Koopman-mode decomposition of the cylinder wake

2013 ◽  
Vol 726 ◽  
pp. 596-623 ◽  
Author(s):  
Shervin Bagheri
Keyword(s):  
Dynamic Mode Decomposition ◽  
Nonlinear Flow ◽  
Single Frequency ◽  
Dynamic Mode ◽  
Cylinder Wake ◽  
Koopman Operator ◽  
Mode Decomposition ◽  
Analytical Construction ◽  
Vortex Shedding Frequency ◽  
Global Modes

AbstractThe 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.

scholarly journals Recursive dynamic mode decomposition of transient and post-transient wake flows

2016 ◽  
Vol 809 ◽  
pp. 843-872 ◽  
Author(s):  
Bernd R. Noack ◽  
Witold Stankiewicz ◽  
Marek Morzyński ◽  
Peter J. Schmid
Keyword(s):  
Dynamic Mode Decomposition ◽  
Attractive Alternative ◽  
Dynamic Mode ◽  
Cylinder Wake ◽  
Frequency Content ◽  
Mode Decomposition ◽  
Nonlinear Transient ◽  
Galerkin Models ◽  
Nonlinear Transient Dynamics ◽  
Proper Orthogonal

A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The $N$th mode is defined recursively in an analogous manner based on the residual of an expansion using the first $N-1$ modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.

scholarly journals A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

2015 ◽  
Vol 25 (6) ◽  
pp. 1307-1346 ◽  
Author(s):  
Matthew O. Williams ◽  
Ioannis G. Kevrekidis ◽  
Clarence W. Rowley
Keyword(s):  
Dynamic Mode Decomposition ◽  
Data Driven ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Mode Decomposition

scholarly journals Detecting regime transitions in time series using dynamic mode decomposition

2020 ◽  
Author(s):  
Georg Gottwald ◽  
Federica Gugole
Keyword(s):  
Time Series ◽  
Reconstruction Error ◽  
Reanalysis Data ◽  
Dynamic Mode Decomposition ◽  
Transient Dynamics ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Regime Changes ◽  
Fast Relaxation ◽  
Mode Decomposition

<p>We employ the framework of the Koopman operator and dynamic mode decomposition to devise a computationally cheap and easily implementable method to detect transient dynamics and regime changes in time series. We argue that typically transient dynamics experiences the full state space dimension with subsequent fast relaxation towards the attractor. In equilibrium, on the other hand, the dynamics evolves on a slower time scale on a lower dimensional attractor. The reconstruction error of a dynamic mode decomposition is used to monitor the inability of the time series to resolve the fast relaxation towards the attractor as well as the effective dimension of the dynamics. We illustrate our method by detecting transient dynamics in the Kuramoto-Sivashinsky equation. We further apply our method to atmospheric reanalysis data; our diagnostics detects the transition from a predominantly negative North Atlantic Oscillation (NAO) to a predominantly positive NAO around 1970, as well as the recently found regime change in the Southern Hemisphere atmospheric circulation around 1970.</p>

scholarly journals Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

2018 ◽  
Vol 847 ◽  
pp. 821-867 ◽  
Author(s):  
Aaron Towne ◽  
Oliver T. Schmidt ◽  
Tim Colonius
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Landau Equation ◽  
Orthogonal Decomposition ◽  
Dynamic Mode Decomposition ◽  
Single Frequency ◽  
Dynamic Mode ◽  
Stationary Flows ◽  
Mode Decomposition ◽  
Expansion Coefficients ◽  
Proper Orthogonal

We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

scholarly journals Dynamic reconstruction and data reconstruction for subsampled or irregularly sampled data

2017 ◽  
Vol 825 ◽  
pp. 133-166
Author(s):  
Jakub Krol ◽  
Andrew Wynn
Keyword(s):  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Decomposition Algorithms ◽  
Cylinder Wake ◽  
Sampled Data ◽  
Mode Decomposition ◽  
Image Velocimetry ◽  
Flow Evolution ◽  
Irregularly Sampled Data ◽  
Low Order

The Nyquist–Shannon criterion indicates the sample rate necessary to identify information with particular frequency content from a dynamical system. However, in experimental applications such as the interrogation of a flow field using particle image velocimetry (PIV), it may be impracticable or expensive to obtain data at the desired temporal resolution. To address this problem, we propose a new approach to identify temporal information from undersampled data, using ideas from modal decomposition algorithms such as dynamic mode decomposition (DMD) and optimal mode decomposition (OMD). The novel method takes a vector-valued signal, such as an ensemble of PIV snapshots, sampled at random time instances (but at sub-Nyquist rate) and projects onto a low-order subspace. Subsequently, dynamical characteristics, such as frequencies and growth rates, are approximated by iteratively approximating the flow evolution by a low-order model and solving a certain convex optimisation problem. The methodology is demonstrated on three dynamical systems, a synthetic sinusoid, the cylinder wake at Reynolds number $Re=60$ and turbulent flow past the axisymmetric bullet-shaped body. In all cases the algorithm correctly identifies the characteristic frequencies and oscillatory structures present in the flow.

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