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Cluster-based reduced-order modelling of a mixing layer

Published online by Cambridge University Press:  06 August 2014

Eurika Kaiser*
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43, rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Bernd R. Noack
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43, rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Laurent Cordier
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43, rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Andreas Spohn
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEAT, 43, rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Marc Segond
Ambrosys GmbH, Albert Einstein Strasse 1–5, D-14469 Potsdam, Germany
Markus Abel
Ambrosys GmbH, Albert Einstein Strasse 1–5, D-14469 Potsdam, Germany LEMTA, 2 Avenue de la Forêt de Haye, F-54518 Vandoeuvre-lès-Nancy, France Potsdam University, Institute for Physics and Astrophysics, Karl-Liebknecht Strasse 24–25, D-14476 Potsdam, Germany
Guillaume Daviller
CERFACS, 42 Avenue Gaspard Coriolis, F-31057 Toulouse CEDEX 01, France
Jan Östh
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Siniša Krajnović
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Robert K. Niven
School of Engineering and Information Technology, The University of New South Wales at ADFA, Canberra, Australian Capital Territory, 2600, Australia
Email address for correspondence:


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.

© 2014 Cambridge University Press 

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