Anomalous dissipation in fluids, deterministic turbulence, and intermittency

流体中的反常耗散、确定性湍流和间歇性

• 批准号: 1210896
• 负责人: Roman Shvydkoy
• 金额: $ 30.84万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1210896&HistoricalAwards=false
• 关键词: Anomalous; dissipation; fluids; deterministic; turbulence

ShvydkoyDMS-1210896 The most pressing mathematical issues arising in fluid dynamics are becoming increasingly intertwined with the statistical theory of turbulence. The Navier-Stokes and Euler equations have served as the primary tools for numerical modelling, yet the very basic question of whether they can actually produce solutions satisfying laws of turbulence has not been settled. In recent years, with the advances of topological methods and better understanding of the nonlinear structure of the equations, this problem, commonly known as the Onsager conjecture, has come within our reach. The underlying theme of this project is to develop an analytical approach for studying statistical properties of deterministic solutions. The focus is on finding stationary energy dissipative weak solutions of the Euler and active scalar equations in the Onsager-critical regularity class, developing rigorous fractal analysis of the empirical concept of intermittency as a volumetric measure of non-uniformity of the Richardson cascade, excluding extreme deviations from the classical Kolmogorov laws as shown by the vast experimental data, and establishing links between intermittency and the global regularity problem. Turbulence is a complex chaotic process of fluid motion that commonly happens when a stirring force, like heat from the sun or an airplane cutting fast through air, injects a lot of energy into the system. This motion gets so complicated and rough that a part of the kinetic energy gets lost, creating what is known as anomalous dissipation. In our lives we can see this dissipation responsible, for example, for creation of additional drag in the turbulent wake of an airplane or our cars. In order to improve energy efficiency it is therefore essential to understand the mechanisms behind the process of anomalous energy loss and the reasons why it arises. In this project the investigator develops mathematical foundations of this aspect of turbulence based directly on the governing equations of fluid motion. A particular emphasis is given to studying non-uniformity of energy dissipation, which allows identifying regions where it occurs the most. An integral part of the project is developing the analytical and computational skills of graduate and undergraduate students.

流体动力学中出现的最紧迫的数学问题正日益与湍流统计理论交织在一起。纳维-斯托克斯方程和欧拉方程已经成为数值模拟的主要工具，但它们是否能真正产生满足湍流定律的解这一最基本的问题还没有解决。近年来，随着拓扑学方法的进步和对方程非线性结构的更好理解，这个通常被称为Onsager猜想的问题已经成为我们可以解决的问题。这个项目的基本主题是开发一种分析方法来研究确定性解决方案的统计特性。重点是在onsager临界正则类中寻找欧拉方程和有源标量方程的平稳能量耗散弱解，对间歇性的经验概念进行严格的分形分析，作为Richardson级联非均匀性的体积度量，排除大量实验数据所显示的与经典Kolmogorov定律的极端偏差，并建立间歇性与全局正则问题之间的联系。湍流是流体运动的一个复杂的混沌过程，通常发生在一个搅拌力，如来自太阳的热量或飞机快速穿过空气，向系统注入大量能量时。这种运动变得如此复杂和粗糙，以至于一部分动能丢失了，产生了所谓的反常耗散。在我们的生活中，我们可以看到这种耗散负责，例如，在飞机或汽车的湍流尾迹中产生额外的阻力。因此，为了提高能源效率，必须了解异常能量损失过程背后的机制及其产生的原因。在这个项目中，研究者直接基于流体运动的控制方程来发展湍流这方面的数学基础。特别强调的是研究能量耗散的非均匀性，这可以识别出它发生最多的区域。该项目的一个组成部分是培养研究生和本科生的分析和计算技能。