Experiments in Fluids

, Volume 41, Issue 3, pp 401–413 | Cite as

Experimental characterization of the instability of the vortex rings. Part II: Non-linear phase

  • Antoine Dazin
  • Patrick Dupont
  • Michel StanislasEmail author
Research Article


The first stage of the instability of a vortex ring is linear and characterized by the growth of an azimuthal stationary wave which develops around the ring. Theoretical works predict its origin, shape, number of waves and growth rate. Apart for the growth rate, experimental and numerical results in viscous fluids fit well with the predictions based on an ideal fluid hypothesis. On the other hand, the next stages of the development of the instability (which are non-linear) are not well known. Only few phenomena are described, in an isolated way, in various partial contributions. The aim of this paper is to report on a complete experimental investigation of the non-linear phase of the instability of the vortex ring. The vortices were produced in water and their Reynolds number Re p was varied from 2,650 to 6,100. Visualizations were performed using planar laser induced fluorescence and measurements with 2D2C and 2D3C particle image velocimetry. Based on a Fourier analysis of the results, it appears that the non-linear phase begins with the development of harmonics of the linear modes (first unstable modes). But the growth of those harmonics is rapidly stopped by the development of low order modes. Then appears an m=0 mode, which corresponds to a mean azimuthal velocity around the vortex. Simultaneously, secondary vortical structures develop all around the vortex in its peripheral zone. These vortical structures are linked with the ejection of vorticity in the wake of the ring and they appear just before the transition towards turbulence. A tentative is made here to place all these phenomena chronologically, in order to propose a scenario for the transition from the linear phase to turbulence.


Vortex Particle Image Velocimetry Vortex Ring Vortex Core Fourier Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



vortex core radius


effective core radius, defined by \({V}=\frac{\Gamma }{{4\pi {R}}}{\left({\ln \frac{{8{R}}}{{{a}_{\rm e} }} - \frac{1}{4}} \right)}\) (Saffman 1978)


inner core radius (distance from the core centre to the points where the tangential velocity is maximal) (Saffman 1978)


amplitude of an unstable mode


pipe inner diameter


distance between the laser sheet and the median plane of the vortex




piston stroke


number of unstable waves in the radial direction


number of unstable waves in the azimuthal direction


radial coordinate in the cylindrical coordinate system centred on the vortex ring


vortex radius


size of secondary structures

\(Re _{\rm p} = \frac{{D_{\rm p} U_{\rm p} }}{\nu }\)

Piston Reynolds number based on the piston velocity and the tube diameter

\(Re _{0} = \frac{{2RV}}{\nu }\)

Vortex Reynolds number based on the vortex velocity and diameter






average piston velocity


propagation speed of the ring




growth rate of the instability


circulation of the ring

Γ st

circulation of the secondary structures




azimuthal coordinate in the cylindrical coordinate system centred on the vortex ring

ρ st

distance of the secondary structure to the vortex core


phase of an unstable mode



radial component in the cylindrical coordinate system centred on the vortex ring


azimuthal component in the cylindrical coordinate system centred on the vortex ring


  1. Allen JJ, Auvity B (2002) Interaction of a vortex ring with a piston vortex. J Fluid Mech 465:353–378zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. Dazin A (2003) Caractérisation de l’instabilité du tourbillon torique par des méthodes optiques quantitatives. PhD thesis, Université des Sciences et Technologies de LilleGoogle Scholar
  3. Dazin A, Dupont P, Stanislas M (2006) Experimental characterization of the instability of the vortex ring. Part I: linear phase. Exp Fluids 40(3):383–399CrossRefGoogle Scholar
  4. Didden N (1977) Untersuchung laminärer, instabiler Ringwirbel mittels laser-Doppler-anemometer. Mitt MPI und AVA, Göttingen, Nr 64Google Scholar
  5. Foucaut JM, Miliat B, Perenne N, Stanislas M (2004) Characterization of different PIV algorithms using the EUROPIV synthetic image generator and real images from a turbulent boundary layer. In: Proceeding of the EUROPIV 2 workshop on particle image velocimetry. Springer, Berlin Heidelberg New York, pp 163–186Google Scholar
  6. Glezer A (1988) The formation of vortex rings. Phys Fluids A 31:3532–3542CrossRefADSGoogle Scholar
  7. Maxworthy T (1972) The structure and stability of vortex rings. J Fluid Mech 51:15–32CrossRefADSGoogle Scholar
  8. Maxworthy T (1977) Some experimental studies of vortex rings. J Fluid Mech 81:465–495CrossRefADSGoogle Scholar
  9. Naitoh T, Fukuda N, Gotoh T, Yamada H, Nakajima K (2002) Experimental study of axial flow in a vortex ring. Phys Fluids 14(1):143–148CrossRefADSGoogle Scholar
  10. Peck B, Sigurdson L (1995) The vortex ring velocity resulting from an impacting water drop. Exp fluids 18(5):351–357CrossRefGoogle Scholar
  11. Saffman PG (1978) The number of waves on unstable vortex rings. J Fluid Mech 84:721–733MathSciNetCrossRefGoogle Scholar
  12. Schneider PEM (1980) Sekundärerwirbelbildung bei Ringwirbeln und in Freistrahlen. Z FlugwissWeltraumforsch 4, 307–318Google Scholar
  13. Shariff K, Verzicco R, Orlandi P (1994) A numerical study of three-dimensional vortex ring instabilities: viscous correction and early non-linear stage. J Fluid Mech 279:351–375zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. Sullivan JP, Widnall SE, Ezekiel S (1973) Study of vortex rings using a laser Doppler velocimeter. AIAA J 11:1384–1389CrossRefGoogle Scholar
  15. Weigand A, Gharib M (1994) On the decay of a turbulent vortex ring. Phys Fluids 6(12):3806–3808CrossRefADSGoogle Scholar
  16. Widnall SE, Tsai CY (1977) The instability of the thin vortex ring of constant vorticity. Philos Trans Roy Soc Lond A 287:273–305zbMATHMathSciNetADSGoogle Scholar
  17. Widnall SE, Bliss DB, Tsai CY (1974) The instability of short waves on a vortex ring. J Fluid Mech 66:35–47zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Antoine Dazin
    • 1
  • Patrick Dupont
    • 1
  • Michel Stanislas
    • 1
    Email author
  1. 1.Laboratoire de Mécanique de Lille, UMR CNRS 8107Villeneuve d’Ascq CedexFrance

Personalised recommendations