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Vortex bursting near a free surface

Published online by Cambridge University Press:  11 February 2020

Promode R. Bandyopadhyay Open the ORCID record for Promode R. Bandyopadhyay [Opens in a new window]
Promode R. Bandyopadhyay*
Affiliation:
Naval Undersea Warfare Center, Newport, RI 02841, USA
*
†Present address: 153 Chases Ln., Middletown, RI 02842, USA. Email address for correspondence: [email protected]
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Abstract

When the cross-section is reduced, a vortex displays spiralling and elasticity, bursting when the external velocity drops. How are these properties affected near a free surface? To answer, a visualization experiment in water is considered where an oscillating obstruction is pitched orthogonally to a rolling oscillation expending minimum energy. We show that short aerated vortices in adverse pressure gradient and shear remain stable during stretching, bursting into bubbles when relaxed after maximum stretching. A vertical aerated double helix (DH) root vortex of contra-rotating vortex tubes is produced near a Rossby number of 0.20. The vortex approaches an inclination of $45^{\circ }$ to the vertical, where stretching intensifies the vorticity to the maximum extent. Vortex inclinations up to $45^{\circ }$ are stable and unstable thereafter. Vortex bursting commences only when the inclination crosses $45^{\circ }$ by the slightest amount. When relaxed, oscillations are produced, breaking the vortex into arrays of bubbles, sometimes precisely at $45^{\circ }$ inclination. The bubble diameter, modelled by equating the effects of the centrifugal force (CF) on the vortex core pressure, to surface tension scales with the inverse of the square of the rotational velocity. When a high CF is withdrawn, the bursting DH aerated vortex cone is crushed due to surface tension. A root vortex and a coiled-up trailing edge vortex together form a compact Hill’s vortex when the highest CF is relieved, bursting into bubbles spectacularly scattered in the unstable sector. The DH vortex breakup is the connection point of vortices in proximity. At high CF, the DH has stacks of Taylor air tubes at $45^{\circ }$ to the external flow bursting when relaxed. Kelvin waves abound when bursting.

JFM classification

Vortex Flows: Vortex breakdown Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Elastic waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A27
Copyright
© United States Navy, 2020. This is a work of the U.S. Government and is not subject to copyright protection within the United States. Published by Cambridge University Press

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References

Adrian, R. J. & Marusic, I. 2012 Coherent structures in flow over hydraulic engineering surfaces. J. Hydraul. Res. 50 (5), 451–464.CrossRefGoogle Scholar
Adrian, R. J., Meinhart, C. D. & Tomkins, C. D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.CrossRefGoogle Scholar
Akilli, H., Sahin, B. & Rockwell, D. 2003 Control of vortex breakdown by a coaxial wire. Phys. Fluids 15 (1), 123–133.CrossRefGoogle Scholar
Andereck, C. D., Liu, S. S. & Swinney, H. L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Andreopolous, J. & Agui, J. 1996 Wall-vorticity flux dynamics in a two-dimensional turbulent boundary layer. J. Fluid Mech. 309, 45–84.CrossRefGoogle Scholar
Baker, C. J. 1979 The laminar horseshoe vortex. J. Fluid Mech. 95 (2), 347–367.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 1980 Large structure with a characteristic upstream interface in turbulent boundary layers. Phys. Fluids 23 (11), 2326–2327.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 1986 Aspects of the equilibrium puff in transitional pipe flow. J. Fluid Mech. 163, 439–458.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 1995 A low-dimensional model of an intermittently separating turbulent boundary layer. In ASME Symp. on Separated and Complex Flows, FED, vol. 217, pp. 75–82. American Society of Mechanical Engineers.Google Scholar
