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Singularity formation in three-dimensional motion of a vortex sheet

Published online by Cambridge University Press:  26 April 2006

Takashi Ishihara  and
Yukio Kaneda
Takashi Ishihara
Affiliation:
Department of Applied Physics, Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan Present address: Department of Mathematics, Faculty of Science, Toyama University, Gofuku, Toyama 930, Japan.
Yukio Kaneda
Affiliation:
Department of Applied Physics, Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan
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Abstract

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 300 , 10 October 1995 , pp. 339 - 366
Copyright
© 1995 Cambridge University Press

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