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.

Potential Impact:
A growing body of recent evidence has been supporting that the dynamics of turbulent flows at high Reynolds numbers may well be approached with suitable extensions of the notions of dynamical systems theory previously used to study transition and turbulence at low Reynolds number. Given the nature of the proposed research, which exercises these notions for high-Reynolds number turbulence, its main output will be knowledge in the form of scientific papers to be published in leading fluid mechanics journals. Therefore, the primary and immediate beneficiaries from the proposed research will be academics working on fluid dynamics (applied mathematicians, statistical physicists and engineers).

The fundamental research of wall-bounded shear flows, such as channel flow, pipe flow and boundary layers, underpins the technologies concerning development of aircrafts, ships, trains, turbo-machineries and heat exchangers. It is also crucial to understand atmospheric surface layer, a popular location for wind energy harvesting. With this nature, the proposed research will deliver an improved description on the mechanisms of turbulent skin-friction generation, heat transfer and noise generation. As such, it would be highly valuable for the early-stage design processes of next-generation transportation products and wind-energy farms, ultimately offering a long-term benefit for manufacturing and energy industries.