@article{oai:nagoya.repo.nii.ac.jp:00008573,
 author = {ISHIHARA, TAKASHI and KANEDA, YUKIO},
 journal = {Journal of Fluid Mechanics Digital Archive},
 month = {Oct},
 note = {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+δmλ-2],where λ-1 and λ-2 are Langrangian 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 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 valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time t-c=O(1nε^{-1}) where ε is the amplitude of the initial disturbance. The singularity is such that A-n,-o=O(t-c^{-1}) behaves like n^{-5/2}, while A-n,±1=O(εt-c) behaves like n^{-3/2} for large n. The evolution of A-0,-m(spanwise mode) is also studied by an asysmptotic 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 A-0,-2 ∝K^{-5/2} for large K.},
 pages = {339--366},
 title = {Singularity formation in three-dimensional motion of a vortex sheet},
 volume = {300},
 year = {1995}
}