Bandyopadhyay, P. R. 2015 A novel large slosh-or-spin low-speed underwater propulsor bridges the unsteady and steady propulsion mechanisms of nature and engineering. IEEE J. Ocean. Engng 41 (4), 868–881.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 2016 Highly maneuverable biorobotic underwater vehicles – Part B: 11.1 Autonomous ocean vehicles, subsystems and control. In Springer Handbook of Ocean Engineering (ed. Dhanak, M. R. & Nikolas, X. I.), pp. 281–300. Springer.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 2019 A hemispherical motor oscillator for experiments on swimming and flying of small animals. Trans ASME J. Fluids Engng 141, 021102.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Beal, D. N., Hrubes, J. D. & Mangalam, A. 2012 Relationship of roll and pitch oscillations in a fin of low aspect ratio and rounded leading edge flapping at transitional to high Reynolds numbers. J. Fluid Mech. 702, 298–331.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Beal, D. N. & Menozzi, A. 2008a Biorobotic insights into how animals swim. J. Expl Biol. 211 (Pt 2), 206–214.CrossRefGoogle Scholar
Bandyopadhyay, P. R. & Gad-El-Hak, M. 1996 Rotating gas-liquid flows in finite cylinders: Sensitivity of standing vortices to end effects. Exp. Fluids 21, 124.CrossRefGoogle Scholar
Bandyopadhyay, P. R. & Hellum, A. M. 2014 Modeling how shark and dolphin skin patterns control transitional wall-turbulence vorticity patterns using spatiotemporal phase reset mechanisms. Sci. Rep. 4 (1), 6650.CrossRefGoogle ScholarPubMed
Bandyopadhyay, P. R. & Leinhos, H. A. 2013 Propulsion efficiency of bodies appended with multiple flapping fins – when more is less. Phys. Fluids 25, 041902.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Leinhos, H. A. & Hellum, A. M. 2013 Handedness helps homing in swimming and flying animals. Sci. Rep. 3, 1128.CrossRefGoogle ScholarPubMed
Bandyopadhyay, P. R., Leinhos, H. A., Hrubes, J. D., Toplosky, N. & Hansen, J. 2011 Turning of a short-length cable using flapping fin propulsion. IEEE J. Ocean. Engng 36 (4), 571–585.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Riester, J. E. & Ash, R. L. 1992 Helical-perturbation device for cylinder-wing vortex generators. AIAA J. 30 (4), 988–992.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Singh, S. N., Thivierge, D. P., Annaswamy, A. M., Leinhos, H. A., Fredette, A. R. & Beal, D. N. 2008b Synchronization of animal-inspired multiple fins in an underwater vehicle using olivo-cerebellar dynamics. IEEE J. Ocean. Engng 33 (4), 563–578.CrossRefGoogle Scholar
Bandyopadhyay, P. R., Stead, D. J. & Ash, R. L. 1991 Organized nature of a turbulent trailing vortex. AIAA J. 29 (10), 1627–1633.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1949 The nature of turbulent motion at large wave-numbers. Proc. R. Soc. Lond. A 199, 238–255.Google Scholar
Beal, D. N. & Bandyopadhyay, P. R. 2007 A harmonic model of hydrodynamic forces produced by a flapping fin. Exp. Fluids 43, 675–682.CrossRefGoogle Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593.CrossRefGoogle Scholar
Billant, P., Chomaz, J. & Huerre, P. 1998 Experimental study of vortex breakdown in swirling jets. J. Fluid Mech. 376, 183–219.CrossRefGoogle Scholar
Boersma, I. & Wood, D. H. 1999 On the self-induction of a helical vortex. J. Fluid Mech. 384, 263–279.CrossRefGoogle Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. J. Fluid Mech. 254, 447–466.CrossRefGoogle Scholar
Brown, G. L. & Lopez, R. M. 1990 Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech. 221, 553–576.CrossRefGoogle Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids 2 (5), 765.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.CrossRefGoogle Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (6), 532–540.CrossRefGoogle Scholar
Cuypers, Y., Maurel, A. & Petitjeans, P. 2003 Vortex burst as a source of turbulence. Phys. Rev. Lett. 91 (19), 194502.CrossRefGoogle ScholarPubMed
Currie, I. G. 1993 Fundamental Mechanics of Fluids, 2nd edn. McGraw-Hill.Google Scholar
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418, 839–844.CrossRefGoogle ScholarPubMed
