A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow

A particle method is presented for computing vortex sheet motion in three-dimensional flow. The particles representing the sheet are advected by a regularized Biot–Savart integral in which the exact singular kernel is replaced by the Rosenhead–Moore kernel. New particles are inserted to maintain resolution as the sheet rolls up. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor approximation in Cartesian coordinates, and the necessary Taylor coefficients are computed by a recurrence relation. The adaptive features include a divide-and-conquer evaluation strategy, nonuniform rectangular clusters, variable-order approximation, and a run-time choice between Taylor approximation and direct summation. Tests are performed to document the treecodeʹs accuracy and efficiency. The method is applied to simulate the roll-up of a circular-disk vortex sheet into a vortex ring. Two examples are presented, azimuthal waves on a vortex ring and the merger of two vortex rings.