2nd AIAA Flow Control Conference, June 28 – July 1, 2004, Portland, OR, USA Low-Dimensional Models For Feedback Flow Control. Part I: Empirical Galerkin models * Bernd R. Noack † Technical University of Berlin HF1, Straße des 17. Juni, 10623 Berlin, Germany Gilead Tadmor ‡ Northeastern University, 440 Dana Research Building, Boston, MA 02115, U.S.A. Marek Morzy´ nski § Poznan University of Technology, Piotrowo 3, 60-965 Poznan, Poland A flow model which is accessible to control design must combine low dimension, robust- ness and a simple structure with an ample dynamic range to cover controlled transients. Key enablers are reviewed in the context of empirical Galerkin models and are exemplified for incompressible shear flows. These enablers include ‘subgrid’ turbulence and pressure representations, hybrid models that combine multiple operating points, and actuation mod- els. The range of model validity is identified in terms of invariant manifolds which can be exploited by observer design and has to be respected by controller design. Nomenclature x Location ˆ e x ,ˆ e y ,ˆ e z unit vectors in x-, y-, and z-direction x,y,z Cartesian coordinate system Ω Domain y cyl Transverse position of the cylinder t Time u, v Velocity fields u 0 Base flow (mostly: averaged velocity field) u ′ Fluctuation of the velocity field u ′ inhom Wall-imposed fluctuation u ′ hom Free fluctuation (flow response) p Pressure field g α α-th volume force Γ α Amplitude of α-th volume force N Γ Number of volume forces Re Reynolds number ν Reciprocal of Reynolds number u i i-th actuation mode for i< 0; i-th expansion mode for i> 0 u [N] Galerkin approximation with N modes ∗ This paper was invited by the AIAA Architecture and Algorithms Working Group headed by Prof. David Williams and Dr. Andrzej Banaszuk † Research Engineer, Hermann-F¨ ottinger-Institute of Fluid Mechanics, AIAA member ‡ Professor, Communication & Digital Signal Processing Center, Department of Electrical and Computer Engineering, AIAA member § Professor, Institute of Combustion Engines and Basics of Machine Design Copyright c  2004 by B.R. Noack, G. Tadmor and M. Morzy´ nski. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 1 of 11 American Institute of Aeronautics and Astronautics Paper 2004-2408