Numerical simulation of vortex interactions using a fast multipole discrete particle method

Abstract

The discrete vortex method (DVM) is based on a Lagrangian description of the vorticity transport equation. In order to numerically solve the DVM, one can split the vorticity equation into separate diffusive and convective effects. Several formulations can be used to model the diffusive effect, e.g. The random walk method, the core spreading method and the velocity diffusion method. The convection effect can be treated using the material derivative to avoid the solution of a non-linear term; this is the major advantage of the method since each discrete vortex is convected with the fluid velocity field. However, the solution of the fluid velocity field requires the contributions from the incident flow, the perturbation due to the body and the particle interactions. The latter contribution is computationally expensive since the Biot-Savart law is used to compute the induced velocity by all discrete vortices in the cloud. The fast multipole method is an attractive algorithm used to accelerate the expensive interactions of the discrete vortices. It reduces the computational cost of the Biot-Savart law from O(N²) to O(N), where N is the number of discrete vortices in the cloud for a particular time step. The present FMM algorithm is based on the original ideas of Greengard and Rokhlin, with modifications to further accelerate the solution. In the present work, both FMM and Biot-Savart law solutions are compared by calculating the vortex-vortex interactions for different cylinder wakes, previously generated by a DVM. The present numerical tool will be used in future computational simulations of aerodynamic flows past airfoils in pitching and plunging motions and vortex induced vibration problems.