A new quadrature for vortex sheet evolution

The motion of a vortex sheet can be effectively computed using trapezoidal integration combined with a blob regularization. Such a method experiences competing trade-offs between the blob parameter and the resolvable spatial scales and simulation run time. Here, a different method is proposed that eliminates the blob parameter by utilizing the evolution of the vortex sheet strength. This addresses the effects of curvature and sheet deformation with momentum conservation. The quadrature expression is obtained with two assumptions. First, a linear variation of the vortex sheet strength is used rather than a linear variation of the complete integrand (strength and kernel) as in the trapezoid rule. Second, the sheet is taken to be comprised of straight panels. A Kutta condition removes the logarithmic singularities in the local contribution to the principal value (i.e. self-induced velocity) as would be the case for an infinitely-resolved smooth sheet. It is hoped that this method will allow more detailed study of the small-scale structure of vortex cores.