Fluid Simulation on Vortex Particle Flow Maps

TL;DR

VPFM presents a hybrid vortex–particle flow map framework that advances long-term vorticity preservation by evolving vorticity, flow-map Jacobians, and Hessians on moving particles while reconstructing velocity on a background grid. The method introduces a novel Hessian evolution, a SPSD cut-cell approach for no-through boundaries, and a simplified Brinkmann penalization to approximate no-slip, enabling flow maps 3–12× longer than prior state-of-the-art and up to ~30× longer stability in 3D benchmarks. Validation across 2D/3D tests, including Hopf link, trefoil knot, vortex rings, and flows around complex geometries, demonstrates improved vorticity preservation, reduced numerical dissipation, and faithful boundary behavior. The work revitalizes VIC-type methods for graphics applications by leveraging flow-map theory, providing a robust, scalable path for simulating intricate vortex dynamics with dynamic solid boundaries. Potential impact includes more accurate, visually compelling fluid animations in graphics pipelines and improved physical fidelity in vortex-dominated flows, with open directions toward full harmonic dynamics, free-surface handling, and two-way coupling.

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.

Figures (32)

• Figure 1: A plesiosaur propels through water by flapping its flippers. The top left images display the vorticity of the fluid during the plesiosaur's movement. Bubbles are generated around each of the four flippers of the plesiosaur. The bottom left images show side views of the bubble flow during the plesiosaur's movement, while the four images on the right provide a top-down view.
• Figure 2: Fluid flows upward from directly beneath the head of Michelangelo's David. The two images on the left illustrate the vorticity as the fluid moves past the sculpture. Bubbles are generated around the base of the sculpture and at the base of the hair, and the two images on the right depict the flow of these bubbles past the sculpture.
• Figure 3: Illustration of VPFM (left) and the adaptive flow map (right).
• Figure 3: Summary of 2D and 3D leapfrog explosion times under a challenging flow map length (240 for 2D and 100 for 3D). The Hessian term vanishes in 2D (see Section \ref{['para:vpfm_2D']}).
• Figure 4: The propeller rotates, with the inflow passing from left to right. The images on the left depict the propeller rotating counterclockwise, while those on the right show clockwise rotation. The upper images on both sides illustrate the fluid vorticity during the propeller's motion, and the lower images display the bubbles generated by the propeller's rotation. Notably, a spiral vortex is formed during counterclockwise rotation, whereas the clockwise rotation generates turbulence phenomenon.