Study of dynamics in post-transient flows using Koopman mode decomposition

Abstract

The Koopman Mode Decomposition (KMD) is a data-analysis technique which is often used to extract the spatio-temporal patterns of complex flows. In this paper, we use KMD to study the dynamics of the lid-driven flow in a two-dimensional square cavity based on theorems related to the spectral theory of the Koopman operator. We adapt two algorithms, from the classical Fourier and power spectral analysis, to compute the discrete and continuous spectrum of the Koopman operator for the post-transient flows. Properties of the Koopman operator spectrum are linked to the sequence of flow regimes occurring between Re = 10000 and Re = 30000, and changing the flow nature from steady to aperiodic. The Koopman eigenfunctions for different flow regimes, including flows with mixed spectra, are constructed using the assumption of ergodicity in the state space. The associated Koopman modes show remarkable robustness even as the temporal nature of the flow is changing substantially. We observe that KMD outperforms the proper orthogonal decomposition in reconstruction of the flows with strong quasi-periodic components.

Koopman analysis of Burgers equation

“…In response, several alternative data processing techniques have been proposed to extract Koopman modes from simulation/experimental data. For example, Sharma et al[10] demonstrated that a purely oscillatory Koopman mode with frequency m (a Fourier component of a limit cycle) is well described by the first response mode of the resolvent operator at the same frequency, while Arbabi and Mezić [11] have adapted techniques from signal processing to extract Koopman modes from chaotic, statistically stationary flows. Williams et al[12] modified the DMD algorithm ("extended dynamic mode decomposition" -EDMD) to make it better suited to identifying Koopman modes by populating the vectors ψ i with functionals of the state u. EDMD is a robust method for determining unknown Koopman eigenfunctions provided that (i) they can be expressed as a linear combination of the user-specified observables that make up the ψ i and (ii) sufficient data is available [8,9,12].Choosing a suitable set of functionals for ψ is non-trivial, and is made more challenging by the fact that Koopman eigenfunctions have only been determined analytically for low-dimensional ODEs [e.g.…”

“…[10] demonstrated that a purely oscillatory Koopman mode with frequency m (a Fourier component of a limit cycle) is well described by the first response mode of the resolvent operator at the same frequency, while Arbabi and Mezić [11] have adapted techniques from signal processing to extract Koopman modes from chaotic, statistically stationary flows. Williams et al…”

Searching turbulence for periodic orbits with dynamic mode decomposition

“…Applying a Koopman decomposition to a turbulent flow yields a representation of the state as a superposition of a set of harmonic averages and a broadband continuous spectrum (Mezic & Banaszuk 2004;Mezić 2005Mezić , 2013Arbabi & Mezić 2017), but individual simple invariant sets also possess their own local Koopman decompositions. For example, Mezic (2017) has shown that the Koopman eigenvalues for a nonlinear system collapsing onto a limit cycle consists of a set of repeated neutral harmonics (the limit cycle's fundamental frequency and higher harmonics) and an infinite lattice of decaying eigenvalues which can be determined from linear combinations of the cycle's Floquet multipliers.…”

Koopman mode expansions between simple invariant solutions

“…Alongside DMD, other related methods have been proposed to extract Koopman modes from turbulent flows that may circumvent some of these issues. For example, Arbabi & Mezić (2017) proposed an approach based on harmonic averaging to extract Koopman modes in high-Reynolds number lid-driven cavity flow, while Sharma et al (2016) demonstrated a connection between Koopman modes and modes of the resolvent operator.…”