Geometric Structure of the Turbulent Cascade

自然界和工程应用中的大多数流体流动都是湍流的。尽管已经司空见惯，但对湍流的理解和建模仍然是一个巨大的挑战。湍流与更一般的非定常流动不同，因为它们显示出从产生能量的大长度尺度到通过粘性耗散能量的小长度尺度的特征级联能量。通常，这一过程是以抽象的方式描述的，这使得理解潜在的物理学变得困难。这项拟议研究的目标是将湍流能量梯级重塑为一个机械过程，其中大尺度确实在小尺度上起作用。这个模型立即强调了几何学所起的重要作用，因为如果大尺度提供的力与流动方向不一致，它们就不能做功。在这项拟议的研究中，将详细研究湍流叶栅的几何形状，并阐明其与其他湍流过程的联系。最终，这项研究的结果可能会为湍流模拟提供新的策略。拟议的研究将支持研究生的教育和培训，其结果将纳入现有的研究生课程。此外，受资助的科学家将参与通过斯坦福大学鲍勃和诺玛街环境流体力学实验室协调的教育外展活动。这项研究的目的是为了更深入地了解尺度相关的湍流应力和应变率的几何特性，特别是它们的相对排列如何控制湍流叶栅。具体地说，将描述湍流应力和应变率的排列和可能的空间有序性，将发展对这种排列如何受平流调制的理解，并将研究排列和间歇之间的潜在联系。这些问题将通过理论工作和对三维湍流、二维湍流以及非定常、随机和多尺度但非湍流速度场的数据集的分析来研究。通过将这项研究建立在功和能量传递的健全和透明的力学原理上，结果将为湍流详细结构的起源带来新的清晰，这种结构比多重分形法等抽象方法更易于解释。

The majority of fluid flows in nature and in engineering applications are turbulent. But despite being so commonplace, understanding and modeling turbulence remains a significant challenge. Turbulent flows are distinct from more general unsteady flows because they display a characteristic cascade of energy from the large length scales at which energy is generated to the small length scales on which it is dissipated by viscosity. Typically, this process is described in an abstract way that makes it difficult to appreciate the underlying physics. The goal of this proposed research is to recast the turbulent energy cascade as a mechanical process, where large scales do work on the small scales. This model immediately emphasizes the important role played by geometry, since if the forces provided by the large scales are misaligned with the flow direction, they can do no work. In this proposed research, the geometry of the turbulent cascade will be studied in detail and its links to other turbulent processes will be clarified. Ultimately, the results of this research may lead to new strategies for turbulence modeling. The proposed research will support the education and training of graduate students, and the results will be folded into existing graduate courses. Additionally, the supported scientists will participate in educational outreach activities coordinate through Stanford's Bob and Norma Street Environmental Fluid Mechanics Laboratory. The objective of this proposed research is to gain a deeper understanding of how the geometric properties of scale-dependent turbulent stresses and strain rates and, in particular, their relative alignment control the turbulent cascade. Specifically, the alignment and possible spatial ordering of the turbulent stress and strain rate will be characterized, an understanding of how this alignment is modulated by advection will be developed, and potential links between alignment and intermittency will be investigated. These questions will be studied via theoretical work and analysis of data sets for three-dimensional turbulence, two-dimensional turbulence, and unsteady, random, and multiscale but non-turbulent velocity fields. By basing this research on sound and transparent mechanical principles of work and energy transfer, the results will bring new clarity to the origins of the detailed structure of turbulence that is more interpretable than abstract approaches such as multifractality.