The shadow of turbulence: algorithms and applications

在从空气动力学到燃烧和热声学的许多领域，设计优化问题传统上都是使用用于流动预测的定常雷诺平均N-S(RANS)求解器和用于灵敏度分析的伴随方法相结合来解决的。然而，当尺度分辨湍流模拟用于流动预测时，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，它在相空间中产生指数级发散的轨迹，这种现象通常被称为“蝴蝶效应”，它导致非物理上的巨大敏感性。计算资源的趋势表明，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RANS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，因为现有的伴随设计和优化方法无法应用。然而，在上个十年开始，基于动力系统理论的基本结果--影子引理，提出了一种适合于混沌动力学的灵敏度分析的范例转换方法。基于跟踪的伴随方法可以防止轨迹的指数发散，因此可以在混沌系统中提供准确和现实的灵敏度。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，现有的为一般混沌系统开发的实现随着雷诺数的变化而扩展得很差，阻碍了它们在实际工程流中的应用。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的收敛的基本假设，这些进展对目前的湍流系统的相关性仍然是值得怀疑的。该项目的目标是打破这些挑战，开发下一代工具，使研究人员和实践者能够使用尺度分辨模拟来解决设计问题，用于流动预测和优化。为此，该项目将利用伦敦帝国理工学院和南安普顿大学各小组的专业知识和过往记录。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。将最新开发的湍流分析工具与基于阴影的敏感性分析相结合，以提供基本动力学的紧凑描述，这在很大程度上是未知的，但它具有提供范式转换的潜力。为了展示新开发工具的潜力，我们将考虑两个壁面边界流中的优化问题，即在过渡边界层中使用数据同化进行流动重建，以及为压力驱动流动中的湍流摩擦阻力减阻设计最佳的非均质顺应性涂层。这些流动表现出广泛的流动机制和不稳定性，与各种工程应用相关。我们的期望是，综合从这些应用程序中获得的见解，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.