High-Precision Calculations of Vortex Sheet Motion

Abstract

The motion of a vortex sheet undergoing Kelvin-Helmholtz instability is known to be ill-posed, causing deterioration in numerical calculations from the rapid growth of round-off errors. In particular, it is the smallest scales (introduced by round-off) that grow the fastest. Krasny ([12]) introduced a spectral filter to suppress the growth of round-off errors of the smallest scales. He was then able to detect evidence supporting asymptotic studies that indicate the formation of a curvature singularity in finite time. We use high precision interval arithmetic, coded in C++, to re-examine the evolution of a vortex sheet from initial conditions used previously by several researchers. Most importantly, our results are free from the influence of round-off errors. We show excellent agreement between results obtained through high precision interval arithmetic and through the use of Krasny's spectral filter. In particular, our results support the formation of a curvature singularity in finite time, After the time of singularity formation, the markers move in peculiar patterns. We rule out any possibility of this motion resulting from round-off errors, but it does depend on the level of resolution. We find no consistent behavior in the motion of the markers as we improve the resolution of the vortex sheet. Also, we find some disagreement between the results obtained through high precision interval arithmetic and through the use of the spectral filter.