- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Study of dynamics in post-transient flows using Koopman mode decomposition

Hassan Arbabi* and Igor Mezić†

  • University of California, Santa Barbara, Santa Barbara, California 93106, USA

  • *harbabi@engineering.ucsb.edu
  • †mezic@engineering.ucsb.edu
PDF

Phys. Rev. Fluids 2, 124402 – Published 29 December, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.124402

Abstract

The Koopman mode decomposition (KMD) is a data-analysis technique which is often used to extract the spatiotemporal patterns of complex flows. In this paper, we use KMD to study the dynamics of the lid-driven flow in a two-dimensional square cavity based on theorems related to the spectral theory of the Koopman operator. We adapt two algorithms, from the classical Fourier and power spectral analysis, to compute the discrete and continuous spectrum of the Koopman operator for the post-transient flows. Properties of the Koopman operator spectrum are linked to the sequence of flow regimes occurring between Re=10000 and Re=30000, and changing the flow nature from steady to aperiodic. The Koopman eigenfunctions for different flow regimes, including flows with mixed spectra, are constructed using the assumption of ergodicity in the state space. The associated Koopman modes show remarkable robustness even as the temporal nature of the flow is changing substantially. We observe that KMD outperforms the proper orthogonal decomposition in reconstruction of the flows with strong quasiperiodic components.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (110)

