Reduced-Order Models for Optimal Control of Vortex Shedding

Abstract

For an active flow control, proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the flow past a circular cylinder. Snapshots of the flo w field are recorded from the two-dimensional simulations of incompressible Navier-Stokes equations. Suction on the cylinder surface is the control mechanism to suppress vortex shedding. Feedback linearization approach is used to reduce the complex model to a linear-time-invariant system. Optimal control theory is implemented on the reduced-order model using a control function method. Appropriate gains are computed to reduce the hydrodynamic forces on the cylinder. 1 Introduction Feedback control of fluid-structure interaction is of pract ical importance from the perspective of wake modification and vortex-induced vibrations (VI V) reduction. Therefore, the development of design models and computational schemes for controlling vortex production and its shedding pattern has received considerable int erest in research as well as in industry. Despite recent progress in CFD and parallel computing, flow control through a computational approach still remains a formidable task. Application of control on systems described by PDEs is a challenging problem. A typical example is the control of fluid dynamical systems in which the Navier-Stokes equations are the state equations. To apply a control technique, one often reduces the partial-differen tial equations (PDEs) to ordinarydifferential equation (ODEs) to simplify the complexity of the dynamical system. For control design purposes, the POD approach enables modeling the Navier-Stokes equations as a set of ODEs. We use the CFD data of the flow past a circular cyl inder and compute the POD eigenfunctions from the velocity and pressure fields. Th e Navier-Stokes equations are then projected onto a low-dimensional space to develop a reduced-order model. Control of the flow over a circular cylinder has received consi derable attention because of its canonical nature and being a typical unstable flow. The re are various control mechanisms which have been employed on a circular cylinder to suppress vortex shedding. These methods include cylinder rotation, transverse motion, blowing/suction on the cylinder surface, and acoustic actuation. The focus of this work is to develop a reduced-order model and apply an optimal control strategy to suppress the vortex shedding. POD based reduced-order models have been successfully implemented for various control strategies. Graham et al. (1999a,b) developed a reduced-order model for the flow past a circular cylinder at Re = 100 using numerical simulations and the control action was achieved by cylinder rotation. They suggested two approaches to incorporate variable input control in the reduced-order model. In the first approach, kn own as the “control function method”, a suitable control function is included in the velocity expansion to account for the inhomogeneous boundary conditions on the cylinder surface. The POD modes, used in the modified expansion, retain the homogeneous boundary condit ions. In the second approach, the “penalty method”, the velocity expansion remains the same as for the unactuated flow and the essential boundary condition is enforced in an integral “weak” fashion. Thus, to