Numerical study of a three-dimensional vortex method

A three-dimensional vortex method based on the discretization of the vorticity field into vortex vector elements of finite spherical cores is constructed for the simulation of inviscid incompressible flow. The velocity is obtained by summing the contribution of individual elements using the Biot-Savart law desingularized according to the vorticity cores. Vortex elements are transported in Lagrangian coordinates, and vorticity is redistributed, when necessary, among larger number of elements arranged along its direction. The accuracy and convergence of the method are investigated by comparing numerical solutions to analytical results on the propagation and stability of vortex rings. Accurate discretization of the initial vorticity field is shown to be essential for the prediction of the linear growth of azimuthal instability waves on vortex rings. The unstable mode frequency, growth rate and shape are in agreement with analytical results. The late stages of evolution of the instability show the generation of small scales in the form of bair-pin vortex structures. The behavior of the turbulent vortex ring is in good qualitative agreement with experimental data.