The shadow of turbulence: algorithms and applications

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.