  1. B. O. Koopman, Hamiltonian systems and transformation in hilbert space, Proc. Natl. Acad. Sci. USA 17, 315 (1931).
  2. I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Phys. D (Amsterdam, Neth.) 197, 101 (2004).
  3. I. Mezić, Spectral properties of dynamical systems, model reduction, and decompositions, Nonlinear Dyn. 41, 309 (2005).
  4. Y. Susuki and I. Mezić, Nonlinear Koopman modes and power system stability assessment without models, IEEE Trans. Power Syst. 29, 899 (2014).
  5. D. Giannakis, J. Slawinska, and Z. Zhao, Spatiotemporal feature extraction with data-driven Koopman operators, J. Machine Learning Res. 44, 103 (2015).
  6. M. Georgescu and I. Mezić, Building energy modeling: A systematic approach to zoning and model reduction using Koopman mode analysis, Energy Build. 86, 794 (2015).
  7. N. B. Erichson, S. L. Brunton, and J. N. Kutz, Compressed dynamic mode decomposition for background modeling, J. Real-Time Image Process., 1 (2016).
  8. B. W. Brunton, L. A. Johnson, J. G. Ojemann, and J. N. Kutz, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition, J. Neurosci. Methods 258, 1 (2016).
  9. J. Mann and J. N. Kutz, Dynamic mode decomposition for financial trading strategies, Quant. Finance 16, 1643 (2016).
  10. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech. 641, 115 (2009).
  11. P. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, in Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, San Antonio, Texas, USA (2008).
  12. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  13. P. J. Schmid, L. Li, M. P. Juniper, and O. Pust, Applications of the dynamic mode decomposition, Theor. Comput. Fluid Dyn. 25, 249 (2011).
  14. C. Pan, D. Yu, and J. Wang, Dynamical mode decomposition of gurney flap wake flow, Theor. Appl. Mech. Lett. 1, 012002 (2011).
  15. A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations, Int. J. Heat Fluid Flow 32, 1098 (2011).
  16. T. W. Muld, G. Efraimsson, and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition, Comp. Fluids 57, 87 (2012).
  17. J.-C. Hua, G. H. Gunaratne, D. G. Talley, J. R. Gord, and S. Roy, Dynamic-mode decomposition based analysis of shear coaxial jets with and without transverse acoustic driving, J. Fluid Mech. 790, 5 (2016).
  18. K. K. Chen, J. H. Tu, and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci. 22, 887 (2012).
  19. S. Bagheri, Koopman-mode decomposition of the cylinder wake, J. Fluid Mech. 726, 596 (2013).
  20. T. Sayadi, P. J Schmid, J. W. Nichols, and P. Moin, Reduced-order representation of near-wall structures in the late transitional boundary layer, J. Fluid Mech. 748, 278 (2014).
  21. P. K. Subbareddy, M. D. Bartkowicz, and G. V. Candler, Direct numerical simulation of high-speed transition due to an isolated roughness element, J. Fluid Mech. 748, 848 (2014).
  22. T. Sayadi, P. J. Schmid, F. Richecoeur, and D. Durox, Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems, Phys. Fluids 27, 037102 (2015).
  23. B. Kramer, P. Grover, P. Boufounos, S. Nabi, and M. Benosman, Sparse sensing and DMD-based identification of flow regimes and bifurcations in complex flows, SIAM J. Appl. Dynam. Syst. 16, 1164 (2017).
  24. H. L. Swinney and J. P. Gollub, The transition to turbulence, Phys. Today 31, 41 (1978).
  25. D. Farmer, J. Crutchfield, H. Froehling, N. Packard, and R. Shaw, Power spectra and mixing properties of strange attractors, Ann. NY Acad. Sci. 357, 453 (1980).
  26. G. E. Karniadakis and G. S. Triantafyllou, Three-dimensional dynamics and transition to turbulence in the wake of bluff objects, J. Fluid Mech. 238, 1 (1992).
  27. A. G. Tomboulides and S. A. Orszag, Numerical investigation of transitional and weak turbulent flow past a sphere, J. Fluid Mech. 416, 45 (2000).
  28. Y.-F. Peng, Y.-H. Shiau, and R. R. Hwang, Transition in a 2-d lid-driven cavity flow, Comp. Fluids 32, 337 (2003).
  29. J. Basley, L. R. Pastur, F. Lusseyran, T. M. Faure, and N. Delprat, Experimental investigation of global structures in an incompressible cavity flow using time-resolved piv, Exp. Fluids 50, 905 (2011).
  30. I. Mezić, Koopman operator spectrum and data analysis, arXiv:1702.07597.
  31. D. Giannakis, Data-driven spectral decomposition and forecasting of ergodic dynamical systems, Appl. Comput. Harmonic Anal. (2017), doi:10.1016/j.acha.2017.09.001.
  32. M. Korda and I. Mezić, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, arXiv:1611.03537.
  33. A. Surana and A. Banaszuk, Linear observer synthesis for nonlinear systems using Koopman operator framework, IFAC-PapersOnLine 49, 716 (2016).
  34. A. Surana, Koopman operator based observer synthesis for control-affine nonlinear systems, in 2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, USA (IEEE, 2016), p. 6492.
  35. E. Kaiser, J. N. Kutz, and S. L. Brunton, Data-driven discovery of Koopman eigenfunctions for control, arXiv:1707.01146.
  36. K. Taira, S. L. Brunton, S. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley, Modal analysis of fluid flows: An overview, arXiv:1702.01453.
