Geometry and Topology of Fluid Turbulence: Theory and Experiment

该研究项目探索并实验测试了一个全新的数学框架，用于理解和预测科学、工程和医学中许多基础和实际问题中的复杂行为（例如，天气预报、心律失常的表征等）。在许多这类问题中，复杂的行为通常是由短暂但反复出现的模式控制的。该研究开发了通用的、强大的技术来识别和量化关键模式，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”。本研究将着重于通过构建在实验室实验中观察到的湍流流体流动的复杂行为的路线图来展示“原理证明”。如果成功，这项研究的结果将直接导致更快、更准确的方法的发展，以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重要的模式和序列的能力应该使天气预报比目前可能的更好、更快。研究活动产生的所有软件和有用的解决方案数据都将公开提供。该研究项目与本科和研究生阶段的教学紧密结合，并包括增加代表性不足群体参与的活动。本研究计划的主要目标是发展一种新的几何/拓扑方法来建模和预测湍流，并进行实验验证。研究将集中在浅电解质流体层中的弱湍流。现有的数值方法和作为该计划一部分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状态）和这些状态之间的连接网络，这些状态由流动数学模型的数值精确解给出。将时间平均与状态平均在实验和模拟中进行比较，以验证周期轨道理论的统计预测。基于连接网络拓扑的动态低维预测模型也将通过实验和模拟进行验证。理解、预测和控制时空混沌动力学，一般来说，特别是湍流流体的流动，在很大程度上是一个既具有实际意义又具有根本意义的开放性问题。将在该项目下开发和测试的几何/拓扑框架将为湍流流体流动提供一种新颖的降阶、预测性和动态描述。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，该框架应作为在广泛应用中控制复杂动态机制的全新方法的基础。

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.