Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T05:29:54.123Z Has data issue: false hasContentIssue false
  • English
  • Français

The inviscid nonlinear instability of parallel shear flows

Published online by Cambridge University Press:  29 March 2006

J. L. Robinson
J. L. Robinson
Affiliation:
Physics and Engineering Laboratory, D.S.I.R., Lower Hutt, New Zealand
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.

A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 4 , 15 May 1974 , pp. 723 - 752
Copyright
© 1974 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

Bispijnohoff, R. L., Ashley, H. & Halfman, R. L. 1955 Aeroehticity, pp. 67–71, 81–87. Addison-Wesley.
Frei, E. H., Leibinzohn, S., Neufeld, H. N. & Ashkenazy, H. N. 1963 The Pod, a new magnetic device for medical application. Proc. 16th Conf. Eng. Med. Biol., Baltimore, p. 156.Google Scholar
Frei, E. H., Leibinzohn, S. & Driller, I. 1966 The Pod - a magnetically remote con-trolled device. Am. J. Med. Electron.Google Scholar
Kaplwn, S. 1957 Low Reynolds numbers flow past a circular cylinder. J. Math. Mech. 6, 595–603.Google Scholar
Kaplun, S. & Lagerstrom, P. A. 1957 Asymptotic expansion of Navier-Stokes solutions for small Reynolds numbers. J. Math. Mech. 6, 585–593.Google Scholar
Kely, H. R., Rentz, A. W. & Siekxan, J. 1964 Experimental studies on the motion of a flexible hydrofoil. J. Pluid Mech. 19, 30–48.Google Scholar
Lavie, A. M. 1970a The forces on a slender flexible cylinder swimming in viscous flow. Israel J. Tech. 8, 51.Google Scholar
Lavie, A. M. 1970b The swimming of the Pod: theoretical analysis and experimental results. I.E.E.E. Tram. on Magnetics, MAG-6, no. 2.Google Scholar
Lighthill, M. J. 1960a Mathematics and aeronautics. J. Roy. Aero. Soc. 64, 375–395.Google Scholar
Lighthill, M. J. 1960b Note on the swimming of slender fish. J. Fluid Mech. 9, 305–317.Google Scholar
Longman, I. M. & Lavie, A. M. 1966 On the swimming of a Pod. J. Inst. Appl. 2, 273–282.Google Scholar
Segel, L. A. 1961 A uniform-valid asymptotic expansion of the solution on unsteady boundary-layer problem. Quart. Appl. Math, 23, 180–197.Google Scholar
Stuart, J. T. & Woodgate, L. 1955 Experimental determination of the aerodynamical damping on a vibrating circular cylinder. Phil. Mug. 46, 40–46.Google Scholar
Taylor, G. I. 1952 Analysis of the swimming of long end narrow animals. Pmc. Roy. Soc. A 214, 168–183.Google Scholar
Watson, G. N. 1962 A Tmmtise of the Theory of Bessd Fmctions. Cambridge University Press.