### 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 - Jan 1 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Fluid Mechanics*,

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

**Singularity Formation in Three-Dimensional Motion of a Vortex Sheet.** / Ishihara, Takashi; Kaneda, Yukio.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 300, pp. 339-366. https://doi.org/10.1017/S0022112095003715

}

TY - JOUR

T1 - Singularity Formation in Three-Dimensional Motion of a Vortex Sheet

AU - Ishihara, Takashi

AU - Kaneda, Yukio

PY - 1995/1/1

Y1 - 1995/1/1

N2 - 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) + ∑n, m An, m exp[i(nλ1 + γmλ2)], where λ1 and λ2 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 An, m, The behaviour of An, mis 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(lne ⋲-1) where e is the amplitude of the initial disturbance. The singularity is such that An,0 = 0(tc-1) behaves like n-5/2, while An,± 1 = O(⋲tc) 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(⋲-1) and the singularity is characterized by A0,2k ∞ k-5/2 for large k.

AB - 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) + ∑n, m An, m exp[i(nλ1 + γmλ2)], where λ1 and λ2 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 An, m, The behaviour of An, mis 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(lne ⋲-1) where e is the amplitude of the initial disturbance. The singularity is such that An,0 = 0(tc-1) behaves like n-5/2, while An,± 1 = O(⋲tc) 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(⋲-1) and the singularity is characterized by A0,2k ∞ k-5/2 for large k.

UR - http://www.scopus.com/inward/record.url?scp=0029390198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029390198&partnerID=8YFLogxK

U2 - 10.1017/S0022112095003715

DO - 10.1017/S0022112095003715

M3 - Article

AN - SCOPUS:0029390198

VL - 300

SP - 339

EP - 366

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -