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Generalized helical vortex pairs

  • E. Durán Venegas (a1) and S. Le Dizès (a1)


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.


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Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471479.
Betz, A. 1926 Windenergie und ihre Ausnützung durch Windmühlen. Vandenhoeck und Ruprecht.
Blanco-Rodríguez, F. J. & Le Dizès, S. 2016 Elliptic instability of a curved Batchelor vortex. J. Fluid Mech. 804, 224247.
Blanco-Rodríguez, F. J. & Le Dizès, S. 2017 Curvature instability of a curved Batchelor vortex. J. Fluid Mech. 814, 397415.10.1017/jfm.2017.34
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, 219247.
Boersma, J. & Wood, D. H. 1999 On the self–induced motion of a helical vortex. J. Fluid Mech. 384, 263280.
Cottet, G.-H. & Koumoutsakos, P. D. 2000 Vortex Methods: Theory and Applications. Cambridge University Press.
Da Rios, L. S. 1916 Vortici ad elica. Il Nuovo Cimento 11, 419431.10.1007/BF02960988
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, 107111.
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.
Froude, W. 1878 On the elementary relation between pitch, slip and propulsive efficiency. Trans. Inst. Naval Arch. 19, 2233.
Fukumoto, Y. & Okulov, V. L. 2005 The velocity induced by a helical vortex tube. Phys. Fluids 17, 107101.10.1063/1.2061427
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), 467486.10.1115/1.4031964
Goldstein, M. A. 1929 On the vortex theory of screw propellers. Proc. R. Soc. Lond. A 123, 440465.10.1098/rspa.1929.0078
Gupta, B. P. & Loewy, R. G. 1974 Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12, 13811387.
Gupta, S. & Leishman, J. G. 2005 Accuracy of the induced velocity from helical vortices using straight-line segmentation. AIAA J. 43, 2940.
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 19491952.
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222249.10.1017/jfm.2013.591
Joukowski, N. 1929 Théorie tourbillonnaire de l’hélice propulsive. Gauthier-Villars.
Kawada, S. 1936 Induced velocity by helical vortices. J. Aero. Sci. 3 (3), 8687.
Kelvin, L. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397409.
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201248.
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, 607614.10.1063/1.869587
Kwiecinski, J. A. & Van Gorder, R. A. 2018 Dynamics of nearly parallel interacting vortex filaments. J. Fluid Mech. 835, 575623.
Leishman, J. G. 2006 Principles of Helicopter Aerodynamics. Cambridge University Press.
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670690.10.1098/rspa.1928.0174
Lucas, D. & Dritschel, D. G. 2009 A family of helically symmetric vortex equilibria. J. Fluid Mech. 634, 245268.
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.10.1017/S0022112004001934
Okulov, V. L., Sørensen, J. N. & Wood, D. H. 2015 The rotor theories by Professor Joukowsky: vortex theories. Prog. Aero. Sci. 73, 1946.10.1016/j.paerosci.2014.10.002
Okulov, V. N. 2016 An acentric rotation of two helical vortices of the same circulations. Regular Chaotic Dyn. 21, 267273.10.1134/S1560354716030035
Quaranta, H. U., Bolnot, H. & Leweke, T. 2015 Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687716.
Quaranta, U.2017 Instabilities in a swirling rotor wake. PhD thesis, Aix Marseille Université.
Rankine, W. J. M. 1865 On the mechanical principles of the action of propellers. Trans. Inst. Naval Arch. 6, 1339.
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241259.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Selçuk, C., Delbende, I. & Rossi, M. 2017a Helical vortices: linear stability analysis and nonlinear dynamics. Fluid Dyn. Res. 50, 011411.
Selçuk, C., Delbende, I. & Rossi, M. 2017b Helical vortices: Quasi-equilibrium states and their time evolution. Phys. Rev. Fluids 2, 084701.
Sørensen, J. N. 2016 General Momentum Theory for Horizontal Axis Wind Turbines, Springer Series: Research Topics in Wind Energy, vol. 4. Springer.
Velasco Fuentes, O. 2018 Motion of a helical vortex. J. Fluid Mech. 836, R1.10.1017/jfm.2017.845
Vermeer, L. J., Sørensen, J. N. & Crespo, A. 2003 Wind turbine wake aerodynamics. Prog. Aero. Sci. 39, 467510.10.1016/S0376-0421(03)00078-2
Wald, Q. R. 2006 The aerodynamics of propellers. Prog. Aero. Sci. 42, 85128.
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.
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641663.
Winckelmans, G., Cocle, R., Dufresne, L. & Capart, R. 2005 Vortex methods and their application to trailing wake vortex simulations. C. R. Physique 6, 467486.
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Journal of Fluid Mechanics
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