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Long-wave instability of a helical vortex

Published online by Cambridge University Press:  09 September 2015

Hugo Umberto Quaranta ,
Hadrien Bolnot  and
Thomas Leweke
Hugo Umberto Quaranta
Affiliation:
IRPHE UMR 7342, Aix-Marseille Université, CNRS, Centrale Marseille, 13384 Marseille, France Aerodynamics Department, Airbus Helicopters, 13725 Marignane, France
Hadrien Bolnot
Affiliation:
IRPHE UMR 7342, Aix-Marseille Université, CNRS, Centrale Marseille, 13384 Marseille, France Aerodynamics Department, Airbus Helicopters, 13725 Marignane, France
Thomas Leweke*
Affiliation:
IRPHE UMR 7342, Aix-Marseille Université, CNRS, Centrale Marseille, 13384 Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the instability of a single helical vortex filament of small pitch with respect to displacement perturbations whose wavelength is large compared to the vortex core size. We first revisit previous theoretical analyses concerning infinite Rankine vortices, and consider in addition the more realistic case of vortices with Gausssian vorticity distributions and axial core flow. We show that the various instability modes are related to the local pairing of successive helix turns through mutual induction, and that the growth rate curve can be qualitatively and quantitatively predicted from the classical pairing of an array of point vortices. We then present results from an experimental study of a helical vortex filament generated in a water channel by a single-bladed rotor under carefully controlled conditions. Various modes of displacement perturbations could be triggered by suitable modulation of the blade rotation. Dye visualisations and particle image velocimetry allowed a detailed characterisation of the vortex geometry and the determination of the growth rate of the long-wave instability modes, showing good agreement with theoretical predictions for the experimental base flow. The long-term (downstream) development of the pairing instability leads to a grouping and swapping of helix loops. Despite the resulting complicated three-dimensional structure, the vortex filaments surprisingly remain mostly intact in our observation interval. The characteristic distance of evolution of the helical wake behind the rotor decreases with increasing initial amplitude of the perturbations; this can be predicted from the linear stability theory.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 780 , 10 October 2015 , pp. 687 - 716
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© 2015 Cambridge University Press 

