Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T16:02:58.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Cluster-based reduced-order modelling of a mixing layer

Published online by Cambridge University Press:  06 August 2014

Eurika Kaiser ,
Bernd R. Noack ,
Laurent Cordier ,
Andreas Spohn ,
Marc Segond ,
Markus Abel ,
Guillaume Daviller ,
Jan Östh ,
Siniša Krajnović  and
Robert K. Niven
Eurika Kaiser*
Affiliation:
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
Affiliation:
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
Affiliation:
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
Affiliation:
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
Affiliation:
Ambrosys GmbH, Albert Einstein Strasse 1–5, D-14469 Potsdam, Germany
Markus Abel
Affiliation:
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
Affiliation:
CERFACS, 42 Avenue Gaspard Coriolis, F-31057 Toulouse CEDEX 01, France
Jan Östh
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Siniša Krajnović
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Robert K. Niven
Affiliation:
School of Engineering and Information Technology, The University of New South Wales at ADFA, Canberra, Australian Capital Territory, 2600, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Wakes/Jets: Shear layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 754 , 10 September 2014 , pp. 365 - 414
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, M., Biferale, L., Cencini, M., Falcioni, M., Vergni, D. & Vulpiani, A. 2000a Exit-time approach to $\epsilon $ -entropy. Phys. Rev. Lett. 84, 6002–6005.Google Scholar
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, 12–35.Google Scholar
Afraimovich, V. S. & Shil’nikov, L. P. 1983 Nonlinear Dynamics and Turbulence. Pitmen.Google Scholar
Ahmed, S. R., Ramm, G. & Faltin, G. 1984 Some Salient Features of the Time Averaged Ground Vehicle Wake. Society of Automotive Engineers; 840300.CrossRefGoogle Scholar
Antoulas, A. C. 2005 Approximation of Large-Scale Dynamical Systems. Society for Industrial and Applied Mathematics.Google Scholar
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. 1027–1035. Society for Industrial and Applied Mathematics.Google Scholar
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, 115–173.Google Scholar
Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623.Google Scholar
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, 1–27.Google Scholar
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.Google Scholar
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, 7813–7840.Google Scholar
Bishop, C. M. 2007 Pattern Recognition and Machine Learning. Springer.Google Scholar
Bollt, E. M. & Santitissadeekorn, N. 2013 Applied and Computational Measurable Dynamics. Society for Industrial and Applied Mathematics.Google Scholar
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76 (1), 127–144.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
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, 337–355.CrossRefGoogle Scholar
Burkardt, J., Gunzburger, M. & Lee, H.-C. 2006b Centroidal Voronoi tessellation-based reduced-order modeling of complex systems. SIAM J. Sci. Comput. 28 (2), 459–484.Google Scholar
Cacuci, D. G., Navon, I. M. & Ionescu-Bujor, M. 2013 Computational Methods for Data Evaluation and Assimilation. Chapman and Hall/CRC.Google Scholar
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), 4098–4113.CrossRefGoogle Scholar
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, 3–40.Google Scholar
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), 269–296.Google Scholar
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.).CrossRefGoogle Scholar
Cox, T. F. & Cox, M. A. A. 2000 Multidimensional Scaling, 2nd edn, Monographs on Statistics and Applied Probability, vol. 88. Chapman and Hall.Google Scholar
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G. 2012 Chaos: Classical and Quantum. Niels Bohr Institute.Google Scholar
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.Google Scholar
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, 2337–2354.CrossRefGoogle Scholar
Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78 (3), 535–560.Google Scholar
Du, Q., Faber, V. & Gunzburger, M. 1999 Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637–676.CrossRefGoogle Scholar
Fletcher, C. A. J. 1984 Computational Galerkin Methods, 1st edn. Springer.Google Scholar
Fowler, R. A. 1929 Statistical Mechanics. Cambridge University Press.Google Scholar
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), 223–247.CrossRefGoogle Scholar
Gaspard, P., Nicolis, G., Provata, A. & Tasaki, S. 1995 Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51, 74–94.CrossRefGoogle ScholarPubMed
Gonchenko, S. V., Shil’nikov, L. P. & Turaev, D. V. 1997 Quasiattractors and homoclinic tangencies. Comput. Maths Applics. 34, 195–227.Google Scholar
Grandemange, M., Cadot, O. & Gohlke, M. 2012 Reflectional symmetry breaking of the separated flow over three-dimensional bluff bodies. Phys. Rev. E 86, 035302.CrossRefGoogle ScholarPubMed
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, 51–84.CrossRefGoogle Scholar
Hartigan, J. A. 1975 Clustering Algorithms. John Wiley & Sons.Google Scholar
