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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 m...

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Autores principales: Huang, D., Chernyshenko, S., Goulart, P., Lasagna, D., Tutty, O., Fuentes, F.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society Publishing 2015
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4685881/
https://www.ncbi.nlm.nih.gov/pubmed/26730219
http://dx.doi.org/10.1098/rspa.2015.0622
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author Huang, D.
Chernyshenko, S.
Goulart, P.
Lasagna, D.
Tutty, O.
Fuentes, F.
author_facet Huang, D.
Chernyshenko, S.
Goulart, P.
Lasagna, D.
Tutty, O.
Fuentes, F.
author_sort Huang, D.
collection PubMed
description 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, 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.
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spelling pubmed-46858812016-01-04 Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application Huang, D. Chernyshenko, S. Goulart, P. Lasagna, D. Tutty, O. Fuentes, F. Proc Math Phys Eng Sci Research Articles 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, 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. The Royal Society Publishing 2015-11-08 /pmc/articles/PMC4685881/ /pubmed/26730219 http://dx.doi.org/10.1098/rspa.2015.0622 Text en © 2015 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Research Articles
Huang, D.
Chernyshenko, S.
Goulart, P.
Lasagna, D.
Tutty, O.
Fuentes, F.
Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title_full Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title_fullStr Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title_full_unstemmed Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title_short Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
title_sort sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
topic Research Articles
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4685881/
https://www.ncbi.nlm.nih.gov/pubmed/26730219
http://dx.doi.org/10.1098/rspa.2015.0622
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