A dynamical systems analysis of high-Reynolds-number wall turbulence

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Turbulence in fluid flows over a solid surface (i.e. wall turbulence) is ubiquitous and central to the design of many aeronautical- and mechanical-engineering devices, such as aircraft wings, ship hulls, trains, cars, turbine blades, pipelines, and heat exchangers. The momentum transfer in wall turbulence is dominated by highly-organised energy-containing fluid motions, often referred to as coherent structures. There is a growing body of recent evidence that wall turbulence at high Reynolds numbers is organised into a hierarchy of self-similar, self-sustaining coherent structures, the size of which is proportional to their distance from the wall. Recently, the group of the applicant has discovered a set of exact solutions of the Navier-Stokes equations, which are directly linked with these self-similar coherent structures. In dynamical systems theory, such exact solutions form a skeleton of chaotic dynamics of turbulence in `state space'. Motivated by this recent discovery, this proposal aims to formulate and examine a dynamical-systems-theory-based description of wall turbulence at high Reynolds numbers. To this end, the present proposal sets out two work packages based on the state-of-the-art understanding of wall turbulence: 1) Computation of self-similar time-periodic solutions (periodic orbits) for the dynamics of individual coherent structures; 2) Dynamical systems analysis of minimal multi-scale (two-scale) wall turbulence. The outcome of this proposal will provide fundamental physical insight into the individual and collective dynamics of coherent structures in high-Reynolds-number wall turbulence. In particular, it will form a key building-block knowledge in a low-dimensional description of high-Reynolds-number wall turbulence. Ultimately, this will play a pivotal role in illuminating the precise `dynamical' mechanisms of turbulent skin-friction generation, heat transfer, and noise generation, the central processes underpinning many industrial designs.