Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T08:49:35.658Z Has data issue: false hasContentIssue false
  • English
  • Français

On the secondary instability of Taylor-Görtler vortices to Tollmien-Schlichting waves in fully developed flows

Published online by Cambridge University Press:  21 April 2006

James Bennett  and
Philip Hall
James Bennett
Affiliation:
Mathematics Department, Exeter University, Exeter EX4 4QE, UK
Philip Hall
Affiliation:
Mathematics Department, Exeter University, Exeter EX4 4QE, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Görtler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor-Görtler vortices on the growth of small-amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor-Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 445 - 469
Copyright
© 1988 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

Abramowitz, M. & Stegun, I. A. 1965 A Handbook of Mathematical Functions. Dover.
Bennett, J. 1986 Theoretical properties of three dimensional interactive boundary layers. Ph.D. Thesis, University of London.
Dean, W. R. 1928 Fluid motion in a curved channel. Phil. Mag. 5, 673.Google Scholar
Fasel, H. & Booz, I. 1984 Numerical investigation of supercritical Taylor vortices in the wide gap limit. J. Fluid Mech. 138, 21–52.Google Scholar
Hall, P. 1982a Taylor-Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–490.Google Scholar
Hall, P. 1982b On the non-linear evolution of Görtler vortices in non-parallel boundary layers. IMA J. Appl. Maths 29, 173–196.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–59.Google Scholar
Hall, P. & Bennett, J. 1986 Taylor-Görtler instabilities of Tollmien-Schlichting waves and other flows governed by the interactive boundary-layer equations. J. Fluid Mech. 171, 441–457.Google Scholar
Nayfeh, A. H. 1981 Effect of streamwise vortices on Tollmien-Schlichting waves. J. Fluid Mech. 107, 441–453.Google Scholar
Rogers, E. H. & Beard, D. W. 1969 A numerical study of wide-gap Taylor vortices. J. Comp. Phys. 4, 1.Google Scholar
Seminara, G. 1976 Instability of some unsteady viscous flows. Ph.D. Thesis, University of London.
Smith, F. T. 1979a Instability of flow through pipes of general cross-section. Mathematika 26, 187–210.Google Scholar
Smith, F. T. 1979b On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 368, 573.Google Scholar