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Experiments on the elliptic instability in vortex pairs with axial core flow

Published online by Cambridge University Press:  11 April 2011

CLÉMENT ROY ,
THOMAS LEWEKE ,
MARK C. THOMPSON  and
KERRY HOURIGAN
CLÉMENT ROY
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHÉ), CNRS/Universités Aix-Marseille/École Centrale Marseille, 49 rue Frédéric Joliot-Curie, B.P. 146, F-13384 Marseille CEDEX 13, France Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
THOMAS LEWEKE*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHÉ), CNRS/Universités Aix-Marseille/École Centrale Marseille, 49 rue Frédéric Joliot-Curie, B.P. 146, F-13384 Marseille CEDEX 13, France
MARK C. THOMPSON
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
KERRY HOURIGAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Division of Biological Engineering, Monash University, Melbourne, Victoria 3800, Australia
*
†Email address for correspondence: thomas.leweke@irphe.univ-mrs.fr
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Abstract

Results are presented from an experimental study on the dynamics of pairs of vortices, in which the axial velocity within each core differs from that of the surrounding fluid. Co- and counter-rotating vortex pairs at moderate Reynolds numbers were generated in a water channel from the tips of two rectangular wings. Measurement of the three-dimensional velocity field was accomplished using stereoscopic particle image velocimetry, revealing significant axial velocity deficits in the cores. For counter-rotating pairs, the long-wavelength Crow instability, involving symmetric wavy displacements of the vortices, could be clearly observed using dye visualisation. Measurements of both the axial wavelength and the growth rate of the unstable perturbation field were found to be in good agreement with theoretical predictions based on the full experimentally measured velocity profile of the vortices, including the axial flow. The dye visualisations further revealed the existence of a short-wavelength core instability. Proper orthogonal decomposition of the time series of images from high-speed video recordings allowed a precise characterisation of the instability mode, which involves an interaction of waves with azimuthal wavenumbers m = 2 and m = 0. This combination of waves fulfils the resonance condition for the elliptic instability mechanism acting in strained vortical flows. A numerical three-dimensional stability analysis of the experimental vortex pair revealed the same unstable mode, and a comparison of the wavelength and growth rate with the values obtained experimentally from dye visualisations shows good agreement. Pairs of co-rotating vortices evolve in the form of a double helix in the water channel. For flow configurations that do not lead to merging of the two vortices over the length of the test section, the same type of short-wave perturbations were observed. As for the counter-rotating case, quantitative measurements of the wavelength and growth rate, and comparison with previous theoretical predictions, again identify the instability as due to the elliptic mechanism. Importantly, the spatial character of the short-wave instability for vortex pairs with axial flow is different from that previously found in pairs without axial flow, which exhibit an azimuthal variation with wavenumber m = 1.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 383 - 416
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Alkislar, M. B., Krothapali, A. & Lourenco, L. M. 2003 Structure of a screeching rectangular jet: a stereoscopic particle image velocimetry study. J. Fluid Mech. 489, 121–154.CrossRefGoogle Scholar
Antkowiak, A. & Brancher, P. 2004 Transient energy growth for the Lamb–Oseen vortex. Phys. Fluids 16, L1–L4.CrossRefGoogle Scholar
Arendt, S., Fritts, D. C. & Andreassen, Ø. 1997 The initial-value problem for Kelvin vortex waves. J. Fluid Mech. 344, 181–212.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.CrossRefGoogle Scholar
