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Metric for attractor overlap

Published online by Cambridge University Press:  12 July 2019

Rishabh Ishar
Affiliation:
Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 160012, India
Eurika Kaiser
Affiliation:
University of Washington, Department of Mechanical Engineering, Stevens Way, Box 352600, Seattle, WA 98195, USA
Marek Morzyński
Affiliation:
Poznań University of Technology, Chair of Virtual Engineering, Jana Pawla II 24, PL 60-965 Poznań, Poland
Daniel Fernex
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany
Richard Semaan
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany
Marian Albers
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany
Pascal S. Meysonnat
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany
Wolfgang Schröder
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstr. 5a, 52062 Aachen, Germany Forschungszentrum Jülich, JARA-High-Performance Computing, 52425 Jülich, Germany
Bernd R. Noack
Affiliation:
Institut für Strömungsmechanik, Technische Universität Braunschweig, Hermann-Blenk-Str. 37, 38108 Braunschweig, Germany LIMSI, CNRS, Université Paris-Saclay, Bât 507, rue du Belvédère, Campus Universitaire, F-91403 Orsay, France Institut für Strömungsmechanik und Technische Akustik (ISTA), Technische Universität Berlin, Müller-Breslau-Straße 8, 10623 Berlin, Germany Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology, Shenzhen Campus, 518055 Shenzhen, China

Abstract

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e. ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshot-to-snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by a few representative centroids. We employ MAO for two quite different actuated flow configurations: a two-dimensional wake with vortices in a narrow frequency range and three-dimensional wall turbulence with a broadband spectrum. In the first application, seven control laws are applied to the fluidic pinball, i.e. the two-dimensional flow around three circular cylinders whose centres form an equilateral triangle pointing in the upstream direction. These seven operating conditions comprise unforced shedding, boat tailing, base bleed, high- and low-frequency forcing as well as two opposing Magnus effects. In the second example, MAO is applied to three-dimensional simulation data from an open-loop drag reduction study of a turbulent boundary layer. The actuation mechanisms of 38 spanwise travelling transversal surface waves are investigated. MAO compares and classifies these actuated flows in agreement with physical intuition. For instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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Footnotes

Present address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. Email address for correspondence: rishabhishar.bemech14@pec.edu.in

