Stabilisation of exact coherent structures in fluid turbulence
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
The unpredictable and multi-scale nature of fluid turbulence makes it a significant modelling challenge for many industries and natural systems. From designing planes, trains and ships, renewable energy technologies to sustainable ventilation systems, the list of situations where turbulent flows are important is almost endless. This means that there are considerable economic opportunities associated with understanding and influencing turbulent flows. After many years of intense research there is yet to emerge a rigorous predictive theory for turbulence, despite progress in modelling and analysing the governing equations.
The main reason for this difficulty lies in the complexity such flows exhibit in space and time. Temporal complexity is manifest via the chaotic nature of the solutions to the governing equations; the solutions never repeat and are sensitive to initial conditions. Spatial complexity appears as turbulence being more intense in some regions than others, for no appreciable reason. It is possible, though currently difficult, to extract from this complexity, simpler solutions which underly the turbulent state. These solutions may be steady in time or time periodic but crucially they are unstable, and therefore never fully realised. The hypothesis is that these solutions are good proxies for the turbulence from which we can glean new insights and make rigorous predictions of the flow.
This project will provide a new computationally efficient method for predicting fluid turbulence by developing a control method to stabilise simple unstable solutions embedded in the chaos. Until now these solutions require careful convergence which is a two-step process; first a guess needs to be found, then the solution converged using a sophisticated numerical algorithm. Our approach is to include new terms into the governing equations which will 'passively' stabilise solutions enabling the evolution to tend towards them without the need for guesses or convergences. The increases in efficiency afforded by our new method will allow us to tackle vastly more complex scenarios than previously possible enabling us to approach the true "spatiotemporal complexity" of real world turbulence.
The main reason for this difficulty lies in the complexity such flows exhibit in space and time. Temporal complexity is manifest via the chaotic nature of the solutions to the governing equations; the solutions never repeat and are sensitive to initial conditions. Spatial complexity appears as turbulence being more intense in some regions than others, for no appreciable reason. It is possible, though currently difficult, to extract from this complexity, simpler solutions which underly the turbulent state. These solutions may be steady in time or time periodic but crucially they are unstable, and therefore never fully realised. The hypothesis is that these solutions are good proxies for the turbulence from which we can glean new insights and make rigorous predictions of the flow.
This project will provide a new computationally efficient method for predicting fluid turbulence by developing a control method to stabilise simple unstable solutions embedded in the chaos. Until now these solutions require careful convergence which is a two-step process; first a guess needs to be found, then the solution converged using a sophisticated numerical algorithm. Our approach is to include new terms into the governing equations which will 'passively' stabilise solutions enabling the evolution to tend towards them without the need for guesses or convergences. The increases in efficiency afforded by our new method will allow us to tackle vastly more complex scenarios than previously possible enabling us to approach the true "spatiotemporal complexity" of real world turbulence.
Planned Impact
Understanding, predicting and controlling turbulent fluid flows has the potential to impact an enormous number of industrial and societal problems.
Many industries encounter difficulties when interacting with turbulent flows. Important examples are in aerodynamics and the oil and gas sector, but also in building ventilation and renewable technologies. As society moves towards sustainable energy sources and more energy efficient transport, unravelling the complicated, unpredictable behaviour of turbulent flows will become increasingly important. By increasing the visibility of fundamental turbulence research to industrial communities and by developing relationships with industrial partners we aim to disseminate the mathematical and fundamental research ideas toward practical deployment. Following success at the 102nd European Study Group with Industry (www.scm.keele.ac.uk/staff/d_lucas/esgi102/) we aim to develop further industrial contacts with participation at more study groups during the course of the project. In addition we are in discussions to develop a Knowledge Transfer Partnership to perform some modelling in collaboration with a local pump manufacturer. Funding from the EPSRC will allow for the fundamental research to progress while contacts are developed with these industrial partners with the aim of filtering some of the optimisation and control ideas from this research into a real-world application.
As mentioned above, understanding fluid turbulence will have impact on innumerable societal problems, from sustainable living to efficient energy use and production. Climate change possibly presents the most severe societal challenge in the post-industrial era; having increasingly accurate climate predictions will enable society to adapt to new and hitherto unforeseen changes. We will seek to engage with the public to increase awareness of the scientific challenges involving fluid turbulence. We will do this by attending local science festivals (e.g. mmems.org/potteries) and give talks and demonstrations. We will also perform outreach with local schools in order to increase students choosing STEM subjects at tertiary education, with particular focus on encouraging and supporting minority groups into these subjects. During the course of the grant we will write an article for "The Conversation" following attendance at the workshop "Writing for a Lay Audience" to be held as part of the Keele "Writefest" in November 2018. This article will outline some of the challenges associated with understanding and controlling turbulence with a mathematical perspective. We will also seek further public outreach training to enable us to more effectively communicate our research to a broader audience.