Billant, P., Brancher, P. & Chomaz, J.-M. 1999 Three-dimensional stability of a vortex pair. Phys. Fluids 11, 2069–2077.CrossRefGoogle Scholar
Boulanger, N., Meunier, P. & LeDizès, S. Dizès, S. 2007 Structure of a tilted stratified vortex. J. Fluid Mech. 583, 443–458.CrossRefGoogle Scholar
Boulanger, N., Meunier, P. & LeDizès, S. Dizès, S. 2008 Instability of a tilted vortex in stratified fluid. J. Fluid Mech. 596, 1–20.CrossRefGoogle Scholar
Bristol, R. L., Ortega, J. M., Marcus, P. S. & Savaş, Ö. 2004 On cooperative instabilities of parallel vortex pairs. J. Fluid Mech. 517, 331–358.CrossRefGoogle Scholar
Carlier, J. & Stanislas, M. 2005 Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188.CrossRefGoogle Scholar
Chatterjee, A. 2000 An introduction to the proper orthogonal decomposition. Curr. Sci. India 78, 808–817.Google Scholar
Chen, A. L., Jacob, J. D. & Savaş, Ö. 1999 Dynamics of corotating vortex pairs in the wakes of flapped airfoils. J. Fluid Mech. 382, 155–193.CrossRefGoogle Scholar
Crouch, J. D. & Jacquin, L. 2005 Aircraft trailing vortices/ tourbillons de sillages d'avions. C. R. Phys. 6 (Special Issue), 393–565.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.CrossRefGoogle Scholar
Devenport, W. J., Rife, M. C., Liapis, S. I. & Follin, G. J. 1996 The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 67–106.CrossRefGoogle Scholar
Devenport, W. J., Vogel, C. M. & Zsoldos, J. S. 1999 Flow structure produced by the interaction and merger of a pair of co-rotating wing-tip vortices. J. Fluid Mech. 394, 357–377.CrossRefGoogle Scholar
Devenport, W. J., Zsoldos, J. S. & Vogel, C. M. 1997 The structure and development of a counter-rotating wing-tip vortex pair. J. Fluid Mech. 332, 71–104.CrossRefGoogle Scholar
Eloy, C. & LeDizès, S. Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660–676.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 145–166.CrossRefGoogle ScholarPubMed
Fabre, D. 2002 Instabilitités et instationnarités dans les tourbillons: application aux sillages d'avions. PhD thesis, ONERA/Université Paris VI.Google Scholar
Fabre, D., Cossu, C. & Jacquin, L. 2000 Spatio-temporal development of the long and short-wave vortex-pair instabilities. Phys. Fluids 12, 1247–1250.CrossRefGoogle Scholar
Fabre, D. & Jacquin, L. 2004 Short-wave cooperative instabilities in representative aircraft vortices. Phys. Fluids 16, 1366–1378.CrossRefGoogle Scholar
Fontane, J., Brancher, P. & Fabre, D. 2008 Stochastic forcing of the Lamb–Oseen vortex. J. Fluid Mech. 613, 233–254.CrossRefGoogle Scholar
Gerz, T., Holzäpfel, F. & Darracq, D. 2002 Commercial aircraft wake vortices. Prog. Aerosp. Sci. 38, 181–208.CrossRefGoogle Scholar
Hoerner, S. F. 1965 Fluid Dynamic Drag. Hoerner Fluid Dynamics.Google Scholar
Jacob, J. D. 1995 Experimental investigation of the trailing vortex wake of rectangular airfoils. PhD thesis, University of California at Berkeley.CrossRefGoogle Scholar
Jiménez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18 (11), 1580–1581.CrossRefGoogle Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Klein, R. & Knio, O. M. 1995 Asymptotic vorticity structure and numerical simulation of slender vortex filaments. J. Fluid Mech. 284, 275–321.CrossRefGoogle 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.CrossRefGoogle Scholar
Krutzsch, C.-H. 1939 Über eine experimentell beobachtete Erscheinung an Wirbelringen bei ihrer translatorischen Bewegung in wirklichen Flüssigkeiten. Annu. Phys. Leipzig 427, 497–523.CrossRefGoogle Scholar
Lacaze, L., Birbaud, A.-L. & Le Dizès, S. 2005 Elliptic instability in a Rankine vortex with axial flow. Phys. Fluids 17, 017101.CrossRefGoogle Scholar
Lacaze, L., Le Gal, P. & Le Dizès, S. 2004 Elliptical instability in a rotating spheroid. J. Fluid Mech. 505, 1–22.CrossRefGoogle Scholar
Lacaze, L., Le Gal, P. & Le Dizès, S. 2005 Elliptical instability of a flow in a rotating shell. Phys. Earth Planet. Inter. 151, 194–205.CrossRefGoogle Scholar
Lacaze, L., Ryan, K. & Le Dizès, S. 2007 Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341–361.CrossRefGoogle Scholar
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.CrossRefGoogle Scholar
