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Cluster-based reduced-order modelling of a mixing layer

  • Eurika Kaiser (a1), Bernd R. Noack (a1), Laurent Cordier (a1), Andreas Spohn (a1), Marc Segond (a2), Markus Abel (a2) (a3) (a4), Guillaume Daviller (a5), Jan Östh (a6), Siniša Krajnović (a6) and Robert K. Niven (a7)...

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.

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Abel, M., Biferale, L., Cencini, M., Falcioni, M., Vergni, D. & Vulpiani, A. 2000a Exit-time approach to $\epsilon $ -entropy. Phys. Rev. Lett. 84, 60026005.
Abel, M., Biferale, L., Cencini, M., Falcioni, M., Vergni, D. & Vulpiani, A. 2000b Exit-times and $\epsilon $ -entropy for dynamical systems, stochastic processes, and turbulence. Physica 147, 1235.
Afraimovich, V. S. & Shil’nikov, L. P. 1983 Nonlinear Dynamics and Turbulence. Pitmen.
Ahmed, S. R., Ramm, G. & Faltin, G. 1984 Some Salient Features of the Time Averaged Ground Vehicle Wake. Society of Automotive Engineers; 840300.
Antoulas, A. C. 2005 Approximation of Large-Scale Dynamical Systems. Society for Industrial and Applied Mathematics.
Arthur, D. & Vassilvitskii, S. 2007 $k$ -means $++$ : the advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, pp. 10271035. Society for Industrial and Applied Mathematics.
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115173.
Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596623.
Bagheri, S., Hoepffner, J., Schmid, P. J. & Henningson, D. S. 2008 Input–output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62, 127.
Ball, G. & Hall, D.1965 ISODATA, a novel method of data anlysis and pattern classification. Tech. Rep. NTIS AD 699616. Stanford Research Institute, Stanford, CA.
Bergmann, M. & Cordier, L. 2008 Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced order models. J. Comput. Phys. 227, 78137840.
Bishop, C. M. 2007 Pattern Recognition and Machine Learning. Springer.
Bollt, E. M. & Santitissadeekorn, N. 2013 Applied and Computational Measurable Dynamics. Society for Industrial and Applied Mathematics.
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76 (1), 127144.
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775816.
Burkardt, J., Gunzburger, M. & Lee, H. C. 2006a POD and CVT-based reduced-order modeling of Navier–Stokes flows. Comput. Meth. Appl. Mech. Engng 196, 337355.
Burkardt, J., Gunzburger, M. & Lee, H.-C. 2006b Centroidal Voronoi tessellation-based reduced-order modeling of complex systems. SIAM J. Sci. Comput. 28 (2), 459484.
Cacuci, D. G., Navon, I. M. & Ionescu-Bujor, M. 2013 Computational Methods for Data Evaluation and Assimilation. Chapman and Hall/CRC.
Cavalieri, A., Daviller, G., Comte, P., Jordan, P., Tadmor, G. & Gervais, Y. 2011 Using large eddy simulation to explore sound-source mechanisms in jets. J. Sound Vib. 330 (17), 40984113.
Chiang, M. M.-T. & Mirkin, B. 2010 Intelligent choice of the number of clusters in K-means clustering: an experimental study with different cluster spreads. J. Classification 27, 340.
Cordier, L., Abou El Majd, B. & Favier, J. 2010 Calibration of POD reduced-order models using Tikhonov regularization. Intl J. Numer. Meth. Fluids 63 (2), 269296.
Cordier, L., Noack, B. R., Tissot, G., Lehnasch, G., Delvile, J., Balajewicz, M., Daviller, G. & Niven, R. K. 2013 Control-oriented model identification strategy. Exp. Fluids 54 (8), 1580 Invited paper for the Special Issue ‘Flow Control’ (ed. Bonnet, J.-P. & Cattafesta, L.).
Cox, T. F. & Cox, M. A. A. 2000 Multidimensional Scaling, 2nd edn, Monographs on Statistics and Applied Probability, vol. 88. Chapman and Hall.
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G. 2012 Chaos: Classical and Quantum. Niels Bohr Institute.
Daviller, G.2010 Étude numérique des effets de température dans les jets simples et coaxiaux. PhD thesis, École Nationale Supérieure de Mécanique et d’Aérotechnique.
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3, 23372354.
Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78 (3), 535560.
Du, Q., Faber, V. & Gunzburger, M. 1999 Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637676.
Fletcher, C. A. J. 1984 Computational Galerkin Methods, 1st edn. Springer.
Fowler, R. A. 1929 Statistical Mechanics. Cambridge University Press.
Froyland, G., Junge, O. & Koltai, P. 2013 Estimating long-term behavior of flows without trajectory integration: the infinitesimal generator approach. SIAM J. Numer. Anal. 51 (1), 223247.
Gaspard, P., Nicolis, G., Provata, A. & Tasaki, S. 1995 Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51, 7494.
Gonchenko, S. V., Shil’nikov, L. P. & Turaev, D. V. 1997 Quasiattractors and homoclinic tangencies. Comput. Maths Applics. 34, 195227.
Grandemange, M., Cadot, O. & Gohlke, M. 2012 Reflectional symmetry breaking of the separated flow over three-dimensional bluff bodies. Phys. Rev. E 86, 035302.
Grandemange, M., Gohlke, M. & Cadot, O. 2013 Turbulent wake past a three-dimensional blunt body. Part 1. Global modes and bi-stability. J. Fluid Mech. 722, 5184.
Hartigan, J. A. 1975 Clustering Algorithms. John Wiley & Sons.
Hastie, T., Tibshirani, R. & Friedman, J. 2009 The Elements of Statistical Learning. Data Mining, Inference, and Prediction, 2nd edn. Springer.
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 2658.
