Efficient capture of the dominant periodic orbits underlying turbulent fluid flow.

We are familiar with turbulence, through its affect on the stability of aircraft during flight. Fluids, in this case air, are generally regarded as exhibiting two states of flow - a 'laminar' state and a 'turbulent' state. Turbulence is characterised by chaotic variations in the direction of the flow, through the appearance of whirls or 'eddies'. In industrial applications, turbulence typically leads to a loss of performance, as significant energy can be lost to the generation of eddies. A typical example is in pipelines, important for domestic water supply, irrigation, cooling systems, oil and gas supply. Rather than energy being expended in moving fluid directly from A to B, almost all the energy is lost to the creation and sustenance of turbulence! The question of how to model turbulence, therefore, is consistently listed among the most important outstanding problems of applied mathematics and theoretical physics
(e.g. http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics).

This work builds on recent progress in understanding turbulence, made possible by the recent discovery of solutions to the equations governing flow in pipes and channels. These solutions are in the form of waves. Although they travel with the flow, their structure is otherwise static in time. Turbulence is chaotic in time, however. A radical step-change in this approach will be to model turbulence in terms of solutions that vary in time and that repeat after a period of time. Substantially new computational methods will be required to isolate such solutions in the future. There is strong motivation for isolating repeating cycles, otherwise called periodic orbits - from dynamical systems theory they are known to efficiently capture complex dynamics, filtering out activity that is otherwise a distraction. Often only a handful of periodic orbits are required to reproduce the statistical properties of a seemingly complex system.

By extracting periodic orbits directly from simulations of turbulence itself, this project aims to capture those periodic orbits that are dynamically most important. So far it has only been possible to find orbits via numerical continuation methods, where there is no clear link between the orbits and the actual dynamics of the system. Capturing periodic cycles in a 'large' system such as turbulence, however, has been a challenging task. In this work, a new symmetry projection method will be developed to enable meaningful visualisations of the underlying dynamics. It has been shown that this particular method dramatically improves our ability to spot recurring cycles, i.e. periodic orbits. Collaboration with a leading European experimental facility will enable further application of these methods, plus theoretically guided searches to be performed more rapidly than is possible in simulation.

This work will have great impact on our understanding of dynamical processes underlying turbulence, where periodic orbits will provide a basis for describing and predicting fluid flow patterns. This will open new avenues of future research in methods of prediction and control.

The proposed research is of a fundamental mathematical nature, and therefore impact on industry is likely to be indirect, via academic impact on applied disciplines. Examples of research areas that have already taken an interest in the techniques being developed in fundamental fluid dynamics include complex fluids and magnetohydrodynamics.

The symmetry reduction method is currently being developed for applications as diverse as rotating flows and cardiac dynamics. There are wide range of target areas for applications of symmetry reductions methods, from industrial processes to the life sciences. The spatial evolution of chemical fronts and models of predator-prey interactions are specific target examples.

Turbulence is familiar to the majority of people through commercial flights, and the question of how to model turbulence is consistently listed among the top outstanding issues in applied mathematics and theoretical physics. This public familiarity makes turbulence a topic well suited to public engagement. The specific subject matter of the project, and recent developments based on dynamical systems, makes extensive use of easy-to-interpret phase diagrams, schematic and from data. There is a substantial visual element, which is again suited to outreach. This research will benefit the interested general public through a number of public engagement channels.