Instability of an elliptical flow: Weakly nonlinear analysis

Y. Hattori, Y. Fukumoto, K. Fujimura

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Elliptical flows are of interest since they appear in various vortical flows in which circular vortices are deformed by straining flow. 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 evolution of bending waves is considered. Nonlinear interaction of the two base Kelvin waves results in cubic nonlinear terms, which leads to saturation of the elliptical instability. Next, the secondary instability is considered. Three Kelvin waves, one of which is a bending wave, 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 previous numerical results on the secondary instability obtained by Kerswell J. Fluid Mech., 382, pp. 283-386, 1999.

Original languageEnglish
Title of host publicationFluid Structure Interaction and Moving Boundary Problems IV
Number of pages9
Publication statusPublished - 2007
Externally publishedYes
Event4th International Conference on Fluid Structure Interaction (Incorporating the Free and Moving Boundary Problems Seminar) - , United Kingdom
Duration: 2007 May 142007 May 16

Publication series

NameWIT Transactions on the Built Environment
ISSN (Print)1743-3509


Other4th International Conference on Fluid Structure Interaction (Incorporating the Free and Moving Boundary Problems Seminar)
Country/TerritoryUnited Kingdom


  • Elliptical flow
  • Secondary instability
  • Triad resonance
  • Weakly nonlinear analysis

ASJC Scopus subject areas

  • Architecture
  • Civil and Structural Engineering
  • Building and Construction
  • Automotive Engineering
  • Safety, Risk, Reliability and Quality
  • Arts and Humanities (miscellaneous)
  • Transportation
  • Safety Research
  • Computer Science Applications


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