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Evolution of an Elliptical Flow in Weakly Nonlinear Regime

  • Yuji Hattori
  • Yasuhide Fukumoto
  • Kaoru Fujimura
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 4)

Abstract

We study the nonlinear evolution of an elliptical flow by weakly nonlinear analysis. Two sets of amplitude equations are derived for different situations. First, the weakly nonlinear evolution of helical modes is considered. Nonlinear selfinteraction of the two base Kelvin waves results in cubic nonlinear terms, which causes saturation of the elliptical instability. Next, the case of triad interaction is considered. Three Kelvin waves, one of which is a helical mode, form a resonant triad thanks to freedom of wavenumber shift. As a result three-wave equations augmented with linear terms are obtained as amplitude equations. They explain the numerical results on the secondary instability obtained by Kerswell (1999).

Keywords

elliptical flow elliptical instability weakly nonlinear analysis amplitude equations secondary instability 

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Copyright information

© Springer 2008

Authors and Affiliations

  • Yuji Hattori
    • 1
  • Yasuhide Fukumoto
    • 2
  • Kaoru Fujimura
    • 3
  1. 1.Division of Computer Aided ScienceKyushu Institute of TechnologyKitakyushuJapan
  2. 2.Graduate School of Mathematics and Space Environment Research CenterKyushu UniversityFukuokaJapan
  3. 3.Department of Applied Mathematics and PhysicsTottori UniversityTottoriJapan

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