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Convection in a rotating cylinder. Part 1 Linear theory for moderate 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:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, E 08034 Barcelona, Spain
M. Net
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, E 08034 Barcelona, Spain
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Abstract

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.

The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 248 , March 1993 , pp. 583 - 604
Copyright
© 1993 Cambridge University Press

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