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The decay of confined vortex rings

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Abstract

Vortex rings are produced during the ejection of fluid through a nozzle or orifice, which occurs in a wide range of biological conditions such as blood flow through the valves of the heart or through arterial constrictions. Confined vortex ring dynamics, such as these, have not been previously studied despite their occurrence within the biological flow conditions mentioned. In this work, we investigate laminar vortex rings using particle image velocimetry and develop a new semi-empirical model for the evolution of vortex ring circulation subject to confinement. Here we introduce a decay parameter β which exponentially grows with increasing vortex ring confinement ratio, the ratio of the vortex ring diameter (D VR) to the confinement diameter (D), with the relationship \(\beta=4.38 \exp(9.5D_{\rm VR}/D),\) resulting in a corresponding increase in the rate of vortex ring circulation decay. This work enables the prediction of circulation decay rate based on confinement, which is important to understanding naturally occurring confined vortex ring dynamics.

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References

  • Anderson EJ, Grosenbaugh MA (2005) Jet flow in steadily swimming adult squid. J Exp Biol 208(Pt 6):1125–1146

    Article  Google Scholar 

  • Chakraborty P, Balachandar S, Adrian RJ (2005) On the relationships between local vortex identification schemes. J Fluid Mech 535:189–214

    Article  MathSciNet  MATH  Google Scholar 

  • Chang TY, Hertzberg JR, Kerr RM (1997) Three dimensional vortex/wall interaction: entrainment in numerical simulation and experiment. Phys Fluids 9(1):57

    Article  Google Scholar 

  • Dabiri JO, Colin SP, Costello JH, Gharib M (2005) Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses. J Exp Biol 208(Pt 7):1257–1265

    Article  Google Scholar 

  • Didden N (1979) On the formation of vortex rings: rolling-up and production of circulation. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 30(1):101–116

    Article  Google Scholar 

  • Doligalski TL, Smith CR, Walker JDA (1994) Vortex interactions with walls. Annu Rev Fluid Mech 26(1):573–616

    Article  MathSciNet  Google Scholar 

  • Eckstein A, Vlachos PP (2009a) Assessment of advanced windowing techniques for DPIV. Meas Sci Technol 20:75,402

    Google Scholar 

  • Eckstein A, Vlachos PP (2009b) Digital particle image velocimetry (DPIV) robust phase correlation. Meas Sci Technol 20:55,401

    Google Scholar 

  • Eckstein AC, Charonko J, Vlachos P (2008) Phase correlation processing for DPIV measurements. Exp Fluids 45(3):485–500

    Article  Google Scholar 

  • Etebari A, Vlachos PP (2005) Improvements on the accuracy of derivative estimation from DPIV velocity measurements. Exp Fluids 39(6):1040–1050

    Article  Google Scholar 

  • Gharib M, Rambod E, Shariff K (1998) A universal time scale for vortex ring formation. J Fluid Mech 360:121–140

    Article  MathSciNet  MATH  Google Scholar 

  • Gharib M, Weigand A (2006) Experimental studies of vortex disconnection and connection at a free surface. J Fluid Mech 321:59

    Article  Google Scholar 

  • Glezer A (1988) The formation of vortex rings. Phys Fluids 31:3532

    Article  Google Scholar 

  • Glezer A, Coles D (1990) An experimental study of a turbulent vortex ring. J Fluid Mech 211:243–283

    Article  Google Scholar 

  • Hall MG (1972) Vortex breakdown. Annu Rev Fluid Mech 4(1):195–218

    Article  Google Scholar 

  • Harvey JK (1962) Some observations of the vortex breakdown phenomenon. J Fluid Mech 14:585–592

    Article  MATH  Google Scholar 

  • Helmholtz H (1867) On integrals of the hydrodynamical equations, which express vortex-motion. Lond Edinb Dublin Philos Mag J Sci 33(226):485–512

    Google Scholar 

  • Hershberger RE, Bolster D, Donnelly RJ (2010) Slowing of vortex rings by development of Kelvin waves. Phys Rev E 82(3):036309-1–036309-4

    Article  MathSciNet  Google Scholar 

  • Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285(1):69–94

    Article  MathSciNet  MATH  Google Scholar 

  • Kida S, Takaoka M, Hussain F (1989) Reconnection of two vortex rings. Phys Fluids A Fluid Dyn 1(4):630

    Article  Google Scholar 

  • Kilner PJ, Yang GZ, Wilkes AJ, Mohiaddin RH, Firmin DN, Yacoub MH (2000) Asymmetric redirection of flow through the heart. Nature 404(6779):759–761

    Article  Google Scholar 

  • Krueger PS (2008) Circulation and trajectories of vortex rings formed from tube and orifice openings. Phys D Nonlinear Phenom 237(14–17):2218–2222

    Article  MATH  Google Scholar 

  • Leonard A (1985) Computing three-dimensional incompressible flows with vortex elements. Annu Rev Fluid Mech 17(1):523–559

    Article  Google Scholar 

  • Lim TT (1989) An experimental study of a vortex ring interacting with an inclined wall. Exp Fluids 7(7):453–463

    Article  Google Scholar 

  • Luton JA, Ragab SA (1997) The three-dimensional interaction of a vortex pair with a wall. Phys Fluids 9:2967

