Coherent Structure, Chaos, and Turbulence in Fluid Mechanics

我们每天与之相互作用的流体(如水和空气)表现出惊人的多样性。一个极端是混沌和湍流，其中流体形成复杂的分形状图案，由于对初始条件的微小变化非常敏感，这些图案的细节在实验中不能准确重复。湍流经常在流体中观察到，例如在飞机、车辆的尾迹中，甚至在建筑物和桥梁等障碍物中，在海洋和大气的大规模动力学中。另一个极端是涡丝的运动，最显著的例子是龙卷风和通常从机翼和直升机叶片上脱落的翼尖涡流。涡旋细丝倾向于以可预测的方式移动，并在较长一段时间内保持其结构完整性。这个项目的目标是发展对流体力学中这些相反现象的更好的数学理解。对涡丝和湍流的准确预测在各种科学和工业应用中都是至关重要的，包括在空中、陆地和海洋飞行器的设计中，以及在理解复杂的流体系统如气候和天气方面。拥有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，对这些极端情况的理解有助于为更好地理解相互作用和中间制度铺平道路，在这些制度中，流动既有结构，也有混乱。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。这些研究项目还与数学和STEM方面的研究生和年轻科学家的培训相结合。PI将在数学上更严格地理解在高雷诺数下观察到的不可压缩流体中的两种行为：(1)涡丝的相干运动；(2)在统计上稳定的“一般”强迫下的湍流。推动PI的基本问题是：(A)涡量集中在光滑曲线上的流体中涡丝运动的常用几何演化模型(如局部感应近似(LIA))的精度有多高？(B)我们能否证明在高雷诺数极限下，实验观察到的正Lyapunov指数和反常耗散(如著名的柯尔莫戈洛夫4/5定律)来自随机强迫的三维N-S方程？这些需要一些未被探索的数学思想，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受Gross-Pitaevskii方程支配的量子流体中的涡旋细丝。由于涡量的量子化和稍微更易于线性化的算符，量子情况预计比经典情况更容易。首先，PI和合作者将研究二维Gross-Pitaevskii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准确地描述了近直的、可能更一般的量子涡旋细丝的运动，其长度足以做出有用的预测。对于(B)，PI和他的合作者将：(1)从随机动力系统的想法得到关于随机偏微分方程中Lyapunov指数的新的定性理论；(2)开发更好的工具来定量地估计高维系统中的亚椭圆正则性和Lyapunov指数；以及(3)研究拉格朗日混沌的定量和非线性方面，即如何将流体中粒子的混沌动力学转化为流体本身的非线性动力学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.