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Spatio—temporal behaviour in a rotating annulus with a source—sink flow

Published online by Cambridge University Press:  26 April 2006

E. Crespo Del Arco ,
P. Maubert ,
A. Randriamampianina  and
P. Bontoux
E. Crespo Del Arco
Affiliation:
Departamento de Física Fundamental, UNED, Apdo. 60.141, 28080 Madrid, Spain
P. Maubert
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
A. Randriamampianina
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
P. Bontoux
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
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Abstract

The axisymmetric flows arising in a rotating annulus with a superimposed forced flow are investigated with a pseudo-spectral numerical method. The flow enters the annulus at the inner radius with a radial velocity, then develops into a geostrophic flow azimuthally directed and flanked by two Ekman (nonlinear) boundary layers, and finally exits the outer radius, with a radially directed velocity. In this study the rotation rate of the cavity is fixed and very high. When the forced flow is weak, the flow is steady. On increasing the mass flow rate, the flow evolves to a chaotic temporal behaviour through several bifurcations, which perturbs the basic spatial configuration of the flow. The first bifurcation drives the steady state into an oscillatory regime, associated with a break of symmetry with respect to the midheight of the annulus. The entry flow travels radially through the cavity as in the steady flow, but it wavers and then is alternately sucked towards each Ekman layer. The frequency of this oscillation is close to the rotation rate frequency of the cavity, which is characteristic of inertial waves in rotating flows. A second transition to a quasi-periodic regime is characterized by the appearance of a second frequency. Further increases in the flow rate lead to a period-five state, via a locking of both frequencies, and then to a chaotic motion. This second frequency is of the order of the inverse of the Ekman spin-up characteristic time, suggesting that this instability is originated by the relaxation of the perturbations in the flow field. These perturbations of the unsteady flow field are corotating vortices along the rigid boundary walls. They are excited by the entry flow and their strength diminishes with increasing radius due to the low value of the Reynolds number. The parameters characterizing the unstable flows are also consistent with this explanation. The conclusion is that in this configuration, the origin of the described dynamical behaviour is not the instability of the Ekman boundary layers, as could be expected, but the instability of the entry flow. The reason is the importance of the nonlinear inertial terms in cavities with small radius of curvature.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 328 , 10 December 1996 , pp. 271 - 296
Copyright
© 1996 Cambridge University Press

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