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Linear stability of natural convection in spherical annuli

Published online by Cambridge University Press:  26 April 2006

David R. Gardner ,
Rod W. Douglass  and
Steven A. Trogdon
David R. Gardner
Affiliation:
Nuclear Energy Science and Materials Technology, Severe Accident Phenomenology Division, Sandia National Laboratories, Albuquerque, NM 87185–5800, USA
Rod W. Douglass
Affiliation:
Computational Mechanics Laboratory, Department of Mechanical Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588–0656, USA
Steven A. Trogdon
Affiliation:
Department of Mathematics and Statistics, University of Minnesota-Duluth, Duluth, MN 55812–2496, USA
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Abstract

Natural convection in a Boussinesq fluid filling the narrow gap between two isothermal, concentric spheres at different temperatures depends strongly on radius ratio, Prandtl number, and Grashof number. When the inner sphere has a higher temperature than the outer sphere, and for fixed values of radius ratio and Prandtl number, experiments show the flow to be steady and axisymmetric for sufficiently small Grashof number and quasi-periodic and axisymmetric for Grashof numbers greater than a critical value. It is our hypothesis that the observed transition is a flow bifurcation. This hypothesis is examined by solving an appropriate eigenvalue problem. The critical Grashof number, critical eigenvalues, and corresponding eigenvectors are obtained as functions of the radius ratio, Prandtl number, and longitudinal wavenumber. Critical Grashof numbers range from 1.18 × 104 to 2.63 × 103 as Prandtl number Pr increases from zero to 0.7, for radius ratios of 0.900 and 0.950. A transitional Prandtl number Prt exists such that for Pr < Prt the bifurcation is time-periodic and axisymmetric. For Pr > Prt the bifurcation is steady and non-axisymmetric with wavenumber two

A first approximation to the bifurcated flow is obtained using the critical eigenvectors. For Pr < Prt the bifurcation sets in as a cluster of relatively strong cells with alternating directions of rotation. The cells remain fixed in location, but pulsate with time. The cluster moves toward the top of the annulus as Pr increases toward Prt. An important feature of the non-axisymmetric bifurcation for Pr > Prt is a set of four cells located at each pole of the annulus in which the radial velocity alternates direction in moving from any one cell to an adjacent one. For fixed radius ratio, the average Nusselt number at criticality varies only slightly with Prandtl number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 221 , December 1990 , pp. 105 - 129
Copyright
© 1990 Cambridge University Press