  37. M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci. 25, 1307 (2015).
  38. J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, On dynamic mode decomposition: Theory and applications, J. Comp. Dyn. 1, 391 (2014).
  39. J. N. Kutz, X. Fu, and S. L. Brunton, Multiresolution dynamic mode decomposition, SIAM J. Appl. Dyn. Syst. 15, 713 (2016).
  40. J. L. Proctor, S. L. Brunton, and J. N. Kutz, Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst. 15, 142 (2016).
  41. M. S. Hemati, M. O. Williams, and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Phys. Fluids 26, 111701 (2014).
  42. F. Guéniat, L. Mathelin, and L. R. Pastur, A dynamic mode decomposition approach for large and arbitrarily sampled systems, Phys. Fluids 27, 025113 (2015).
  43. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Compressive sampling and dynamic mode decomposition, arXiv:1312.5186.
  44. S. T. M. Dawson, M. S. Hemati, M. O. Williams, and C. W. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Exp. Fluids 57, 1 (2016).
  45. M. S. Hemati, C. W. Rowley, E. A. Deem, and L. N. Cattafesta, De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets, Theor. Comput. Fluid Dyn. 31, 349 (2017).
  46. J. Laskar, The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones, Icarus 88, 266 (1990).
  47. J. Laskar, C. Froeschlé, and A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies: Application to the standard mapping, Phys. D (Amsterdam, Neth.) 56, 253 (1992).
  48. P. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Trans. Audio Electroacoust. 15, 70 (1967).
  49. A. Wynn, D. S. Pearson, B. Ganapathisubramani, and P. J. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech. 733, 473 (2013).
  50. M. R. Jovanović, P. J. Schmid, and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids 26, 024103 (2014).
  51. G. Tissot, L. Cordier, N. Benard, and B. R Noack, Model reduction using dynamic mode decomposition, Compt. Rendus Méc. 342, 410 (2014).
  52. I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech. 45, 357 (2013).
  53. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer Science & Business Media, Berlin, 2003), Vol. 2.
  54. J. von Neumann, Zur operatorenmethode in der klassischen mechanik, Ann. Math. 33, 587 (1932).
  55. B. MacCluer, Elementary Functional Analysis, Graduate Texts in Mathematics Vol. 253 (Springer Science & Business Media, Berlin, 2008).
  56. S. Luzzatto, I. Melbourne, and F. Paccaut, The Lorenz attractor is mixing, Commun. Math. Phys. 260, 393 (2005).
  57. K. E. Petersen, Ergodic Theory (Cambridge University Press, Cambridge, UK, 1989), Vol. 2.
  58. J. L. Doob, Stochastic Processes (Wiley, New York, 1953), Vol. 7.
  59. P. Z. Peebles and B. E. Shi, Probability, Random Variables, and Random Signal Principles, Vol. 3 (McGraw-Hill, New York, NY, 2001).
  60. Y.-h. Tseng and J. H. Ferziger, Mixing and available potential energy in stratified flows, Phys. Fluids 13, 1281 (2001).
  61. H. Gildor, E. Fredj, and A. Kostinski, The Gulf of Eilat/Aqaba: A natural driven cavity? Geophys. Astrophys. Fluid Dyn. 104, 301 (2010).
  62. R. Chella and J. M. Ottino, Fluid mechanics of mixing in a single-screw extruder, Ind. Eng. Chem. Fundam. 24, 170 (1985).
  63. M. Gharib and P. Derango, A liquid film (soap film) tunnel to study two-dimensional laminar and turbulent shear flows, Phys. D (Amsterdam, Neth.) 37, 406 (1989).
  64. J. R. Koseff and R. L. Street, The lid-driven cavity flow: A synthesis of qualitative and quantitative observations, J. Fluids Eng. 106, 390 (1984).
  65. J. Shen, Hopf bifurcation of the unsteady regularized driven cavity flow, J. Comput. Phys. 245, 228 (1991).
  66. M. Poliashenko and C. K. Aidun, A direct method for computation of simple bifurcations, J. Comput. Phys. 121, 246 (1995).
  67. W. Cazemier, R. W. C. P. Verstappen, and A. E. P. Veldman, Proper orthogonal decomposition and low-dimensional models for driven cavity flows, Phys. Fluids 10, 1685 (1998).
  68. F. Auteri, N. Parolini, and L. Quartapelle, Numerical investigation on the stability of singular driven cavity flow, J. Comput. Phys. 183, 1 (2002).
  69. M. J. Balajewicz, E. H. Dowell, and B. R. Noack, Low-dimensional modeling of high-reynolds-number shear flows incorporating constraints from the navier–stokes equation, J. Fluid Mech. 729, 285 (2013).
  70. O. Botella, On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time, Comp. Fluids 26, 107 (1997).
  71. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Vol. 68 (Springer-Verlag, Berlin, 1988).
  72. L. N. Trefethen, Spectral Methods in MATLAB (Siam, Philadelphia, 2000), Vol. 10.
  73. P. Stoica and R. L. Moses, Spectral Analysis of Signals (Pearson Prentice Hall, Upper Saddle River, NJ, 2005).
  74. M. Ghil, M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi et al., Advanced spectral methods for climatic time series, Rev. Geophys. 40, 3-1 (2002).
  75. J. Li and P. Stoica, Efficient mixed-spectrum estimation with applications to target feature extraction, IEEE Trans. Signal Proc. 44, 281 (1996).
