From Self-similar Solutions to Turbulent Cascades

湍流流动在现代社会中无处不在，而且非常重要。它们塑造气候和天气，控制许多传染病的传播和污染的扩散，并决定我们的汽车、飞机和轮船的燃油效率。然而，尽管经过了几个世纪的系统研究，许多湍流的性质仍然是神秘的。特别是，我们还不能完全理解在非常小的尺度上控制空气和水等流体运动的原理。因此，尽管计算能力不断进步，但解决湍流小尺度结构的数值模拟仍然遥不可及，限制了我们在众多民用和军事应用中进行持续改进的能力。该项目利用最新的数值和理论进展来描述一个尺度上的流体运动如何在另一个尺度上（通常大得多或小得多）产生运动，从而产生一个独特而美丽的流体湍流结构的漩涡层次。本研究项目结合了新颖的数值方法和创新的理论方法来解决流体湍流中的几个基本问题，尽管近一个世纪的共同努力，这些问题在很大程度上仍未得到解决。在流体湍流中，最古老、最基本和最不为人所知的问题之一是高雷诺数流动的多尺度结构。这种结构是由于描述流体流动的能量、涡度和螺旋度等不同量的级联而出现的。我们对各个级联的物理机制的了解非常有限。例如，决定级联方向的因素并不完全清楚，即某一特定量的通量是朝向大尺度（逆级联）还是朝向小尺度（正级联）。本项目旨在在两个空间维度上对正级联和逆级联进行动态描述。采用先进的数值方法来寻找无粘控制方程的自相似解，以揭示描述流体流动中的尺度相互作用的基本物理机制，这是造成发达湍流多尺度性质的原因。利用来自动力系统理论的已建立的工具来开发湍流级联的全面定量理论，该理论不依赖于现有湍流统计理论基础上未经证实的假设和近似。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Turbulent fluid flows are both ubiquitous and immensely important in the modern society. They shape climate and weather, control transmission of many infectious diseases and the spreading of pollution, and determine the fuel-efficiency of our cars, airplanes, and ships. Yet, many of turbulent flows’ properties remain mysterious despite centuries of systematic research. In particular, we do not fully understand the principles that control the motion of fluids such at air and water at very small scales. As a result, despite continuing advances in computing power, numerical simulations resolving the small-scale structure of turbulent flows remains out of reach, limiting our ability to make sustained improvements in numerous civilian and military applications. This project harnesses recent numerical and theoretical advances to describe how fluid motions at one scale can generate motions at another (often much larger or much smaller) scale, yielding a hierarchy of eddies responsible for the unique and beautiful structure of fluid turbulence. This research project uses a combination of novel numerical methods and innovative theoretical approaches to addresses several fundamental problems in fluid turbulence that remain largely unsolved despite almost a century of concerted effort. One of the oldest, most fundamental, and least understood issues in fluid turbulence is the multi-scale structure of the flow at high Reynolds numbers. Such structure emerges due to cascades of various quantities, such as energy, vorticity, and helicity, describing the fluid flow. Our understanding of the physical mechanisms of respective cascades is very limited. For instance, it is not entirely clear what determines the direction of the cascades, i.e., whether the flux of a particular quantity is towards large scales (inverse cascade) or small scales (direct cascade). This project aims to develop a dynamical description of direct and inverse cascades in two spatial dimensions. Advanced numerical methods for finding self-similar solutions to inviscid governing equations are used to uncover the fundamental physical mechanisms describing scale interaction in fluid flows that is responsible for the multi-scale nature of developed turbulence. Established tools from dynamical systems theory are leveraged to develop a comprehensive quantitative theory of turbulent cascades that does not rely on unproven assumptions and approximations that underlie existing statistical theories of turbulence.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.