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Numerical simulations of multiple flow transitions in axisymmetric annulus convection

Published online by Cambridge University Press:  26 April 2006

P. Le Quéré  and
J. Pécheux
P. Le Quéré
Affiliation:
LIMSI-CNRS, BP 30, 91406 Orsay Cedex, France
J. Pécheux
Affiliation:
LEA-UA CNRS 191, 40 Av. Recteur Pineau, 86022 Poitiers Cedex, France
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Abstract

We examine numerically the behaviour of the solutions of the axisymmetric Boussinesq equations in a tall, differentially heated, air-filled annulus. The numerical algorithm integrates the time-dependent equations in primitive variables and combines a pseudospectral Chebyshev spatial expansion with a second-order time-stepping scheme. The instability of the conduction regime is found to be unsteady cross-rolls. By assuming Hopf bifurcation, we can accurately determine the critical Rayleigh number. As the Rayleigh number increases, the solution is monoperiodic at first. Then it undergoes a period-doubling bifurcation. When the Rayleigh number is further increased, the solution reverts to a monocellular steady state through suberitical bifurcations with hysteresis. At even higher Rayleigh number, boundary-layer instability sets in, in the form of travelling waves. This instability has the characteristics of a supercritical Hopf bifurcation. We examine the space-time structure of the two types of unsteady solutions. We have presented the basic periods of the steady oscillations as functions of the Rayleigh number in the vicinity of the Hopf bifurcation points, and have also computed the Nusselt numbers for the various flow regimes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 206 , September 1989 , pp. 517 - 544
Copyright
© 1989 Cambridge University Press

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