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Motion of a helical vortex

  • Oscar Velasco Fuentes (a1)

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

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Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263280.
Da Rios, L. S. 1906 Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rendiconti del Circolo Matematico di Palermo 22, 117137.
Da Rios, L. S. 1916 Vortici ad elica. Il Nuovo Cimento 11, 419431.
Fitzgerald, G. F. 1899 On a hydro-dynamical hypothesis as to electro-magnetic actions. Sci. Proc. Dublin R. Soc. 9, 5559.
Fukumoto, Y., Okulov, V. L. & Wood, D. H. 2015 The contribution of Kawada to the analytical solution for the velocity induced by a helical vortex filament. Appl. Mech. Rev. 67, 060801.
Greenwell, D. I. 2003 Comment on ‘Assessment of the accuracy of representing a helical vortex by straight segments’. AIAA J. 41, 332.
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 19491952.
Joukowsky, N. E.1912 Vihrevaja teorija grebnogo vinta. Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest. 16, 1–31. French translation in Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929) pp. 1–47.
Kuibin, P. A. & Okulov, V. L. 1998 Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament. Phys. Fluids 10, 607614.
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670690.
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics: Compressible and Incompressible Flows (ed. Mueller, U., Roesner, K. G. & Schmidt, B.), pp. 309321. Springer.
Mezić, I., Leonard, A. & Wiggins, S. 1998 Regular and chaotic particle motion near a helical vortex filament. Physica D 111, 179201.
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.
Okulov, V. L. & Sørensen, J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 125.
Okulov, V. L. & Sørensen, J. N. 2010 Maximum efficiency of wind turbine rotors using Joukowsky and Betz approaches. J. Fluid Mech. 649, 497508.
Parsons, C. A. 1901 The marine steam turbine and its application to fast vessels. Trans. Inst. Engrs. Shipbuilders Scotland 44, 175221.
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241259.
Saffman, P. G. 1995 Vortex Dynamics. Cambridge University Press.
Schwedoff, T. 1887 Les mouvements cycloniques. Revue Scientifique 39, 7881.
Suaza Jaque, R. & Velasco Fuentes, O. 2017 Reconnection of orthogonal cylindrical vortices. Eur. J. Mech. (B/Fluids) 62, 5156.
Thomson, W. (Lord Kelvin) 1880 Vibrations of a columnar vortex. Proc. R. Soc. Edin. 10, 443456.
Velasco Fuentes, O.2015 Helical vortex in a moving frame. Available at:
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641663.
Wood, D. H. & Boersma, J. 2001 On the motion of multiple helical vortices. J. Fluid Mech. 447, 149171.
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