A dynamical systems analysis of high-Reynolds-number wall turbulence

流体在固体表面上流动时的湍流（即壁面湍流）无处不在，是许多航空和机械工程设备设计的核心，如飞机机翼、船体、火车、汽车、涡轮叶片、管道和热交换器。壁面湍流中的动量传递是由高度有组织的含能流体运动主导的，通常被称为相干结构。最近有越来越多的证据表明，高雷诺数的壁面湍流被组织成一个自相似、自维持的连贯结构的层次结构，其大小与它们与壁面的距离成正比。最近，申请人小组发现了一组与这些自相似相干结构直接相关的Navier-Stokes方程的精确解。在动力系统理论中，这样的精确解构成了“状态空间”中湍流混沌动力学的骨架。受这一最新发现的启发，本提案旨在制定和检验基于动力系统理论的高雷诺数壁面湍流描述。为此，本提案基于对壁面湍流的最新理解，提出了两个工作包：1)计算单个相干结构动力学的自相似时间周期解（周期轨道）；最小多尺度（双尺度）壁面湍流的动力系统分析。这一建议的结果将为高雷诺数壁湍流中相干结构的个体和集体动力学提供基本的物理见解。特别是，它将在高雷诺数壁面湍流的低维描述中形成关键的构建块知识。最终，这将在阐明湍流表面摩擦产生、传热和噪声产生的精确“动力学”机制方面发挥关键作用，这些是支撑许多工业设计的核心过程。

Turbulence in fluid flows over a solid surface (i.e. wall turbulence) is ubiquitous and central to the design of many aeronautical- and mechanical-engineering devices, such as aircraft wings, ship hulls, trains, cars, turbine blades, pipelines, and heat exchangers. The momentum transfer in wall turbulence is dominated by highly-organised energy-containing fluid motions, often referred to as coherent structures. There is a growing body of recent evidence that wall turbulence at high Reynolds numbers is organised into a hierarchy of self-similar, self-sustaining coherent structures, the size of which is proportional to their distance from the wall. Recently, the group of the applicant has discovered a set of exact solutions of the Navier-Stokes equations, which are directly linked with these self-similar coherent structures. In dynamical systems theory, such exact solutions form a skeleton of chaotic dynamics of turbulence in `state space'. Motivated by this recent discovery, this proposal aims to formulate and examine a dynamical-systems-theory-based description of wall turbulence at high Reynolds numbers. To this end, the present proposal sets out two work packages based on the state-of-the-art understanding of wall turbulence: 1) Computation of self-similar time-periodic solutions (periodic orbits) for the dynamics of individual coherent structures; 2) Dynamical systems analysis of minimal multi-scale (two-scale) wall turbulence. The outcome of this proposal will provide fundamental physical insight into the individual and collective dynamics of coherent structures in high-Reynolds-number wall turbulence. In particular, it will form a key building-block knowledge in a low-dimensional description of high-Reynolds-number wall turbulence. Ultimately, this will play a pivotal role in illuminating the precise `dynamical' mechanisms of turbulent skin-friction generation, heat transfer, and noise generation, the central processes underpinning many industrial designs.