Fast multipole method applied to Lagrangian simulations of vortical flows

Abstract

Lagrangian simulations of unsteady vortical flows are accelerated by the multi-level fast multipole method, FMM. The combination of the FMM algorithm with a discrete vortex method, DVM, is discussed for free domain and periodic problems with focus on implementation details to reduce numerical dissipation and avoid spurious solutions in unsteady inviscid flows. An assessment of the FMM-DVM accuracy is presented through a comparison with the direct calculation of the Biot-Savart law for the simulation of the temporal evolution of an aircraft wake in the Trefftz plane. The role of several parameters such as time step restriction, truncation of the FMM series expansion, number of particles in the wake discretization and machine precision is investigated and we show how to avoid spurious instabilities. The FMM-DVM is also applied to compute the evolution of a temporal shear layer with periodic boundary conditions. A novel approach is proposed to achieve accurate solutions in the periodic FMM. This approach avoids a spurious precession of the periodic shear layer and solutions are shown to converge to the direct Biot-Savart calculation using a cotangent function.

A fast algorithm for simulation of periodic flows using discrete vortex particles

“…The singular term of the cotangent series is solved using the FMM series expansion truncated with 40 and 75 terms for double and quadruple precision, respectively. These values guarantee machine precision [20]. In Fig.…”

“…Hence, these two methodologies are coupled to solve the approximation through the power series expansion. The methodology of the FMM used here is the standard one presented in the literature and the authors refer to the following references for details [10,17,20]. For particles with large separation along the yaxis, the exponential series expansion is employed.…”

“…One should remind that it is possible to avoid the singular values of the cotangent function (z jk ¼ np, for n 2 Z) using the Lamb-Oseen vortex with a viscous core r [20] as…”

“…These authors studied several aspects of the problem including the regularization of the kernel and the development of instabilities in the solution. Fast algorithms such as the FMM have been applied in combination with vortex methods for the solution of flows with periodic boundary conditions [7,20,[25][26][27]. In general, periodicity is implemented in the FMM through replication of the multipole expansions of the computational domain.…”

“…In general, periodicity is implemented in the FMM through replication of the multipole expansions of the computational domain. Ricciardi [20] presents an error analysis of such implementation for a vortex sheet roll-up problem. Other authors have also applied fast algorithms together with Lagrangian methods to solve problems in a different context, such as in Stokes flow simulations with periodic boundary conditions [9,16].…”