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


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  • Flow stability
  • Nonlinear systems and SOS
  • SOS and Flow Stability
  • Rotating Couette flow
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    • Deqing Huang
    • Giorgio Valmorbida
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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 = T_c = 1708. Accordingly, the critical Reynolds number R_l for the corresponding loss of stability with respect to infinitesimal perturbation depends on Ω: R_l = Ω + T_c/Ω. In particular, in case of no rotation the circular Couette flow remains linearly stable for any R. On the other hand, R_l as a function of Ω has a minimum R_l = 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 R_l 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 < R_e = 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.

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