Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-20T11:33:25.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability of strongly nonlinear waves in vortex flows

Published online by Cambridge University Press:  26 April 2006

A. Kribus  and
S. Leibovich
A. Kribus
Affiliation:
Weizmann Institute of Science, Rehovot, Israel
S. Leibovich
Affiliation:
Cornell University, Ithaca, NY 14853–7501, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Weakly nonlinear descriptions of axisymmetric cnoidal and solitary waves in vortices recently have been shown to have strongly nonlinear counterparts. The linear stability of these strongly nonlinear waves to three-dimensional perturbations is studied, motivated by the problem of vortex breakdown in open flows. The basic axisymmetric flow varies both radially and axially, and the linear stability problem is therefore nonseparable. To regularize the generalization of a critical layer, viscosity is introduced in the perturbation problem. In the absence of the waves, the vortex flows are linearly stable. As the amplitude of a wave constituting the basic flow increases owing to variation in the level of swirl, stability is first lost to non-axisymmetric ‘bending’ modes. This instability occurs when the wave amplitude exceeds a critical value, provided that the Reynolds number is larger enough. The critical wave amplitudes for instability typically are large, but not large enough to create regions of closed streamlines. Examination of the most-amplified eigenvectors shows that the perturbations tend to be concentrated downstream of the maximum streamline displacement in the wave, in a position consistent with the observed three-dimensional perturbations in the interior of a bubble type of vortex breakdown.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 247 - 264
Copyright
© 1994 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

Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.Google Scholar
Benjamin, T. B. 1967 Some developments in the theory of vortex breakdown. J. Fluid Mech. 28, 65–84.Google Scholar
Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153–183.Google Scholar
Benjamin, T. B. 1974 Lectures on nonlinear wave motion. In Nonlinear Wave Motion (ed. A. C. Newell). Lectures in Applied Mathematics, vol. 15, pp. 3–47. Am. Math. Soc.
Beran, P. S. 1987 Numerical simulations of trailing vortex bursting. AIAA Paper 87-1313.
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400.Google Scholar
Faler, J. H. & Leibovich, S. 1978 An experimental map of the internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335.Google Scholar
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown flow fields. Phys. Fluids 22, 2053–2064.Google Scholar
Goldhirsch, I., Orszag, S. A. & Maulik, K. 1987 An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. J. Sci. Comput. 2, 33–58.Google Scholar
Hafez, M., Kuruvila, G. & Salas, M. D. 1986 Numerical study of vortex breakdown. Appl. Numer. Maths 2, 291–302.Google Scholar
Howard, L. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Kribus, A. 1990 Computation of leading eigenspaces for generalized eigenvalue problems. Trans. Seventh Army Conf. on Applied Mathematics and Computing, ARO 90–1.
Kribus, A. 1991 Large-amplitude waves, instabilities, and breakdown in vortex flows. PhD thesis, Cornell University.
Leibovich, S. 1970 Weakly nonlinear waves in rotating fluids. J. Fluid Mech. 42, 803–822.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1983 Vortex stability and breakdown. In Aerodynamics of Vortical Type Flows in Three Dimensions (ed. A. D. Young). AGARD Conf. Proc. 342 (NATO), Paper 23.
Leibovich, S. 1984 Vortex stability and breakdown: Survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. 1990 Vortex breakdown: A coherent transition trigger in concentrated vortices. In Turbulence and Coherent Structures (ed. M. Lesieur & O Métais), pp. 285–302. Kluwer.
Leibovich, S. & Kribus, A. 1990 Large-amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459–504 (referred to herein as L&K).Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Mac Giolla Mhuiris, N. 1986 a Numerical calculations of the stability of some axisymmetric flows proposed as a model for vortex breakdown. PhD thesis, Cornell University.
Mac Giolla Mhuiris, N. 1986b Calculations of the stability of some axisymmetric flows proposed as a model of vortex breakdown. Appl. Numer. Maths 2, 273–290.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G. 1985 Wave motions on vortex cores. J. Fluid Mech. 151, 141–165.Google Scholar
Maxworthy, T., Mory, M. & Hopfinger, E. J. 1983 Waves on vortex cores and their relation to vortex breakdown. In Aerodynamics of Vortical Type Flows in Three Dimensions (ed. A. D. Young). AGARD Conf. Proc. 342 (NATO), Paper 29.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes, the Art of Scientific Computing. Cambridge University Press.
Pritchard, W. G. 1970 Solitary waves in rotating fluids. J. Fluid Mech. 42, 61–83.Google Scholar
Salas, M. D. & Kuruvila, G. 1989 Vortex breakdown simulation: a circumspect study of the steady, laminar, axisymmetric model. Computers & Fluids 17, 247–262.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.Google Scholar
Stewart, G. W. 1976 Simultaneous iterations for computing invariant subspaces of non-hermitian matrices. Numer. Maths 25, 123–136.Google Scholar
Ta'asan, S. 1986 Multigrid method for a vortex breakdown simulation. Appl. Numer. Maths 2, 303–311.Google Scholar
Yang, Z. 1990 Two theoretical studies of stability of swirling flows. PhD thesis, Cornell University.