The University of Southampton
University of Southampton Institutional Repository

Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application

Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterizing the magnitude of the Coriolis force. By converting the original Navier–Stokes equations to a finite- dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares of polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study, several results meaningful in the context of the method used were also obtained. Overall, 2 the results obtained demonstrate the applicability of the recently proposed approach to global
stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach.
flow stability, rotating couette flow, lyapunov function, sum-of-squares of polynomials, semi-definite programming
1364-5021
1-18
Huang, Deqing
96e466d6-59e1-4428-a6f6-4c1cecd45d00
Chernyshenko, Sergei
0381c775-bd7f-4fb9-862d-d6d378cd9b5d
Goulart, Paul
b19a7bb9-29c3-46ca-954e-b8ea389b148a
Lasagna, Davide
0340a87f-f323-40fb-be9f-6de101486b24
Tutty, Owen
c9ba0b98-4790-4a72-b5b7-09c1c6e20375
Fuentes, Federico
656d44ff-5ca1-4e52-bc55-113c5ec25e0b
Huang, Deqing
96e466d6-59e1-4428-a6f6-4c1cecd45d00
Chernyshenko, Sergei
0381c775-bd7f-4fb9-862d-d6d378cd9b5d
Goulart, Paul
b19a7bb9-29c3-46ca-954e-b8ea389b148a
Lasagna, Davide
0340a87f-f323-40fb-be9f-6de101486b24
Tutty, Owen
c9ba0b98-4790-4a72-b5b7-09c1c6e20375
Fuentes, Federico
656d44ff-5ca1-4e52-bc55-113c5ec25e0b

Huang, Deqing, Chernyshenko, Sergei, Goulart, Paul, Lasagna, Davide, Tutty, Owen and Fuentes, Federico (2015) Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2183), 1-18, [20150622]. (doi:10.1098/rspa.2015.0622).

Record type: Article

Abstract

With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterizing the magnitude of the Coriolis force. By converting the original Navier–Stokes equations to a finite- dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares of polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study, several results meaningful in the context of the method used were also obtained. Overall, 2 the results obtained demonstrate the applicability of the recently proposed approach to global
stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach.

Text
20150622.full.pdf - Version of Record
Available under License Creative Commons Attribution.
Download (507kB)

More information

Accepted/In Press date: 26 October 2015
Published date: 25 November 2015
Keywords: flow stability, rotating couette flow, lyapunov function, sum-of-squares of polynomials, semi-definite programming
Organisations: Aerodynamics & Flight Mechanics Group

Identifiers

Local EPrints ID: 384893
URI: http://eprints.soton.ac.uk/id/eprint/384893
ISSN: 1364-5021
PURE UUID: 6263788f-4777-416d-a776-c3e6de4fe878
ORCID for Davide Lasagna: ORCID iD orcid.org/0000-0002-6501-6041

Catalogue record

Date deposited: 13 Jan 2016 15:48
Last modified: 15 Mar 2024 03:47

Export record

Altmetrics

Contributors

Author: Deqing Huang
Author: Sergei Chernyshenko
Author: Paul Goulart
Author: Davide Lasagna ORCID iD
Author: Owen Tutty
Author: Federico Fuentes

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×