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Derivation and analysis of the analytical velocity and vortex stretching expressions for an O(N log N)-FMM

: Berdowski, T.; Walther, J.; Ferreira, C.; Meng, F.

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TU München; European Academy of Wind Energy -EAWE-:
6th International Conference "The Science of Making Torque from Wind", TORQUE 2016 : 5-7 October 2016, Munich, Germany; Proceedings
Bristol: IOP Publishing, 2016 (Journal of physics. Conference series 753)
Art. 082023, 12 pp.
International Conference "The Science of Making Torque from Wind" (TORQUE) <6, 2016, Munich>
Conference Paper, Electronic Publication
Fraunhofer IWES ()

In the current paper, a method for deriving the analytical expressions for the velocity and vortex stretching terms as a function of the spherical multipole expansion approximation of the vector potential is presented. These terms are essential in the context of 3D Lagrangian vortex particle methods combined with fast summation techniques. The convergence and computational efficiency of this approach is assessed in the framework of an O(N log N)-type Fast Multipole Method (FMM), by using vorticity particles to simulate a system of coaxial vortex rings for which also the exact results are known. It is found that the current implementation converges rapidly to the exact solution with increasing expansion order and acceptance factor. An investigation into the computational efficiency demonstrated that the O(N log N)-type FMM is already viable for a particle size of only several thousands and that this speedup increases significantly with the number of particles. Finally, i t is shown that the implementation of the FMM with the current analytical expressions is at least twice as fast as when opting for using even the simplest implementation of finite differences instead.