Sum-of-Squares Approach to Global Stability and Control of Fluid Flows
Lead Research Organisation:
Imperial College London
Department Name: Aeronautics
Abstract
This project aims at developing new methods of analysis of the stability of fluid flows and flow control. Flow control is among the most promising routes for reducing drag, thus reducing carbon emissions, which is the strongest challenge for aviation today. However, the stability analysis of fluid flows poses significant mathematical and computational challenges. The project is based on a recent major breakthrough in mathematics related to positive-definiteness of polynomials. Positive-definiteness is important in stability and control theory because it is an essential property of a Lyapunov function, which is a powerful tool for establishing stability of a given system. For more than a century since their introduction in 1892 constructing Lyapunov functions was dependent on ingenuity and creativity of the researcher. In 2000 a systematic and numerically tractable way of constructing polynomials that are sums of squares and that satisfy a set of linear constraints was discovered. If a polynomial is a sum of squares of other polynomials then it is positive-definite. Thus, systematic, computer-aided construction of Lyapunov functions became possible for systems described by equations with polynomial non-linearity. In the last decade the Sum-of-Squares approach became widely used with significant impact in several research areas.
The Navier-Stokes equations governing motion of incompressible fluid have a polynomial nonlinearity. This project will achieve its goals by applying sum-of-squares approach to stability and control of the fluid flows governed by these equations. This will require development of new advanced analytical techniques combined with extensive numerical calculations. The project has a fundamental nature, with main expected outcomes being applicable to a large variety of fluid flows. The rotating Taylor-Couette flow will be the first object to which the developed methods will be applied. Taylor-Couette flow, encountered in a wide range of industrial application, for a variety of reasons has an iconic status in the stability theory, traditionally serving as a test-bench for new methods.
In order to maximise the impact of the research, the project collaborators will conduct targeted dissemination activities for industry and academia in the form of informal and formal workshops, in addition to traditional dissemination routes of journal papers and conferences. Selected representatives from industry will be invited to attend the workshops. Wider audience will be reached via a specially created and continuously maintained web page.
The Navier-Stokes equations governing motion of incompressible fluid have a polynomial nonlinearity. This project will achieve its goals by applying sum-of-squares approach to stability and control of the fluid flows governed by these equations. This will require development of new advanced analytical techniques combined with extensive numerical calculations. The project has a fundamental nature, with main expected outcomes being applicable to a large variety of fluid flows. The rotating Taylor-Couette flow will be the first object to which the developed methods will be applied. Taylor-Couette flow, encountered in a wide range of industrial application, for a variety of reasons has an iconic status in the stability theory, traditionally serving as a test-bench for new methods.
In order to maximise the impact of the research, the project collaborators will conduct targeted dissemination activities for industry and academia in the form of informal and formal workshops, in addition to traditional dissemination routes of journal papers and conferences. Selected representatives from industry will be invited to attend the workshops. Wider audience will be reached via a specially created and continuously maintained web page.
Planned Impact
Apart from academia, the potential beneficiaries of this project are all industries interested in developing new methods of analysis of the stability of fluid flows and flow control. This includes aviation and, more widely, all transport. Via these industries the entire society will receive economical and environmental benefits.
Flow control is among the most promising routes for reducing drag, which gives immediate economical benefits and also reduces carbon emissions, this improving the environment and contributing to the solution of the problem of global warming.
These benefits will be achieved by developing new methods of flow stability analysis and flow control, and by training researchers in application of these new methods.
The project includes activities directly aimed at achieving these goals, namely:
1) Organisation of two industry-oriented workshops, for informing industry about new opportunities offered thanks to this project.
2) Supporting an electronic mailing list notifying the interested parties in industry about the project deliverables as they are produced.
3) Delivering conference talks for rapid dissemination of the results.
4) Training researchers to use the methods developed in the project, including sending them to visit the word centres doing related research.
5) Supporting a web site for informing wide public about the project.
There is also a possibility that the project will generate commercializable output in the form of software.
Flow control is among the most promising routes for reducing drag, which gives immediate economical benefits and also reduces carbon emissions, this improving the environment and contributing to the solution of the problem of global warming.
These benefits will be achieved by developing new methods of flow stability analysis and flow control, and by training researchers in application of these new methods.
The project includes activities directly aimed at achieving these goals, namely:
1) Organisation of two industry-oriented workshops, for informing industry about new opportunities offered thanks to this project.
2) Supporting an electronic mailing list notifying the interested parties in industry about the project deliverables as they are produced.
3) Delivering conference talks for rapid dissemination of the results.
4) Training researchers to use the methods developed in the project, including sending them to visit the word centres doing related research.
5) Supporting a web site for informing wide public about the project.
There is also a possibility that the project will generate commercializable output in the form of software.
