Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T07:06:00.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers

Published online by Cambridge University Press:  26 April 2006

H. F. Goldstein ,
E. Knobloch ,
I. Mercader  and
M. Net
H. F. Goldstein
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
E. Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
I. Mercader
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
M. Net
Affiliation:
Department de Física Aplicada, Universitat Politècnica de Catalunya, E 08034 Barcelona, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

The onset of convection in a low-Prandtl-number fluid confined in a uniformly rotating vertical cylinder heated from below is studied. The linear stability problem is solved for perfectly conducting stress-free or rigid boundary conditions at the top and bottom; the sidewalls are taken to be insulating and rigid. For these Prandtl numbers axisymmetric overstability leads to an oscillating concentric pattern of rolls. When the instability breaks axisymmetry the resulting pattern must in addition precess. The relationship between these two types of oscillatory behaviour is explored in detail. The complex interaction between different types of neutrally stable modes is traced out as a function of the Prandtl and Taylor numbers, as well as the aspect ratio. A qualitative explanation is provided for the multiplicity of modes of a given azimuthal wavenumber and its dependence on the parameters. Specific predictions are made for the Prandtl numbers 0.025, 0.49 and 0.78, corresponding to mercury, liquid helium 4 and compressed carbon dioxide gas.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 293 - 324
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bestehorn, M., Fantz, M., Friedrich, R., Haken, H. & Pérez-García, C. 1992 Spiral patterns in thermal convection. Z. Phys. B 88, 93–94.Google Scholar
Buell, J. C. & Catton, I. 1983 Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below. Phys. Fluids 26, 892–896.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.
Clune, T. & Knobloch, E. 1993 Pattern selection in rotating convection with experimental boundary conditions. Phys. Rev. E 47, 2536–2550.Google Scholar
Da Costa, L. N., Knobloch, E. & Weiss, N. O. 1981 Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar
Dangelmayr, G. & Knobloch, E. 1987 The Takens–Bogdanov bifurcation with O(2) symmetry. Phil. Trans. R. Soc. Lond. A 322, 243–279.Google Scholar
Ecke, R. E., Zhong, F. & Knobloch, E. 1992 Hopf bifurcation with broken reflection symmetry in rotating Rayleigh–Bénard convection. Europhys. Lett. 19, 177–182.Google Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.Google Scholar
Golubitsky, M. & Schaeffer, D. 1984 Singularities and Groups in Bifurcation Theory, Vol. 1. Springer.
Guckenheimer, J. & Knobloch, E. 1983 Nonlinear convection in a rotating layer: amplitude expansions and normal forms. Geophys. Astrophys. Fluid Dyn. 23, 247–272.Google Scholar
Jones, C. A. 1988 Multiple eigenvalues and mode classification in plane Poiseuille flow. Q. J. Mech. appl. Maths 41, 363–382.Google Scholar
Jones, C. A. & Moore, D. R. 1979 The stability of axisymmetric convection. Geophys. Astrophys. Fluid Dyn. 11, 245–270.Google Scholar
Knobloch, E. 1992 Bifurcations in rotating systems. In Theory of Solar and Planetary Dynamos: Introductory Lectures (ed. M. R. E. Proctor & A. D. Gilbert). Cambridge University Press, in press.
Lucas, P. G. J., Pfotenhauer, J. M. & Donnelly, R. J. 1983 Stability and heat transfer in rotating cryogens. Part 1. Influence of rotation on the onset of convection in liquid 4Helium. J. Fluid Mech. 129, 251–264.Google Scholar
Marqués, F., Net, M., Massaguer, J. M. & Mercader, I. 1993 Thermal convection in vertical cylinders: A method based on potentials of velocity. Comput. Math. Appl. Mech. Engng, in press.Google Scholar
Matthews, P. C., Hurlburt, N. E., Proctor, M. R. E. & Brownjohn, D. P. 1992 Compressible magnetoconvection in oblique fields: linearized theory and simple nonlinear models. J. Fluid Mech. 240, 559–569.Google Scholar
Pfotenhauer, J. M., Lucas, P. G. J. & Donnelly, R. J. 1984 Stability and heat transfer of rotating cryogens. Part 2. Effects of rotation on heat-transfer properties of convection in liquid 4Helium. J. Fluid Mech. 145, 239–252.Google Scholar
Pfotenhauer, J. M., Niemela, J. J. & Donnelly, R. J. 1987 Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection. J. Fluid Mech. 175, 85–96.Google Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 19–74.Google Scholar
Soward, A. M. 1979 Thermal and magnetically driven convection in a rapidly rotating fluid layer. J. Fluid Mech. 90, 669–684.Google Scholar
Thual, O. 1992 Zero-Prandtl-number convection. J. Fluid Mech. 240, 229–258.Google Scholar
Thurlow, M. S. 1993 Rayleigh–Bénard convection in a rotating liquid 3He-4He mixture: the effects of Coriolis forces and a finite geometry. PhD thesis, University of Manchester.
Zhang, K.-K. & Busse, F. H. 1987 On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 39, 119–147.Google Scholar
Zhong, F., Ecke, R. E. & Steinberg, V. 1991 Asymmetric modes and the transition to vortex structures in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 67, 2473–2476.Google Scholar
Zhong, F., Ecke, R. E. & Steinberg, V. 1993 Rotating Rayleigh-Bénard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.Google Scholar