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Experiments in Fluids

, 40:383 | Cite as

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

  • Antoine Dazin
  • Patrick Dupont
  • Michel Stanislas
Research Article

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.

Keywords

Vortex Particle Image Velocimetry Vortex Ring Vortex Core Unstable Mode 
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.

Nomenclature

a

vortex core radius

ae

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

ai

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

A

initial perturbation

Dp

pipe inner diameter

h

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

k

wavenumber

Lp

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

Up

average piston velocity

uρ

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

uϕ

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

V

propagation speed of the ring

zc

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

Subscripts

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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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Antoine Dazin
    • 1
  • Patrick Dupont
    • 1
  • Michel Stanislas
    • 1
  1. 1.Laboratoire de Mécanique de Lille, UMR CNRS 8107, Boulevard Paul LangevinVilleneuve d’Ascq CedexFrance

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