Eggers, J. 2001 Air entrainment through free-surface cusps. Phys. Rev. Lett. 86 (19), 4290–4293.CrossRefGoogle ScholarPubMed
Einstein, H. A. & Li, H. 1958 The viscous sublayer along a smooth boundary. Trans. Am. Soc. Civil Engng 123, 293–313.Google Scholar
Ellington, C. P. 1999 The novel aerodynamics of insect flight: applications to micro-air vehicles. J. Expl Biol. 202 (23), 3439–3448.Google ScholarPubMed
Escudier, M. P. & Zehnder, N. 1982 Vortex-flow regimes. J. Fluid Mech. 115, 105–121.CrossRefGoogle Scholar
Faler, J. H. & Leibovich, S. L. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20 (9), 1385–1400.CrossRefGoogle Scholar
Fiedler, B. H. 2009 Suction vortices and spiral breakdown in numerical simulations of tornado-like vortices. Atmos. Sci. Lett. 10, 109–114.CrossRefGoogle Scholar
Fuentes, O. V. 2018 Motion of a helical vortex. J. Fluid Mech. 836, R1.CrossRefGoogle Scholar
Fuster, D. & Colonius, T. 2011 Modelling bubble clusters in compressible liquids. J. Fluid Mech. 688, 352–389.CrossRefGoogle Scholar
Gollub, J. P. & Sweeny, H. L. 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927.CrossRefGoogle Scholar
Gordon, J. L. 1970 Vortices at intakes. Water Power 22 (4), 137–138.Google Scholar
Haberman, W. L. & Morton, R. K.1953 An experimental investigation of the drag and shape of air bubbles rising in various liquids. Report 802, U.S. Navy Department: The David W. Taylor Model Basin.CrossRefGoogle Scholar
Hall, M. G. 1972 Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–218.CrossRefGoogle Scholar
Hama, F. R., Long, J. D. & Hegarty, J. C. 1957 On transition from laminar to turbulent flow. J. Appl. Phys. 28, 388–394.CrossRefGoogle Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.CrossRefGoogle Scholar
Heron, I. & Myose, R. Y. 2009 Delta wing vortex-burst behavior under a dynamic freestream. J. Aircraft 46 (5), 1500–1512.CrossRefGoogle Scholar
Hill, M. G. 1977 Vortex breakdown. Annu. Rev. Fluid Mech. 4, 195–218.CrossRefGoogle Scholar
Hite, J. E. & Mih, W. C. 1994 Velocity of air-core vortices at hydraulic intakes. J. Hydraul. Engng 120 (3), 284–297.CrossRefGoogle Scholar
Hopfinger, E. J., Browand, F. K. & Gagne, Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.CrossRefGoogle Scholar
Hugo, R. & Veer, C. 2003 An experimental investigation of vortex breakdown in multiphase pipe flow. In 33rd AIAA Fluid Dynamics Conference and Exhibit. American Institute of Aeronautics and Astronautics.Google Scholar
Humphreys, H. W., Sigurdsson, G. & Owen, J. H.1970 Model test results of circular, square, and rectangular forms of drop-inlet entrance to closed-conduit spillways. Illinois State Water Survey, Urbana, Rep. Investigation 65.Google Scholar
Joukowsky, N. E. 1912 Vihrevaja teorija grebnogo vinta. Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest. 16, 1–31 (French translation in Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929), 1–47).Google Scholar
Kazantsev, V. B., Nekorkin, V. I., Makarenko, V. & Llinas, R. 2004 Self-referential phase reset based on inferior olive oscillator dynamics. Proc. Natl Acad. Sci. USA 101 (52), 18183–18188.CrossRefGoogle ScholarPubMed
Kerr, R. M. & Hussain, F. 1989 Simulation of vortex reconnection. Physica D 37, 474–484.CrossRefGoogle Scholar
Khalil, H. K. 1996 Nonlinear Systems. Prentice-Hall.Google Scholar
Khoo, B. C., Yeo, K. S., Lim, D. F. & He, X. 1997 Vortex breakdown in an unconfined vortical flow. Exp. Therm. Fluid Sci. 14, 131–148.CrossRefGoogle Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Konno, A., Wakabayashi, K., Yamaguchi, H., Maeda, M., Ishii, N., Soejima, S. & Kimura, K. 2002 On the mechanism of the bursting phenomena of propeller tip vortex cavitation. J. Mar. Sci. Technol. 6, 181–192.CrossRefGoogle Scholar
Kuo, A. Y.-S. & Corrsin, S. 1971 Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J. Fluid Mech. 50, 285–319.CrossRefGoogle Scholar
Kway, J. H. L., Lo, Y.-S. & Chan, E.-S. 1998 Laboratory study of deep-water breaking waves. Ocean Engng 25 (8), 657–676.CrossRefGoogle Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 211–240.CrossRefGoogle Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.CrossRefGoogle Scholar
