Evolution of an elliptical flow in weakly nonlinear regime

Yuji Hattori; Yasuhide Fukumoto; Kaoru Fujimura

doi:10.1007/978-1-4020-6472-2_66

Evolution of an elliptical flow in weakly nonlinear regime

Yuji Hattori, Yasuhide Fukumoto, Kaoru Fujimura

IFS - Complex Flow Research Division

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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
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	https://doi.org/10.1007/978-1-4020-6472-2_66
Publication status	Published - 2008
Event	IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Nagoya, Japan Duration: 2006 Sept 11 → 2006 Sept 14

Publication series

Name	Solid Mechanics and its Applications
Volume	4
ISSN (Print)	1875-3507

Conference

Conference	IUTAM Symposium on Computational Physics and New Perspectives in Turbulence
Country/Territory	Japan
City	Nagoya
Period	06/9/11 → 06/9/14

Keywords

Amplitude equations
Elliptical flow
Elliptical instability
Secondary instability
Weakly nonlinear analysis

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10.1007/978-1-4020-6472-2_66

Cite this

Hattori, Y., Fukumoto, Y., & Fujimura, K. (2008). Evolution of an elliptical flow in weakly nonlinear regime. In 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

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title = "Evolution of an elliptical flow in weakly nonlinear regime",

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 = "Amplitude equations, Elliptical flow, Elliptical instability, Secondary instability, Weakly nonlinear analysis",

author = "Yuji Hattori and Yasuhide Fukumoto and Kaoru Fujimura",

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doi = "10.1007/978-1-4020-6472-2_66",

language = "English",

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booktitle = "IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence",

note = "IUTAM Symposium on Computational Physics and New Perspectives in Turbulence ; Conference date: 11-09-2006 Through 14-09-2006",

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Hattori, Y, Fukumoto, Y & Fujimura, K 2008, Evolution of an elliptical flow in weakly nonlinear regime. in IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence. Solid Mechanics and its Applications, vol. 4, pp. 433-438, IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, Nagoya, Japan, 06/9/11. https://doi.org/10.1007/978-1-4020-6472-2_66

Evolution of an elliptical flow in weakly nonlinear regime. / Hattori, Yuji; Fukumoto, Yasuhide; Fujimura, Kaoru.
IUTAM Symposium on Computational Physics and New Perspectives in Turbulence - Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence. 2008. p. 433-438 (Solid Mechanics and its Applications; Vol. 4).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Evolution of an elliptical flow in weakly nonlinear regime

AU - Hattori, Yuji

AU - Fukumoto, Yasuhide

AU - Fujimura, Kaoru

PY - 2008

Y1 - 2008

N2 - 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).

AB - 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).

KW - Amplitude equations

KW - Elliptical flow

KW - Elliptical instability

KW - Secondary instability

KW - Weakly nonlinear analysis

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Evolution of an elliptical flow in weakly nonlinear regime

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