Dynamics and scattering of vortices and vortex rings

涡流和涡环的动力学和散射

• 批准号: 2206016
• 负责人: Roy Goodman
• 金额: $ 30万
• 依托单位: New Jersey Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206016&HistoricalAwards=false
• 关键词: Dynamics; scattering; vortices; vortex; rings

This project focuses on a classical subject that has found new relevance due to recent experimental results. Nineteenth century researchers including Helmholtz and Kirchhoff wrote down a system of equations describing how point vortices---infinitesimal whirlpools---induce motion in a fluid, which in turn affects the motion of the vortices. A similar system exists that describes the more complex problem of vortex rings. Both Bose-Einstein Condensates (BEC), an exotic phase of matter first produced experimentally in the 1990s, and superfluids including liquid helium, can support point vortices and vortex rings. Recent experimental work has imaged interacting point vortices in a BEC, showing strong agreement with mathematical predictions. This project will mathematically investigate a number of phenomena in the interactions of point vortices, and in the closely-related problem of interacting vortex rings. In particular, there is a motion called the leapfrogging orbit in which a pair of vortex rings propagate along a line, periodically expanding and contracting while moving through each other (this is a well-known trick performed by cigar smokers with smoke rings). This is well understood in the case of two vortex rings, and computer simulations have generalized this phenomenon to three or more rings, but have not performed a full mathematical study. This project will perform the first detailed mathematical study of the generalized problem. It is likely impossible to write down exact formulas for these solutions, so numerical techniques will be essential, especially numerical continuation and bifurcation methods and a recently-developed method called Lagrangian descriptors, which allows us to see structure in otherwise opaque chaotic systems. A second problem will study a phenomenon called chaotic scattering of point vortices: A pair of vortices can be made to propagate at a constant speed along a straight line. Collisions between such pairs can lead to extremely complex dynamics which will be investigated using dynamical systems methods.This project will apply techniques from dynamical systems to two classes of problems in the classical subject of the dynamics of interacting point vortices or vortex rings. The first is several generalizations of the problem of leapfrogging vortices, in which two pairs of vortices travel along a straight line while repeatedly and periodically widening, then narrowing, with one pair passing through the other, and the lead changing. This has been very well-studied for point vortices, but less well for vortex rings. The study aims to generalize this system to systems of three or more pairs of vortices (or three or more vortex rings), and to leapfrogging orbits in other geometries such as on the surface of a sphere. While limited numerical studies exist of such orbits, they remain unexplored mathematically. Hamiltonian reduction is the technique at the center of our understanding of the four-vortex system, leading two a two-degree-of-freedom Hamiltonian that is amenable to further analytical simplification. With larger sets of vortices, Hamiltonian reduction does not lead to systems of small enough dimension for the method to provide much insight, and other methods are required, including numerical continuation and bifurcation methods. The project will also make use of the newly-developed method of Lagrangian descriptors, although the size of the system means that choices need to be made about how to find surfaces in initial-condition space that will lead allow insight into the dynamics. The second class of problems is the chaotic scattering of colliding point vortex pairs. This system has some features in common with chaotic scattering studied in earlier NSF-sponsored research: a two degree-of-freedom system consisting of a slow dynamical system with a separatrix coupled to a fast dynamical system. However, the dynamics in this case are far more complex: there is no small parameter. Instead, the separation of time scales arises due to the dynamics but may not hold for all time. In the previous system, a separatrix describes how solutions escape to infinity, while the mechanism of escape here is still unknown.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目侧重于一个经典主题，由于最近的实验结果，该主题发现了新的相关性。十九世纪的研究人员，包括亥姆霍兹和基尔霍夫，写下了一套方程组，描述了点涡流（无限小漩涡）如何引起流体运动，进而影响涡流的运动。存在一个类似的系统来描述更复杂的涡环问题。 玻色-爱因斯坦凝聚态 (BEC)（一种在 20 世纪 90 年代首次通过实验产生的奇异物质相）和包括液氦在内的超流体都可以支持点涡流和涡环。最近的实验工作对 BEC 中的相互作用点涡旋进行了成像，显示出与数学预测的高度一致。该项目将从数学角度研究点涡相互作用中的许多现象，以及密切相关的涡环相互作用问题。特别是，有一种称为“蛙跳轨道”的运动，其中一对涡环沿着一条线传播，在相互穿过时周期性地膨胀和收缩（这是吸烟者用烟环表演的众所周知的技巧）。这在两个涡环的情况下很好理解，计算机模拟已将这种现象推广到三个或更多环，但尚未进行完整的数学研究。该项目将对广义问题进行首次详细的数学研究。写出这些解决方案的精确公式可能是不可能的，因此数值技术至关重要，特别是数值连续和分岔方法以及最近开发的称为拉格朗日描述符的方法，它使我们能够看到不透明混沌系统中的结构。第二个问题将研究一种称为点涡旋混沌散射的现象：可以使一对涡旋沿直线以恒定速度传播。这些对之间的碰撞可能会导致极其复杂的动力学，将使用动力系统方法对其进行研究。该项目将应用动力系统技术来解决相互作用的点涡或涡环动力学的经典主题中的两类问题。第一个是跳跃涡问题的几种概括，其中两对涡沿着直线行进，同时反复地、周期性地变宽，然后变窄，其中一对涡穿过另一对，并且超前发生变化。这对于点涡旋已经得到了很好的研究，但对于涡环的研究还不够深入。该研究旨在将该系统推广到三对或更多对涡旋（或三个或更多涡环）的系统，以及其他几何形状（例如球体表面）的跨越轨道。虽然对此类轨道的数值研究有限，但它们仍未在数学上得到探索。哈密​​顿量约简是我们理解四涡系统的核心技术，导致两个二自由度哈密顿量可以进一步简化分析。对于较大的涡集，哈密顿约简不会导致系统的维数足够小，无法使该方法提供更多洞察力，并且需要其他方法，包括数值连续和分岔方法。该项目还将利用新开发的拉格朗日描述符方法，尽管系统的规模意味着需要选择如何在初始条件空间中找到表面，这将有助于深入了解动力学。第二类问题是碰撞点涡旋对的混沌散射。该系统与早期 NSF 资助的研究中研究的混沌散射具有一些共同特征：二自由度系统，由慢速动力系统和与快速动力系统耦合的分界线组成。然而，这种情况下的动态要复杂得多：没有小参数。相反，时间尺度的分离是由于动态而产生的，但可能不会一直保持。在之前的系统中，分界线描述了解决方案如何逃逸到无穷大，而这里的逃逸机制仍然未知。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Transition to instability of the leapfrogging vortex quartet

跨越式涡四重奏向不稳定的转变

• DOI: 10.1016/j.mechrescom.2023.104068
• 发表时间: 2023
• 期刊: Mechanics Research Communications
• 影响因子: 2.4
• 作者: Goodman, Roy H.;Behring, Brandon M.
• 通讯作者: Behring, Brandon M.