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References

Astill, K. N., Leong, H. & Martorana, R., 1980 A numerical solution for natural convection in concentric spherical annuli. In Natural Convection in Enclosures, pp. 105–113. ASME.
Bishop, E. H., Kolflat, R. S., Mack, L. R. & Scanlan, J. A., 1964a Convective heat transfer between concentric spheres. In Proc. 1964 Heat Transfer and Fluid Mechanics Institute, UC Berkeley, CA, 10–12 June 1964 (ed. W. H. Giedt & S. Levy), pp. 69–80, Stanford University Press, Stanford, CA.
Bishop, E. H., Kolflat, R. S., Mack, L. R. & Scanlan, J. A., 1964b Photographic studies of convective patterns between concentric spheres. 9th SPIE Technical Symp., Miami Beach, FL, 28 August 1964, pp. 47–49.Google Scholar
Bishop, E. H., Mack, L. R. & Scanlan, J. A., 1966 Heat transfer by natural convection between concentric spheres. Intl J. Heat Mass Transfer 9, 649–652.Google Scholar
Brown, R. J.: 1967 Natural convection heat transfer between concentric spheres. Ph.D. dissertation, University of Texas, Austin.
Caltagirone, J.-P., Combarnous, M. & Mojtabi, A., 1980 Natural convection between two concentric spheres: transition toward a multicellular flow. Numer. Heat Transfer 3, 107–114.Google Scholar
Chandrasekhar, S.: 1961 Hydrodynamic and Hydromagnetic Stability, pp. 220–226, 622–625, Oxford University Press.
Douglass, R. W., Tebeest, K. G., Trogdon, S. A. & Gardner, D. R., 1990 Prandtl number effects on the stability of natural convection between spherical shells. Intl J. Heat Transfer (in press).Google Scholar
Farmer, M. T., Douglass, R. W. & Trogdon, S. A., 1986 The stability of natural convection in narrow-gap spherical annuli to axisymmetric disturbances. Intl J. Heat Mass Transfer 29, 1575–1584.Google Scholar
Gardner, D. R.: 1988 Linear stability and bifurcation of natural convection flows in narrow-gap, concentric spherical annulus enclosures. Ph.D. dissertation, University of Nebraska, Lincoln.
Gardner, D. R., Trogdon, S. A. & Douglass, R. W., 1989 A modified tau spectral method that eliminates spurious eigenvalues. J. Comput. Phys. 80, 137–167.Google Scholar
Gottlieb, D. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods: Theory and Application, pp. 159–161. SIAM.
Ingham, D. B.: 1981 Heat transfer by natural convection between spheres and cylinders. Numer. Heat Transfer 4, 53–67.Google Scholar
Joseph, D. D.: 1976 Stability of Fluid Motions I & II. Springer.
Kloeden, P. E.: 1986 Minimal truncated spectral systems which preserve bifurcations. Z. Angew. Math. Mech. 66, 51–54.Google Scholar
Macdonald, J. R.: 1964 Accelerated convergence, divergence, iteration, extrapolation and curve fitting. J. Appl. Phys. 35, 3034–3041.Google Scholar
Mack, L. R. & Hardee, H. C., 1968 Natural convection between concentric spheres at low Rayleigh numbers. Intl J. Heat Mass Transfer 11, 387–396.Google Scholar
Mojtabi, A. & Caltagirone, J.-P. 1982 Natural convection between two concentric spheres for high Prandtl number. Proc. 7th Intl Heat Transfer Conf. Munich, vol. 2, pp. 245–249.Google Scholar
Nelsen, J. M. & Douglass, R. W., 1984 Non-uniform energy generation effects on natural convection in a spherical annulus enclosure. J. Heat Mass Transfer 27, 1928–1932.Google Scholar
Nelsen, J., Douglass, R. & Alexander, D., 1982 Natural convection in a spherical annulus filled with heat generating fluid. Proc. 7th Intl Heat Transfer Conf., vol. 2. pp. 171–176.Google Scholar
Orszag, S. A.: 1971 Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Sherman, M.: 1968 Toroidal and poloidal field representation for convective flow within a sphere. Phys. Fluids 11, 1895–1900.Google Scholar
Singh, S. N. & Chen, J., 1980 Numerical solution for free convection between concentric spheres at moderate Grashof numbers. Numer. Heat Transfer 3, 441–459.Google Scholar
Singh, S. N. & Elliot, J. M., 1981 Natural convection between concentric spheres in a slightly-thermally stratified medium. Intl J. Heat Mass Transfer 24, 395–406.Google Scholar
Walton, I. C.: 1980 The stability of free convection in a horizontal cylindrical annulus. Q. J. Mech. Appl. Maths 33, 125–139.Google Scholar
Warner, W. H.: 1972 Poloidal and toroidal potentials for solenoidal fields. Z. Angew. Math. Phys. 23, 221–230.Google Scholar
Wright, J. L.: 1984 Natural convective flow in narrow gap spherical annuli. M. S. thesis, University of Nebraska. Lincoln.
Wright, J. L. & Douglass, R. W., 1986 Natural convection in narrow-gap, spherical annuli. Intl J. Heat Mass Transfer 29, 725–739.Google Scholar
Yin, S. H., Powe, R. E., Scanlan, J. A. & Bishop, E. H., 1973 Natural convective flow patterns in spherical annuli. J. Heat Mass Transfer 16, 1785–1795.Google Scholar