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.
|Number of pages||28|
|Journal||Journal of Fluid Mechanics|
|Publication status||Published - Oct 1995|
ASJC Scopus subject areas
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering