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

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## Abstract

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.

## Keywords

Vortex Particle Image Velocimetry Vortex Ring Vortex Core Fourier Spectrum## Nomenclature

*a*vortex core radius

*a*_{e}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)

*a*_{i}inner core radius (distance from the core centre to the points where the tangential velocity is maximal) (Saffman 1978)

*A*amplitude of an unstable mode

*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 in the radial direction

*m*number of unstable waves in the azimuthal direction

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

*R*vortex radius

*r*_{st}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

*t*time

*u*velocity

*U*_{p}average piston velocity

*V*propagation speed of the ring

*z*_{c}*h*/*R*- α
growth rate of the instability

- Γ
circulation of the ring

- Γ
_{st} circulation of the secondary structures

- ν
viscosity

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

- ρ
_{st} distance of the secondary structure to the vortex core

- φ
phase of an unstable mode

## Subscript

*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

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