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The stability of a helical vortex filament

Published online by Cambridge University Press:  29 March 2006

Sheila E. Widnall
Sheila E. Widnall
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
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Abstract

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.

The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 54 , Issue 4 , 22 August 1972 , pp. 641 - 663
Copyright
© 1972 Cambridge University Press

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References

Betchov, R. 1965 On the curvature and torsion of a vortex filament J. Fluid Mech. 22, 471–479.Google Scholar
Bliss, D. B. 1970 The dynamics of curved rotational vortex lines. M.S. Thesis, Massachusetts Institute of Technology.
Krutzsch, C. H. 1939 Über eine experimentell beobachtete Erscheining an Werbelringen bei ehrer translatorischen Beivegung in Werklechin. Flussigheiter Ann. Phys. 5. Folge Band 35, 497–523.Google Scholar
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. Roy. Soc A 120, 670–690.Google Scholar
Loukakis, T. A. 1971 A theory for the wake of Marine propellers. ScD. thesis, Department of Naval Architecture and Marine Engineering, Massachusetts Institute of Technology.
Saffman, P. G. 1970 The velocity of viscous vortex rings Studies in Appl. Math. 49, 370–380.Google Scholar
Widnall, S. E., Bliss, D. B. & Zalay, A. 1970 Theoretical and experimental study of the stability of a vortex pair. Proceedings of the Symposium on Aircraft Wake Turbulence, Seattle. Plenum Press.