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Generalized helical vortex pairs

Published online by Cambridge University Press:  20 February 2019

E. Durán Venegas Open the ORCID record for E. Durán Venegas [Opens in a new window]  and
S. Le Dizès Open the ORCID record for S. Le Dizès [Opens in a new window]
E. Durán Venegas*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
S. Le Dizès
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

New solutions describing the interaction of helical pairs of counter-rotating vortices are obtained using a vortex filament approach. The vortices are assumed to have a small core size allowing the calculation of the self-induced velocities from the Biot–Savart law using the cutoff theory. These new vortex structures do not possess any helical symmetry but they exhibit a spatial periodicity and are stationary in a rotating and translating frame. Their properties, such as radial deformation, frame velocity and induced flow, are provided as a function of the four geometric parameters characterizing each solution. Approximate solutions are also obtained when the mutual interaction is weak. This allows us to provide explicit expressions for the rotation and translation velocities of the structure in this limit. First-order corrections describing helix deformation are also calculated and used for comparison with the numerical results. The variation of the vortex core size induced by the helix deformation is also analysed. We show that these variations have a weak effect on the shape and characteristics of the solutions, for the range of parameters that we have considered. The results are finally applied to rotor wakes. It is explained how these solutions could possibly describe the far wake of an helicopter rotor in vertical flight.

JFM classification

Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 523 - 545
Copyright
© 2019 Cambridge University Press 

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References

Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Betz, A. 1926 Windenergie und ihre Ausnützung durch Windmühlen. Vandenhoeck und Ruprecht.Google Scholar
Blanco-Rodríguez, F. J. & Le Dizès, S. 2016 Elliptic instability of a curved Batchelor vortex. J. Fluid Mech. 804, 224–247.Google Scholar
Blanco-Rodríguez, F. J. & Le Dizès, S. 2017 Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397–415.10.1017/jfm.2017.34Google Scholar
Blanco-Rodríguez, F. J., Le Dizès, S., Selçuk, C., Delbende, I. & Rossi, M. 2015 Internal structure of vortex rings and helical vortices. J. Fluid Mech. 785, 219–247.Google Scholar
Boersma, J. & Wood, D. H. 1999 On the self–induced motion of a helical vortex. J. Fluid Mech. 384, 263–280.Google Scholar
Cottet, G.-H. & Koumoutsakos, P. D. 2000 Vortex Methods: Theory and Applications. Cambridge University Press.Google Scholar
Da Rios, L. S. 1916 Vortici ad elica. Il Nuovo Cimento 11, 419–431.10.1007/BF02960988Google Scholar
Drees, J. M. & Hendal, W. P. 1951 The field of flow through a helicopter rotor obtained from wind tunnel smoke tests. J. Aircraft Eng. 23, 107–111.Google Scholar
Durán Venegas, E. & Le Dizès, S. 2018 Structure and stability of rotor generated vortices. European Fluid Mechanics Conference 12. 9–13 September 2018, Vienna, Austria.Google Scholar
Froude, W. 1878 On the elementary relation between pitch, slip and propulsive efficiency. Trans. Inst. Naval Arch. 19, 22–33.Google Scholar
Fukumoto, Y. & Okulov, V. L. 2005 The velocity induced by a helical vortex tube. Phys. Fluids 17, 107101.10.1063/1.2061427Google Scholar
Fukumoto, Y., Okulov, V. L. & Wood, D. 2015 The contribution of Kawada to the analytical solution for the velocity induced by a helical vortex filament. Appl. Mech. Rev. 67 (6), 467–486.10.1115/1.4031964Google Scholar
Goldstein, M. A. 1929 On the vortex theory of screw propellers. Proc. R. Soc. Lond. A 123, 440–465.10.1098/rspa.1929.0078Google Scholar
Gupta, B. P. & Loewy, R. G. 1974 Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12, 1381–1387.Google Scholar
Gupta, S. & Leishman, J. G. 2005 Accuracy of the induced velocity from helical vortices using straight-line segmentation. AIAA J. 43, 29–40.Google Scholar
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.Google Scholar
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222–249.10.1017/jfm.2013.591Google Scholar
Joukowski, N. 1929 Théorie tourbillonnaire de l’hélice propulsive. Gauthier-Villars.Google Scholar
Kawada, S. 1936 Induced velocity by helical vortices. J. Aero. Sci. 3 (3), 86–87.Google Scholar
Kelvin, L. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409.Google Scholar
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.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, 607–614.10.1063/1.869587Google Scholar
Kwiecinski, J. A. & Van Gorder, R. A. 2018 Dynamics of nearly parallel interacting vortex filaments. J. Fluid Mech. 835, 575–623.Google Scholar
Leishman, J. G. 2006 Principles of Helicopter Aerodynamics. Cambridge University Press.Google Scholar
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670–690.10.1098/rspa.1928.0174Google Scholar
Lucas, D. & Dritschel, D. G. 2009 A family of helically symmetric vortex equilibria. J. Fluid Mech. 634, 245–268.Google Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.10.1017/S0022112004001934Google Scholar
Okulov, V. L., Sørensen, J. N. & Wood, D. H. 2015 The rotor theories by Professor Joukowsky: vortex theories. Prog. Aero. Sci. 73, 19–46.10.1016/j.paerosci.2014.10.002Google Scholar
Okulov, V. N. 2016 An acentric rotation of two helical vortices of the same circulations. Regular Chaotic Dyn. 21, 267–273.10.1134/S1560354716030035Google Scholar
Quaranta, H. U., Bolnot, H. & Leweke, T. 2015 Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687–716.Google Scholar
Quaranta, U.2017 Instabilities in a swirling rotor wake. PhD thesis, Aix Marseille Université.Google Scholar
Rankine, W. J. M. 1865 On the mechanical principles of the action of propellers. Trans. Inst. Naval Arch. 6, 13–39.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. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Selçuk, C., Delbende, I. & Rossi, M. 2017a Helical vortices: linear stability analysis and nonlinear dynamics. Fluid Dyn. Res. 50, 011411.Google Scholar
Selçuk, C., Delbende, I. & Rossi, M. 2017b Helical vortices: Quasi-equilibrium states and their time evolution. Phys. Rev. Fluids 2, 084701.Google Scholar
Sørensen, J. N. 2016 General Momentum Theory for Horizontal Axis Wind Turbines, Springer Series: Research Topics in Wind Energy, vol. 4. Springer.Google Scholar
Velasco Fuentes, O. 2018 Motion of a helical vortex. J. Fluid Mech. 836, R1.10.1017/jfm.2017.845Google Scholar
Vermeer, L. J., Sørensen, J. N. & Crespo, A. 2003 Wind turbine wake aerodynamics. Prog. Aero. Sci. 39, 467–510.10.1016/S0376-0421(03)00078-2Google Scholar
Wald, Q. R. 2006 The aerodynamics of propellers. Prog. Aero. Sci. 42, 85–128.Google Scholar
Walther, J. H., Guénot, M., Machefaux, E., Rasmussen, J. T., Chatelain, P., Okulov, V. L., Sørensen, J. N., Bergdorf, M. & Koumoutsakos, P. 2007 A numerical study of the stabilitiy of helical vortices using vortex methods. J. Phys.: Conf. Ser. 75, 012034.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Winckelmans, G., Cocle, R., Dufresne, L. & Capart, R. 2005 Vortex methods and their application to trailing wake vortex simulations. C. R. Physique 6, 467–486.Google Scholar