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The linear inviscid secondary instability of longitudinal vortex structures in boundary layers

Published online by Cambridge University Press:  26 April 2006

Philip Hall  and
Nicola J. Horseman
Philip Hall
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Nicola J. Horseman
Affiliation:
Mathematics Department, Exeter University, North Park Road, Exeter EX4 4QE, UK
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Abstract

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has wavelength comparable with the boundary-layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalization of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Görtler vortices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 357 - 375
Copyright
© 1991 Cambridge University Press

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References

Aihara, Y. & Kohama, H. 1981 Secondary instability of Görtler vortices: Formation of periodic three-dimensional coherent structure. Trans. Japan Soc. Aero. Sci. 24, 78.Google Scholar
Bassom, A. P. & Seddougui, S. O. 1990 The onset of time-dependence and three-dimensionality in Görtler vortices: neutrally stable wavy modes. J. Fluid Mech. 220, 661.Google Scholar
Bippes, H. 1972 Experimental study of the laminar-turbulent transition of a concave wall in a parallel flow. NASA TM 75243.
Davey, A., Diprima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17.Google Scholar
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. 0. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335, 51.Google Scholar
Görtler, H. 1940 Uber eine dreidimensionale instabilität laminare Grenzschubten an Konkaven Wänden. NACA TM 1357.
Hall, P. 1982a Taylor-Görtler vortices in fully developed or boundary-layer flows. J. Fluid Mech. 124, 475.Google Scholar
Hall, P. 1982b On the nonlinear evolution of Görtler vortices in non-parallel boundary layers. J. Inst. Maths Applies 29, 173.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41.Google Scholar
Hall, P. 1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 247.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematica 37, 151.Google Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Proc. R. Soc. Lond. A 415, 421.Google Scholar
Hall, P. & Seddougui, S. 1989 On the onset of three-dimensionality and time-dependence in Görtler vortices. J. Fluid Mech. 204, 405.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641.Google Scholar
Honji, H. 1981 Streaked flow around an oscillating cylinder. J. Fluid Mech. 107, 509.Google Scholar
Horseman, N. J. 1991 Some centrifugal instabilities in viscous flows. Ph.D. thesis, Exeter University.
Liepmann, H. W. 1943 Investigations on laminar boundary layer stability and transition on curved boundaries. NACA Wartime Rep. W107.Google Scholar
Liepmann, H. W. 1945 Investigation of boundary layer transition on concave walls. NACA Wartime Rep. W87.Google Scholar
Peerhossaini, H. & Wesfreid, J. E. 1988a On the inner structure of streamwise Görtler rolls. Intl J. Heat Fluid Flow 9, 12.Google Scholar
Peerhossaini, H. & Wesfreid, J. E. 1988 Experimental study of the Taylor—Görtler instability. In Propagation in Systems Far from, Equilibrium (ed. J. E. Wesfreid, H. R. Brand, P. Manneville, G. Albinet & N. Boccara). Springer Series in Synergetics, vol. 41, pp. 399. Springer.
Seminara, G. & Hall, P. 1975 Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 299.Google Scholar
Soward, A. M. & Jones, C. A. 1983 The linear stability of the flow in the narrow gap between concentric rotating spheres. Q. J. Mech. Appl. Maths 36, 19.Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. A 223, 289.Google Scholar
Walton, I. C. 1978 The linear stability of the flow in a narrow spherical annulus. J. Fluid Mech. 86, 673.Google Scholar