A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method

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Abstract
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 behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.

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