A Case of Singularity Formation in Vortex Sheet Motion Studied by a Spectrally Accurate Method

Moore’s asymptotic analysis of vortex sheet motion predicts that the Kelvin-Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron, Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff-Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behavior of the spectrum, the calculations must be performed in very high precision. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the evolution of a large amplitude initial condition, the results indicate that Moore’s asymptotic analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs significantly away from that predicted by Moore. Convergence of the numerical solution beyond the singularity time is not observed.