Coherent Structure, Chaos, and Turbulence in Fluid Mechanics

流体力学中的相干结构、混沌和湍流

• 批准号: 2348453
• 负责人: Jacob Bedrossian
• 金额: $ 36.48万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-03-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348453&HistoricalAwards=false
• 关键词: Coherent; Structure; Chaos; Turbulence; Fluid

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.

我们每天与之互动的流体（如水和空气）显示出各种各样的动力学。一个极端是混沌和湍流，其中流体形成复杂的分形图案，由于对初始条件的微小变化非常敏感，因此在实验中无法精确重复其细节。湍流经常在流体中观察到，例如在飞机、车辆的尾流中，甚至在建筑物和桥梁等障碍物中，以及在海洋和大气的大尺度动力学中。另一个极端是涡丝的运动，最著名的例子是龙卷风和通常从机翼和直升机桨叶上脱落的翼尖涡。涡丝倾向于以可预测的方式移动，并在很长一段时间内保持其结构完整性。这个项目的目标是发展一个更好的数学理解这些相反的现象在流体力学。涡丝和湍流的准确预测在各种科学和工业应用中至关重要，包括空中，陆地和海上车辆的设计以及对气候和天气等复杂流体系统的理解。有一个坚实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，并导致更好的建模。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间状态铺平道路，其中流动具有结构和混乱的混合。最后，克服这些问题的数学挑战将需要更广泛的数学界感兴趣的创新。研究项目还与数学和STEM研究生和年轻科学家的培训相结合。PI将对不可压缩流体在高雷诺数下观察到的两种行为进行更严格的数学理解：（1）涡丝的相干运动;（2）“通用”统计稳定强迫下的湍流。激发PI的基本问题是：（A）常用的几何演化模型，如局部诱导近似（LIA），对于涡量集中在光滑曲线上的流体中的涡丝运动，其准确性如何？(B)我们能否为实验观察到的正李雅普诺夫指数和异常耗散（例如，著名的Kolmogorov 4/5定律）在高雷诺数极限下的随机强迫三维Navier-Stokes方程的解？这需要一些未经探索的数学思想，目前还没有明确的方法来攻击它们。相反，PI已经确定了几个独立有趣的问题，以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方程控制的量子流体中的涡丝。量子的情况下，预计将比经典的情况下，由于涡量的量子化和稍微更顺从的线性算子。首先，PI和合作者将研究二维Gross-Pitaevskii中涡旋解的稳定性，为理解细丝核心提供必要的基础。接下来，PI和合作者将证明LIA准确地描述了近直线的运动，并且可能更普遍，量子涡旋细丝足够长，可以做出有用的预测。对于（B），PI及其合作者将：（1）受随机动力系统思想的启发，开发随机偏微分方程中Lyapunov指数的新定性理论;（2）开发高维系统中亚椭圆正则性和Lyapunov指数定量估计的更好工具;（3）研究拉格朗日混沌的定量和非线性方面，即该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。