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On the stability of a columnar vortex to disturbances with large azimuthal wavenumber: the lower neutral points

Published online by Cambridge University Press:  21 April 2006

K. Stewartson  and
S. Leibovich
K. Stewartson
Affiliation:
Department of Mathematics, University College, London, UK
S. Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

The inviscid instability of a columnar trailing-line vortex at large values of the azimuthal wavenumber n near neutral conditions is considered. This extends an earlier analysis (Leibovich & Stewartson 1983), which is not accurate near the limiting values of the axial wavenumber for which instabilities exist. Here an asymptotic expansion is derived for the solution in the neighbourhood of the lower neutral point and the results compared with existing computations a t moderate values of n. For these weak instabilities disturbances are centred near the axis of the vortex and the relevant equation is solved in the complex plane by a generalized saddle-point method. In addition, the marginal stability of the vortex is examined, and an estimate obtained of the value of the swirl parameter above which the vortex is stable at large values of n.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 178 , May 1987 , pp. 549 - 566
Copyright
© 1987 Cambridge University Press

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References

Batchelor, G. K. 1964 Axial flow in trailing line vortiees. J. Fluid Mech. 20, 645–658.Google Scholar
Duck, P. W. & Foster, M. R. 1980 The inviscid stability of a trailing line vortex. Z. angew. Math. Phys. 31, 523–530.Google Scholar
Emanuel, K. A. 1984 A note on the stability of column vortices. J. Fluid Mech. 145, 235–238.Google Scholar
Foster, M. R. & Duck, P. W. 1982 The inviscid stability of Long's vortex. Phys. Fluids 25, 1715–1718.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Leibovich, S. 1983 Vortex stability and breakdown. In Aerodynamics of Vortical Type Flows in Three Dimensions (ed. A. D. Young), AGARD CP-342, pp. 23.1–23.22.
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1205.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. J. Fluid Mech. 63, 753–763.Google Scholar
Ludwieg, H. 1960 Stabilität der Strömung in einem zylindrischen Ringraum, Z. Flugwiss. 8, 135–140.Google Scholar
Ludwieg, H. 1961 Ergänzung zu der Arbeit: ‘Stabität der Strömung in einem zylindrishchen Ringraum’. Z. Flugwiss. 9, 359–361.Google Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Stewartson, K. & Capell, K. 1985 Marginally stable ring modes in a trailing line vortex: the upper neutral points. J. Fluid Mech. 156, 369–386.Google Scholar
Synge, J. L. 1933 The stability of heterogeneous liquids. Trans. R. Soc. Can. 27, 1–18.Google Scholar