From Self-similar Solutions to Turbulent Cascades

湍流流动在现代社会中无处不在，也是极其重要的。它们塑造气候和天气，控制许多传染病的传播和污染的传播，并决定我们汽车、飞机和轮船的燃油效率。然而，尽管进行了几个世纪的系统研究，湍流的许多特性仍然是个谜。特别是，我们没有完全了解在非常小的尺度上控制空气和水中的流体运动的原理。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，限制了我们在许多民用和军事应用中取得持续改进的能力。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，从而产生一系列涡流，从而形成独特而美丽的流体湍流结构。这一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，尽管经过近一个世纪的共同努力，这些问题在很大程度上仍然没有得到解决。流体湍流中最古老、最基本、也是最不为人所知的问题之一是高雷诺数下流动的多尺度结构。这种结构是由于描述流体流动的各种量，如能量、涡度和螺旋度的级联而产生的。我们对各个级联的物理机制的了解非常有限。例如，不完全清楚是什么决定了级联的方向，即特定数量的通量是朝向大尺度(逆级联)还是小尺度(正级联)。这个项目的目的是在两个空间维度上开发正向和反向叶栅的动态描述。用先进的数值方法寻找无粘性控制方程的自相似解，揭示了描述流体流动中尺度相互作用的基本物理机制，这种相互作用是导致发展的湍流的多尺度性质的原因。这一奖项反映了NSF的法定使命，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，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.