﻿ Sum-Of-Squares Approach to Global Stability and Control of Fluid Flows - project background - flow stability

# Sum-Of-Squares Approach to Global Stability and Control of Fluid Flows

## Flow Stability

Flow stability theory is a huge and well-established part of fluid dynamics, with Helmholtz, Kelvin, Rayleigh and Reynolds among its pioneers. The following is limited to what is relevant here.

In the majority of applications steady flows are better than unsteady flows. Steady flows are usually associated with smaller fuel consumption, less fatigue, and less noise. Theoretically, a steady flow always exists as a solution of the governing equations. However, in engineering applications fluid flows are usually unsteady, or even turbulent, because the corresponding steady flow is unstable, that is, if disturbed it will never return to the steady state. Flow control aims at stabilising fluid flows. Technology progress is making more complicated control approaches, like, for example, feedback control, more and more feasible. The bulk of the work on hydrodynamic stability is concerned with infinitesimal perturbations. This allows to represent the solution of the governing equations as a sum of the steady solution and a small perturbation, and neglect the nonlinear terms. The resulting linear problem is much easier to solve.

Linear stability of canonical flows is now largely a closed area of research, with the centre of gravity shifted to developing efficient numerical methods applicable to complex flows encountered in practice. Importantly, linear stability analysis can reveal instability but cannot prove stability. This is because steady flows are often stable with respect to infinitesimally small perturbations but unstable with respect to finite perturbations. Moreover, the finite amplitude required to destabilise the flow is often small. Hence, it is the stability with respect to finite disturbances, which is also called global stability, and the control of finite disturbances that represent major practical interest. Typically, flows are stable if the Reynolds number R is small enough, but there is a value Rl above which the flow is unstable with respect to arbitrary small perturbations, i.e. if R > Rl . On the other hand, there is a critical Reynolds number Rc such that if R < Rc the flow is stable with respect to disturbances of any amplitude. Finding Rc is difficult since this is a nonlinear problem.

Progress in understanding instability with respect to finite disturbances was made in the late eighties in the works on non-modal stability, but no methods for determining Rc have yet emerged from these works. Serrin (1959) demonstrated that for each incompressible flow in a closed domain there is Re such that the energy of an arbitrary perturbation decreases monotonously if R < Re. This, of course, means that the flow is globally stable. Moreover, Serrin demonstrated that the problem of determining Re can be reduced to a linear eigenvalue problem similar to the eigenvalue problems arising in the linear stability theory. Naturally, Re ≤ Rc ≤ Rl . In many cases the difference between Re and Rl is large: for example, for a pipe flow Rl is infinite. Therefore, while finding Re and Rl is relatively easy, more powerful methods are required for estimating Rc.

### 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.