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

Lead Research Organisation: University of Sheffield
Department Name: Mathematics and Statistics

### Abstract

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.

### Planned Impact

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.

### ORCID iD

Ashley Phillip Willis (Principal Investigator)

### Publications

10 25 50
Budanur N (2017) Relative periodic orbits form the backbone of turbulent pipe flow in Journal of Fluid Mechanics

Willis A (2017) The Openpipeflow Navier-Stokes solver in SoftwareX

Description The 'method of slices' has been developed to automatically remove the
redundant freedom of spatial shifts of a pattern in fluid flow.
Patterns that are
identical up to a shift are automatically reduced to a single state that
lives within a reduce space, the 'slice'. Sliced dynamics reveals dynamically important features, such as recurring periodic orbits, that
are otherwise impossible to visualise.

Determining shifts using multiple templates has proven difficult, as there
are innumerable options in how and when to apply the switch between
templates. With appropriate manipulation, however, it has been shown
that it is possible to implement the symmetry reduction with a single
template. Features of the symmetry reduction, e.g. rapid changes in
shift, have been explained in terms of properties of the projection onto
the slice.

Using the refined method, it has been possible to isolate many more
orbits direct from simulation data than in any previous study. Through
visualisation of the state space, it has been found that orbits may be grouped into distinct families.

It has proven difficult, so far, to get sufficiently good quality
experimental data from which to extract orbits. Therefore our research
is directed to optimising small adjustments to the system that enable their
realisation and detection.
Exploitation Route The key development is in the very general method. This has been shown to enable vastly improved visualisation of paths taken by a chaotic dynamical system, and thereby can be employed to reveal its fundamental features, e.g. its dominant periodic orbits.

Beyond fluid flow, the method is directly applicable to any system that includes a homogeneous direction. This includes e.g. population models in ecology, models of fronts in chemical processes, models of cardiac dynamics.
Sectors Education,Manufacturing, including Industrial Biotechology

URL http://www.openpipeflow.org/index.php?title=Periodic_orbits

Description The symmetry reduction method enables substantially more informative visualisation of the dynamics of a system. The method is a fundamental mathematical tool, and is applicable to applications as diverse as fluid flows and cardiac dynamics. In the current research, these methods have been adapted and in particular, shown that they be applied even to 'large' dynamical systems, here the case of a turbulent flow. Many repeating patterns that underlie the dynamics were extracted than has previously been possible, and relationships between families of repeating patterns identified. As turbulence is familiar to many people through commercial flight, the notion that turbulent flow is chaotic is easily understood. Our research has shown that this chaotic dynamics can be explained in terms of just a few sets of repeating patterns, and this has been conveyed through public speaking (e.g. Cafe Scientifique).
Sector Education,Other

Description EPSRC standard grant
Amount £269,704 (GBP)
Funding ID EP/P000959/1
Organisation Engineering and Physical Sciences Research Council (EPSRC)
Sector Public
Country United Kingdom
Start 01/2017
End 01/2020

Description GeorgiaTech
Organisation Georgia Institute of Technology
Country United States
PI Contribution We have discussed several forms of implementation of symmetry reduction methods. The method is crucial for identifying periodic orbits in turbulent flow, and in comprehending their properties. I have implemented the numerical method and trained several PhD students in the theory and use of the code.
Collaborator Contribution Prof. P. Cvitanovic has provided insight on properties of periodic orbits, how they are distributed, generated, and in methods developed in chaos theory, that we are applying together on turbulent flow
Impact We have developed code that implements the 'method of slices' (a 'symmetry-reduction' method) that is essential for visualisation of the dynamical system of turbulence. An article has been published, showing that vastly improved visualisations of dynamics is possible with the symmetry reduction method, and that it may be applied to high-dimensional systems (Willis, Short & Cvitanovic 2016, Phys. Rev. E) A website www.openpipeflow.org has been created, which aims to make accessible the tools for a range of modelling techniques.
Start Year 2010

Description IST-Austria
Organisation Institute of Science and Technology Austria
Country Austria
PI Contribution Our aim is to combine numerical modelling and laboratory experiments for the isolation and comprehension of periodic orbits and travelling waves in shear flow. I bring expertise of numerical modelling, in particular, nonlinear optimisation techniques.
Collaborator Contribution Prof. B. Hof leads a team who has constructed shear flow apparatus. We discuss what can be implemented experimentally.
Impact Laboratory flows are not subject to symmetries that in numerical simulations can be imposed for computational convenience. We have searched and identified periodic orbits without the artificial rotational symmetry that is routinely used in simulation. We find that the type of orbits that tend to dominate in these more realistic domains have multiple frequencies, including long period oscillations. These render the standard techniques for identifying orbits, via recurrences, in effective. An approach that alleviates this difficulty has been proposed. An article on this topic is submitted and currently under review.
Start Year 2008

Title Double double-pendulum, Android App.
Description This is an Android application, designed as a demonstration piece for showing fundamental properties of chaotic systems. With two copies of the pendula side-by-side, it may be used to demonstrate sensitivity to initial conditions (SIC) for example. What this application also shows, in particular, is that embedded within the chaos are recurrent patterns, periodic orbits (POs), that underlie the dynamics.
Type Of Technology Physical Model/Kit
Year Produced 2014
Impact (latest update 2016.)

Title openpipeflow.org
Description A simulation code and documentation on techniques and methods are provided. The majority of mathematical techniques described in the manual pages are applicable to a huge range of problems beyond pipe flow. The core code is designed to be flexible, and subroutines for some well-known methods are designed to be problem-independent. Openpipeflow.org Aims: - To make accessible a range of modelling techniques. - To facilitate rapid entry into the world of numerical simulation and fluid dynamics. - To provide flexible modules for more the use and development of advanced techniques in research. Methods being developed under current funding are among those documented.
Type Of Technology Physical Model/Kit
Year Produced 2014
Impact The website front page receives around 1000 visitors per month. Several research groups are now using the resource, not just among my collaborators, but externally. I receive requests for assistance using the resource, which feedback into the online manual and online tutorial. PhD projects beyond my current collaborators are using this software. Permission has been requested and granted to use an adaptation of the code in a summer school on fluid instabilities.
URL http://www.openpipeflow.org

Description CafeScientifique2014
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? Yes
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Approximately 40 people (room capacity) came to first hear a talk on 'Unravelling chaos in turbulent flows', approx 40mins. This was followed by approximately an hour of discussion, mostly regarding the modelling and applications.
Unravelling and visualisation of the chaotic dynamics for these flows specifically requires methods being developed under current funding.

Considerable interest was expressed in an 'App' that was produced to demonstrate some of the key points regarding chaos and periodic orbits. I plan to release this as a tool for educational use.
Year(s) Of Engagement Activity 2014
URL http://www.sciencecafesheffield.org/