Application of Sum-of-Squares approach to flow stability

It is well known that the Galerkin approximation of the incompressible Navier-Stokes equations leads to an ODE with quadratic non-linearity. The SOS approach can be applied to such a system since the dynamics are represented by polynomials, and this application is one of the topics of the proposed research. The proposers have already used SOS in a study of a hydrodynamic-type ODE system. In this study the method of exploiting in the SOS approach the energy-invariance property of the bilinear component of the systems of hydrodynamic type was developed. This allowed the construction of a Lyapunov function for the value of R about seven times larger than Re, thus demonstrating the strong potential of applying SOS in fluid dynamics.

Although these results were very promising, in the sense that very substantial improvements in the computed stability bounds for the flow were achieved, significant technical hurdles remain if this approach is to gain widespread acceptance. In particular, the method should be able to handle systems with significantly larger numbers of states, since accurate approximation of a fluid flow requires considering systems with a large number of Galerkin modes. However, application of conventional SOS methods to very large systems is problematic, since the SDP problems to be solved in such cases are prohibitively large. Therefore, the proposers developed a method of reducing the order of the system in SOS studies, based on representing the higher modes with their energy only. The method is guaranteed always to give results at least as good as the standard energy stability analysis. Moreover, it has also been proven mathematically that if the flow remains globally stable for R > Re given by the energy stability analysis, then there exists a polynomial Lyapunov function(al), which could be sought for using a SOS approach.

An alternative would be to search for a Lyapunov functional for a fluid system directly without recourse to a finite-dimensional approximation. This is more difficult since in this case the arising polynomials contain powers of the partial derivatives of the flow variables in addition to the variables themselves. Some preliminary results in this direction have been published, and it was shown that SOS optimization methods can be applied directly to analysis of infinite dimensional systems characterized by PDEs, and in particular to stability of 2D channel flow.

Latest:

  • The project has ended.
  • Paolo Luchini visited us on February 2016.
  • Mihailo Jovanovic and Armin Zade visited us during the two weeks 2-15 November 2015.
  • All our postdocs secured positions which will allow them to contninue working in cooperation with us: Deqing Huang will be a full professor at the Southwest Jiaotong University, China, Davide Lasagna will be a Frontier Fellow at the Universiy of Southampton, UK, and Giorgio Valmorbida will move to a faculty position at the Supelec and Paris-Sud University, France. Well done!
  • Next progress meeting on February 23.2015.
  • Remote conference was held on 5.02.2015.
  • Sergei was at UCLA attending the Mathematics of Turbulence programme. He gave two presentations about the project.
  • Remote conference was held on 30.10.2014.
  • Remote conference was held on 09.06.2014.
  • A progress meeting was held at Imperial College on 27.03.2014.
  • Charles Doering is at Imperial 25-27.03.2014 and in Oxford 27-28.03.2014 He is giving a talk on 26.03, see "Miscellaneous".
  • Remote conference was held on 14.03.2014.
  • Sergei gave a talk at the NeZaTeGiUs conference NeZaTeGiUs conference on 02.03.2014.
  • Deqing went to China 26-30 December and gave a talk at a workshop organized by the State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou.
  • Remote conference was held on 20.12.2014 The record is available from Sergei.
  • Paul visited London and Oxford for two weeks in October, and took part in the project meeting on 21.10.
  • Sergei gave a talk at ICNAAM 2013.
  • Remote conference was held on 19.09.2013 The record is available from Sergei.
  • Remore conferences were video recorded on 14.05, 06.06, 25.06, 11.07, 16.08, and 21.08.2013 The records are available for the members of the team from Sergei, but the records are not in the repository, to save space.
  • July 2013. Deqing gave a talk at the School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China.
  • June 5 and 6, 2013. Our remote conference was spread over 2 days this time.
  • May 19-31, 2013. Sergei was in Stockholm, taking part in NORDITA Stability and Transition programme, and the follow-on SIG-33 workshop, where he gave a talk on using SoS in fluid dynamics.
  • May 14, 2013. We had our first remote conference. It was interesting, and useful. Giorgio, Davide, and Deqing presented, and we all had a good chat.
  • April 22, 2013. SVN repository for the project was created by Paul, hosted at ETH.
  • Kick-off meeting on April 18 went fine.
  • Paul is here!
  • Paul Goulart will be visiting Imperial from April 15 to May 3.
  • 30.03.2013 This website opens.
  • 28.03.2013 Project mailing list created.
  • 11-15.03.2013 Wynn visited Doering in Michigan.
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  • 19.02.2013 Chernyshenko visited Southampton team.