Reviving the Vortex Particle Method: A Stable Formulation for Meshless Large Eddy Simulation

Abstract

The vortex particle method (VPM) is a mesh-free approach to computational fluid dynamics (CFD) solving the Navier-Stokes equations in their velocity-vorticity form. The VPM uses a Lagrangian scheme, which not only avoids the hurdles of mesh generation, but it also conserves vortical structures over long distances with minimal numerical dissipation while being orders of magnitude faster than conventional mesh-based CFD. However, VPM is known to be numerically unstable when vortical structures break down close to the turbulent regime. In this study, we reformulate the VPM as a large eddy simulation (LES) in a scheme that is numerically stable, without increasing its computational cost. A new set of VPM governing equations are derived from the LES-filtered Navier-Stokes equations. The new equations reinforce conservation of mass and angular momentum by reshaping the vortex elements subject to vortex stretching. In addition to the VPM reformulation, a new anisotropic dynamic model of subfilter-scale (SFS) vortex stretching is developed. This SFS model is well suited for turbulent flows with coherent vortical structures where the predominant cascade mechanism is vortex stretching. Advection, viscous diffusion, and vortex stretching are validated through simulation of isolated and leapfrogging vortex rings. Mean and fluctuating components of turbulent flow are validated through simulation of a turbulent round jet, where Reynolds stresses are resolved directly and compared to experimental measurements. Finally, the computational efficiency of the scheme is showcased in the simulation of an aircraft rotor in hover, showing our meshless LES to be 100x faster than a mesh-based LES with similar fidelity, while being 10x faster than a low-fidelity unsteady Reynolds-average Navier-Stokes simulation and 1000x faster than a high-fidelity detached-eddy simulation.