Coherent Structure, Chaos, and Turbulence in Fluid Mechanics

流体力学中的相干结构、混沌和湍流

基本信息

批准号：
2108633
负责人：
Jacob Bedrossian
金额：
$ 36.48万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2023-10-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108633&HistoricalAwards=false
关键词：
Coherent Structure Chaos Turbulence Fluid

项目摘要

Fluids that we interact with on a daily basis (such as water and air) display a remarkable variety of dynamics. At one extreme is chaos and turbulence, wherein the fluid forms complicated fractal-like patterns, the details of which cannot be exactly repeated in experiments due to being very sensitive to small changes in the initial conditions. Turbulence is observed often in fluids, such as in the wakes of aircraft, vehicles, and even obstacles such as buildings and bridges and in the large-scale dynamics of the ocean and atmosphere. At the opposite extreme is the motion of vortex filaments, the most notable examples being tornadoes and the wing-tip vortices commonly shed from wings and helicopter blades. Vortex filaments tend to move in a predictable manner and maintain their structural integrity for extended periods of time. The goal of this project is to develop a better mathematical understanding of these opposite phenomena in fluid mechanics. The accurate prediction of vortex filaments and turbulence is crucial in a variety of scientific and industrial applications, including in the design of air, land, and sea vehicles and in the understanding of complex fluid systems such as the climate and weather. Having a firm mathematical foundation could help other applied researchers obtain deeper insights and lead to better modeling. Further, an understanding of these extremes helps pave the way for a better understanding of the interactions and intermediate regimes, where flows have a mix of structure and chaos. Finally, overcoming the mathematical challenges to these questions will require innovations that will be of interest to the wider mathematical community. The research projects are also integrated with the training of graduate students and younger scientists in mathematics and STEM. The PI will develop a more mathematically rigorous understanding of two behaviors observed in incompressible fluids at high Reynolds numbers: (1) the coherent motion of vortex filaments; (2) turbulence under "generic", statistically steady forcing. The fundamental questions motivating the PI are: (A) how accurate are the commonly used geometric evolution models such as the Local Induction Approximation (LIA) for the motion of a vortex filament in a fluid with vorticity concentrated on a smooth curve? (B) can we provide a proof for the experimentally observed positive Lyapunov exponents and anomalous dissipation (e.g., as the celebrated Kolmogorov 4/5 law) from the stochastically forced 3D Navier-Stokes in the high Reynolds number limit? These require a number of unexplored mathematical ideas and currently, there exists no clear way to attack them yet. Instead, the PI has identified several independently interesting problems to build necessary mathematical foundations. For (A), the PI and his collaborators will study vortex filaments in quantum fluids governed by the Gross-Pitaevskii equation. The quantum case is expected to be easier than the classical case due to the quantization of vorticity and slightly more amenable linearized operators. First, the PI and collaborators will study the stability of vortex solutions in 2d Gross-Pitaevskii, providing necessary ground for understanding the filament core. Next, the PI and collaborators will show that the LIA accurately describes the motion of nearly-straight, and potentially more general, quantum vortex filaments long enough to make useful predictions. For (B), the PI and his collaborators will: (1) develop novel qualitative theory for Lyapunov exponents in stochastic PDEs inspired by ideas from random dynamical systems; (2) develop better tools for quantitative hypoelliptic regularity and Lyapunov exponent estimation in high dimensional systems; and (3) study quantitative and nonlinear aspects of Lagrangian chaos, that is, how the chaotic dynamics of particles in a fluid can be translated into nonlinear dynamics of the fluid itself.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.

我们每天与之相互作用的流体（例如水和空气）表现出了不同的动态。在一个极端是混乱和湍流，其中流体形成复杂的分形图案，由于对初始条件中的小变化非常敏感，因此在实验中无法精确地重复其细节。经常在流体中观察到湍流，例如飞机，车辆，甚至是建筑物和桥梁等障碍物，以及海洋和大气的大规模动态。相反的极端是涡旋细丝的运动，最值得注意的例子是龙卷风和翼尖涡流通常是从机翼和直升机叶片中脱落的。涡旋细丝倾向于以可预测的方式移动，并长时间保持其结构完整性。该项目的目的是对流体力学中这些相反现象有更好的数学理解。在各种科学和工业应用中，包括空气，陆地和海上车辆的设计以及对复杂的流体系统（如气候和天气）的理解，对涡流和湍流的准确预测至关重要。拥有坚定的数学基础可以帮助其他应用研究人员获得更深入的见解，并带来更好的建模。此外，对这些极端的理解有助于为更好地理解相互作用和中级政权铺平道路，其中流动混合了结构和混乱。最后，克服这些问题的数学挑战将需要更广泛的数学社区感兴趣的创新。研究项目还与研究生和年轻科学家的数学和STEM进行培训。 PI将对在高雷诺数字上不可压缩的流体中观察到的两种行为产生更严格的了解：（1）涡旋丝的相干运动；（2）“通用”下的湍流，统计上稳定的强迫。激发PI的基本问题是：（a）常用的几何进化模型（例如局部诱导近似（LIA））在涡流中涡流丝运动的运动集中在平滑曲线上？（b）我们是否可以为实验观察到的积极的Lyapunov指数和异常耗散（例如，作为著名的kolmogorov 4/5 Law）提供了异常的耗散，从而在高雷诺数高雷诺数中被迫使3D Navier-Stokes中强迫3D Navier-Stokes？这些需要许多未开发的数学思想，目前尚无清晰的攻击方法。取而代之的是，PI确定了几个独立有趣的问题来建立必要的数学基础。对于（a），PI和他的合作者将研究由Gross-Pitaevskii方程的量子流体中的涡旋细丝。由于涡度的量化量和可正约的线性化运算符，预计量子案例比经典案例更容易。首先，PI和合作者将研究2D Gross-Pitaevskii中Vortex解决方案的稳定性，为了解细丝核心提供必要的理由。接下来，PI和合作者将表明LIA准确地描述了几乎连续的，可能更通用的量子涡流丝的运动，足以做出有用的预测。对于（b），PI和他的合作者将：（1）在随机动力学系统中启发的随机PDES中为Lyapunov指数开发新颖的定性理论；（2）为高维系统中的定量性低纤维规律性和Lyapunov指数估计而开发更好的工具；（3）研究拉格朗日混乱的定量和非线性方面，也就是说，流体中颗粒的混乱动态如何被转化为流体本身的非线性动力学。该奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛影响来评估NSF的法定任务，并被认为是值得的。