Optimal control applied to plane couette flow: (towards the) full-information state-feedback stabilization of the Nagata Lower-Branch

Abstract

Turbulence can be seen as deterministic chaos evolving within a finite dimensional dynamical state-space, where each invariant solution (IS) of the Navier-Stokes Equations (NSE) acts as an unstable attractor of the turbulent state. The mechanism by which the turbulent state remains/leaves the neighborhood of an IS is still not completely known. Supposedly, the turbulent dynamical state escapes the neighborhood of an IS along its unstable eigen-space, although recent work suggests that the non-normality of its stable eigen-space may help the turbulent trajectory to leave along stable directions.

To elucidate this process, we present a procedure to stabilize via linear optimal control the least-unstable IS of the NSE within a Plane Couette Flow (PCF) configuration, the Nagata lower-branch (EQ1).

Linear optimal control requires a linearized state-space model. Around an IS, this model is very high-dimensional, which prevents the solution of the associated Riccati equation and the finding of the optimal control law. Therefore, a new divergence-free model is derived and validated: the Orr-Sommerfeld Squire model Extended for an IS as baseflow. It resulted in a boundary actuated full-matrix state-space model. This model depicts faithfully the dynamical evolution of the flow in the neighborhood of an IS, reduces the dimension of the state and enables access to linear control theory.

It is now possible to build a full-information optimal control actuating via wall-transpiration and targeting the unstable eigenmodes of EQ1. Analytically, it was demonstrated that these modes are controllable with this actuation type, and that consequently, EQ1 is stabilizable. Within linear simulations, EQ1 was successfully stabilized. Yet, the stabilization was not achieved for the non-linear case. Further research would be needed to conclude on this limitation.

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Identifiers

ORCID for Ati Sharma: orcid.org/0000-0002-7170-1627

ORCID for Davide Lasagna: orcid.org/0000-0002-6501-6041

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