A reduced-order approach for optimal control of fluids using proper orthogonal decomposition

Abstract

In this article, a reduced-order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced-order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced-order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward-facing step is considered. Reduced-order models/low-dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.