Spectral analysis of nonlinear flows

2009/11/18 by CLARENCE W. ROWLEY, Clarence W. Rowley, Igor Mezić +7 · 79 citations
Physics and Astronomy · Engineering · #Model Reduction and Neural Networks #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis

paper · doi:10.1017/s0022112009992059

Abstract

We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn ( Sixty-First Annual Meeting of the APS Division of Fluid Dynamics , 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

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