Hastie, T., Tibshirani, R. & Friedman, J. 2009 The Elements of Statistical Learning. Data Mining, Inference, and Prediction, 2nd edn. Springer.Google Scholar
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 26–58.Google Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.Google Scholar
Hopf, E. 1952 Statistical hydromechanics and functional analysis. J. Rat. Mech. Anal. 1, 87–123.Google Scholar
Huang, S.-C. & Kim, J. 2008 Control and system identification of a separated flow. Phys. Fluids 20 (10), 101509.CrossRefGoogle Scholar
Hyvärinen, A. 2012 Independent component analysis: recent advances. Phil. Trans. R. Soc. A 371 (1984).Google Scholar
Kolmogorov, A. N. & Tikhomirov, V. M. 1959 $\varepsilon $ -entropy and $\varepsilon $ -capacity of sets in function spaces. Usp. Mat. Nauk 86, 3–86.Google Scholar
Kullback, S. 1959 Information Theory and Statistics, 1st edn. John Wiley & Sons.Google Scholar
Kullback, S. & Leibler, R. A. 1951 On information and sufficiency. Ann. Math. Statist. 22, 79–86.CrossRefGoogle Scholar
Kutz, J. N. 2013 Data-Driven Modeling and Scientific Computation: Methods for Complex Systems and Big Data. Oxford University Press.Google Scholar
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow, 1st edn. Gordon and Breach.Google Scholar
Laizet, S., Lardeau, L. & Lamballais, E. 2010 Direct numerical simulation of a mixing layer downstream a thick splitter plate. Phys. Fluids 22 (1).Google Scholar
Langville, A. N. & Meyer, C. D. 2012 Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press.Google Scholar
Lasota, A. & Mackey, M. C. 1994 Chaos, Fractals, and Noise, 2nd edn. Springer.Google Scholar
Li, T. Y. 1976 Finite approximation for the Frobenius–Perron operator: a solution to Ulam’s conjecture. J. Approx. Theory 17 (2), 177–186.CrossRefGoogle Scholar
Lienhart, H. & Becker, S.2003 Flow and turbulent structure in the wake of a simplified car model. SAE Paper (2003-01-0656).Google Scholar
Lloyd, S. 1956 Least squares quantization in PCM. IEEE Trans. Inf. Theory 28, 129–137; originally as an unpublished Bell laboratories Technical Note (1957).Google Scholar
Lorenz, E. N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
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, 283–316.Google Scholar
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.Google Scholar
Mardia, K. V., Kent, J. T. & Bibby, J. M. 1979 Multivariate Analysis. Academic Press.Google Scholar
Meyer, C. D. 2000 Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics.Google Scholar
Mezic, I. & Wiggins, S. 1999 A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9 (1), 213–218.Google Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223–252.Google Scholar
Moore, B. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 17–32.Google Scholar
Murphy, K. P. 2012 Machine Learning: A Probabilistic Perspective. MIT Press.Google Scholar
Niven, R. K. 2009 Combinatorial entropies and statistics. Eur. Phys. J. B 70 (1), 49–63.CrossRefGoogle Scholar
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, 335–363.Google Scholar
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528. Springer-Verlag.CrossRefGoogle Scholar
Noack, B. R. & Niven, R. K. 2012 Maximum-entropy closure for a Galerkin system of incompressible shear flow. J. Fluid Mech. 700, 187–213.CrossRefGoogle Scholar
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, 339–365.Google Scholar
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.Google Scholar
Norris, J. R. 1998 Markov Chains. Cambridge University Press.Google Scholar
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, 518–544.Google Scholar
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, 161–196.Google Scholar
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, 1–29.Google Scholar
Rempfer, D. 2006 On boundary conditions for incompressible Navier–Stokes problems. Appl. Mech. Rev. 59, 107–125.CrossRefGoogle Scholar
Rempfer, D. & Fasel, F. H. 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257–283.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 997–1013.Google Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115–127.Google Scholar
Santitissadeekorn, N. & Bollt, E. M. 2007 Identifying stochastic basin hopping by partitioning with graph modularity. Physica D 231, 95–107.Google Scholar
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, 367–398.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition for numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Schneider, T. M., Eckhardt, B. & Vollmer, J. 2007 Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.Google Scholar
Simovici, D. A. & Djeraba, C. 2008 Mathematical Tools for Data Mining – Set Theory, Partial Orders, Combinatorics. Springer-Verlag.CrossRefGoogle Scholar
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, 251–276.CrossRefGoogle Scholar
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, 1st edn. Applied Mathematical Sciences, vol. 41. Springer-Verlag.Google Scholar
Steinhaus, H. 1956 Sur la division des corps matériels en parties. Bull. Acad. Polon. Sci. 4 (12), 801–804.Google Scholar
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, 411–423.Google Scholar
Vishik, M. I. & Fursikov, A. V. 1988 Mathematical Problems of Statistical Hydrodynamics. Kluwer.Google Scholar