A comparison of blob methods for vortex sheet roll-up

Abstract

The motion of vortex sheets is susceptible to the onset of the Kelvin–Helmholz instability. There is now a large body of evidence that the instability leads to the formation of a curvature singularity in finite time. Vortex blob methods provide a regularization for the motion of vortex sheets. Instead of forming a curvature singularity in finite time, the curves generated by vortex blob methods form spirals. Theory states that these spirals will converge to a classical weak solution of the Euler equations as the blob size vanishes. This theory assumes that the blob method is the result of a convolution of the sheet velocity with an appropriate choice of a smoothing function. We consider four different blob methods, two resulting from appropriate choices of smoothing functions and two not. Numerical results indicate that the curves generated by these methods form different spirals, but all approach the same weak limit as the blob size vanishes. By scaling distances and time appropriately with blob size, the family of spirals generated by different blob sizes collapse almost perfectly to a single spiral. This observation is the next step in developing an asymptotic theory to describe the nature of the weak solution in detail.