Cluster-based reduced-order modelling of a mixing layer

By Eurika Kaiser, Bernd R. Noack, Laurent Cordier, A. Spohn, Marc Segond, Markus Abel, Guillaume Daviller, J. Osth, Sinivsa Krajnovi'c, R. K. N. I. P. A. GmbH and 5 more. Journal of Fluid Mechanics, 2013

315 citations recorded by the source corpus.

Abstract We propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger’s group (Burkardt, Gunzburger & Lee, Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337–355) and transition matrix models introduced in fluid dynamics in Eckhardt’s group (Schneider, Eckhardt & Vollmer, Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam–Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron–Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Subfields: Aerospace Engineering, Civil and Structural Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/59577b1d3b215d473413281a9491c8a09053052b

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Citations in the corpus

Machine Learning Control – Taming Nonlinear Dynamics and Turbulence (2016, unknown): “Cluster-based reduced-order models (CROMs) are especially promising and have recently been developed in fluids [154], building on cluster analysis [49] and transition matrix models [240].”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, methodology): “Alternatively, a probabilistic approach can be adopted to model and control fluid flows, such as cluster-based reduced order modeling (CROM) (Kaiser et al. 2014, 2017).”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “…operator (Froyland and Padberg, 2009), cluster reduced-order models based on probabilistic transition between various system behaviors (Kaiser et al., 2014), and methods for uncertainty quantification and subspace analysis in turbulent flows and the climate (Majda and Harlim, 2007;…”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Data-driven approaches have been used to characterize turbulent systems [9], including clustering [90], network theory [91], [92], DMD-based model reduction [93], [94], and local POD subspaces [95], to name a few.”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, unknown): “184 The transition dynamics between these centroids can be modeled as a probabilistic Markov model 185–187 or a deterministic-stochastic network model.”
Enhancing computational fluid dynamics with machine learning (2021, methodology): “Cluster reduced-order modeling (CROM) [114] is a simple and powerful unsupervised-learning approach that de-composes a time series into a few representative clusters, and then models the transition-state probability between clusters.”
Constrained sparse Galerkin regression (2016, background): “In contrast, data-driven approaches are becoming increasingly popular and encompass a variety of different techniques such as the eigensystem realization algorithm (ERA) (Juang & Pappa 1985), dynamic mode decomposition (DMD) (Rowley et al. 2009; Schmid 2010; Kutz et al. 2016), Koopman theory (Mezić”
A localized reduced-order modeling approach for PDEs with bifurcating solutions (2018, methodology): “In [21, 31], k-means clustering and nearest neighbor classifier with respect to parameters or a low-dimensional representation of the current state are used.”
Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems (2013, methodology): “A recent method that combines transition matrix models with dynamic regimes that cluster, the so-called cluster reduced-order modeling (CROM), is also interesting for these multiregime problems [29].”