Many industries encounter difficulties when interacting with turbulent flows. Important examples are in aerodynamics and the oil and gas sector, but also in building ventilation and renewable technologies. As society moves towards sustainable energy sources and more energy efficient transport, unravelling the complicated, unpredictable behaviour of turbulent flows will become increasingly important. By increasing the visibility of fundamental turbulence research to industrial communities and by developing relationships with industrial partners we aim to disseminate the mathematical and fundamental research ideas toward practical deployment. Following success at the 102nd European Study Group with Industry (www.scm.keele.ac.uk/staff/d_lucas/esgi102/) we aim to develop further industrial contacts with participation at more study groups during the course of the project. In addition we are in discussions to develop a Knowledge Transfer Partnership to perform some modelling in collaboration with a local pump manufacturer. Funding from the EPSRC will allow for the fundamental research to progress while contacts are developed with these industrial partners with the aim of filtering some of the optimisation and control ideas from this research into a real-world application.
As mentioned above, understanding fluid turbulence will have impact on innumerable societal problems, from sustainable living to efficient energy use and production. Climate change possibly presents the most severe societal challenge in the post-industrial era; having increasingly accurate climate predictions will enable society to adapt to new and hitherto unforeseen changes. We will seek to engage with the public to increase awareness of the scientific challenges involving fluid turbulence. We will do this by attending local science festivals (e.g. mmems.org/potteries) and give talks and demonstrations. We will also perform outreach with local schools in order to increase students choosing STEM subjects at tertiary education, with particular focus on encouraging and supporting minority groups into these subjects. During the course of the grant we will write an article for "The Conversation" following attendance at the workshop "Writing for a Lay Audience" to be held as part of the Keele "Writefest" in November 2018. This article will outline some of the challenges associated with understanding and controlling turbulence with a mathematical perspective. We will also seek further public outreach training to enable us to more effectively communicate our research to a broader audience.
Organisations
People |
ORCID iD |
| Dan Lucas (Principal Investigator) |
Publications
Lucas D
(2022)
Stabilization of exact coherent structures in two-dimensional turbulence using time-delayed feedback
in Physical Review Fluids
| Description | The first findings of this project are that it is possible to stabilise nonlinear equilibrium and travelling wave solutions embedded in two-dimensional turbulence at relatively high Reynolds number (large flow rate and/or low viscosity), and with quite high instability, using the method of time-delayed feedback. Importantly this is possible without any advance knowledge of solution properties; for instance the phase-speed of travelling waves is able to be computed as part of the stabilisation. We find that symmetries help avoid the notorious "odd-number" limitation of this method and provide a targeting of solution type; 8 solutions are able to be stabilised in our first work, 3 of which are newly reported. Following on from the above success we find the method is transferrable to the more challenging, but physically relevant, geometry of the three-dimensional pipe. Again symmetries are helpful and relatively large instability can be handled, however we find that with high instability comes a new, unforeseen, issue of multiscale temporal behaviour. In order to attenuate the large number of frequencies responsibly for instability we devised a novel multiple term approach to the control method. We have a strict criterion for stabilisation to ensure that we find true asymptotic stability of our solutions. We are able to stabilise five different nonlinear states, up to relatively high Reynolds numbers (Re=3000) and beginning control from the turbulent state. We have also conducted a rigorous analysis of various alternative ways to avoid the odd number limitation for unstable periodic solutions and in doing so have shown the stabilisation of periodic orbits in a partial differential equation (Kuramoto-Sivashinsky) for the first time, again without using knowledge of the solution in advance and obtaining the period as part of the stabilisation method. |
| Exploitation Route | This project has reported the stabilisation, or control, of relatively complex and a diverse range of flow configurations from fluid turbulence. This new control methodology may prove of interest to a variety of other researchers, either interested directly in the so-called "dynamical systems approach" to fluid turbulence (for instance moving to larger system sizes or obtaining time-periodic solutions without the need of root-finding) or even engineers looking for novel ways to control an environmental or industrial flow with specific properties. It would be of interest to see if this method might be blended with a data-driven approach to add to the automation of discovering new solutions. |
| Sectors | Aerospace Defence and Marine Environment Manufacturing including Industrial Biotechology |