Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models

In this paper, optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime ( Re = 200 ). The control parameters are the amplitude and the frequency of the time-harmonic cylinder rotation. In order to reduce the size of the discretized optimality system, a proper orthogonal decomposition (POD) reduced-order model (ROM) is derived to be used as state equation. We then propose to employ the trust-region proper orthogonal decomposition (TRPOD) approach, originally introduced by Fahl [M. Fahl, Trust-region methods for flow control based on reduced order modeling, Ph.D. Thesis, Trier University, 2000], to update the reduced-order models during the optimization process. A lot of computational work is saved because the optimization process is now based only on low-fidelity models. A particular care was taken to derive a POD ROM for the pressure and velocity fields with an appropriate balance between model accuracy and robustness. The key enablers are the extension of the POD basis functions to the pressure data, the use of calibration methods for the POD ROM and the addition in the POD expansion of several non-equilibrium modes to describe various operating conditions. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% at reduced cost.