An Eulerian Vortex Method on Flow Maps

Sinan Wang, Yitong Deng, Molin Deng, Hong-Xing Yu, Junwei Zhou, Duowen Chen, Taku Komura, Jiajun Wu, Bo Zhu
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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 numerically 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.
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流图上的欧拉旋涡法
我们提出了一种基于流图理论的欧拉旋涡方法,用于模拟不可压缩流体的复杂旋涡运动。我们方法的核心是新颖地加入了线元的流图传输方程,结合流图的双向行进方案,实现了涡度变量的高保真欧拉平流。其根本原因在于,与脉冲$\mathbf{m}$相比,涡度$\boldsymbol\{omega}$在数值稳定性和物理可解释性方面更胜一筹。为了充分发挥这种新颖形式的潜力,我们开发了一种新的泊松求解方案,用于涡度-速度重建,既高效又能准确处理固体边界附近的耦合。我们用一系列涡旋模拟实例证明了我们方法的有效性,包括跃迁涡旋、涡旋碰撞、空腔流以及固液相互作用形成的复杂涡旋结构。
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