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.

这个项目的重点是一个经典的主题，发现了新的相关性，由于最近的实验结果。19世纪世纪的研究人员，包括亥姆霍兹和基尔霍夫，写下了一个方程组，描述了点涡--无穷小的涡池--如何在流体中引起运动，这反过来又影响了涡的运动。还有一个类似的系统描述更复杂的涡环问题。 玻色-爱因斯坦凝聚体（BEC），一种在20世纪90年代首次实验产生的物质的奇异相，以及包括液氦在内的超流体，都可以支持点涡和涡环。最近的实验工作已经在BEC中成像了相互作用的点涡，显示出与数学预测的高度一致。本计画将从数学上探讨点涡相互作用中的许多现象，以及与之密切相关的相互作用涡环问题。特别是，有一种运动称为蛙跳轨道，其中一对涡环沿沿着一条线传播，在相互移动的同时周期性地膨胀和收缩（这是吸烟者用烟圈表演的众所周知的把戏）。这在两个涡环的情况下是很好理解的，并且计算机模拟已经将这种现象推广到三个或更多个环，但还没有进行完整的数学研究。该项目将对广义问题进行第一次详细的数学研究。很可能不可能写出这些解的精确公式，因此数值技术将是必不可少的，特别是数值延拓和分叉方法以及最近开发的称为拉格朗日描述符的方法，它使我们能够看到其他不透明混沌系统的结构。第二个问题将研究一种称为点涡混沌散射的现象：可以使一对涡以恒定速度沿着直线传播。这种对之间的碰撞会导致极其复杂的动力学，这将使用动力系统方法进行研究。本项目将把动力系统的技术应用于相互作用的点涡或涡环的动力学这一经典课题中的两类问题。第一个是蛙跳涡问题的几种概括，其中两对涡沿沿着直线运动，同时重复地和周期性地变宽，然后变窄，其中一对穿过另一对，并且导程改变。对于点涡，这一点已经得到了很好的研究，但对于涡环，研究得较少。该研究旨在将该系统推广到三对或更多对涡（或三个或更多涡环）的系统，以及在其他几何形状（如球体表面）中的蛙跳轨道。虽然有限的数值研究存在这样的轨道，他们仍然没有探索数学。哈密尔顿约化是我们理解四涡系统的核心技术，导致两个自由度的哈密尔顿算子，可以进一步分析简化。对于较大的涡集，哈密顿约化并不导致系统的足够小的维度的方法，以提供更多的见解，和其他方法是必需的，包括数值延拓和分叉方法。该项目还将利用新开发的拉格朗日描述符方法，尽管系统的大小意味着需要选择如何在初始条件空间中找到表面，这将导致对动态的深入了解。第二类问题是碰撞点涡对的混沌散射。这个系统有一些共同的特点与混沌散射研究在早期NSF赞助的研究：一个两个自由度的系统组成的一个缓慢的动力系统与一个分界线耦合到一个快速的动力系统。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。在以前的系统中，分界线描述了解决方案如何逃逸到无限，而逃逸的机制仍然是未知的。该奖项反映了NSF的法定使命，并通过使用基金会的知识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.