Short-wavelength stability analysis of thin vortex rings

2003 ◽  
Vol 15 (10) ◽  
pp. 3151 ◽  
Author(s):  
Y. Hattori ◽  
Y. Fukumoto
Keyword(s):  
Stability Analysis ◽  
Short Wavelength ◽  
Vortex Rings

Short-wavelength stability analysis of a helical vortex tube

2009 ◽  
Vol 21 (1) ◽  
pp. 014104 ◽  
Author(s):  
Y. Hattori ◽  
Y. Fukumoto
Keyword(s):  
Stability Analysis ◽  
Short Wavelength ◽  
Vortex Tube ◽  
Helical Vortex

Short-wavelength stability analysis of Hill’s vortex with/without swirl

2010 ◽  
Vol 22 (7) ◽  
pp. 074104 ◽  
Author(s):  
Y. Hattori ◽  
K. Hijiya
Keyword(s):  
Stability Analysis ◽  
Short Wavelength

Evolution of localized disturbances in the elliptic instability

2014 ◽  
Vol 755 ◽  
pp. 603-627 ◽  
Author(s):  
Yuji Hattori ◽  
Mohd Syafiq bin Marzuki
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Energy Spectrum ◽  
Short Wavelength ◽  
Isotropic Turbulence ◽  
Inertial Range ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Elliptical Flow ◽  
The Mean

AbstractThe time evolution of localized disturbances in an elliptical flow confined in an elliptical cylinder is studied by direct numerical simulation (DNS). The base flow is subject to the elliptic instability. The unstable growth of localized disturbances predicted by the short-wavelength stability analysis is captured. The time evolution can be divided into four stages: linear, weakly nonlinear, nonlinear and turbulent. In the linear stage a single wavepacket grows exponentially without changing its shape. The exponential growth is accompanied by large oscillations which have time period half that of the fluid particles in the elliptical flow. An averaged wavepacket, which is a train of bending waves that has a finite spatial extent, also grows exponentially, while the oscillations of the growth rate are small. The averaged growth rate increases as the kinematic viscosity decreases; the inviscid limit is close to the value predicted by the short-wavelength stability analysis. In the weakly nonlinear stage the energy stops growing. The vortical structure of the initial disturbances is deformed into wavy patterns. The energy spectrum loses the peak at the initial wavenumber, developing a broad spectrum, and the flow goes into the next stage. In the nonlinear stage weak vorticity is scattered in the whole domain although strong vorticity is still localized. The probability density functions (p.d.f.) of a velocity component and its longitudinal derivative are similar to those of isotropic turbulence; however, the energy spectrum does not have an inertial range showing the Kolmogorov spectrum. Finally in the turbulent stage fine-scale structures appear in the vorticity field. The p.d.f. of the longitudinal derivative of velocity shows the strong intermittency known for isotropic turbulence. The energy spectrum attains an inertial range showing the Kolmogorov spectrum. The turbulence is not symmetric because of rotation and strain; the component of vorticity in the compressing direction is smaller than the other two components. The energy of the mean flow as well as the total energy decreases. The ratio of the lost energy to the initial energy of the mean flow is large in the core region.

scholarly journals Linear stability of inviscid vortex rings to axisymmetric perturbations

2019 ◽  
Vol 874 ◽  
pp. 1115-1146
Author(s):  
Bartosz Protas
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Equilibrium Configuration ◽  
Three Dimensional ◽  
Vortex Rings ◽  
Meridional Plane ◽  
The Family ◽  
A Free Boundary Problem ◽  
Shape Differentiation ◽  
Axisymmetric Perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (J. Fluid Mech., vol. 57, 1973, pp. 417–431). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ‘fat’. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine’s vortex and Hill’s vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.

A generalized treatment of the electrostatic instability of rippled non-neutral magnetically focused electron beams

1981 ◽  
Vol 26 (2) ◽  
pp. 267-281 ◽  
Author(s):  
S. Cuperman ◽  
F. Petran
Keyword(s):  
Stability Analysis ◽  
Surface Waves ◽  
Growth Rates ◽  
Short Wavelength ◽  
Electron Beams ◽  
Hill Equation ◽  
Long Wavelength ◽  
Azimuthal Symmetry ◽  
The Stability ◽  
Brillouin Flow

A generalization of the stability analysis of rippled non-neutral electron beams is presented. The treatment covers regimes ranging between those known as ‘Brillouin flow’ (ωpe = ωc/2½) and ‘immersed flow’ (ωpe≪ωc). It considers electrostatic surface waves with azimuthal symmetry (l = 0) as well as harmonic modes (l≠0). Both long-wavelength and short-wavelength domains are analysed. A general discussion of instability is given in terms of the solution of the Mathieu-Hill equation. New unstable modes are found and their growth rates are derived and compared.

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