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Experimental characterization of the instability of the vortex ring. Part I: Linear phase

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Abstract

The results of experiments performed to study the linear phase of the instability of vortex rings are presented. The experiments were performed in water. The vortex rings are generated by pushing water through the cylindrical nozzle of a pipe submerged in an aquarium. The experiments were made with the help of planar laser induced fluorescence as well as 2D2C and 2D3C particle image velocimetry. They show the straining field causing the instability, and for the first time experimentally the growth of a band of linear unstable modes. They also confirm previous studies concerning the shape of the instability and theories predicting the number of waves and the bandwidth of unstable modes. However, the measurement of the growth rate shows the influence of viscous damping, and consequently, the limit of the theories based on the hypothesis of an ideal fluid.

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Abbreviations

a :

vortex core radius

a e :

effective core radius; defined by \(V = \frac{\Gamma}{{4\pi R}}{\left({\ln \frac{{8R}}{{a_{{\text{e}}}}} - \frac{1}{4}} \right)}\) (Saffman 1978)

a i :

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

A :

initial perturbation

D p :

pipe inner diameter

h :

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

k :

wavenumber

L p :

piston stroke

n :

number of unstable waves

r :

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

R :

vortex radius

\(Re_{{\text{p}}} = \frac{{D_{{\text{p}}} U_{{\text{p}}}}}{\nu}\) :

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

\(Re_{{\text{s}}} = \frac{{\sigma a^{2}_{{\text{i}}}}}{\nu}\) :

instability Reynolds number based on the intensity of the straining field

U p :

average piston velocity

u ρ :

radial velocity in the polar coordinate system centred on the vortex core

u ϕ :

azimuthal velocity in the polar coordinate system centred on the vortex core

V :

propagation speed of the ring

z c :

h/R

α:

growth rate of the instability

Γ:

circulation of the ring

ɛ=a/R :

core radius to diameter radius ratio

ν:

viscosity

ξ:

characteristic parameter of the hypergeometric profile defined by Saffman (1978)

θ:

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

ρ:

radial coordinate in the polar coordinate system centred on the vortex core

σ:

strength of the straining field

ϕ:

azimuthal coordinate in the polar coordinate system centred on the vortex core

Φ:

velocity potential

χ:

strength of the perturbation in the geometrical model

ω:

vorticity

0:

zero-order term in ɛ

1:

first-order term in ɛ

2:

second-order term in ɛ

exp:

related to the experiments presented in this paper

saf:

Saffman model (Saffman 1978)

wid:

Widnall model (Widnall and Tsai 1977)

cor:

viscous correction to Widnall model introduced by Shariff et al. (1994)

r :

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

θ:

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

ρ:

radial component in the polar coordinate system centred on the vortex core

φ:

azimuthal component in the polar coordinate system centred on the vortex core

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Authors and Affiliations

  1. Laboratoire de Mécanique de Lille, UMR CNRS 8107, Boulevard Paul Langevin, 59655, Villeneuve d’Ascq Cedex, France

    Antoine Dazin, Patrick Dupont & Michel Stanislas

Authors
  1. Antoine Dazin
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  2. Patrick Dupont
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  3. Michel Stanislas
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Correspondence to Michel Stanislas.

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Dazin, A., Dupont, P. & Stanislas, M. Experimental characterization of the instability of the vortex ring. Part I: Linear phase. Exp Fluids 40, 383–399 (2006). https://doi.org/10.1007/s00348-005-0075-8

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  • DOI: https://doi.org/10.1007/s00348-005-0075-8

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