Lepercq-Bost, E., Giorgi, M., Isambert, A. & Arnaud, C. 2008 Use of the capillary number for the prediction of droplet size in membrane emulsification. J. Membr. Sci. 314, 76–89.CrossRefGoogle Scholar
Lighthill, M. J. 1963 Introduction (to) boundary layer theory. In Laminar Boundary Layers (ed. Rosenhead, L.), pp. 46–113. Dover Publishers.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lubin, P. & Glockner, S. 2015 Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments. J. Fluid Mech. 767, 364–393.CrossRefGoogle Scholar
Lucas, M. C., María, S. F., Patricia, P. S., Laura, M. F. & Fasce, A. 2015 Determination of elastic modulus of gelatin gels by indentation experiments. Procedia Materials Sci. 8, 287–296.Google Scholar
Luca-Negro, L. & O’Doherty, T. 2001 Vortex breakdown: a review. Prog. Energy Combust. Sci. 27 (4), 431–481.CrossRefGoogle Scholar
Lugt, H. J. 1989 Vortex breakdown in atmospheric columnar vortices. Bull. Am. Meteorol. Soc. 70, 1526–1537.2.0.CO;2>CrossRefGoogle Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193.CrossRefGoogle Scholar
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 1998 Fluid ‘rope trick’ investigated. Nature 392, 140.CrossRefGoogle Scholar
Medwin, H. & Beaky, M. M. 1989 Bubble sources of the Knudsen sea noise spectrum. J. Acoust. Soc. Am. 83, 1124–1130.CrossRefGoogle Scholar
Medwin, H., Nystuen, J. A., Jacobus, P. W., Ostwald, L. H. & Snyder, D. E. 1992 The anatomy of underwater rain noise. J. Acoust. Soc. Am. 92, 1613–1623.CrossRefGoogle Scholar
Menozzi, A., Leinhos, H., Beal, D. N. & Bandyopadhyay, P. R. 2008 Open-loop control of a multi-fin biorobotic underwater vehicle. IEEE J. Ocean. Engng 33 (2), 59–68.CrossRefGoogle Scholar
Misaka, T., Holzäpfel, F., Hennemann, I., Gerz, T., Manhart, M. & Schwertfirm, F. 2012 Vortex bursting and tracer transport of a counter-rotating vortex pair. Phys. Fluids 24, 025104.CrossRefGoogle Scholar
Moet, H., Laporte, F., Chevalier, G. & Poinsot, T. 2005 Wave propagation in vortices and vortex bursting. Phys. Fluids 17 (5), 054109.CrossRefGoogle Scholar
Moore, D. W. & Pullin, D. I. 1998 On steady compressible flows with compact vorticity; the compressible Hill’s spherical vortex. J. Fluid Mech. 374, 285–303.CrossRefGoogle Scholar
Mowbray, D. E. 1967 The use of schlieren and shadowgraph techniques in the study of flow patterns in density stratified liquids. J. Fluid Mech. 27, 595–608.CrossRefGoogle Scholar
Nolan, D. S. 2012 Three-dimensional instabilities in tornado-like vortices with secondary circulations. J. Fluid Mech. 711, 61–100.CrossRefGoogle Scholar
Nystuen, J. A.2002 Using ambient sound to passively monitor sea surface processes. In Keynote speech, Pan Ocean Remote Sensing Conference (PORSEC 2002), Bali, Indonesia 3–6 September 2002, special vol. 9–14.Google Scholar
Odgaard, A. J. 1986 Free-surface air core vortex. ASCE J. Hydraul. Engng 112, 610–620.CrossRefGoogle Scholar
Paterson, O., Wang, B. & Mao, X. 2018 Coherent structures in the breakdown bubble of a vortex flow. AIAA J. 56 (5), 1812–1817.CrossRefGoogle Scholar
Pauley, R. L. & Snow, J. T. 1988 On the kinematics and dynamics of the 18 July 1986 Minneapolis tornado. Mon. Weath. Rev. 116, 2731–2736.Google Scholar
Peckham, D. H. & Atkinson, S. A.1957 Preliminary results of low speed wind tunnel tests on a Gothic wing of aspect ratio 1.0. Aero. Res. Counc. Tech. Rep. CP-508, TN no. Aero. 2504.Google Scholar
Perry, A. E & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A. E., Lim, T. T. & Teh, E. W. 1981 A visual study of turbulent spots. J. Fluid Mech. 104, 387–405.CrossRefGoogle Scholar
Phillips, W. R. C. 1985 On vortex boundary layers. Proc. R. Soc. Lond. A 400, 253–261.Google Scholar
Pierrehumbert, R. & Widnall, S. 1982 The two-dimensional and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.CrossRefGoogle Scholar
Pipe, C. & Monkewitz, P. A.2005 Experiments investigating the effects of fluid elasticity on laminar vortex shedding from a cylinder. EPFL Lausanne. doi:10.5075/epfl-thesis-3347.CrossRefGoogle Scholar
