一种基于流映射的欧拉涡旋方法
An Eulerian Vortex Method on Flow Maps
AI总结：
本文提出一种基于流映射的欧拉涡旋方法，通过引入线元素流映射输运方程和双向推进格式实现涡量高保真平流，并开发新的泊松求解方案处理边界耦合，在多种涡旋模拟中验证了有效性。
摘要：
我们提出了一种基于流映射理论的欧拉涡旋方法，用于模拟不可压缩流体的复杂涡旋运动。该方法的核心是创新性地引入线元素的流映射输运方程，并结合流映射的双向推进格式，实现了涡量变量的高保真欧拉平流。其根本动机在于，与最近已与流映射结合并取得令人鼓舞结果的冲量 $\mathbf{m}$ 相比，涡量 $\boldsymbol\omega$ 因其数值稳定性和物理可解释性而更具优势。为充分发挥这一新公式的潜力，我们开发了一种新的涡量到速度重构的泊松求解方案，该方案既高效又能准确处理固体边界附近的耦合。我们通过一系列涡旋模拟示例证明了该方法的有效性，包括蛙跳涡旋、涡旋碰撞、空腔流动以及由固流相互作用形成的复杂涡旋结构。
英文摘要：
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.
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