Laporte, F. & Corjon, A. 2000 Direct numerical simulations of the elliptic instability of a vortex pair. Phys. Fluids 12, 1016–1031.CrossRefGoogle Scholar
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169–201.CrossRefGoogle Scholar
Le Dizès, S. & Verga, A. 2002 Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389–410.CrossRefGoogle Scholar
Leibovich, S., Brown, S. N. & Patel, T. 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595–624.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 2011 Experiments on long-wavelength instability and reconnection of a vortex pair. Phys. Fluids 23, 024101.CrossRefGoogle Scholar
Liang, Y. C., Lee, H. P., Lim, S. P., Lin, W. Z., Lee, K. H. & Wu, C. G. 2002 Proper orthogonal decomposition and its applications. Part 1. Theory. J. Sound Vib. 252, 527–544.CrossRefGoogle Scholar
Liu, H.-T. 1992 Effects of ambient turbulence on the decay of a trailing vortex wake. J. Aircraft 29, 255–263.CrossRefGoogle Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.CrossRefGoogle Scholar
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2001 Three-dimensional instability during vortex merging. Phys. Fluids 13, 2747–2750.CrossRefGoogle 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.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1973 Axial flow in laminar trailing vortices. Proc. R. Soc. Lond. A 333, 491–508.Google Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Ortega, J. M., Bristol, R. L. & Savaş, Ö. 2003 Experimental study of the stability of unequal-strength counter-rotating vortex pairs. J. Fluid Mech. 474, 35–84.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2160.CrossRefGoogle Scholar
Prasad, A. K. & Jensen, K. 1995 Scheimpflug stereocamera for particle image velocimetry in liquid flows. Appl. Optics 34, 7092–7099.CrossRefGoogle ScholarPubMed
Rennich, C. & Lele, S. K. 1997 Numerical method for incompressible vortical flows with two unbounded directions. J. Comp. Phys. 137, 101–129.CrossRefGoogle Scholar
Rossow, V. J. 1999 Lift-generated vortex of subsonic transport aircraft. Prog. Aerosp. Sci. 35, 507–660.CrossRefGoogle Scholar
Roy, C. & Leweke, T. 2008 Experiments on vortex meandering. European project ‘FAR-Wake’ (AST4-CT-2005-012238), Tech. Rep. TR 1.1.1-4. Available at: http://www.far-wake.org/.Google Scholar
Roy, C., Schaeffer, N., Le Dizès, S. & Thompson, M. C. 2008 Stability of a pair of co-rotating vortices with axial flow. Phys. Fluids 20, 094101.CrossRefGoogle Scholar
Sarpkaya, T. 1983 Trailing vortices in homogeneous and density-stratified media. J. Fluid Mech. 136, 85–109.CrossRefGoogle Scholar
Scheimpflug, T. 1904 Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes. Great Britain Patent no. 1196.Google Scholar
Scorer, R. S. & Davenport, L. J. 1970 Contrails and aircraft downwash. J. Fluid Mech. 43, 451–464.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 1861–1874.CrossRefGoogle Scholar
Spalart, P. R. 1998 Airplane trailing vortices. Annu. Rev. Fluid Mech. 30, 107–138.CrossRefGoogle Scholar
Taylor, J. R. 1997 An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books.Google Scholar
Thomas, P. J. & Auerbach, D. 1994 The observation of the simultaneous development of a long- and a short-wave instability mode on a vortex pair. J. Fluid Mech. 265, 289–302.CrossRefGoogle Scholar
Tombach, I. 1973 Observations of atmospheric effects on vortex wake behavior. J. Aircraft 10, 641–647.CrossRefGoogle Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76–80.CrossRefGoogle Scholar
Widnall, S. E., Bliss, D. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66 (1), 35–47.CrossRefGoogle 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.), p. 305. Plenum.CrossRefGoogle Scholar
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex ring. Proc. R. Soc. Lond. A 332, 335–353.Google Scholar
Willert, C. 1997 Stereoscopic digital particle image velocimetry for application in wind tunnel flows. Meas. Sci. Technol. 8, 1465–1479.CrossRefGoogle Scholar
Zang, W. & Prasad, A. K. 1997 Performance evaluation of a Scheimpflug stereocamera for particle image velocimetry. Appl. Optics 36, 8738–8744.CrossRefGoogle ScholarPubMed