Piquet, J. & Patel, V. C. 1999 Transverse curvature effects in turbulent boundary layer. Prog. Aerosp. Sci. 35 (7), 661–672.CrossRefGoogle Scholar
Polhamus, E. C.1966 A concept of the vortex lift of sharp-edge delta wings based on a leading-edge suction-analogy. NASA Tech. Note D-3767.Google Scholar
Prosperetti, A. 1988 Bubble-related ambient noise in the ocean. J. Acoust. Soc. Am. 84 (3), 1042–1054.CrossRefGoogle Scholar
Rahm, L.1953 Flow problems with respect to intakes and tunnels of Swedish hydroelectric power plants. Bulletin no. 36, Hydraulic Institute, Swedish Royal Institute of Technology, Stockholm.Google Scholar
Rotunno, R. 2013 The fluid dynamics of tornado. Annu. Rev. Fluid Mech. 45, 59–84.CrossRefGoogle Scholar
Ruppert-Felsot, J., Farge, M. & Petitjeans, P. 2009 Wavelet tools to study intermittency: application to vortex bursting. J. Fluid Mech. 636, 427–453.CrossRefGoogle Scholar
Saffman, P. G. 1990 A model of vortex reconnection. J. Fluid Mech. 212, 395–402.CrossRefGoogle Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.CrossRefGoogle Scholar
Sarpkaya, T.1995 Vortex breakdown and turbulence. In 33rd Aerospace Sciences Meeting. AIAA-95-0433. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Schatzl, P. R.1987 An experimental study of fusion of vortex rings. PhD thesis, California Institute of Technology, Pasadena, CA.Google Scholar
Sibulkin, M. 1962 A note on the bathtub vortex. J. Fluid Mech. 14 (1), 21–24.CrossRefGoogle Scholar
Smits, A. J. & Lim, T. T.(Eds) 2012 Flow Visualization: Techniques and Examples, 2nd edn. World Scientific.CrossRefGoogle Scholar
Spall, R. E. & Gatski, T. B. 1991 A computational study of the topology of vortex breakdown. Proc. R. Soc. Lond. A 435, 321–337.Google Scholar
Stevenson, T. N. 1968 Some two-dimensional internal waves in a stratified fluid. J. Fluid Mech. 33, 715–720.CrossRefGoogle Scholar
Stokes, G. G. 1880 Mathematical and Physical Papers, vol. 1. Cambridge University Press.Google Scholar
Talaia, M. A. R. 2007 Terminal velocity of a bubble rise in a liquid column. World Acad. Sci. Engng. Tech. 28, 264–268.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans R. Soc. Lond. A 223, 289–343; Containing Papers of a Mathematical or Physical Character.Google Scholar
Theodorsen, T. 1952 Mechanisms of turbulence. In Proceedings of the 2nd Midwestern Conference on Fluid Mechanics, Ohio State University, Columbus, Ohio. Ohio State University.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Trapp, R. J. 2000 A clarification of vortex breakdown and tornado genesis. Mon. Weath. Rev. 128, 888–895.2.0.CO;2>CrossRefGoogle Scholar
Tricoche, X., Garth, C., Kindlmann, G., Deines, E., Scheuermann, G., Ruetten, M. & Hansen, C. 2004 Visualization of intricate flow structures for vortex breakdown analysis. In IEEE Visualization ’04 Proceedings, pp. 187–194. Institute of Electrical and Electronics Engineers.CrossRefGoogle Scholar
Tritton, D. J. 1989 Physical Fluid Dynamics. Oxford Science Publications.Google Scholar
Tsoy, M. A., Skripkin, S. G., Kuibin, P. A., Shtork, S. I. & Alekseenko, S. V. 2018 Kelvin waves on helical vortex tube in swirling flows. IOP J. Phys.: Conf. Ser. 980, 012003.Google Scholar
Wakeling, J. M. & Ellington, C. P. 1997 Dragonfly flight. III. Lift and power requirements. J. Expl. Biol. 200, 583–600.Google ScholarPubMed
Werle, H.1984 On vortex bursting. NASA-TM-77587, NASA Technical Reports Server.Google Scholar
Wereley, S. T. & Lueptow, R. M. 1999 Velocity field for Taylor–Couette flow with an axial flow. Phys. Fluids 11, 3637–3649.CrossRefGoogle Scholar
White, F. M. 1994 Fluid Mechanics, 3rd edn. McGraw-Hill.Google Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat plate boundary layer. J. Fluid Mech. 630, 5–41.CrossRefGoogle Scholar
Yamada, K. & Suzuki, K. 2016 Dynamical system analysis on the onset of recirculating flow region associated with vortex breakdown. In 46th AIAA Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics.Google Scholar
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