Efficient capture of the dominant periodic orbits underlying turbulent fluid flow.

我们熟悉湍流，因为它影响飞机在飞行过程中的稳定性。流体，在这种情况下是空气，通常被认为表现出两种流动状态--‘层流’状态和‘湍流’状态。湍流的特征是流动方向上的混沌变化，表现为漩涡或“漩涡”的出现。在工业应用中，湍流通常会导致性能损失，因为涡流的产生可能会损失大量能量。一个典型的例子是管道，这对家庭供水、灌溉、冷却系统、石油和天然气供应非常重要。几乎所有的能量都损失在湍流的产生和维持上，而不是将能量直接从A移动到B！因此，如何对湍流建模的问题一直被列为应用数学和理论物理中最重要的突出问题之一(例如http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics).这项工作建立在理解湍流的最新进展的基础上，这是由于最近发现了控制管道和通道中流动的方程的解。这些解决方案是以波浪的形式出现的。尽管它们随波逐流，但它们的结构在时间上是静态的。然而，湍流在时间上是混乱的。这种方法的一个根本改变将是根据随时间变化的解来模拟湍流，并在一段时间后重复。未来将需要大量新的计算方法来分离此类解决方案。有强烈的动机来隔离重复的周期，也就是所谓的周期轨道--从动力系统理论来看，它们被认为可以有效地捕捉复杂的动力学，过滤掉原本令人分心的活动。通常只需要几个周期轨道就可以再现一个看似复杂的系统的统计特性。通过直接从湍流本身的模拟中提取周期轨道，该项目旨在捕获那些在动力学上最重要的周期轨道。到目前为止，只能通过数值延拓方法找到轨道，在这种方法中，轨道和系统的实际动力学之间没有明确的联系。然而，捕捉像湍流这样的“大”系统中的周期周期一直是一项具有挑战性的任务。在这项工作中，将开发一种新的对称投影方法，以使潜在的动力学能够有意义的可视化。事实证明，这种特殊的方法极大地提高了我们识别循环周期，即周期轨道的能力。与一家领先的欧洲实验机构的合作将使这些方法的进一步应用，加上理论上的指导搜索可以比在模拟中更快地执行。这项工作将对我们理解湍流背后的动力学过程产生重大影响，其中周期性轨道将为描述和预测流体流动模式提供基础。这将为未来预测和控制方法的研究开辟新的途径。

We are familiar with turbulence, through its affect on the stability of aircraft during flight. Fluids, in this case air, are generally regarded as exhibiting two states of flow - a 'laminar' state and a 'turbulent' state. Turbulence is characterised by chaotic variations in the direction of the flow, through the appearance of whirls or 'eddies'. In industrial applications, turbulence typically leads to a loss of performance, as significant energy can be lost to the generation of eddies. A typical example is in pipelines, important for domestic water supply, irrigation, cooling systems, oil and gas supply. Rather than energy being expended in moving fluid directly from A to B, almost all the energy is lost to the creation and sustenance of turbulence! The question of how to model turbulence, therefore, is consistently listed among the most important outstanding problems of applied mathematics and theoretical physics (e.g. http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics). This work builds on recent progress in understanding turbulence, made possible by the recent discovery of solutions to the equations governing flow in pipes and channels. These solutions are in the form of waves. Although they travel with the flow, their structure is otherwise static in time. Turbulence is chaotic in time, however. A radical step-change in this approach will be to model turbulence in terms of solutions that vary in time and that repeat after a period of time. Substantially new computational methods will be required to isolate such solutions in the future. There is strong motivation for isolating repeating cycles, otherwise called periodic orbits - from dynamical systems theory they are known to efficiently capture complex dynamics, filtering out activity that is otherwise a distraction. Often only a handful of periodic orbits are required to reproduce the statistical properties of a seemingly complex system.By extracting periodic orbits directly from simulations of turbulence itself, this project aims to capture those periodic orbits that are dynamically most important. So far it has only been possible to find orbits via numerical continuation methods, where there is no clear link between the orbits and the actual dynamics of the system. Capturing periodic cycles in a 'large' system such as turbulence, however, has been a challenging task. In this work, a new symmetry projection method will be developed to enable meaningful visualisations of the underlying dynamics. It has been shown that this particular method dramatically improves our ability to spot recurring cycles, i.e. periodic orbits. Collaboration with a leading European experimental facility will enable further application of these methods, plus theoretically guided searches to be performed more rapidly than is possible in simulation.This work will have great impact on our understanding of dynamical processes underlying turbulence, where periodic orbits will provide a basis for describing and predicting fluid flow patterns. This will open new avenues of future research in methods of prediction and control.