulations were carried out for wide ranges of annulus radius ratio and modulation frequency. It was shown that the fluctuating gravitational field induces recirculating flows in the annulus that reverse direction of rotation in response to the gravitational reversals. At high frequencies, the strength of the flow remains

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