References

Alkishriwi, N., Meinke, M. & Schröder, W. 2006 A large-eddy simulation method for low Mach number flows using preconditioning and multigrid. Comput. Fluids 35 (10), 11261136.CrossRefGoogle Scholar
Arthur, D. & Vassilvitskii, S. 2007 k-means + +: the advantages of careful seeding. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 10271035. Society for Industrial and Applied Mathematics.Google Scholar
Bansal, M. S. & Yarusevych, S. 2017 Experimental study of flow through a cluster of three equally spaced cylinders. Exp. Therm. Fluid Sci. 80, 203217.CrossRefGoogle Scholar
Barros, D., Borée, J., Noack, B. R., Spohn, A. & Ruiz, T. 2016 Bluff body drag manipulation using pulsed jets and Coanda effect. J. Fluid Mech. 805, 442459.CrossRefGoogle Scholar
Bearman, P. W. 1967 The effect of base bleed on the flow behind a two-dimensional model with a blunt trailing edge. Aeronaut. Q. 18 (03), 207224.CrossRefGoogle Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539575.CrossRefGoogle Scholar
Boris, J. P., Grinstein, F. F., Oran, E. S. & Kolbe, R. L. 1992 New insights into large eddy simulation. Fluid Dyn. Res. 10 (4–6), 199228.CrossRefGoogle Scholar
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.Google Scholar
Burkardt, J., Gunzburger, M. & Lee, H. C. 2006 POD and CVT-based reduced-order modeling of Navier–Stokes flows. Comput. Meth. Appl. Mech. Engng 196, 337355.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
Du, Y., Symeonidis, V. & Karniadakis, G. E. 2002 Drag reduction in wall-bounded turbulence via a transverse travelling wave. J. Fluid Mech. 457, 134.CrossRefGoogle Scholar
Duriez, T., Brunton, S. L. & Noack, B. R. 2016 Machine Learning Control – Taming Nonlinear Dynamics and Turbulence, (Fluid Mechanics and Its Applications) , vol. 116. Springer.Google Scholar
Endres, D. M. & Schindelin, J. E. 2003 A new metric for probability distributions. IEEE Trans. Inf. Theory 49, 18581860.CrossRefGoogle Scholar
García-Mayoral, R. & Jiménez, J. 2011 Hydrodynamic stability and breakdown of the viscous regime over riblets. J. Fluid Mech. 678, 317347.CrossRefGoogle Scholar
Geropp, D.1995 Process and device for reducing the drag in the rear region of a vehicle, for example, a road or rail vehicle or the like. United States Patent US 5407245 A.Google Scholar
Geropp, D. & Odenthal, H.-J. 2000 Drag reduction of motor vehicles by active flow control using the Coanda effect. Exp. Fluids 28 (1), 7485.CrossRefGoogle Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.CrossRefGoogle Scholar
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1997 An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 135 (2), 203216.CrossRefGoogle Scholar
Hu, J. & Zhou, Y. 2008a Flow structure behind two staggered circular cylinders. Part 1. Downstream evolution and classification. J. Fluid Mech. 607, 5180.CrossRefGoogle Scholar
Hu, J. & Zhou, Y. 2008b Flow structure behind two staggered circular cylinders. Part 2. Heat and momentum transport. J. Fluid Mech. 607, 81107.CrossRefGoogle Scholar
Itoh, M., Tamano, S., Yokota, K. & Taniguchi, S. 2006 Drag reduction in a turbulent boundary layer on a flexible sheet undergoing a spanwise traveling wave motion. J. Turbul. 7, N27.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.CrossRefGoogle Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wallbounded flows by highfrequency spanwise oscillations. Phys. Fluids A 4 (8), 16051607.CrossRefGoogle Scholar
Kaiser, E., Li, R. & Noack, B. R. 2017a On the control landscape topology. In The 20th World Congress of the International Federation of Automatic Control (IFAC), pp. 14.Google Scholar
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M. W., Daviller, G., Östh, J., Krajnović, S. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365414.CrossRefGoogle Scholar
Kaiser, E., Noack, B. R., Spohn, A., Cattafesta, L. N. & Morzyński, M. 2017b Cluster-based control of nonlinear dynamics. Theor. Comput. Fluid Dyn. 31 (5–6), 15791593.Google Scholar
Kasten, J., Reininghaus, J., Hotz, I., Hege, H.-C., Noack, B. R., Daviller, G., Comte, P. & Morzyński, M. 2016 Acceleration feature points of unsteady shear flows. Arch. Mech. 68, 5580.Google Scholar
Klumpp, S., Meinke, M. & Schröder, W. 2010a Numerical simulation of riblet controlled spatial transition in a zero-pressure-gradient boundary layer. Flow Turbul. Combust. 85 (1), 5771.CrossRefGoogle Scholar
Klumpp, S., Meinke, M. & Schröder, W. 2010b Drag reduction by spanwise transversal surface waves. J. Turbul. 11, N22.Google Scholar
Klumpp, S., Meinke, M. & Schröder, W. 2011 Friction drag variation via spanwise transversal surface waves. Flow Turbul. Combust. 87 (1), 3353.CrossRefGoogle Scholar
Koh, S. R., Meysonnat, P., Statnikov, V., Meinke, M. & Schröder, W. 2015a Dependence of turbulent wall-shear stress on the amplitude of spanwise transversal surface waves. Comput. Fluids 119, 261275.CrossRefGoogle Scholar
Koh, S. R., Meysonnat, P., Meinke, M. & Schröder, W. 2015b Drag reduction via spanwise transversal surface waves at high Reynolds numbers. Flow Turbul. Combust. 95 (1), 169190.CrossRefGoogle Scholar
Kullback, S. 1959 Information Theory and Statistics, 1st edn. John Wiley.Google Scholar
