Representation of core dynamics in slender vortex filament simulations

The numerical description of slender vortex motion faces several major obstacles: (i) The stiffness induced by the rapid rotatory motion in the vortex core, where peak verticities are an order of magnitude larger than the filament velocity. In a vortioty-velocity formulation, this stiffness is reflected by the singular behavior of the line-Biot-Savart integral as one approaches the vortex geometry. Regularization occurs physically by viscous smoothing of the vorticity, (ii) The vortex core vorticity distribution has a crucial influence on the vortex filament motion. Thus, an accurate description of the core structure evolution due to vortex stretching and vorticity diffusion is necessary. We propose a numerical scheme that allows an accurate description of the effects of axial flow in the core, viscosity and vortex stretching on slender vortex filament motion. The approach is based on incorporating the detailed asymptotic analyses of the vortex core structure evolution by Callegari and Ting [SIAM J. Appl. Math. 15. 148 (1978)] and Klein and Ting [Appl. Math. Lett. 8. 45 (1995)] for stretched viscous slender vortices into the improved thin-tube vortex element schemes of Klein and Knio (1995). The resulting schemes overcome the difficulties mentioned above except for the issue of temporal stiffness, which we leave for future work.