People |
ORCID iD |
| Sergei I. Chernyshenko (Principal Investigator) |
Publications
Ahmadi M
(2019)
A framework for input-output analysis of wall-bounded shear flows
in Journal of Fluid Mechanics
Chernyshenko S
(2013)
Nonlinear stability analysis of fluid flow using sum of squares of polynomials
Chernyshenko SI
(2014)
Polynomial sum of squares in fluid dynamics: a review with a look ahead.
in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Fantuzzi G
(2016)
Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization
in SIAM Journal on Applied Dynamical Systems
Fuentes F
(2022)
Global Stability of Fluid Flows Despite Transient Growth of Energy.
in Physical review letters
Huang D
(2017)
Expensive Control of Long-Time Averages Using Sum of Squares and Its Application to A Laminar Wake Flow
in IEEE Transactions on Control Systems Technology
| Description | 1). The main goal of this grant was to verify an idea about a particular way, in which a breakthrough in the area of semi-algebraic geometry, namely, in the numerically tractable ways of computing representation of multivariate polynomials as sum of squares and solving related optimisation problems, could be used in fluid dynamics. The breakthrough was made in 2000 in a PhD thesis of P. Parrilo, and received much attention. However, it applies most naturally to finite-dimensional dynamical systems, while fluid flows are described by infinite-dimensional dynamical systems. The idea of how to overcome this difficulty was put forward by P.Goulart and S.Chernyshenko in 2012, who then initiated the current project. The first key finding of the project was a confirmation of that idea: the proof of concept was provided by successfully applying the idea to study the global stability of a particular version of rotating Couette flow. 2). There exists a nontrivial link between the problems of global stability of fluid flows and the problem of finding bounds for time-averaged characteristics of fluid flows. Such bounds are potentially of large practical interest, because the bounds might be easier to calculate than the average values itself. At the same time, if both an upper and lower bound are found, say, for the lift or the drag force in a turbulent flow past an object, and if the difference between these bounds is not large, this might well be sufficient for all practical purposes. The second key finding of the project was that the sum-of-squares of polynomials methodology can be applied to finding bounds for time-averaged quantities. The classical methods of determining such bound for fluid flows was shown to be a very special case of the newly proposed approach. One of the key advantages of the proposed approach is the possibility of a trade-off between the complexity of calculations and the tightness of the bound, allowing in principle to find a bound as close as desired to the actual time-averaged value. 3). Nonlinear systems, such as fluid flows, often have non-unique solutions, for example, under the same conditions both turbulent flow and a laminar flow are possible. In practice the laminar flow would often be unstable, and, would not affect the actual time-averages. However, the classical methods of obtaining bounds do not distinguish stable and unstable solutions. As a result, the gap between the upper and lower bound inevitably would be greater than the gap between the averages in the turbulent and the laminar flow. The third key finding of the project was that the newly proposed method of obtaining bounds can be easily extended to systems with noise. Noise removes unstable solutions, and as a result, the new proposed method can, and on a simple example was shown to, reduce the difference between the upper and the lower bound to very small values. 4). The second original goal of the project was an application of the sum-of-squares of polynomials methodology to control of fluid flows. The project findings allowed to achieve this goal in a rather unexpected way. The fourth key finding of the project is the possibility of using the upper bound of the time-averaged cost function as the objective function for designing a feedback controller. It turned out that in this case the natural assumption of expensive control resolves the well-known difficulty of non-convexity of optimizing over both the control law and a Lyapunov function. As of 2020, of the above key findings, the second one, now called the 'auxiliary function(al) method', proved to be the most significant, leading to a number of further studies. |
| Exploitation Route | The results can be directly used in further research, in particular in fluid dynamics, and in developing control strategy for various systems. |
| Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Education,Energy,Transport,Other |
| URL | https://www.imperial.ac.uk/aeronautics/fluiddynamics/sumofsquares/index.php |
| Description | The project is of fundamental nature. Its results are purely theoretical and mathematical. For this reason its main outputs will take a significant time to make explicit societal or economical impact. The only current impact that is not purely academic stems from the project contribution to training students. Over the duration of the project 12 MSc and final year master students worked on their research theses on the topics related to the project under supervision of the project PI, helped by the project post-doctoral researcher. Two of the students because co-authors of the journal papers. The new area of research created by the project attracts more and more people, which now include PhD students, post-doctoral researchers and full-time academics. |
| First Year Of Impact | 2014 |
| Sector | Education,Other |
| Impact Types | Societal |
| Description | Joint research with a group at the University of Michigan |
| Organisation | University of Michigan |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Actively participated in research leading to a paper "Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization" by G. Fantuzzi, D. Goluskin, D. Huang, S. I. Chernyshenko, arXiv:1512.05599 [math.DS] |
| Collaborator Contribution | Actively participated in research leading to a paper "Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization" by G. Fantuzzi, D. Goluskin, D. Huang, S. I. Chernyshenko, arXiv:1512.05599 [math.DS] |
| Impact | "Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization" by G. Fantuzzi, D. Goluskin, D. Huang, S. I. Chernyshenko, arXiv:1512.05599 [math.DS] |
| Start Year | 2014 |
| Description | SoS project website |
| Form Of Engagement Activity | Engagement focused website, blog or social media channel |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Undergraduate students |
| Results and Impact | Creating and supporting a large website devoted to the project. |
| Year(s) Of Engagement Activity | 2013,2014,2015,2016 |
| URL | https://www.imperial.ac.uk/aeronautics/fluiddynamics/sumofsquares/index.php |
| Description | Web page describing the main ideas behind the project for the undergraduate audience, developed by a student. |
| Form Of Engagement Activity | Engagement focused website, blog or social media channel |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Undergraduate students |
| Results and Impact | A page was added to the project website. The page describing the main ideas behind the project for the undergraduate audience. The page was developed by a 1st-year mathematics student of the Saint Petersburg University, and has two language versions, English and Russian. |
| Year(s) Of Engagement Activity | 2015,2016 |
| URL | https://www.imperial.ac.uk/aeronautics/fluiddynamics/sumofsquares/LChStudentCorner.php |