An Eulerian Vortex Method on Flow Maps

TL;DR

The paper addresses the challenge of building a robust, purely Eulerian vortex solver by evolving vorticity on long-range flow maps with a bi-directional transport framework. The core method uses vorticity as a line-element transported via flow-map Jacobians, coupled with a novel velocity reconstruction from a revised, boundary-aware velocity–vorticity Poisson solver implemented on GPUs. Key contributions include the flow-map-based vorticity advection, a matrix-free MGPCG solver that enforces solid boundaries without a streamfunction, and a detailed evaluation across 2D and 3D vortex phenomena demonstrating improved boundary handling, reduced divergence, and favorable performance. This work advances graphics-oriented vortex simulations by combining accurate flow-map advection with stable, interpretable vorticity dynamics and efficient implicit velocity solves, enabling complex solid-fluid interactions with reduced numerical dissipation.

Abstract

We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations for line elements, which, in combination with a bi-directional marching scheme for flow maps, enables the high-fidelity Eulerian advection of vorticity variables. The fundamental motivation is that, compared to impulse $\mathbf{m}$, which has been recently bridged with flow maps to encouraging results, vorticity $\boldsymbolω$ promises to be preferable for its numerical stability and physical interpretability. To realize the full potential of this novel formulation, we develop a new Poisson solving scheme for vorticity-to-velocity reconstruction that is both efficient and able to accurately handle the coupling near solid boundaries. We demonstrate the efficacy of our approach with a range of vortex simulation examples, including leapfrog vortices, vortex collisions, cavity flow, and the formation of complex vortical structures due to solid-fluid interactions.

Figures (19)

• Figure 1: The development of a vortex trefoil knot over time leads to its division into two vortex rings of varying sizes.
• Figure 2: The transformation of oblique vortex rings starts with two vortices colliding and merging into a single entity. It then undergoes several structural changes, eventually splitting into three distinct vortex rings.
• Figure 3: The picture depicts an eagle under the behavior of gliding (256 x 256 x 128). Vortices are created at the tail and back of the wings.
• Figure 4: Head-on vortex collision. Two vortex rings are positioned opposite to each other. They merge as one ring and expand upon collision, which ultimately leads to the ring breaking up into a series of smaller, radially arranged secondary vortices.
• Figure 5: Motivational Experiment. The increase factor comparison of the maximum norm of impulse and vorticity in leapfrog experiments. For impulse, the maximum increase factors in 100 steps are 20.35 and 3124.27 for 3D and 2D, respectively, while the maximum increase factors for vorticity are 1.22 and 1.01, indicating large instability induced by using impulse.