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

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Rapid Communication
  • Access by Xinjiang University

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Ati S. Sharma*

Igor Mezić†

Beverley J. McKeon‡

  • University of Southampton, Southampton SO17 1BJ, United Kingdom

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

  • California Institute of Technology, Pasadena, California 91125, USA

  • *a.sharma@soton.ac.uk
  • †mezic@ucsb.edu
  • ‡mckeon@caltech.edu
PDF

Phys. Rev. Fluids 1, 032402(R) – Published 18 July, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.032402

Abstract

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. G. Kawahara, M. Uhlmann, and L. van Veen, The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech. 44, 203 (2012).
  2. B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech. 39, 447 (2007).
  3. R. R. Kerswell, Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity 18, R17 (2005).
  4. I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech. 45, 357 (2013).
  5. M. Budisić, R. Mohr, and I. Mezić, Applied Koopmanism, Chaos 22, 047510 (2012).
  6. I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dyn. 41, 309 (2005).
  7. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech. 641, 115 (2009).
  8. P. Schmid and J. Sesterhenn, Bulletin of the American Physical Society, 61st Meeting of the Division of Fluid Dynamics (American Physical Society, College Park, 2008), Vol. 53, p. 208.
  9. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  10. A. Wynn, D. S. Pearson, B. Ganapathisubramani, and P. J. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech. 733, 473 (2013).
  11. J. N. Kutz, S. L. Brunton, D. M. Luchtenburg, C. W. Rowley, and J. H. Tu, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn. 1, 391 (2014).
  12. M. R. Jovanović, P. J. Schmid, and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids 26, 024103 (2014).
  13. B. J. McKeon and A. S. Sharma, A critical-layer framework for turbulent pipe flow, J. Fluid Mech. 658, 336 (2010).
  14. A. S. Sharma and B. J. McKeon, On coherent structure in wall turbulence, J. Fluid Mech. 728, 196 (2013).
  15. A. S. Sharma, R. Moarref, B. J. McKeon, J. S. Park, M. D. Graham, and A. P. Willis, Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence, Phys. Rev. E 93, 021102(R) (2016).
  16. J. Sesterhenn and A. Shahirpour, A Lagrangian dynamic mode decomposition, arXiv:1603.02539.
  17. B. Bamieh, F. Paganini, and M. A. Dahleh, Distributed control of spatially invariant systems, IEEE Trans. Autom. Control 47, 1091 (2002).
  18. B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Natl. Acad. Sci. USA 17, 315 (1931).
  19. A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise (Springer, New York, 1994).
  20. F. Gómez, H. M. Blackburn, M. Rudman, B. J. McKeon, M. Luhar, R. Moarref, and A. S. Sharma, On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow, Phys. Fluids 26, 101703 (2014).
  21. J.-L. Bourguignon, J. A. Tropp, A. S. Sharma, and B. J. McKeon, Compact representation of wall-bounded turbulence using compressive sampling, Phys. Fluids 26, 015109 (2014).
  22. T. Kreilos, S. Zammert, and B. Eckhardt, Comoving frames and symmetry-related motions in parallel shear flows, J. Fluid Mech. 751, 685 (2014).
  23. A. P. Willis, P. Cvitanović, and M. Avila, Revealing the state space of turbulent pipe flow by symmetry reduction, J. Fluid Mech. 721, 514 (2013).
  24. N. Young, An Introduction to Hilbert Space (Cambridge University Press, Cambridge, 1988).
  25. P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, Cambridge, 1996).
  26. F. Gómez, H. M. Blackburn, M. Rudman, A. S. Sharma, and B. J. McKeon, A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator, J. Fluid Mech. 798, R2 (2016).
  27. R. Moarref, M. R. Jovanović, J. A. Tropp, A. S. Sharma, and B. J. McKeon, A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization, Phys. Fluids 26, 051701 (2014).
  28. R. Moarref, A. S. Sharma, J. A. Tropp, and B. J. McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, J. Fluid Mech. 734, 275 (2013).
  29. B. J. McKeon, A. S. Sharma, and I. Jacobi, Experimental manipulation of wall turbulence: A systems approach, Phys. Fluids 25, 031301 (2013).
  30. J. F. Gibson, J. Halcrow, and P. Cvitanović, Equilibrium and traveling-wave solutions of plane Couette flow, J. Fluid Mech. 638, 243 (2009).
  31. M. Luhar, A. S. Sharma, and B. J. McKeon, Opposition control within the resolvent analysis framework, J. Fluid Mech. 749, 597 (2014).
  32. M. Luhar, A. S. Sharma, and B. J. McKeon, A framework for studying the effect of compliant surfaces on wall turbulence, J. Fluid Mech. 768, 415 (2015).

Outline

Information

Phys. Rev. Fluids 1, 032402(R)– Published 18 July, 2016
  • Received 29 March 2016
Article part of CHORUS

©2016 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation