Abstract
The objective is to demonstrate the use of reduced-order models (ROM) based on proper orthogonal decomposition (POD) to stabilize the flow over a vertically oscillating circular cylinder in the laminar regime (Reynolds number equal to 60). The 2D Navier-Stokes equations are first solved with a finite element method, in which the moving cylinder is introduced via an ALE method. Since in fluid-structure interaction, the POD algorithm cannot be applied directly, we implemented the fictitious domain method of Glowinski et al. [1] where the solid domain is treated as a fluid undergoing an additional constraint. The POD-ROM is classically obtained by projecting the Navier-Stokes equations onto the first POD modes. At this level, the cylinder displacement is enforced in the POD-ROM through the introduction of Lagrange multipliers. For determining the optimal vertical velocity of the cylinder, a linear quadratic regulator framework is employed. After linearization of the POD-ROM around the steady flow state, the optimal linear feedback gain is obtained as solution of a generalized algebraic Riccati equation. Finally, when the optimal feedback control is applied, it is shown that the flow converges rapidly to the steady state. In addition, a vanishing control is obtained proving the efficiency of the control approach.
Highlights
Method, in which the moving cylinder is introduced via an ALE method
The feedback control law that minimizes the value of J linear quadratic regulator (LQR) is c = Kz where the feedback gain K is found after solving a Riccati equation related to (2)
The velocity Vc and the position yc are chosen such that the system rst undergoes from t = 0 to t = 100 a transient regime from the steady state to an established unactuated state
Summary
The cylinder wake is a common generic conguration to test control methods which can be further implemented in more complex engineering applications. The stabilization of this ow consists in targeting and maintaining the unstable steady solution by control. Let u be the velocity eld and usteady be the targeted steady state, the objective is to determine the best cylinder vertical velocity Vc(t), taken as control parameter, such that we minimize. The role of the parameter α is to penalize too large cylinder displacements and to enforce the cylinder to stay near the central position, while β penalizes too strong control actions
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