Control of the channel flow past a cylinder by a piezo-actuated flexible splitter plate

We study the numerical stabilization around an unstable steady solution of a typical fluid-structure interaction problem constituted by a circular cylinder with a flexible splitter plate (Turek and Hron, 2006) actuated by piezoelectric devices and immersed in a fully developed, laminar channel flow. We define a linear feedback control that can locally stabilize the fully coupled nonlinear system. The feedback is based on a spectral decomposition of a non-standard Differential Algebraic Equation resulting from a monolithic Arbitrary Lagrangian Eulerian Finite Element formulation where a simple model of the piezoelectric patches is considered. By projecting the full system on its unstable subspace, a Reduced Order Model is defined. The design of the controlled system exploits the computation of the unstable direct and adjoint subspaces to identify the number and distribution of the patches on the beam. Moreover, the feasibility of such a controller for a real application is assessed by looking at the saturation limit of the control input. This paper is an extension of the methodology presented in Airiau et al. (2017) and Fournié et al. (2019) to control the Navier-Stokes equations to a fluid-structure model actuated by macro-fiber composites. To our knowledge, such active controls are original and the numerical tests presented validate their promising potential.