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Long-wave/short-wave interactions in flow between concentric cylinders

Published online by Cambridge University Press:  26 April 2006

Nicola J. Horseman  and
Andrew P. Bassom
Nicola J. Horseman
Affiliation:
Department of Mathematics, North Park Road, University of Exeter, Exeter, Devon, EX4 4QE, UK
Andrew P. Bassom
Affiliation:
Department of Mathematics, North Park Road, University of Exeter, Exeter, Devon, EX4 4QE, UK
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Abstract

Consider the flow of an incompressible fluid between two infinite concentric circular cylinders. The outer cylinder is at rest whilst the angular velocity of the inner cylinder has a steady part and also a harmonically oscillating component. We examine the situation where, for a suitable choice of parameters, two types of vortex instability can occur simultaneously; first a short-wavelength mode which is essentially trapped in a thin ‘Stokes’ layer near the inner cylinder and, secondly, a long-wavelength mode which fills the whole region between the cylinders. We investigate the problem in which two short-wavelength vortices and one long-wavelength vortex coexist and are such that each pair interacts to drive the third. Additionally, the short-wavelength disturbances are nonlinear in their own right. Coupled amplitude equations for the three modes are derived and their solution discussed.

This form of interaction may also take place in a boundary layer. Such a situation is more complex than that under consideration here as it would be necessary to take into account the growth of the boundary layer. However, this simplified problem gives an insight into the behaviour of the more difficult situation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 215 , June 1990 , pp. 525 - 548
Copyright
© 1990 Cambridge University Press

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