Mathematical underpinnings of stratified turbulence

从水龙头里流出的水到吹在脸上的风，湍流是一种日常体验。尽管它具有广泛的重要性，但湍流的数学描述仍然难以捉摸。在大多数实际关心的湍流中，大量的自由度使得不可能直接求解运动方程。然而，最近取得了重大进展的观点是，湍流可以被视为一个大的动力系统，其中湍流用微分方程表示，原则上，微分方程可以在时间上向前积分，从一种状态演变到下一种状态。虽然该领域以前的工作主要集中在均匀密度流体上，但与密度变化相关的浮力在许多应用中起着重要的动力学作用（例如，热空气的密度小于冷空气，这对于有效的建筑加热和冷却策略至关重要）。这个项目将发展分层湍流的数学基础。该项目建立在动力系统的观点上，为具有恒定密度的流体开发，并将其扩展到流体密度随空间和时间变化的流体。除了将分析扩展到新的实际应用之外，包括与密度变化相关的浮力效应将使我们能够以一种在均匀流体中不可能实现的方式探索湍流的普遍方面。迄今为止，湍流理论的大部分进展都是在过渡性流动中取得的，即流动处于完全层流和完全湍流之间的中间状态。在分层流体中，浮力抑制垂直运动（例如，暖空气倾向于停留在天花板附近），使层流/湍流转变更普遍，其结构更可识别，动力学更丰富，因此有更多的现象需要探索。我们的方法是基于数学、模拟和实验之间的紧密耦合。尽管分层湍流已经进行了多年的实验研究，但没有一种现有的流动几何形状能够满足我们所有的三个主要目标。因此，我们将建立一个新的实验来研究典型几何中的分层湍流，其中湍流由剪切产生而由分层反对。这种新的几何结构将使我们能够观察和测量层流/湍流的转变，所涉及的流动结构及其在时间和空间上的分布。我们还将在另外两种几何形状中进行新的实验，以研究不同构型下流动特征的变化。我们将使用每个实验配置的数值模拟，通过允许额外的诊断来补充实验室实验，并提供数学工具和流动几何形状之间的直接联系。将数学分析、实验室实验和数值模拟相结合的重要成果之一将是建立系统的简化动力学描述。这种简化的动力系统将捕捉分层湍流的关键物理特性，并为理解和模拟不同经济、环境和社会背景下的湍流和混合过程提供通用工具。最终，我们希望提供湍流分层流体中混合和输运的实际估计。

Turbulence is an everyday experience, from the water emerging from a tap to the wind on one's face. Despite its widespread importance, a mathematical description of turbulent flow remains elusive. In most turbulent flows of practical interest, the vast number of degrees of freedom makes it impossible to solve the equations of motion directly. Recently, however, significant progress has been made by taking the view point that turbulence can be treated as a large dynamical system in which the turbulence is represented by differential equations that, in principle, can be integrated forward in time to evolve from one state to the next. While previous work in this area has focused on uniform density fluids, the buoyancy forces associated with density variations play an important dynamical role in many applications (e.g. warm air is less dense than cold air, which is critical for efficient building heating and cooling strategies).This project will develop the mathematical underpinnings of stratified turbulence. The project builds on the dynamical systems viewpoint, developed for a fluid which has a constant density and extends it to one in which the fluid density varies in space and time. In addition to extending the analysis to new practical applications, including buoyancy effects associated with density variations will allow us to probe universal aspects of turbulence in a way that would not be possible in a homogeneous fluid. Much of the progress in turbulence theory to date has been in flows which are transitional so the flow is in an intermediate state between being completely laminar and completely turbulent. In a stratified fluid, the buoyancy force inhibits vertical motions (e.g. warm air tends to stay near the ceiling) and makes laminar/turbulent transitions more prevalent, their structures more identifiable, and the dynamics richer so there are more phenomena to explore.Our approach is based on tight coupling between mathematics, simulation and experimentation. Although stratified turbulence has been studied experimentally for many years, none of the existing flow geometries can address all three of our primary objectives. Consequently, we will build a new experiment to study stratified turbulence in a canonical geometry in which the turbulence is generated by shear and opposed by the stratification. This new geometry will allow us to observe and measure laminar/turbulent transitions, the flow structures that are involved and their distributions in time and space. We will also carry out new experiments in two other geometries to examine how the flow features change in different configurations. We will use numerical simulations of each experimental configuration to complement the laboratory experiments by allowing additional diagnostics, and providing a direct link between the mathematical tools and the flow geometries.One of the important outcomes of combining mathematical analysis, laboratory experiments and numerical simulations will be to develop a simplified dynamical description of the system. This reduced dynamical system will capture the key physics of stratified turbulence, and provide a generic tool for understanding and modelling turbulence and mixing processes in diverse contexts of economic, environmental and societal importance. Eventually we hope to provide practical estimates of mixing and transport in a turbulent stratified fluid.