Elliptical instability of a vortex tube and drift current induced by it

1
Elliptical instability of a vortex tube and drift
current induced by it
Turbulent Mixing and Beyond International
Conference August 18-26, 2007 (Aug. 24) The
Abdus Salam International Centre for Theoretical
Physics (ICTP) Trieste, Italy

• Yasuhide Fukumoto
• and
• Makoto Hirota
• Graduate School of Mathematics,
• Kyushu University, Fukuoka, Japan

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Aircraft trailing vortices (Higuchi 1993)
Cessna Citation IV from B25
3
Instability of trailing vortices (Crow 1970)
from Van Dyke An Album of Fluid Motion
B-47
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Short-wave Instability of trailing vortices
(Leweke Williamson 98)
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Axial flow in a vortex ring
Naitoh, Fukuda, Gotoh, Yamada Nakajima (02)
cf. Maxworthy (77)
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Contents

• 1. Introduction
• 2. Influence of a pure shear on Kelvin waves
• A global stability of the Rankine vortex to
  three-dimensional disturbances"
• Moore Saffman ('75), Tsai Widnall
  ('76)
• Eloy Le Dizès ('01), Y. F. ('03)
• elliptical instability (local
  stability)
• cf. vortex ring Hattori Y. F.
  ('03), Y. F. Hattori ('05)
• 3. Energy of Kelvin waves
• Cairns formula (79), Y. F.
  ('03),
• Hirota Y. F. ('07) for
  continuous spectra
• 4. Weakly nonlinear corrections to Kelvin waves
• kinematically accessible variations (
  isovortical perturbations)
• ? drift current

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Elliptically strained vortex
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Expand infinitesimal disturbance in
Suppose that the core boundary is disturbed to
the linearized Euler equations
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Example of a Kelvin wave m4
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Dispersion relation of Kelvin waves
m-1 (solid lines) and m1 (dashed lines)
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Equations for disturbance of
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Solution of disturbance of
For the m wave, we find, from the Euler equations,
and
(radial wave numbers)
Disturbance field is explicitely written
out.
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Growth rate of helical waves (m1)
Instability occurs at every intersection points
of dispersion curves of (m, m2) waves ???
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Kreins theory of Hamiltonian spectra
Spectra of a finte-dimensional Hamilton system
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Energy of a Kelvin wave
disturbance
base flow
(averaged) Excess energy for generating a Kelvin
wave
(no strain)
Kelvin wave
stationary component ???
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Carins formula (Carins 79)
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Energy of a helical wave (m1)
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Energy signature of helical waves (m1)
m-1 (solid lines) and m1 (dashed lines)
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Difficulty in Eulerian treatment
disturbance
base flow
Excess energy
Complicated calculation would be required for
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Steady Euler flows
Kinematically accessible variation (
preservation of circulation)
iso-vortical sheets
Theorem (Kelvin, Arnold 66) A steady Euler
flow is a coditional extremum of energy H on
an iso-vortical sheet ( w.r.t. kinematically
accessible variations).
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Variational principle for stationary vortical
region
? Volume preserving displacement of fluid
particles
? Iso-vorticity
Then. using
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First and second variations
The first variation
( projection operator )
Given which satisfies
is a solution of
Then
The secobd variation
Further, given which satisfies
Then
is a solution of
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Wave energy in terms of iso-vortical disturbance
Excess energy
by Arnolds theorem
is the wave-energy
It is proved that and that
does not contribute to
are linear disturbances!!
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Drift current
Take the average over a long time
For the Rankine vortex
Substitute the Kelvin wave

• There is no contribution from
• For 2D wave,

genuinly 3D effect !!
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Drift current caused by Kelvin waves
Displacement vector of m wave
Flow-flux, of m wave, in the axial direction
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Axial flow-flux of buldge wave (m0),
elliptic wave (m2)
For the principal mode,
Dispersion relation

• 1.242, -1.242
• 3.370, -0.2443
• 7.058, -0.09046
• 8.882, -0.06828
• 12.521, -0.04564

m0 (dashed lines) and m2 (solid lines)
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Axial flow-flux of a helical wave (m1)
For the principal mode (stationary)

• 2.505
• 4.349

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Axial current of staionary helical modes
For stationary modes time average is not
necessary
Given,
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Summary
Linear stability of an elliptic vortex, a
straight vortex tube subject to a pure shear, to
three-dimensional disturbances is
calculated. This is a parametric resonance
instability between two Kelvin waves caused by a
perturbation breaking S-symmetry of the circular
core.

1. Tsai Widnall ('76) is simplified
   Disturbance field and growth rate are written out
   in terms of the Bessel and modified Bessel
   functions.
2. Energetics Energy of the Kelvin waves is
   calculated by adapting Cairns formula ( black
   box) consistent with Kreins theory

Modification of mean field at 2 nd order
3. Lagrangian approach Energy of the Kelvin
waves is calculated by restricting disturbance to
kinematically accessible field linear
perturbation is sufficient to calcilate energy,
quadratic in amplitude! 4. Axial current For
the Rankine vortex, 2 nd-order drift current
includes not only azimuthal but also axial
component