Stabilisation of exact coherent structures in fluid turbulence

流体湍流的不可预测性和多尺度特性使其成为许多行业和自然系统的重大建模挑战。从飞机、火车和轮船的设计，从可再生能源技术到可持续通风系统，湍流很重要的情况几乎是无穷无尽的。这意味着，有相当多的经济机会与理解和影响湍流有关。经过多年的密集研究，尽管在建模和分析控制方程方面取得了进展，但尚未出现关于湍流的严格预测理论。造成这一困难的主要原因在于这种流动在空间和时间上表现出的复杂性。时间上的复杂性通过控制方程解的混沌性质表现出来；解从不重复，并且对初始条件敏感。空间复杂性表现为一些地区的湍流比其他地区更强烈，没有明显的原因。尽管目前很困难，但从这种复杂性中提取出更简单的解决方案是可能的，这些解决方案低于动荡的状态。这些解决方案可能在时间上或时间周期上是稳定的，但关键是它们是不稳定的，因此从未完全实现。假设这些解是湍流的良好代表，我们可以从中收集新的见解，并对流动做出严格的预测。该项目将通过开发一种控制方法来稳定嵌入在混沌中的简单不稳定解，从而提供一种新的计算高效的方法来预测流体湍流。到目前为止，这些解需要仔细地收敛，这是一个分两步进行的过程；首先需要找到一个猜测，然后使用复杂的数值算法收敛解。我们的方法是在控制方程中加入新的项，这将“被动地”稳定解，使进化能够趋向于它们，而不需要猜测或收敛。我们的新方法带来的效率提高将使我们能够处理比以前更复杂的场景，使我们能够接近真实世界湍流的真正“时空复杂性”。

The unpredictable and multi-scale nature of fluid turbulence makes it a significant modelling challenge for many industries and natural systems. From designing planes, trains and ships, renewable energy technologies to sustainable ventilation systems, the list of situations where turbulent flows are important is almost endless. This means that there are considerable economic opportunities associated with understanding and influencing turbulent flows. After many years of intense research there is yet to emerge a rigorous predictive theory for turbulence, despite progress in modelling and analysing the governing equations. The main reason for this difficulty lies in the complexity such flows exhibit in space and time. Temporal complexity is manifest via the chaotic nature of the solutions to the governing equations; the solutions never repeat and are sensitive to initial conditions. Spatial complexity appears as turbulence being more intense in some regions than others, for no appreciable reason. It is possible, though currently difficult, to extract from this complexity, simpler solutions which underly the turbulent state. These solutions may be steady in time or time periodic but crucially they are unstable, and therefore never fully realised. The hypothesis is that these solutions are good proxies for the turbulence from which we can glean new insights and make rigorous predictions of the flow. This project will provide a new computationally efficient method for predicting fluid turbulence by developing a control method to stabilise simple unstable solutions embedded in the chaos. Until now these solutions require careful convergence which is a two-step process; first a guess needs to be found, then the solution converged using a sophisticated numerical algorithm. Our approach is to include new terms into the governing equations which will 'passively' stabilise solutions enabling the evolution to tend towards them without the need for guesses or convergences. The increases in efficiency afforded by our new method will allow us to tackle vastly more complex scenarios than previously possible enabling us to approach the true "spatiotemporal complexity" of real world turbulence.