An Eulerian Vortex Method on Flow Maps

TL;DR

This paper introduces an Eulerian vortex method leveraging flow maps for high-fidelity simulation of complex vortical flows, emphasizing vorticity's advantages for stability and interpretability.

Contribution

The paper develops a novel Eulerian vortex method based on flow maps, including a new Poisson solver for vorticity-to-velocity reconstruction, improving accuracy and efficiency.

Findings

Accurately simulates leapfrog vortices and vortex collisions.

Effectively models cavity flow and solid-fluid interactions.

Demonstrates high fidelity in complex vortical structure formation.

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 $\boldsymbol{\omega}$ 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.