scholarly journals Koopman mode expansions between simple invariant solutions

2019 ◽  
Vol 879 ◽  
pp. 1-27 ◽  
Author(s):  
Jacob Page ◽  
Rich R. Kerswell
Keyword(s):  
State Space ◽  
Stokes Equations ◽  
Time Window ◽  
Landau Equation ◽  
Dynamic Mode Decomposition ◽  
Navier Stokes ◽  
Dynamic Mode ◽  
Invariant Solutions ◽  
Mode Decomposition ◽  
Heteroclinic Connection

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.

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.

Modal Analysis of Wake behind Stationary and Vibrating Cylinders

2020 ◽  
pp. 1-24
Author(s):  
Marek Janocha ◽  
Guang Yin ◽  
Muk Chen Ong
Keyword(s):  
Reynolds Number ◽  
Modal Analysis ◽  
Reduced Order Model ◽  
Dynamic Mode Decomposition ◽  
Navier Stokes ◽  
Dynamic Mode ◽  
Order Model ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Sst Turbulence Model

Abstract The Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) are used to analyze the coherent structures of turbulent flow around vibrating isolated and piggyback cylinders configurations subjected to a uniform flow at a laminar Reynolds number (Re=200) and a upper transition Reynolds number (Re=3.6×106). Numerical simulations using two-dimensional URANS (Unsteady Reynolds Averaged Navier-Stokes) approach with the k-omega SST turbulence model are used to obtain the flow fields snapshots for the analysis. The wake flows behind the cylinders are decomposed into energy optimal modes (POD modes) and dynamical relevant modes (DMD modes). A reduced-order model for the flow is built based on the modal analysis. A comparison of POD and DMD is performed to characterize their special features. The present study provides new insights into the flow physics of fluid-structure interaction problem of two coupled cylinders. The characteristic vortex shedding frequencies and their harmonics are identified by DMD modes in all the investigated configurations. It is observed that for single cylinder configurations the most energetic and the most dynamically important mode is associated with the fundamental shedding frequency. For the stationary piggyback configuration, the gap flow between the cylinders appears to be a dominant flow feature as evidenced by leading DMD modes. The cylinder vibration increases significantly number of modes necessary to obtain a reduced order model (ROM) at given level of accuracy compared to respective stationary configurations.

Experimental Measurement of In-Plane Rolling Nonpneumatic Tire Vibrations Using High-Speed Imaging

2019 ◽  
Vol 47 (3) ◽  
pp. 196-210
Author(s):  
Meghashyam Panyam ◽  
Beshah Ayalew ◽  
Timothy Rhyne ◽  
Steve Cron ◽  
John Adcox
Keyword(s):  
High Speed ◽  
Image Correlation ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
High Speed Imaging ◽  
Correlation Algorithm ◽  
Mode Decomposition ◽  
Series Of Experiments ◽  
Plane Deformations ◽  
Dominant Frequencies

ABSTRACT This article presents a novel experimental technique for measuring in-plane deformations and vibration modes of a rotating nonpneumatic tire subjected to obstacle impacts. The tire was mounted on a modified quarter-car test rig, which was built around one of the drums of a 500-horse power chassis dynamometer at Clemson University's International Center for Automotive Research. A series of experiments were conducted using a high-speed camera to capture the event of the rotating tire coming into contact with a cleat attached to the surface of the drum. The resulting video was processed using a two-dimensional digital image correlation algorithm to obtain in-plane radial and tangential deformation fields of the tire. The dynamic mode decomposition algorithm was implemented on the deformation fields to extract the dominant frequencies that were excited in the tire upon contact with the cleat. It was observed that the deformations and the modal frequencies estimated using this method were within a reasonable range of expected values. In general, the results indicate that the method used in this study can be a useful tool in measuring in-plane deformations of rolling tires without the need for additional sensors and wiring.

