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工业和自然系统的重大建模挑战。从设计飞机、火车和船舶，可再生能源技术到可持续通风系统，湍流至关重要的情况几乎是无穷无尽的。这意味着与理解和影响湍流有关的经济机会相当大。经过多年的深入研究，尽管在建模和分析控制方程方面取得了进展，但仍未出现一个严谨的湍流预测理论。造成这种困难的主要原因在于这种流在空间和时间上表现出的复杂性。时间复杂性通过控制方程解的混沌性表现出来；解永远不会重复，并且对初始条件很敏感。空间复杂性表现为某些地区的湍流比其他地区更强烈，没有明显的原因。虽然目前很难，但从这种复杂性中提取出隐藏在动荡状态下的更简单的解决方案是可能的。这些解可能在时间上是稳定的，也可能是时间周期的，但关键是它们是不稳定的，因此永远不会完全实现。假设是，这些解决方案是湍流的良好代理，从中我们可以收集新的见解，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混沌中的简单不稳定解，为预测流体湍流提供一种新的计算效率方法。到目前为止，这些解决方案需要小心地收敛，这是一个两步的过程；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，这些项将“被动地”稳定解，使进化趋向于它们，而不需要猜测或收敛。我们的新方法所带来的效率的提高将使我们能够处理比以前可能的更复杂的场景，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.