### 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).

Original language | English |
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Title of host publication | IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence |

Pages | 433-438 |

Number of pages | 6 |

DOIs | |

Publication status | Published - 2008 Dec 1 |

Externally published | Yes |

Event | IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Nagoya, Japan Duration: 2006 Sep 11 → 2006 Sep 14 |

### Publication series

Name | Solid Mechanics and its Applications |
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Volume | 4 |

ISSN (Print) | 1875-3507 |

### Other

Other | IUTAM Symposium on Computational Physics and New Perspectives in Turbulence |
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Country | Japan |

City | Nagoya |

Period | 06/9/11 → 06/9/14 |

### Keywords

- Amplitude equations
- Elliptical flow
- Elliptical instability
- Secondary instability
- Weakly nonlinear analysis

### ASJC Scopus subject areas

- Civil and Structural Engineering
- Automotive Engineering
- Aerospace Engineering
- Acoustics and Ultrasonics
- Mechanical Engineering

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## Cite this

*IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence*(pp. 433-438). (Solid Mechanics and its Applications; Vol. 4). https://doi.org/10.1007/978-1-4020-6472-2_66