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References

Alfredsson, P. H. & Dahlberg, J. A.1979. A preliminary wind tunnel study of windmill wake dispersion in various flow conditions, Technical Note AU-1499, FFA, Stockholm, Sweden.Google Scholar
Ash, R. L. & Khorrami, M. R. 1995 Vortex stability. In Fluid Vortices (ed. Green, S. I.), pp. 317–372. Kluwer.CrossRefGoogle Scholar
Betchov, R. 1965 On the curvature and torsion of a vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Bhagwat, M. J. & Leishman, J. G. 2000 Stability analysis of helicopter rotor wakes in axial flight. J. Am. Helicopter Soc. 45, 165–178.Google Scholar
Bolnot, H.2012. Instabilités des tourbillons hélicoïdaux: application au sillage des rotors. PhD thesis, Aix-Marseille Université, Marseille, France.Google Scholar
Bolnot, H., Le Dizès, S. & Leweke, T. 2014 Spatio-temporal development of the pairing instability in an infinite array of vortex rings. Fluid Dyn. Res. 46, 061405.Google Scholar
Crow, S. C. 1971 Stability theory of a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Fabre, D.2002. Instabilités et instationarités dans les tourbillons: application au sillages des avions. PhD thesis, Université Pierre et Marie Curie–Paris VI, Paris, France.Google Scholar
Fabre, D. & Jacquin, L. 2004 Short-wave cooperative instabilities in representative aircraft vortices. Phys. Fluids 16, 1366–1378.Google Scholar
Felli, M., Camussi, R. & Felice, F. D. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 5–53.Google Scholar
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77–115.Google Scholar
Fukumoto, Y. & Miyazaki, T. 1991 Three-dimensional distortions of a vortex filament with axial velocity. J. Fluid Mech. 222, 369–416.Google Scholar
Gupta, B. P. & Loewy, R. G. 1974 Theoretical analysis of the aerodynamics stability of multiple, interdigitated helical vortices. AIAA J. 12, 1381–1387.Google Scholar
Hattori, Y. & Fukumoto, Y. 2009 Short-wavelength stability analysis of a helical vortex tube. J. Fluid Mech. 21, 014104.Google Scholar
Ivanell, S., Leweke, T., Sarmast, S., Quaranta, H. U., Mikkelsen, R. F. & Sørensen, J. N. 2015 Comparison between experiments and large-eddy simulations of tip spiral structure and geometry. J. Phys.: Conf. Ser. 625, 012018.Google Scholar
Ivanell, S., Mikkelsen, R., Sørensen, J. N. & Henningson, D. 2010 Stability analysis of the tip vortices of a wind turbine. Wind Energy 13, 705–715.Google Scholar
von Kármán, T. & Rubach, H. 1912 Über den Mechanismus des Flüssigkeits- und Luftwiderstandes. Phys. Z. 13, 49–59.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409.Google Scholar
Kida, S. 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655–1662.CrossRefGoogle Scholar
Koch, C. R., Mungal, M. G., Reynolds, W. C. & Powell, J. D. 1989 Helical modes in an acoustically excited round air jet. Phys. Fluids A 1, 1443.Google Scholar
Lamb, H. 1932 Hydrodynamics, § 156. Cambridge University Press.Google Scholar
Leishman, J. G. 2006 Principles of Helicopter Aerodynamics. Cambridge University Press.Google Scholar
Leishman, J. G., Bhagwat, M. J. & Ananthan, S. 2004 The vortex ring state as a spatially and temporally developing wake instability. J. Am. Helicopter Soc. 49, 160–175.Google Scholar
Levy, H. & Forsdyke, A. G. 1927 The stability of an infinite system of circular vortices. Proc. R. Soc. Lond. A 114, 594–604.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
Leweke, T. 2012 Dye visualization – a method for investigating biomechanical flows. Curr. Pharm. Biotechnol. 13, 2141–2152.Google Scholar
Leweke, T., Quaranta, H. U., Bolnot, H., Blanco-Rodríguez, F. J. & Le Dizès, S. 2014 Long- and short-wave instabilities in helical vortices. J. Phys.: Conf. Ser. 524, 012154.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.Google Scholar
Meunier, P. & Leweke, T. 2003 Analysis and optimization of the error caused by high velocity gradients in particle image velocimetry. Exp. Fluids 35, 408–421.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159.Google Scholar
Naumov, I. V., Mikkelsen, R. F., Okulov, V. L. & Sørensen, J. N. 2014 PIV and LDA measurements of the wake behind a wind turbine model. J. Phys.: Conf. Ser. 524, 012168.Google Scholar
Nemes, A., Lo Jacono, D., Blackburn, H. M. & Sheridan, J. 2015 Mutual inductance of two helical vortices. J. Fluid Mech. 774, 298–310.CrossRefGoogle Scholar
Ohanian, C. V., McCauley, G. J. & Savaş, Ö. 2012 A visual study of vortex instabilities in the wake of a rotor in hover. J. Am. Helicopter Soc. 57, 1–8.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.Google Scholar
Robinson, A. C. & Saffman, P. G. 1982 Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411–427.Google Scholar
Rosenhead, L. 1930 The spread in vorticity in the wake behind a cylinder. Proc. R. Soc. Lond. A 127, 590–612.Google Scholar
Roy, C., Leweke, T., Thompson, M. C. & Hourigan, K. 2011 Experiments on the elliptic instability in vortex pairs with axial core flow. J. Fluid Mech. 677, 383–416.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics § 11.2. Cambridge University Press.Google Scholar
Sarmast, S., Dadfar, R., Mikkelsen, R. F., Schlatter, P., Ivanell, S., Sørensen, J. N. & Henningson, D. S. 2014 Mutual inductance instability of the tip vortices behind a wind turbine. J. Fluid Mech. 755, 705–731.Google Scholar
Selig, M. S., Guglielmo, J. J., Broeren, A. P. & Giguere, P. 1995 Summary of Low-Speed Airfoil Data. SoarTech.Google 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.Google Scholar
Sørensen, J. N., Mikkelsen, R., Sarmast, S., Ivanell, S. & Henningson, D. 2014 Determination of wind turbine near-wake length based on stability analysis. J. Phys.: Conf. Ser. 524, 012155.Google Scholar
Sørensen, J. N. & Shen, W. Z. 2002 Numerical modelling of wind turbine wakes. Trans. ASME J. Fluids Engng 124, 393–399.Google Scholar
Stack, J., Caradonna, F. X. & Savaş, Ö. 2005 Flow visualizations and extended thrust time histories of rotor vortex wakes in descent. J. Am. Helicopter Soc. 50, 279–288.Google Scholar
Vermeer, L. J., Sørensen, J. N. & Crespo, A. 2003 Wind turbine wake aerodynamics. Progr. Aerosp. Sci. 39, 467–510.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 instability 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
Widnall, S. E., Bliss, D. B. & Zalay, A. 1971 Theoretical and experimental study of the instability of a vortex pair. In Aircraft Wake Turbulence and its Detection (ed. Olsen, J. H., Goldberg, A. & Rogers, M.), pp. 305–338. Plenum.Google Scholar