Proper evaluation of spherical harmonics-based expressions for the velocity and vortex stretching vectors in three-dimensional grid-free vortex methods

Abstract

We present expressions for approximating the velocity and vortex stretching vectors induced by a far-field collection of point vortices and we propose a set of recurrence relations to properly evaluate these expressions. Expressed as truncated series of spherical harmonics, these approximations are used in the context of O(Nlog⁡N) and O(N)-type fast multipole methods to reduce the computational cost of the advection step in three-dimensional grid-free vortex methods. These methods typically rely on operator splitting to handle diffusion and convection separately. In our implementation, the convection step employs a second order Runge-Kutta time integration scheme, where the particles velocities and vortex stretching vectors are computed using a fast multipole method that employs the proposed expressions. To model diffusion, we introduce an extension of the smoothed redistribution scheme to 3D unbounded flows. We check the accuracy of the expressions by inspecting the convergence of the velocity and the vortex stretching vectors as a function of the expansion order. The performance of the grid-free three-dimensional vortex method is assessed by simulating the collision of two vortex rings, over a long period of time, for different values of Reynolds number covering the range 500−2000. © 2020 Elsevier Inc.