Fluid Simulation on Vortex Particle Flow Maps

TL;DR

The paper introduces Vortex Particle Flow Map (VPFM), a novel hybrid Eulerian-Lagrangian method that simulates complex vortex dynamics with longer flow maps and improved vorticity preservation, especially near solid boundaries.

Contribution

It presents a new vorticity-based flow map framework with an accurate evolution scheme and boundary treatment, enabling significantly longer flow maps than existing methods.

Findings

Longer flow maps (3-12x) than state-of-the-art

Effective capture of vortex dynamics and turbulence

Enhanced vorticity preservation over extended domains

Abstract

We propose the Vortex Particle Flow Map (VPFM) method to simulate incompressible flow with complex vortical evolution in the presence of dynamic solid boundaries. The core insight of our approach is that vorticity is an ideal quantity for evolution on particle flow maps, enabling significantly longer flow map distances compared to other fluid quantities like velocity or impulse. To achieve this goal, we developed a hybrid Eulerian-Lagrangian representation that evolves vorticity and flow map quantities on vortex particles, while reconstructing velocity on a background grid. The method integrates three key components: (1) a vorticity-based particle flow map framework, (2) an accurate Hessian evolution scheme on particles, and (3) a solid boundary treatment for no-through and no-slip conditions in VPFM. These components collectively allow a substantially longer flow map length (3-12 times longer) than the state-of-the-art, enhancing vorticity preservation over extended spatiotemporal domains. We validated the performance of VPFM through diverse simulations, demonstrating its effectiveness in capturing complex vortex dynamics and turbulence phenomena.