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Bending waves on inviscid columnar vortices

Published online by Cambridge University Press:  21 April 2006

S. Leibovich ,
S. N. Brown  and
Y. Patel
S. Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
S. N. Brown
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA Permanent address: Department of Mathematics, University College, Gower Street, London WC1E 6BT, UK.
Y. Patel
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

Bending waves, perturbation modes leading to deflections of the vortex centreline, are considered for an infinitely long straight vortex embedded in an irrotational flow of unlimited extent. We first establish the general form of the dispersion relation for long waves on columnar vortices with arbitrary distributions of axial and azimuthal vorticity by a singular perturbation analysis of the Howard-Gupta equation. The asymptotic results are shown to compare favourably with numerical solutions of the Howard-Gupta equation for wavelengths comparable to the vortex core radius and longer. Dispersion relations are then found numerically for specific models of vortex core structures observed experimentally; here no restrictions are placed on wavelength. The linear dispersion relation has an infinite number of branches, falling into two families; one with infinite phase speed at zero wavenumber (which we call ‘fast’ waves), the other with zero phase speed (‘slow’ waves). In the long-wave limit, slow waves have zero group velocity, while the fast waves may have finite non-zero group speeds that depend on the form of the velocity profiles on the axis of rotation. Weakly nonlinear waves are described under most circumstances by the nonlinear Schrödinger equation. Solitons are possible in certain ‘windows’ of wavenumbers of the carrier waves. An example has already been presented by Leibovich & Ma (1983), who compute solitons and soliton windows on a fast-wave branch for a vortex with a particular core structure. Experimental data of Maxworthy, Hopfinger & Redekopp (1985) reveal solitons which appear to be associated with the slow branch, and these are computed for velocity profiles fitting their data. The nonlinear Schrödinger equation is shown to fail for long waves, and to be replaced by a nonlinear integro-differential equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 595 - 624
Copyright
© 1986 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Davey, A. 1972 The propagation of a weak nonlinear wave. J. Fluid Mech. 53, 769–781.Google Scholar
Hopfinger, E. J., Browand, F. K. & Gagne, Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.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
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155.Google Scholar
Krishnamoorthy, V. 1966 Vortex breakdown and measurements of pressure fluctuations over slender wings. Southampton University Ph.D. dissertation.
Leibovich, S. 1970 Weakly non-linear waves in rotating fluids. J. Fluid Mech. 42, 803–822.Google Scholar
Leibovich, S. 1979 Waves in parallel or swirling stratified shear flows. J. Fluid Mech. 93, 401–412.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: Survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Leibovich, S. 1986 Wave and bifurcations in vortex filaments. In Studies of Vortex Dominated Flows, Proc. Symp. July 9–11, 1985, NASA Langley Research Center (eds M. Y. Hussaini & M. D. Salas), Springer.
Leibovich, S. & Ma, H-Y. 1983 Soliton propagation on vortex cores and the Hasimoto soliton. Phys. Fluids 26, 3173–3179.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G. 1985 Wave motions on vortex cores. J. Fluid Mech. 151, 141–165.Google Scholar
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272, 403–429.Google Scholar
Pocklington, H. C. 1895 The complete system of the periods of a hollow vortex ring. Phil. Trans. R. Soc. Lond. A 186, 603.Google Scholar
Pritchard, W. G. 1970 Solitary waves in rotating fluids. J. Fluid Mech. 42, 61–83.Google Scholar
Stewartson, K. & Brown, S. N. 1985 Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387–399.Google Scholar
Whitham, G. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Widnall, S. E., Bliss, D. & Zalay, A. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence and Its Detection (ed. J. H. Olsen et al.), Plenum.