Singularity Formation in Three-Dimensional Motion of a Vortex Sheet

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} A_{n, 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 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.