Geometry and Topology of Fluid Turbulence: Theory and Experiment

这项研究项目探索并实验测试了一种全新的数学框架，用于理解和预测科学、工程和医学中的许多基本和实际问题(例如，天气预报、心律失常的特征等)中的复杂行为。在许多这样的问题中，复杂的行为通常由短暂但重复出现的模式来管理。这项研究开发了识别和量化关键模式的通用、强大的技术，包括模式可能出现的时间序列；然后可以利用模式和序列的知识来构建预测未来行为的“路线图”。这项研究将侧重于通过构建实验室实验中观察到的湍流中复杂行为的路线图来演示“原则证明”。如果成功，这项研究的结果将直接导致开发出更快、更准确的方法来预测大型现实世界问题中的复杂行为。例如，识别和量化大气湍流中的重要模式和序列的能力应该能够使天气预报比目前可能的预报更好、更快。研究活动产生的所有软件和有用的解决方案数据将公之于众。该研究计划与本科生和研究生水平的教与学紧密结合，并包括增加未被充分代表的群体的参与的活动。该研究计划的主要目标是开发一种新的几何/拓扑方法来模拟和预测湍流，并在实验中对其进行验证。研究将集中在浅电解液流体层中的弱湍流。作为本程序的一部分，现有的数值方法和开发的新方法的组合将用于计算大量的不稳定状态(在流体动力学中称为精确相干态)以及这些状态之间的联系网络，这些网络由流动数学模型的数值精确解给出。在实验和模拟中将时间平均值与状态平均值进行比较，以验证周期轨道理论的统计预测。基于连接网络的拓扑的动力学的低维预测模型将类似地通过实验和模拟来验证。理解、预测和控制时空混沌动力学，特别是湍流动力学，在很大程度上是一个具有实际意义和基本意义的开放问题。将在该项目下开发和测试的几何/拓扑框架将为湍流流动提供一种新的降阶、预测性和动力学描述。该框架还将提供流体湍流的动力学描述和传统统计描述之间的联系。此外，这一框架应该作为一种全新的方法的基础，以控制广泛应用中的复杂动力机制。

This research project explores and experimentally tests a radically new mathematical framework for understanding and predicting complicated behaviors in numerous fundamental and practical problems in science, engineering, and medicine (e.g., weather forecasting, characterization of cardiac arrhythmias, etc.). Complex behaviors in many such problems are often governed by patterns that appear fleetingly but repeatedly. The research develops general, powerful techniques for identifying and quantifying key patterns, including the temporal sequences in which the patterns may appear; knowledge of the patterns and sequences can then be harnessed to construct "road maps" for predicting future behaviors. This study will focus on demonstrating "proof of principle" by constructing road maps of complex behavior observed in turbulent fluid flow in laboratory experiments. If successful, the results of this study will lead directly to the development of faster and more accurate ways to make predictions of complicated behavior in large real world problems. For example, the ability to identify and quantify important patterns and sequences in atmospheric turbulence should enable weather forecasts that are better and more rapid than those currently possible today. All software and useful solution data produced by the research activities will be made publicly available. The research program tightly integrates with teaching and learning at the undergraduate and graduate levels and includes activities to increase participation of underrepresented groups.The primary goal of this research program is to develop a novel geometrical/topological approach to modeling and prediction of turbulent flows and to validate it experimentally. Investigation will focus on a weakly turbulent flow in a shallow electrolyte fluid layer. A combination of existing numerical methods and new methods developed as a part of this program will be used to compute a large set of unstable states (known as exact coherent states in fluid dynamics) and the network of connections between these states, given by the numerically exact solutions of the mathematical model of the flow. Temporal averages will be compared with state averages in experiment and simulations to verify the statistical predictions of periodic orbit theory. A low-dimensional predictive model for the dynamics based on the topology of the network of connections will similarly be validated against experiment and simulations. Understanding, prediction, and control of spatiotemporally chaotic dynamics, in general, and of turbulent fluid flows, in particular, is largely an open problem of both practical and fundamental significance. The geometrical/topological framework that will be developed and tested under this project will provide a novel reduced-order, predictive, dynamical description of turbulent fluid flows. This framework will also provide a connection between the dynamical description and the conventional statistical description of fluid turbulence. In addition, this framework should serve as a foundation for a radically new way to control complex dynamical regimes in a wide range of applications.