Two cylindrical vortex sheets : evolution and singularity formation

TL;DR: Using Rosenhead's point-vortex approximation with correction terms, the evolution of two symmetrical, counter-rotating, initially cylindrical vortex sheets in an incompressible, potential fluid is studied in this paper.

Abstract: Using Rosenhead’s point-vortex approximation with correction terms, the evolution of two symmetrical, counter-rotating, initially cylindrical vortex sheets in an incompressible, potential ‡uid ‡ow is studied Simulations are performed in time up to the occurrence of branch-point curvature singularities in the vortex sheets’geometries The numerical methods employed are discussed Parameters pertaining to the asymptotics of the Fourier coe¢ cients of the vortex sheets’positions are numerically …tted to gain insight into aspects of the singularity formation; these include the order of the branch-point singularities, and the times and locations of singularity formation A smoothing over initial singularity formations is implemented by either the heat equation or through a local application of the vortex blob method in an attempt to gain details into further singularity formations Lastly, the e¤ects of the initially prescribed total circulation around the vortex sheets on their evolutions are studied, both up to the time of singularity formation, and with the implementation of the vortex blob method, past the times of singularity formation