Kullback, S. & Leibler, R. A. 1951 On information and sufficiency. Ann. Math. Statist. 22, 7986.CrossRefGoogle Scholar
Li, W., Jessen, W., Roggenkamp, D., Klaas, M., Silex, W., Schiek, M. & Schröder, W. 2015 Turbulent drag reduction by spanwise traveling ribbed surface waves. Eur. J. Mech. (B/Fluids) 53, 101112.CrossRefGoogle Scholar
Liepmann, H. W. & Roshko, A. 2013 Elements of Gasdynamics. Dover.Google Scholar
Liou, M.-S. & Steffen, C. J. 1993 A new flux splitting scheme. J. Comput. Phys. 107, 2339.CrossRefGoogle Scholar
Lloyd, S. 1982 Least squares quantization in PCM. IEEE Trans. Inf. Theory 28 (2), 129137.CrossRefGoogle Scholar
Loiseau, J.-C., Noack, B. R. & Brunton, S. L. 2018 Sparse reduced-order modeling: sensor-based dynamics to full-state estimation. J. Fluid Mech. 844, 459490.CrossRefGoogle Scholar
Lugt, H. J. 1996 Introduction to Vortex Theory. Vortex Flow Press.Google Scholar
MacQueen, J. 1967 Some methods for classification and analysis of multivariate observations. In Proceedings of the Fifth Berkeley Symposium on Math. Stat. and Prob., vol. 1, pp. 281297.Google Scholar
Meinke, M., Schröder, W., Krause, E. & Rister, T. 2002a A comparison of second-and sixth-order methods for large-eddy simulations. Comput. Fluids 31 (4–7), 695718.CrossRefGoogle Scholar
Meinke, M., Schröder, W., Krause, E. & Rister, T. 2002b A comparison of second-and sixth-order methods for large-eddy simulations. Comput. Fluids 31 (4), 695718.CrossRefGoogle Scholar
Meysonnat, P. S., Roggenkamp, D., Li, W., Roidl, B. & Schröder, W. 2016 Experimental and numerical investigation of transversal traveling surface waves for drag reduction. Eur. J. Mech. (B/Fluids) 55, 313323.CrossRefGoogle Scholar
Noack, B. R. 2016 From snapshots to modal expansions – bridging low residuals and pure frequencies. J. Fluid Mech. 802, 14.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, 335363.CrossRefGoogle Scholar
Noack, B. R. & Morzyński, M.2017 The fluidic pinball – a toolkit for multiple-input multiple-output flow control (version 1.0). Tech. Rep. 02/2017. Chair of Virtual Engineering, Poznan University of Technology, Poland.Google Scholar
Noack, B. R., Stankiewicz, W., Morzyński, M. & Schmid, P. J. 2016 Recursive dynamic mode decomposition of transient and post-transient wake flows. J. Fluid Mech. 809, 843872.CrossRefGoogle Scholar
Oxlade, A. R., Morrison, J. F., Qubain, A. & Rigas, G. 2015 High-frequency forcing of a turbulent axisymmetric wake. J. Fluid Mech. 770, 305318.CrossRefGoogle 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, 161196.CrossRefGoogle Scholar
Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Phil. Trans. R. Soc. Lond. A 369 (1940), 14281442.Google ScholarPubMed
Raibaudo, C., Zhong, P., Martinuzzi, R. J. & Noack, B. R. 2017 Closed-loop control of a triangular bluff body using rotating cylinders. In The 20th World Congress of the International Federation of Automatic Control (IFAC), pp. 16.Google Scholar
Renze, P., Schröder, W. & Meinke, M. 2008 Large-eddy simulation of film cooling flows at density gradients. Intl J. Heat Fluid Flow 29 (1), 1834.CrossRefGoogle Scholar
Roidl, B., Meinke, M. & Schröder, W. 2013 A reformulated synthetic turbulence generation method for a zonal RANS–LES method and its application to zero-pressure gradient boundary layers. Intl J. Heat Fluid Flow 44, 2840.CrossRefGoogle Scholar
Rolland, R.2017 Fluidic pinball – a control study. MS2 Internship Report, LIMSI and ENSAM, Paris, France.Google Scholar
Roussopoulos, K. 1993 Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248, 267296.CrossRefGoogle Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115127.CrossRefGoogle Scholar
Rütten, F., Schröder, W. & Meinke, M. 2005 Large-eddy simulation of low frequency oscillations of the Dean vortices in turbulent pipe bend flows. Phys. Fluids 17 (3), 035107.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition for numerical and experimental data. J. Fluid Mech. 656, 528.CrossRefGoogle Scholar
Schuster, H. G. 1988 Deterministic Chaos, 2nd edn. VCH Verlagsgesellschaft mbH.Google Scholar
Statnikov, V., Meinke, M. & Schröder, W. 2017 Reduced-order analysis of buffet flow of space launchers. J. Fluid Mech. 815, 125.CrossRefGoogle Scholar
Steinhaus, H. 1956 Sur la division des corps matériels en parties. Bull. Acad. Polon. Sci. 4 (12), 801804.Google Scholar
Shinji, T. & Motoyuki, I. 2012 Drag reduction in turbulent boundary layers by spanwise traveling waves with wall deformation. J. Turbul. 13, N9.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.CrossRefGoogle Scholar
Thiria, B., Goujon-Durand, S. & Wesfreid, J. E. 2006 The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123147.CrossRefGoogle Scholar
Venturi, D. 2006 On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate. J. Fluid Mech. 559, 215254.CrossRefGoogle Scholar
Wood, C. J. 1964 The effect of base bleed on a periodic wake. J. R. Aero. Soc. 68 (643), 477482.CrossRefGoogle Scholar
Zhao, H., Wu, J.-Z. & Luo, J.-S. 2004 Turbulent drag reduction by traveling wave of flexible wall. Fluid Dyn. Res. 34 (3), 175198.CrossRefGoogle Scholar
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Metric for attractor overlap
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