  76. P. Stoica, A. Jakobsson, and J. Li, Cisoid parameter estimation in the colored noise case: Asymptotic Cramer-Rao bound, maximum likelihood, and nonlinear least-squares, IEEE Trans. Signal Process. 45, 2048 (1997).
  77. N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Am. J. Math. 63, 415 (1941).
  78. I. Mezić and F. Sotiropoulos, Ergodic theory and experimental visualization of invariant sets in chaotically advected flows, Phys. Fluids 14, 2235 (2002).
  79. U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics Vol. 6 (Walter de Gruyter, Berlin, 1985).
  80. R. W. Schafer, A. V. Oppenheim, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1999).
  81. V. K. Jain, W. L. Collins, and D. C. Davis, High-accuracy analog measurements via interpolated FFT, IEEE Trans. Instrum. Meas. 28, 113 (1979).
  82. G. Andria, M. Savino, and A. Trotta, Windows and interpolation algorithms to improve electrical measurement accuracy, IEEE Trans. Instrum. Meas. 38, 856 (1989).
  83. D. Agrez, Weighted multipoint interpolated DFT to improve amplitude estimation of multifrequency signal, IEEE Trans. Instrum. Meas. 51, 287 (2002).
  84. B. G. R. De Prony, Essai éxperimental et analytique: sur les lois de la dilatabilité de fluides élastique et sur celles de la force expansive de la vapeur de l'alkool, a différentes températures, J. Ec. Polytech. 1, 24 (1795).
  85. Y. Susuki and I. Mezić, A Prony approximation of Koopman mode decomposition, in 2015 IEEE 54th Conference on Decision and Control (CDC), Osaka, Japan (IEEE, 2015), p. 7022.
  86. R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag. 34, 276 (1986).
  87. R. Roy and T. Kailath, Esprit-estimation of signal parameters via rotational invariance techniques, IEEE Trans. Acoustics, Speech Signal Process. 37, 984 (1989).
  88. E. J. Candès and C. Fernandez-Granda, Towards a mathematical theory of super-resolution, Commun. Pure Appl. Math. 67, 906 (2014).
  89. C. Fernandez-Granda, Super-resolution of point sources via convex programming, Infor. Inf. 5, 251 (2016).
  90. J. A. Tropp and A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory 53, 4655 (2007).
  91. S. G. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Process. 41, 3397 (1993).
  92. Y. Wang, Seismic time-frequency spectral decomposition by matching pursuit, Geophysics 72, V13 (2006).
  93. A. Fannjiang and W. Liao, Coherence pattern–guided compressive sensing with unresolved grids, SIAM J. Imaging Sci. 5, 179 (2012).
  94. B. Mamandipoor, D. Ramasamy, and U. Madhow, Newtonized orthogonal matching pursuit: Frequency estimation over the continuum, IEEE Trans. Signal Process. 64, 5066 (2016).
  95. P. Stoica and A. Nehorai, Statistical analysis of two nonlinear least-squares estimators of sine-wave parameters in the colored-noise case, Circuits, Syst. Signal Process. 8, 3 (1989).
  96. M. S. Bartlett, Periodogram analysis and continuous spectra, Biometrika 37, 1 (1950).
  97. R. B. Blackman and J. W. Tukey, The measurement of power spectra from the point of view of communications engineering part I, Bell Labs Tech. J. 37, 185 (1958).
  98. N. Govindarajan, R. Mohr, S. Chandrasekaran, and I. Mezić, A convergent numerical method for computing Koopman spectra of volume-preserving maps on the d-torus (private communication).
  99. M. Korda, M. Putinar, and I. Mezić, Data-driven spectral analysis of the Koopman operator, arXiv:1710.06532.
  100. H. Broer and F. Takens, Mixed spectra and rotational symmetry, Arch. Ration. Mech. Anal. 124, 13 (1993).
  101. D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20, 167 (1971).
  102. F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Warwick 1980 (Springer, Berlin, 1981), pp. 366–381.
  103. N. Mukolobwiez, A. Chiffaudel, and F. Daviaud, Supercritical Eckhaus Instability for Surface-Tension-Driven Hydrothermal Waves, Phys. Rev. Lett. 80, 4661 (1998).
  104. N. Garnier, A. Chiffaudel, F. Daviaud, and A. Prigent, Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows, I: General presentation and periodic solutions, Phys. D (Amsterdam, Neth.) 174, 1 (2003).
  105. M. J. Mercier, N. B. Garnier, and T. Dauxois, Reflection and diffraction of internal waves analyzed with the Hilbert transform, Phys. Fluids 20, 086601 (2008).
  106. B. R. Noack, K. Afanasiev, M. Morzynski, G. Tadmor, and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake, J. Fluid Mech. 497, 335 (2003).
  107. G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech. 25, 539 (1993).
  108. J. M. Lopez and F. Marques, Dynamics of Three-Tori in a Periodically Forced Navier-Stokes Flow, Phys. Rev. Lett. 85, 972 (2000).
  109. Y. Takeda, Quasi-periodic state and transition to turbulence in a rotating Couette system, J. Fluid Mech. 389, 81 (1999).
  110. A. M. Guzmán and C. H. Amon, Transition to chaos in converging-diverging channel flows: Ruelle-Takens-Newhouse scenario, Phys. Fluids 6, 1994 (1994).

Outline

Information

Phys. Rev. Fluids 2, 124402– Published 29 December, 2017
  • Received 4 April 2017
Article part of CHORUS

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation