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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. Elliptical instability of a vortex tube and drift current induced by it. Yasuhide Fukumoto and Makoto Hirota

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

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  1. Turbulent Mixing and Beyond International Conference August 18-26, 2007 (Aug. 24) The Abdus Salam International Centre for Theoretical Physics (ICTP) Trieste, Italy Elliptical instability of a vortex tube and drift current induced by it Yasuhide Fukumoto and Makoto Hirota Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

  2. 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

  4. Short-wave Instability of trailing vortices (Leweke & Williamson ’98)

  5. Axial flow in a vortex ring Naitoh, Fukuda, Gotoh, Yamada & Nakajima (’02) cf. Maxworthy (’77)

  6. 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 Cairn’s 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

  7. Elliptically strained vortex

  8. Expand infinitesimal disturbance in Suppose that the core boundary is disturbed to the linearized Euler equations

  9. Example of a Kelvin wavem=4

  10. Dispersion relation of Kelvin waves m=-1 (solid lines) and m=1 (dashed lines)

  11. Equations for disturbance of

  12. Solution of disturbance of For the m wave, we find, from the Euler equations, and (radial wave numbers) Disturbance field is explicitely written out.

  13. Growth rate of helical waves(m=±1) Instability occurs at every intersection points of dispersion curves of (m, m+2) waves ???

  14. Krein’s theory of Hamiltonian spectra Spectra of a finte-dimensional Hamilton system

  15. Energy of a Kelvin wave disturbance base flow (averaged) Excess energy for generating a Kelvin wave (no strain) Kelvin wave stationary component ???

  16. Carins’ formula(Carins‘79)

  17. Energy of a helical wave (m=1)

  18. Energy signature of helical waves(m=±1) m=-1 (solid lines) and m=1 (dashed lines)

  19. Difficulty in Eulerian treatment disturbance base flow Excess energy Complicated calculation would be required for

  20. 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 Honan iso-vortical sheet (=w.r.t. kinematically accessible variations).

  21. Variational principle for stationaryvortical region ☆Volume preserving displacement of fluid particles: ☆Iso-vorticity: Then. using

  22. 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

  23. Wave energy in terms of iso-vortical disturbance Excess energy by Arnold’s theorem is the wave-energy It is proved that and that does not contribute to are lineardisturbances!!

  24. 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 !!

  25. Drift current caused by Kelvin waves Displacement vector of m wave Flow-flux, of m wave, in the axial direction

  26. Axial flow-flux of buldge wave (m=0),elliptic wave (m=2) 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 m=0 (dashed lines) and m=2 (solid lines)

  27. Axial flow-flux of a helical wave(m=1) For the principal mode　（＝stationary) • 2.505 • 4.349

  28. Axial current of staionary helical modes For stationary modes time average is not necessary : Given,

  29. 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. • Tsai & Widnall ('76) is simplified; Disturbance field and growth rate are written out in terms of the Bessel and modified Bessel functions. • Energetics: Energy of the Kelvin waves is calculated by adapting Cairns’ formula (= black box)consistent with Krein’s 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

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