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Structure, Transport, and Chaos in Volume-Preserving Dynamics

体积保持动力学中的结构、传输和混沌

基本信息

批准号：
1211350
负责人：
James Meiss
金额：
$ 53.7万
依托单位：
University of Colorado at Boulder
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-09-01 至 2018-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211350&HistoricalAwards=false
关键词：
Structure Transport Chaos Volume Preserving

项目摘要

The persistence of quasiperiodic motion on codimension-one tori in nearly-integrable volume-preserving maps is explained by KAM theory. However, the robustness of these tori and the existence of remnants upon destruction are understood only in two-dimensions. The PI proposes to study tori of three-dimensional maps, and to generalize the residue criterion discovered by Greene and the anti-integrable limit discovered by Aubry. Studies will include symmetry reduction, invariance, and the loss of integrability for general, structure-preserving maps and flows. An important application is the optimization of mixing in open duct flows used in the continuous blending of materials. Our current understanding of the mixing process is, for the most part, limited to flows that are in essence two-dimensional and either closed or recycling. Transport in three-dimensional systems can be quantified by the flux through the destroyed structures, computed using a generalized action based on Lagrangian forms, thereby obtaining accurate and computationally efficient volume fluxes. The PI and students will use the concept of transitory dynamics to quantify and optimize transport in open flows. The extension to episodic and more general time-dependence will clarify the definition of Lagrangian coherent structures in aperiodic dynamics.The complexity of patterns obtained by mixing a passive scalar in a fluid can be observed by anyone pouring cream into hot coffee. That this process is not fully understood is perhaps less obvious. If the flow is sufficiently turbulent then mixing is rapid and uniformity is not hard to achieve. If, however, the flow is slow, on a small scale, or viscous, then mixing is much more difficult. Yet, such processes are important to many applications including the development of micrometer scale bioreactors and effective mixing of polymer and granular materials. A predictive theory for laminar mixing would also contribute to the understanding of climate modeling and pollution dispersal in the atmosphere as well as nutrient dispersal and spawning efficiencies for sea life. Mixing in laminar flows proceeds by stretching and folding due to chaotic motion that gives rise to fine-scale structure where diffusion is effective. Any measure of mixing requires quantification of chaos and its concomitant transport. Chaotic motion in incompressible fluids has some similarities to that in conservative dynamics. The later models are used to predict the lifetime of particles in accelerators, obtain rates for simple chemical reactions, calculate confinement times in plasma fusion devices, understand the spectra of highly excited atomic systems, and design efficient spacecraft trajectories. For chaotic dynamics, prediction of specific trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. In this study, chaos will be used to optimize mixing with the goal of obtaining practical designs for open, three-dimensional, mixing devices.

KAM 理论解释了近可积保体积图中余维一圆环上准周期运动的持续性。然而，这些花环的坚固性和破坏后残留物的存在只能在二维上被理解。 PI建议研究三维映射的环面，并推广Greene发现的留数准则和Aubry发现的反可积极限。研究将包括对称性降低、不变性以及一般结构保持地图和流的可积性损失。一个重要的应用是优化连续混合材料中使用的开放管道流中的混合。我们目前对混合过程的理解在很大程度上仅限于本质上是二维的、封闭的或回收的流动。三维系统中的传输可以通过通过被破坏结构的通量来量化，并使用基于拉格朗日形式的广义作用进行计算，从而获得准确且计算高效的体积通量。 PI 和学生将使用瞬态动力学的概念来量化和优化开放流中的运输。扩展到情景和更一般的时间依赖性将澄清非周期动力学中拉格朗日相干结构的定义。任何将奶油倒入热咖啡中的人都可以观察到通过在流体中混合被动标量获得的模式的复杂性。这个过程尚未被完全理解，这一点或许不太明显。如果流动足够湍流，则混合很快并且不难实现均匀性。然而，如果流动缓慢、规模小或粘稠，则混合会困难得多。然而，此类过程对于许多应用都很重要，包括微米级生物反应器的开发以及聚合物和颗粒材料的有效混合。层流混合的预测理论也将有助于理解气候模型和大气中的污染扩散，以及海洋生物的营养物扩散和产卵效率。层流中的混合通过混沌运动的拉伸和折叠进行，从而产生扩散有效的精细结构。任何混合测量都需要对混沌及其伴随的传输进行量化。不可压缩流体中的混沌运动与保守动力学中的混沌运动有一些相似之处。后面的模型用于预测加速器中粒子的寿命，获得简单化学反应的速率，计算等离子体聚变装置中的限制时间，了解高度激发原子系统的光谱，以及设计有效的航天器轨迹。对于混沌动力学来说，预测具体的轨迹是很困难的；然而，混沌可以被有利地利用，例如，通过明智地应用小航向修正来提高航天器轨迹的效率，或者提高限制装置中粒子的寿命和化学反应速率。在本研究中，混沌将用于优化混合，目的是获得开放式三维混合设备的实用设计。