The shadow of turbulence: algorithms and applications

在许多领域，从空气动力学到燃烧和热声学，设计优化问题的传统解决方法是使用稳定的雷诺-平均-纳维-斯托克斯（RANS）求解器进行流量预测和伴随方法进行灵敏度分析。然而，当采用尺度解析湍流模拟进行流动预测时，现有的伴随算法存在分歧，因此不能提供有用的灵敏度。这是由于湍流的混沌性质，它会在相空间中产生指数发散的轨迹，这种现象通常被称为“蝴蝶效应”，会导致非物理上的大灵敏度。计算资源的发展趋势表明，非定常湍流模拟将在未来发挥越来越大的作用，逐渐取代基于ranss的方法。这将提高预测复杂流动中非定常现象的能力，但会在设计周期中留下很大的空白，因为现有的设计和优化伴随方法无法应用。然而，在过去十年的开始，一种适合于混沌动力学的灵敏度分析的范式转换方法被提出，该方法基于动力系统理论的一个基本结果——阴影引理。基于阴影的伴随方法防止了轨迹的指数发散，因此可以在混沌系统中提供准确和真实的灵敏度。这一进展具有改变工程设计实践以及加速基础湍流研究的独特潜力。然而，为一般混沌系统开发的现有实现，与雷诺数的比例非常差，阻碍了它们在实际工程流程中的应用。此外，由Navier-Stokes方程控制的湍流经常违反用于证明当前基于阴影的算法收敛性的基本假设，并且这些进展与湍流系统的相关性目前仍然值得怀疑。该项目的目标是打破这些挑战，开发下一代工具，使研究人员和从业人员能够通过规模解决模拟来解决设计问题，以进行流量预测和优化。为此，该项目将利用伦敦帝国理工学院和南安普顿大学的专业知识和团队的记录。我们计划开发新的基于阴影的伴随算法，利用湍流的结构特性，从而扩展到现实世界的问题。结合最近开发的湍流分析工具，允许对潜在动力学进行紧凑的描述，以及基于阴影的灵敏度分析，这在很大程度上尚未得到探索，但它有可能提供范式转换的进展。为了展示新开发工具的潜力，将考虑壁面流动中的两个优化问题，即在过渡边界层中使用数据同化进行流动重建，以及在压力驱动流动中设计最佳非均质柔性涂层以减少湍流摩擦阻力。这些流动显示了与各种工程应用相关的广泛的流动机制和不稳定性。期望从这些应用程序中获得的综合见解将为现实世界的设计应用程序铺平道路。

In many areas, ranging from aerodynamics to combustion and thermo-acoustics, design optimisation problems have been traditionally solved using a combination of steady Reynolds-Averaged-Navier-Stokes (RANS) solvers for flow prediction and adjoint methods for sensitivity analysis. However, when scale-resolving turbulent flow simulations are employed for flow prediction, existing adjoint algorithms diverge and therefore do not provide useful sensitivities. This is due to the chaotic nature of turbulence that produces exponentially diverging trajectories in phase space, a phenomenon popularly known as the `butterfly effect', which leads to unphysically large sensitivities. Trends in computing resources suggest that unsteady turbulent flow simulations will take an increasingly larger role in the future, gradually replacing RANS-based methods. This will improve the ability to predict unsteady phenomena in complex flows, but will leave a large gap in the design cycle, because existing adjoint methods for design and optimisation cannot be applied.At the start of the previous decade, however, a paradigm-shifting approach for sensitivity analysis suitable for chaotic dynamics was proposed, which is based on the Shadowing Lemma, a fundamental result of dynamical systems theory. Shadowing-based adjoint methods prevent the exponential divergence of trajectories and can therefore provide accurate and realistic sensitivities in chaotic systems. This advance holds the unique potential to transform the engineering design practice as well as to accelerate fundamental turbulence research. However, existing implementations developed for generic chaotic systems, scale very poorly with Reynolds numbers, preventing their application for practical engineering flows. In addition, turbulent flows governed by the Navier-Stokes equations often violate fundamental assumptions utilised to prove the convergence of current shadowing-based algorithms and the relevance of these advances to turbulent fluid systems at present remains questionable.The ambition of this project is to break these challenges and develop the next-generation of tools that will enable researchers and practitioners to tackle design problems using scale-resolving simulations for flow prediction and optimisation. To this end, the project will leverage the expertise and track record of the groups at Imperial College London and University of Southampton. We plan to develop new shadowing-based adjoint algorithms that exploit structural properties of turbulent flows and therefore scale to real-world problems. The combination of recently developed turbulence analysis tools that allow for a compact description of the underlying dynamics together with shadowing-based sensitivity analysis is largely unexplored, yet it carries the potential to provide paradigm-shifting advances.To demonstrate the potential of the newly developed tools, two optimisation problems in wall-bounded flows will be considered, namely flow reconstruction using data assimilation in transitional boundary layers and design of optimal heterogeneous compliant coatings for turbulent friction drag reduction in pressure-driven flows. These flows display a wide range of flow mechanisms and instabilities that are relevant to a variety of engineering applications. The expectation is that synthesising the insight obtained from these applications will pave the path to real-world design applications.