    Article  MathSciNet  MATH  Google Scholar 

  • Maxworthy T (1972) The structure and stability of vortex rings. J Fluid Mech 51(01):15–32

    Article  Google Scholar 

  • Morton BR (1984) The generation and decay of vorticity. Geophys Astrophys Fluid Dyn 28(3):277–308

    Article  MathSciNet  MATH  Google Scholar 

  • Nielsen AH, He X, Juul Rasmussen J, Bohr T, Rasmussen JJ, Bohr T (1996) Vortex merging and spectral cascade in two-dimensional flows. Phys Fluids 8(9):2263–2265

    Article  MathSciNet  MATH  Google Scholar 

  • Ohring S, Lugt H (1991) Interaction of a viscous vortex pair with a free surface. J Fluid Mech 227:47–70

    Article  Google Scholar 

  • Orlandi P, Verzicco R (2006) Vortex rings impinging on walls: axisymmetric and three-dimensional simulations. J Fluid Mech 256:615–646

    Article  Google Scholar 

  • Oshima Y (1978) Head-on collision of two vortex rings. J Phys Soc Japan 44(1):328–331

    Article  Google Scholar 

  • Pullin DI (1978) The large-scale structure of unsteady self-similar rolled-up vortex sheets. J Fluid Mech 88(03):401–430

    Article  MathSciNet  MATH  Google Scholar 

  • Pullin DI (1979) Vortex ring formation at tube and orifice openings. Phys Fluids 22:401

    Article  Google Scholar 

  • Reynolds O (1876) On the resistance encountered by vortex rings and the relation between vortex rings and the stream-lines of a disc. Nature 14:477–479

    Google Scholar 

  • Rockwell D (1998) Vortex-body interactions. Annu Rev Fluid Mech 30(1):199–229

    Article  MathSciNet  Google Scholar 

  • Rosenfeld M, Rambod E, Gharib M (1998) Circulation and formation number of laminar vortex rings. J Fluid Mech 376:297–318

    Article  MathSciNet  MATH  Google Scholar 

  • Saffman PG (1978) The number of waves on unstable vortex rings. J Fluid Mech 81:625–639

    Google Scholar 

  • Saffman PG (1995) Vortex dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  • Scarano F (2002) Iterative image deformation methods in PIV. Meas Sci Technol 13:1–19

    Article  Google Scholar 

  • Shariff K, Lonard A (1992) Vortex Rings. Annu Rev Fluid Mech 24:235–279

  • Sirovich L (1987) Turbulence and the dynamics of coherent structures. I-Coherent structures. II-Symmetries and transformations. III-Dynamics and scaling. Q Appl Math 45:561–571

    MathSciNet  MATH  Google Scholar 

  • Squire HB (1960) Analysis of the vortex breakdown phenomenon. Aero Dept Imp Coll Lond Rep 102:306–312

    Google Scholar 

  • Stanaway SK, Cantwell BJ, Spalart PR (1988) A numerical study of viscous vortex rings using a spectral method. NASA STI/Recon Tech Rep N 89:23,820

  • Sullivan IS, Niemela JJ, Hershberger RE, Bolster D, Donnelly RJ (2008) Dynamics of thin vortex rings. J Fluid Mech 609:319–347

    Article  MathSciNet  MATH  Google Scholar 

  • Troolin DR, Longmire EK (2009) Volumetric velocity measurements of vortex rings from inclined exits. Exp Fluids 48(3):409–420

    Article  Google Scholar 

  • Walker JDA, Smith CR, Cerra AW, Doligalski TL (1987) The impact of a vortex ring on a wall. J Fluid Mech 181:99–140

    Article  Google Scholar 

  • Weigand A (1993) The response of a vortex ring to a transient, spatial cut. PhD thesis, University of California, San Diego

  • Weigand A, Gharib M (1994) On the decay of a turbulent vortex ring. Phys Fluids 6:3806

    Article  Google Scholar 

  • Weigand A, Gharib M (1997) On the evolution of laminar vortex rings. Exp Fluids 22(6):447–457

    Article  Google Scholar 

  • Zawadzki I, Aref H (1991) Mixing during vortex ring collision. Phys Fluids A Fluid Dyn 3(5):1405

    Article  Google Scholar 

  • Zhou J, Adrian RJ, Balachandar S, Kendall TM (1999) Mechanisms for generating coherent packets of hairpin vortices in channel flow. J Fluid Mech 387:353–396

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was partially supported by a National Science Foundation Graduate Research Fellowship as well as National Science Foundation Grant (0547434). Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    K. C. Stewart, C. L. Niebel & P. P. Vlachos

  2. Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA, 24061, USA

    S. Jung

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  1. K. C. Stewart
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  2. C. L. Niebel
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  3. S. Jung
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  4. P. P. Vlachos
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Correspondence to P. P. Vlachos.

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Stewart, K.C., Niebel, C.L., Jung, S. et al. The decay of confined vortex rings. Exp Fluids 53, 163–171 (2012). https://doi.org/10.1007/s00348-012-1277-5

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  • DOI: https://doi.org/10.1007/s00348-012-1277-5

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