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

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

## Rotating Couette flow

The overall goal of this project is to develop general methods of using Sum-of-Squares in the analysis of flow stability and control. The methods have to be tested on a particular flow, which is, hence, our first target application. For this purpose the rotating Couette flow is the best choice.

Rotating Couette flow is a flow between two concentric circular cylinders rotating with different velocities. Due to its simple and common geometry it has numerous industrial prototypes. Some examples (from most common to rather exotic) include flows in bearings, flows in particle separators, flows in rotational viscosity meters, flow between the drillstring that is the inner cylinder to which the drill bit is attached and which rotates rapidly in the drilled hole in the drilling of oil wells, suspension flow in the annular space between an inner rotating cylinder and an outer porous wall where fibres are fractionated in pressure screens in paper making, or flow of foodstaff in rotating heat exchangers used in food industry. At the same time, the stability and dynamics of this flow are very well studied. For the simplest case, when the gap between the cylinders is small relative to the cylinder radii, the flow is controlled by two parameters, the Reynolds number R and a non-dimensional parameter Ω characterising the Coriolis force. Alternatively, the Taylor number T = Ω(R − Ω) can be used. The simplest solution is a circular Couette flow, which is a steady flow with only one non-zero velocity component, namely, the circumferential velocity independent of the azimuthal coordinate. As R increases, other flow regimes appear. The simplest and most widely known alternative regime is the flow with Taylor vortices. It bifurcates supercritically from the circular Couette flow at T = Tc = 1708. Accordingly, the critical Reynolds number Rl for the corresponding loss of stability with respect to infinitesimal perturbation depends on Ω: Rl = Ω + Tc/Ω. In particular, in case of no rotation the circular Couette flow remains linearly stable for any R. On the other hand, Rl as a function of Ω has a minimum Rl = 177.2 at Ω = 82.6 (the Reynolds number here is based on the difference between the velocities of the cylinder surfaces and the gap between them). Such moderate values of the critical Reynolds number make numerical treatment quite feasible, which is one of the reasons for selecting this flow as the target application.

The energy stability limit Re of the circular Couette flow is independent of Ω : Re = 82.6. Note that the Taylor vortex flow is independent of the azimuthal coordinate, as well as the circular Couette flow. If perturbations also are limited to flows independent of the azimuthal coordinate the energy stability limit increases to Re = 177.2. This means that for such, effectively 2D, perturbations the gap between Re and Rl varies with Ω from infinity to zero. This creates a unique opportunity to compare the performance of the new SOS-based approach with the performance of the energy approach, which is the second reason for selecting the rotating Couette flow as the target application. The linear stability limit provides an upper bound for the range of R for which a Lyapunov functional proving stability for arbitrary large disturbances can be found. A tighter bound is implied by the existence of travelling-wave solutions. Notably, such solutions exist even for Ω = 0; they first appear at R ≈ 500. These solutions are fully three-dimensional, and, therefore, for 3D perturbations for Ω = 0 the Lyapunov function (already known for R < Re = 82.6) does not exist for R > 500. Interestingly, it was proved recently that for at least R < 177.2 the flow remains stable with respect to 3D finite, but not arbitrarily large perturbations. This is only a small part of relevant information available for this flow, and the richness of it is yet another reason for selecting the rotating Couette flow as the target application.

### 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.