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The effect of torsion on the motion of a helical vortex filament

Published online by Cambridge University Press:  26 April 2006

Renzo L. Ricca
Renzo L. Ricca
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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Abstract

In this paper we analyse in detail, and for the first time, the rôle of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement of the vortex filament in the fluid, is closely analysed. The study is performed by applying the prescription of Moore & Saffman (1972) to helices of any pitch and a new asymptotic integral formula is derived. We give a rigorous proof that the Kelvin régime and its limit behaviour are obtained as a limit form of that integral asymptotic formula. The results are compared with new calculations based on the re-elaboration of Hardin's (1982) approach and with results obtained by Levy & Forsdyke (1928) and Widnall (1972) for helices of small pitch, here also re-elaborated for the purpose.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 273 , 25 August 1994 , pp. 241 - 259
Copyright
© 1994 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Adebiyi, A. 1981 On the existence of steady helical vortex tubes of small cross-section. Q. J. Mech. Appl. Maths 34, 153–177.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Boratav, O. N., Pelz, R. B. & Zabusky, N. J. 1992 Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications. Phys. Fluids A 4, 581–605.Google Scholar
Callegari, A. J. & Ting, L. 1978 Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Maths 35, 148–175.Google Scholar
Crow, S. C. 1970 Stability theory of a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62.Google Scholar
Fukumoto, Y. & Miyazaki, T. 1991 Three-dimensional distortions of a vortex filament with axial velocity. J. Fluid Mech. 222, 369–416.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series, and Products. Academic Press.
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.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
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerr, R. M. 1985 Higher-order derivative correlation and the alignment of small-scale structures in isotropic turbulence. J. Fluid Mech. 153, 31–58.Google Scholar
Kida, S. 1993 Tube-Like Structures in Turbulence. Lecture Notes in Numerical Applied Analysis. Springer (in press).
Klein, R. & Majda, A. J. 1991 Self-stretching of a perturbed vortex filament. I. The asymptotic equation for deviations from a straight line. Physica D 49, 323–352.Google Scholar
Leibovich, S. & Ma, H. Y. 1983 Soliton propagation on vortex cores and the Hasimoto soliton. Phys. Fluids 26, 3173–3179.Google Scholar
Lesieur, M. (Ed.) 1991 Computational Fluid Dynamics. Institute de Méchanique de Grenoble, CNRS.
Levi-Civita, T. 1932 Attrazione Newtoniana dei tubi sottili e vortici filiformi. (Newtonian attraction of slender tubes and filiform vortices.) Ann. R. Scuola Normale Sup. Pisa 1, 1–54.Google Scholar
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670–690.Google Scholar
Lundgren, T. S. & Ashurst, W. T. 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307.Google Scholar
Marshall, J. S. 1991 A general theory of curved vortices with circular cross-section and variable core area. J. Fluid Mech. 229, 311–338.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G. 1985 Wave motion 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
Ricca, R. L. 1992 Local dynamics of a twisted vortex filament. In XVIII Intl Congr. Theor. Appl. Mech. - Abstracts, p. 123. IUTAM.
Ricca, R. L. 1993 The effect of torsion on the local dynamics of a vortex filament. In Euromech 305 – Dynamics and Geometry of Vortical Structures, pp. 39–40. Università di Roma “La Sapienza”.
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
She, Z.-S., Jackson, E. & Orszag, S. A. 1990 Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226–228.Google Scholar
Siggia, E. D. 1981 Numerical study of small scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375–406.Google Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–25.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Widnall, S. E. & Bliss, D. B. 1971 Slender-body analysis of the motion and stability of a vortex filament containing an axial flow. J. Fluid Mech. 50, 335–353.Google Scholar
Widnall, S. E., Bliss, D. B. & Zalay, A. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence and Its Detection (ed. J. Olsen et al.), pp. 305–338. Plenum.
Willmore, T. J. 1959 An Introduction to Differential Geometry. Oxford University Press.