Arch Comput Methods Eng DOI 10.1007/s11831-012-9074-0 Computational Hydrodynamic Stability and Flow Control Based on Spectral Analysis of Linear Operators

Abstract This paper considers the analysis and control of fluid flows using tools from dynamical systems and con-trol theory. The employed tools are derived from the spec-tral analysis of various linear operators associated with the Navier–Stokes equations. Spectral decomposition of the lin-earized Navier-Stokes operator, the Koopman operator, the spatial correlation operator and the Hankel operator provide a means to gain physical insight into the dynamics of com-plex flows and enables the construction of low-dimensional models suitable for control design. Since the discretization of the Navier-Stokes equations often leads to very large-scale dynamical systems, matrix-free and in some cases iter-ative techniques have to be employed to solve the eigenvalue problem. The common theme of the numerical algorithms is the use of direct numerical simulations. The theory and the algorithms are exemplified on flow over a flat plate and a jet in crossflow, as prototypes for the laminar-turbulent transi-tion and three-dimensional vortex shedding.