An assessment of the fast multipole method applied to a vortex sheet roll-up problem

Aerodynamic wakes of large aircraft and their induced velocity may cause serious hazards to smaller aircraft, affecting the take-off and landing operations in airports. A simplified but representative model of the problem is the evolution of a vortex sheet in the Trefftz plane. In this work, we employ the discrete vortex method (DVM) accelerated by the fast multipole method (FMM) to simulate this problem. Due to the cost associated with vortex clustering, the reduction in time from O(N2) to O(N) by the FMM is achieved when N is sufficiently large and a hierarchical multi-level multipole expansion is used. In the current paper, we show a study of the accuracy of the FMM compared to the Biot-Savart law when applied to the Trefftz plane simulation, which is shown to be a very sensitive non-linear problem where accuracy is a critical factor to avoid spurious instabilities. The analysis discusses the effects of time marching method, time step, level of refinement of the FMM, number of particles in the discretization and the balance between near and far-field computational costs. The DVM-FMM algorithm is also employed to solve the temporal evolution in a shear-layer with periodic boundary conditions.