Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-21T13:51:38.860Z Has data issue: false hasContentIssue false
  • English
  • Français

The self-induced motion of a helical vortex

Published online by Cambridge University Press:  20 November 2019

Valery L. Okulov Open the ORCID record for Valery L. Okulov [Opens in a new window]  and
Jens N. Sørensen Open the ORCID record for Jens N. Sørensen [Opens in a new window]
Valery L. Okulov*
Affiliation:
Department of Wind Energy, Technical University of Denmark, 2800Lyngby, Denmark Kutateladze Institute of Thermophysics, SB RAS, Novosibirsk630090, Russia
Jens N. Sørensen
Affiliation:
Department of Wind Energy, Technical University of Denmark, 2800Lyngby, Denmark
*
†Email address for correspondence: vaok@dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

Helical vortices have been studied for more than a century to understand basic aspects of fluid motion. Helical vortices appear both in nature, e.g. as tornadoes, and in many industrial applications associated with mixing and in wakes behind rotors. Owing to the complexity of the equations governing the self-induced motion of helical vortices, it has up to now not been possible to obtain closed-form solutions describing all aspects of the motion. An important issue concerns the difference between the self-induced motion of the helical structure and the movement of fluid particles located on the helix. Here, we revisit the equations governing both the motion of the helical vortex structure and the motion of material fluid elements on the axis of the helix, and for both cases derive closed-form solutions for the resulting velocities. As a part of the paper, we also devise potential applications of the achieved knowledge.

JFM classification

Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A5
Copyright
© 2019 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alekseenko, S. V., Kuibin, P. A. & Okulov, V. L. 2007 Theory of Concentrated Vortices: An Introduction. Springer.Google Scholar
Alekseenko, S. V., Kuibin, P. A., Okulov, V. L. & Shtork, S. I. 1999 Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Betz, A. 1919 Schraubenpropeller mit geringstem Energieverlust: mit einem Zusatz von L. Prandtl (in German). Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1919, 193–217.Google Scholar
Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280.CrossRefGoogle Scholar
Boersma, J. & Yakubovich, S. B. 1998 Solution to problem 97-18: the asymptotic sum of a Kapteyn series. SIAM Rev. 40, 986–990.Google Scholar
Durán Venegas, E. & Le Dizès, S. 2019 Generalized helical vortex pairs. J. Fluid Mech. 865, 523–545.CrossRefGoogle Scholar
Felli, M., Camussi, R. & Di Felice, F. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 5–53.CrossRefGoogle Scholar
Fuentes, O. V. 2018 Motion of a helical vortex. J. Fluid Mech. 836, R1.Google Scholar
Fukumoto, Y. & Moffatt, H. K. 2000 Motion and expansion of a viscous vortex ring. J. Fluid Mech. 417, 1–45.CrossRefGoogle 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. ASME Appl. Mech. Rev. 67 (6), 060801.CrossRefGoogle Scholar
Goldstein, S. 1929 On the vortex theory of screw propellers. Proc. R. Soc. Lond. A 123, 440–465.CrossRefGoogle Scholar
Joukowsky, N. E. 1912 Vihrevaja teorija grebnogo vinta. Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest 16, 1–31; French translation by Margoulis in Théorie tourbillonnaire de l’hélice propulsive, Gauthier-Villars, 1929, pp. 1–47.Google Scholar
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.CrossRefGoogle Scholar
Hattori, Y. & Fukumoto, Y. 2009 Short-wavelength stability analysis of a helical vortex tube. Phys. Fluids 21, 014104.CrossRefGoogle Scholar
Kawada, S. 1936 Induced velocity by helical vortices. J. Aero. Sci. 3, 86–87.CrossRefGoogle 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.CrossRefGoogle Scholar
Kuibin, P. A., Okulov, V. L. & Pylev, I. M. 2006 Simulation of the flow structure in the suction pipe of a hydroturbine by integral characteristics. Heat Transfer Res. 37 (8), 675–684.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A A272, 403–429.CrossRefGoogle Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.CrossRefGoogle Scholar
Okulov, V. L., Kabardin, I. K., Mikkelsen, R. F., Naumov, I. V. & Sørensen, J. N. 2019 Helical self-similarity of tip vortex cores. J. Fluid Mech. 859, 1084–1097.CrossRefGoogle Scholar
Okulov, V. L., Naumov, I. V., Mikkelsen, R. F., Kabardin, I. K. & Sørensen, J. N. 2014 A regular Strouhal number for large-scale instability in the far wake of a rotor. J. Fluid Mech. 747, 369–380.CrossRefGoogle 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
Okulov, V. L., Sørensen, J. N. & Wood, D. H. 2015 The rotor theories by Professor Joukowsky: Vortex theories. Prog. Aerosp. Sci. 73, 19–46.CrossRefGoogle Scholar
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.CrossRefGoogle Scholar
Quaranta, H. U., Bolnot, H. & Leweke, T. 2015 Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687–716.CrossRefGoogle Scholar
Selçuk, C., Delbende, I. & Rossi, M. 2017 Helical vortices: quasiequilibrium states and their time evolution. Phys. Rev. Fluids 2, 084701.CrossRefGoogle Scholar
Sherry, M., Nemes, A., Lo Jacono, D., Blackburn, H. M. & Sheridan, J. 2013 The interaction of helical tip and root vortices in a wind turbine wake. Phys. Fluids 25, 117102.CrossRefGoogle Scholar
Tung, C. & Ting, L. 1967 The motion and decay of a vortex ring. Phys. Fluids 10, 901–910.CrossRefGoogle Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.CrossRefGoogle Scholar