Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-19T08:23:52.124Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Donnelly, R. J. 1964 Experiments on the stability of viscous flow between rotating cylinders III. Ejtlhancement of stability by modulation. Proc. R. Soc. Lond. A 281, 130–139.Google Scholar
Duck, P. W. 1979 Flow induced by a torsionally oscillating wavy cylinder. Q. J. Mech. Appl. Maths 32, 73–91.Google Scholar
Hall, P. 1973 Some unsteady viscous flows and their stability. Ph.D. thesis, Imperial College, University of London.
Hall, P. 1975 The stability of unsteady cylinder flows. J. Fluid Mech. 67, 29–63 (referred to as H).Google Scholar
Hall, P. 1981 Centrifugal instability of a Stokes layer: subharmonic destabilization of the Taylor vortex mode. J. Fluid Mech. 105, 523–530.Google Scholar
Hall, P. & Smith, F. T. 1988 The nonlinear interaction of Tollmien—Schlichting waves and Taylor—Görtler vortices in curved channel flows. Proc. R. Soc. Lond. A 417, 255–282.Google Scholar
Kumar, K., Bhattacharjee, J. K. & Banerjee, K. 1986 Onset of the first instability in hydrodynamic flows: effect of parametric modulation. Phys. Rev. A 34, 5000–5006.Google Scholar
Park, K. & Donnelly, R. J. 1981 Study of the transition to Taylor vortex flow. Phys. Rev. A 24, 2277–2279.Google Scholar
Seminara, G. 1976 Instability of some unsteady viscous flows. Ph.D. thesis, Imperial College, University of London.
Seminara, G. & Hall, P. 1976 Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 299–316 (referred to as SH).Google Scholar
Seminara, G. & Hall, P. 1977 The centrifugal instability of a Stokes layer: nonlinear theory. Proc. R. Soc. Lond. A 354, 119–126.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar
Stuart, J. T. 1986 Taylor-vortex flow: a dynamical system. SIAM Rev. 28, 315–342.Google Scholar
Walsh, T. J., Wagner, W. T. & Donnelly, R. J. 1987 Stability of modulated couette flow. Phys. Rev. Lett. 58, 2543–2546.Google Scholar