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.
Hopf, E. 1952 Statistical hydromechanics and functional analysis. J. Rat. Mech. Anal. 1, 87123.
Huang, S.-C. & Kim, J. 2008 Control and system identification of a separated flow. Phys. Fluids 20 (10), 101509.
Hyvärinen, A. 2012 Independent component analysis: recent advances. Phil. Trans. R. Soc. A 371 (1984).
Kolmogorov, A. N. & Tikhomirov, V. M. 1959 $\varepsilon $ -entropy and $\varepsilon $ -capacity of sets in function spaces. Usp. Mat. Nauk 86, 386.
Kullback, S. 1959 Information Theory and Statistics, 1st edn. John Wiley & Sons.
Kullback, S. & Leibler, R. A. 1951 On information and sufficiency. Ann. Math. Statist. 22, 7986.
Kutz, J. N. 2013 Data-Driven Modeling and Scientific Computation: Methods for Complex Systems and Big Data. Oxford University Press.
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow, 1st edn. Gordon and Breach.
Laizet, S., Lardeau, L. & Lamballais, E. 2010 Direct numerical simulation of a mixing layer downstream a thick splitter plate. Phys. Fluids 22 (1).
Langville, A. N. & Meyer, C. D. 2012 Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press.
Lasota, A. & Mackey, M. C. 1994 Chaos, Fractals, and Noise, 2nd edn. Springer.
Li, T. Y. 1976 Finite approximation for the Frobenius–Perron operator: a solution to Ulam’s conjecture. J. Approx. Theory 17 (2), 177186.
Lienhart, H. & Becker, S.2003 Flow and turbulent structure in the wake of a simplified car model. SAE Paper (2003-01-0656).
Lloyd, S. 1956 Least squares quantization in PCM. IEEE Trans. Inf. Theory 28, 129137; originally as an unpublished Bell laboratories Technical Note (1957).
Lorenz, E. N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20, 130141.
Luchtenburg, D. M., Günter, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and actuated flows around a high-lift configuration. J. Fluid Mech. 623, 283316.
MacQueen, J.1967 Some methods for classification and analysis of multivariate observations. Proceedings of the Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 281–297.
Mardia, K. V., Kent, J. T. & Bibby, J. M. 1979 Multivariate Analysis. Academic Press.
Meyer, C. D. 2000 Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics.
Mezic, I. & Wiggins, S. 1999 A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9 (1), 213218.
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223252.
Moore, B. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 1732.
Murphy, K. P. 2012 Machine Learning: A Probabilistic Perspective. MIT Press.
Niven, R. K. 2009 Combinatorial entropies and statistics. Eur. Phys. J. B 70 (1), 4963.
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528. Springer-Verlag.
Noack, B. R. & Niven, R. K. 2012 Maximum-entropy closure for a Galerkin system of incompressible shear flow. J. Fluid Mech. 700, 187213.
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339365.
Noack, B. R., Pelivan, I., Tadmor, G., Morzyński, M. & Comte, P.2004 Robust low-dimensional Galerkin models of natural and actuated flows. In Fourth Aeroacoustics Workshop, pp. 0001–0012. RWTH Aachen, 26–27 February 2004.
Norris, J. R. 1998 Markov Chains. Cambridge University Press.
Östh, J., Noack, B. R., Krajnović, S., Barros, D. & Boreé, J. 2014 On the need for a nonlinear subscale turbulence term in pod models as exemplified for a high Reynolds number flow over an ahmed body. J. Fluid Mech. 747, 518544.
Pastoor, M., Henning, L., Noack, B. R., King, R. & Tadmor, G. 2008 Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161196.
Rajaee, M., Karlsson, S. K. F. & Sirovich, L. 1994 Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour. J. Fluid Mech. 258, 129.
Rempfer, D. 2006 On boundary conditions for incompressible Navier–Stokes problems. Appl. Mech. Rev. 59, 107125.
Rempfer, D. & Fasel, F. H. 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257283.
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 9971013.
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115127.
Santitissadeekorn, N. & Bollt, E. M. 2007 Identifying stochastic basin hopping by partitioning with graph modularity. Physica D 231, 95107.
Schlegel, M., Noack, B. R., Jordan, P., Dillmann, A., Gröschel, E., Schröder, W., Wei, M., Freund, J. B., Lehmann, O. & Tadmor, G. 2012 On least-order flow representations for aerodynamics and aeroacoustics. J. Fluid Mech. 697, 367398.
Schmid, P. J. 2010 Dynamic mode decomposition for numerical and experimental data. J. Fluid Mech. 656, 528.
Schneider, T. M., Eckhardt, B. & Vollmer, J. 2007 Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.
Simovici, D. A. & Djeraba, C. 2008 Mathematical Tools for Data Mining – Set Theory, Partial Orders, Combinatorics. Springer-Verlag.
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Rev. Mech. 63, 251276.
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, 1st edn. Applied Mathematical Sciences, vol. 41. Springer-Verlag.
Steinhaus, H. 1956 Sur la division des corps matériels en parties. Bull. Acad. Polon. Sci. 4 (12), 801804.
Tibshirani, R., Walther, G. & Hastie, T. 2001 Estimating the number of clusters in a data set via the gap statistics. J. R. Stat. Soc. B 63, 411423.
Vishik, M. I. & Fursikov, A. V. 1988 Mathematical Problems of Statistical Hydrodynamics. Kluwer.
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