Journal Article

Koopman-mode decomposition of the cylinder wake

Bagheri, Shervin

2013

Overview

The Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow trajectory. In this paper, we perform a Koopman analysis of the first Hopf bifurcation of the flow past a circular cylinder. First, we decompose the flow into a sequence of Koopman modes, where each mode evolves in time with one single frequency/growth rate and amplitude/phase, corresponding to the complex eigenvalues and eigenfunctions of the Koopman operator, respectively. The analytical construction of these modes shows how the amplitudes and phases of nonlinear global modes oscillating with the vortex shedding frequency or its harmonics evolve as the flow develops and later sustains self-excited oscillations. Second, we compute the dynamic modes using the dynamic mode decomposition (DMD) algorithm, which fits a linear combination of exponential terms to a sequence of snapshots spaced equally in time. It is shown that under certain conditions the DMD algorithm approximates Koopman modes, and hence provides a viable method to decompose the flow into saturated and transient oscillatory modes. Finally, the relevance of the analysis to frequency selection, global modes and shift modes is discussed.

• Publisher: Cambridge University Press
• Subject: algorithm / Algorithms / bifurcation / Circular cylinders / cylinder / Dynamic mode decompositions / eigenvalue / Eigenvalues and eigenfunctions / Exact sciences and technology / Fluid dynamics / Frequency selection / Fundamental areas of phenomenology (including applications) / Hopf bifurcation / Hydrodynamic stability / instability / Linear combinations / Linear techniques / Nonlinear analysis / nonlinear dynamical systems / nonlinearity / Nonlinearity (including bifurcation theory) / Oscillating flow / Oscillatory mode / Physics / Plasma stability / Rotational flow and vorticity / Self excited oscillation / Separated flows / Single frequency / trajectory / vortex shedding / Vortex shedding frequency / wake