Abstract
The paper presents an analytical solution for helical vortices with a Gaussian vorticity distribution in the core, which is confirmed by experimental and numerical simulations. This result is obtained by extending the Dyson method to the Biot–Savart law. Previously, analytical solutions were found and studied only for vortices with constant vorticity distribution in the core (a Rankine-type vortex core). One of the important issues raised during the discussion is the difference between self-induced movements of helical structures with both types of vortex core. The proposed solutions are important for the fundamental understanding and description of the behavior of helical eddy flows in various fields of industry and in nature. Examples include tip vortices behind the rotors of wind or hydro turbines, tornadoes, or axial vortices in aerodynamic devices such as vortex apparatuses and generators; cyclone separators, combustion chambers, etc.
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S.V. Alekseenko and V.L. Okulov, Swirl flows in technical applications (A review), Thermophysics and Aeromechanics, 1996, Vol. 3, No. 2, P. 97–128.
N.G. Nygaard, Wakes in very large wind farms and the effect of neighbouring wind farms, J. Physics: Conference Series, 2014, Vol. 524, No. 1, P. 012162.
V.L. Okulov, I.K. Kabardin, I.V. Litvinov, R.F. Mikkelsen, I.V. Naumov, J.N. Sørensen, D.H. Wood, and S.V. Alekseenko, Hydrokinetic energy conversion: the basis of hydro farm, in: Abs. 5th Inter. Workshop on Heat/Mass Transfer Advances for Energy Conservation and Pollution Control August 13–16 2019, Novosibirsk, Russia, P. 134.
S.V. Alekseenko and S.I. Shtork, Swirling flow large-scale structures in a combustor model, Russ. J. Engng Thermophys., 1992, Vol. 2, No. 4, P. 231–266.
R.L. Ricca, The effect of torsion on the motion of a helical vortex filament, J. Fluid Mech., 1994, Vol. 273, P. 241–259.
D.W. Moore and P.G. Saffman, The motion of a vortex filament with axial flow, Phil. Trans. R. Soc. Lond. A, 1972, Vol. 272, P. 403–429.
J. Boersma and D.H. Wood, On the self-induced motion of a helical vortex, J. Fluid Mech., 1999, Vol. 384, P. 263–279.
S. Kawada, Induced velocity by helical vortices, J. Aeronaut. Sci., 1936, Vol. 3, No. 3, P. 86–87.
J.C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluids, 1982, Vol. 25, P. 1949–1952.
Y. Fukumoto, V.L. Okulov, and D.H. Wood, The contribution of Kawada to the analytical solution for the velocity induced by a helical vortex filament, ASME. Appl. Mech. Rev., 2015, Vol. 67, No. 6, P. 060801.
V.L. Okulov, On the stability of multiple helical vortices, J. Fluid Mech., 2004, Vol. 521, P. 319–342.
V.L. Okulov and J.N. Sørensen, The self-induced motion of a helical vortex, J. Fluid Mech., 2020, Vol. 883, A–5.
H.U. Quaranta, H. Bolnot, and T. Leweke, Long-wave instability of a helical vortex, J. Fluid Mech., 2015, Vol. 780, P. 687–716.
V.L. Okulov, I.K. Kabardin, R.F. Mikkelsen, I.V. Naumov, and J.N. Sørensen, Helical self-similarity of tip vortex cores, J. Fluid Mech., 2019, Vol. 859, P. 1084–1097.
M. Ali and M. Abid, Self-similar behavior of a rotor wake vortex core, J. Fluid Mech., 2014, Vol. 740, R1.
C. Selçuk, I. Delbende, and M. Rossi, Helical vortices: quasiequilibrium states and their time evolution, Phys. Rev. Fluids, 2017, Vol. 2, P. 084701.
Y. Fukumoto and V.L. Okulov, The velocity field induced by a helical vortex tube, Phys. Fluids, 2005, Vol. 7, No. 10, P. 107101.
V.L. Okulov and Ya. Fukumoto, Helical dipole, Doklady Physics, 2004, Vol. 49, No. 11, P. 662–667.
V.L. Okulov, E.S. Gesheva, P.A. Kuibin, S.I. Shtork, J. Sørensen, D. Wood, and S.V. Alekseenko, Difference in the movement of helical vortex and particles along its axis, Thermophysics and Aeromechanics, 2020, Vol. 27, No. 4, P. 473–480.
F.W. Dyson, The potential of an anchor ring. II, Philos. Trans. R. Soc. London, Ser. A, 1893, Vol. 184, P. 1041–1106.
Y. Fukumoto and H.K. Moffatt, Motion and expansion of a viscous vortex ring, J. Fluid Mech., 2000, Vol. 417, P. 1–45.
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In this work, V.L. Okulov received support under the contract with the Ministry of Education and Science of the Russian Federation (No. 075-15-2019-1923) and Y. Fukumoto received the grants for research from the Japan Society for the Promotion of Science (No. S17119 and No. 19K03672).
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Okulov, V.L., Fukumoto, Y. Analytical solution for self-induced motion of a helical vortex with a Gaussian core. Thermophys. Aeromech. 27, 481–488 (2020). https://doi.org/10.1134/S0869864320040022
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DOI: https://doi.org/10.1134/S0869864320040022