Spontaneous Singularity Formation in the Shape of Vortex Sheet in Three-Dimensional Flow

The three-dimensional motion of a flat uniform vortex sheet slightly perturbed initially in both streamwise and spanwise directions is studied on the basis of a Lagrangian representation of the motion. The vortex sheet at time t is represented as [formula ommited] where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. The behaviour of An,0 and An,1 is studied by an asymptotic analysis valid at large t. The analysis shows that singularity appears on the sheet at a critical time O (In ([formula omitted])), where ∊1 is the initial amplitude of the streamwise disturbance. The singularity is such that An,0 behaves like [formula omitted] while An,1 behaves like [formula omitted], for large n. The latter is associated with the vortex stretching intrinsic in three-dimensional dynamics of the sheet.