Oscillatory flow around a vertical wall-mounted cylinder: Dynamic mode decomposition

2021 ◽  
Vol 33 (2) ◽  
pp. 025113
Author(s):  
H. K. Jang ◽  
C. E. Ozdemir ◽  
J.-H. Liang ◽  
M. Tyagi
Keyword(s):  
Oscillatory Flow ◽  
Vertical Wall ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Mode Decomposition

Modeling the Turbulent Wake Behind a Wall-Mounted Square Cylinder

2020 ◽  
Author(s):  
Christian Amor ◽  
José M Pérez ◽  
Philipp Schlatter ◽  
Ricardo Vinuesa ◽  
Soledad Le Clainche
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Turbulent Wake ◽  
Orthogonal Decomposition ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Square Cylinder ◽  
Complex Flow ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Proper Orthogonal

Abstract This article introduces some soft computing methods generally used for data analysis and flow pattern detection in fluid dynamics. These techniques decompose the original flow field as an expansion of modes, which can be either orthogonal in time (variants of dynamic mode decomposition), or in space (variants of proper orthogonal decomposition) or in time and space (spectral proper orthogonal decomposition), or they can simply be selected using some sophisticated statistical techniques (empirical mode decomposition). The performance of these methods is tested in the turbulent wake of a wall-mounted square cylinder. This highly complex flow is suitable to show the ability of the aforementioned methods to reduce the degrees of freedom of the original data by only retaining the large scales in the flow. The main result is a reduced-order model of the original flow case, based on a low number of modes. A deep discussion is carried out about how to choose the most computationally efficient method to obtain suitable reduced-order models of the flow. The techniques introduced in this article are data-driven methods that could be applied to model any type of non-linear dynamical system, including numerical and experimental databases.

scholarly journals Analysis of transonic buffet using dynamic mode decomposition

2021 ◽  
Vol 62 (4) ◽  
Author(s):  
Antje Feldhusen-Hoffmann ◽  
Christian Lagemann ◽  
Simon Loosen ◽  
Pascal Meysonnat ◽  
Michael Klaas ◽  
...  
Keyword(s):  
Shock Wave ◽  
Flow Field ◽  
Acoustic Waves ◽  
Feedback Loop ◽  
Measurement Data ◽  
Dynamic Mode Decomposition ◽  
Recirculation Region ◽  
Dynamic Mode ◽  
Mode Decomposition ◽  
Transonic Buffet

AbstractThe buffet flow field around supercritical airfoils is dominated by self-sustained shock wave oscillations on the suction side of the wing. Theories assume that this unsteadiness is driven by a feedback loop of disturbances in the flow field downstream of the shock wave whose upstream propagating part is generated by acoustic waves. High-speed particle-image velocimetry measurements are performed to investigate this feedback loop in transonic buffet flow over a supercritical DRA 2303 airfoil. The freestream Mach number is $$M_{\infty } = 0.73$$ M ∞ = 0.73 , the angle of attack is $$\alpha = 3.5^{\circ }$$ α = 3 . 5 ∘ , and the chord-based Reynolds number is $${\mathrm{Re}}_{c} = 1.9\times 10^6$$ Re c = 1.9 × 10 6 . The obtained velocity fields are processed by sparsity-promoting dynamic mode decomposition to identify the dominant dynamic features contributing strongest to the buffet flow field. Two pronounced dynamic modes are found which confirm the presence of two main features of the proposed feedback loop. One mode is related to the shock wave oscillation frequency and its shape includes the movement of the shock wave and the coupled pulsation of the recirculation region downstream of the shock wave. The other pronounced mode represents the disturbances which form the downstream propagating part of the proposed feedback loop. The frequency of this mode corresponds to the frequency of the acoustic waves which are generated by these downstream traveling disturbances and which form the upstream propagating part of the proposed feedback loop. In this study, the post-processing, i.e., the DMD, is highlighted to substantiate the existence of this vortex mode. It is this vortex mode that via the Lamb vector excites the shock oscillations. The measurement data based DMD results confirm numerical findings, i.e., the dominant buffet and vortex modes are in good agreement with the feedback loop suggested by Lee. Graphic abstract

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