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Motion of a helical vortex

Published online by Cambridge University Press:  11 December 2017

Oscar Velasco Fuentes Open the ORCID record for Oscar Velasco Fuentes [Opens in a new window]
Oscar Velasco Fuentes*
Affiliation:
Departamento de Oceanografía Física, CICESE, Ensenada, B.C. 22860, México
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , R1
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Copyright
© 2017 Cambridge University Press 

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References

Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280.CrossRefGoogle Scholar
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, 117–137.CrossRefGoogle Scholar
Da Rios, L. S. 1916 Vortici ad elica. Il Nuovo Cimento 11, 419–431.Google Scholar
Fitzgerald, G. F. 1899 On a hydro-dynamical hypothesis as to electro-magnetic actions. Sci. Proc. Dublin R. Soc. 9, 55–59.Google Scholar
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.Google Scholar
Greenwell, D. I. 2003 Comment on ‘Assessment of the accuracy of representing a helical vortex by straight segments’. AIAA J. 41, 332.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.Google Scholar
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.Google Scholar
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.Google Scholar
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, 607–614.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
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. 309–321. Springer.Google Scholar
Mezić, I., Leonard, A. & Wiggins, S. 1998 Regular and chaotic particle motion near a helical vortex filament. Physica D 111, 179–201.Google Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.Google Scholar
Okulov, V. L. & Sørensen, J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 1–25.CrossRefGoogle Scholar
Okulov, V. L. & Sørensen, J. N. 2010 Maximum efficiency of wind turbine rotors using Joukowsky and Betz approaches. J. Fluid Mech. 649, 497–508.CrossRefGoogle Scholar
Parsons, C. A. 1901 The marine steam turbine and its application to fast vessels. Trans. Inst. Engrs. Shipbuilders Scotland 44, 175–221.Google Scholar
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.Google Scholar
Saffman, P. G. 1995 Vortex Dynamics. Cambridge University Press.Google Scholar
Schwedoff, T. 1887 Les mouvements cycloniques. Revue Scientifique 39, 78–81.Google Scholar
Suaza Jaque, R. & Velasco Fuentes, O. 2017 Reconnection of orthogonal cylindrical vortices. Eur. J. Mech. (B/Fluids) 62, 51–56.Google Scholar
Thomson, W. (Lord Kelvin) 1880 Vibrations of a columnar vortex. Proc. R. Soc. Edin. 10, 443–456.Google Scholar
Velasco Fuentes, O.2015 Helical vortex in a moving frame. Available at: https://youtu.be/FmrqtHK7wVM.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Wood, D. H. & Boersma, J. 2001 On the motion of multiple helical vortices. J. Fluid Mech. 447, 149–171.Google Scholar