### 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(λ_{1},λ_{2}, t) = (λ_{1},λ_{2},0) + ∑_{n, m} A_{n, m} exp[i(nλ_{1} + γmλ_{2})], where λ_{1} and λ_{2} are Lagrangian parameters in the streamwise and spanwise directions, respectively, and 5 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 A_{n, m}, The behaviour of A_{n, 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 t_{c} = O(lne ⋲^{-1}) where e is the amplitude of the initial disturbance. The singularity is such that A_{n,0} = 0(t_{c}^{-1}) behaves like n^{-5/2}, while A_{n,± 1} = O(⋲t_{c}) behaves like n^{-3/2} for large n. The evolution of A_{0, 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(⋲^{-1}) and the singularity is characterized by A_{0,2k} ∞ k^{-5/2} for large k.

Original language | English |
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Pages (from-to) | 339-366 |

Number of pages | 28 |

Journal | Journal of Fluid Mechanics |

Volume | 300 |

DOIs | |

Publication status | Published - Oct 1995 |

Externally published | Yes |

### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

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## Cite this

*Journal of Fluid Mechanics*,

*300*, 339-366. https://doi.org/10.1017/S0022112095003715