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Effect of helicity on the correlation time of large scales in turbulent flows

Alexandre Cameron*, Alexandros Alexakis†, and Marc-Étienne Brachet‡

  • Laboratoire de Physique Statistique, École Normale Supérieure, PSL Research University; Université Paris Diderot Sorbonne Paris-Cité; Sorbonne Universités UPMC Univ Paris 06; CNRS; 24 rue Lhomond, 75005 Paris, France

  • *alexandre.cameron@ens.fr
  • †alexakis@lps.ens.fr
  • ‡brachet@physique.ens.fr
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Phys. Rev. Fluids 2, 114602 – Published 6 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.114602

Abstract

Solutions of the forced Navier-Stokes equation have been conjectured to thermalize at scales larger than the forcing scale, similar to an absolute equilibrium obtained for the spectrally truncated Euler equation. Using direct numeric simulations of Taylor-Green flows and general-periodic helical flows, we present results on the probability density function, energy spectrum, autocorrelation function, and correlation time that compare the two systems. In the case of highly helical flows, we derive an analytic expression describing the correlation time for the absolute equilibrium of helical flows that is different from the E−1/2k−1 scaling law of weakly helical flows. This model predicts a new helicity-based scaling law for the correlation time as τ(k)∼H−1/2k−1/2. This scaling law is verified in simulations of the truncated Euler equation. In simulations of the Navier-Stokes equations the large-scale modes of forced Taylor-Green symmetric flows (with zero total helicity and large separation of scales) follow the same properties as absolute equilibrium including a τ(k)∼E−1/2k−1 scaling for the correlation time. General-periodic helical flows also show similarities between the two systems; however, the largest scales of the forced flows deviate from the absolute equilibrium solutions.

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References (31)

  1. F. H. Champagne, The fine-scale structure of the turbulent velocity field, J. Fluid Mech. 86, 67 (1978).
  2. Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura, and A. Uno, Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box, Phys. Fluids 15, L21 (2003).
  3. U. Frisch, Turbulence: The Legacy of AN Kolmogorov (AIP, New York, 1996).
  4. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Math. Phys. Sci. 434, 9 (1991).
  5. D. Forster, D. R. Nelson, and M. J. Stephen, Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A 16, 732 (1977).
  6. R. H. Kraichnan and S. Chen, Is there a statistical mechanics of turbulence? Physica D: Nonlinear Phenomena 37, 160 (1989).
  7. R. H. Kraichnan, Helical turbulence and absolute equilibrium, J. Fluid Mech. 59, 745 (1973).
  8. C. Cichowlas, P. Bonaïti, F. Debbasch, and M. Brachet, Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows, Phys. Rev. Lett. 95, 264502 (2005).
  9. G. Krstulovic, P. D. Mininni, M. E. Brachet, and A. Pouquet, Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows, Phys. Rev. E 79, 056304 (2009).
  10. V. Dallas, S. Fauve, and A. Alexakis, Statistical Equilibria of Large Scales in Dissipative Hydrodynamic Turbulence, Phys. Rev. Lett. 115, 204501 (2015).
  11. U. Frisch, Zh. S. She, and P. L. Sulem, Large-scale flow driven by the anisotropic kinetic alpha effect, Physica D: Nonlinear Phenomena 28, 382 (1987).
  12. A. Cameron, A. Alexakis, and M.-E. Brachet, Large-scale instabilities of helical flows, Phys. Rev. Fluids 1, 063601 (2016).
  13. P. Clark di Leoni, P. J. Cobelli, and P. D. Mininni, The spatio-temporal spectrum of turbulent flows, Eur. Phys. J. E 38, 136 (2015).
  14. T. D. Lee, On some statistical properties of hydrodynamical and magneto-hydrodynamical fields, Q. Appl. Math. 10, 69 (1952).
  15. L. D. Landau and E. M. Lifshitz, Statistical Physics (Butterworth-Heinemann, Burlington, 1980).
  16. A. Craya, Contribution a l'analyse de la turbulence associée à des vitesses moyennes, Thesis, Université de Grenoble (1957).
  17. J. R. Herring, Approach of axisymmetric turbulence to isotropy, Phys. Fluids 17, 859 (1974).
  18. M. Lesieur, Turbulence in Fluids (Springer Science, New York, 2012).
  19. M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch, Small-scale structure of the Taylor-Green vortex, J. Fluid Mech. 130, 411 (1983).
  20. A. Pouquet, E. Lee, M. E. Brachet, P. D. Mininni, and D. Rosenberg, The dynamics of unforced turbulence at high Reynolds number for Taylor-Green vortices generalized to MHD, Geophys. Astrophys. Fluid Dyn. 104, 115 (2010).
  21. P. D. Mininni, A. Alexakis, and A. Pouquet, Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: The slow emergence of scaling laws, Phys. Rev. E 77, 036306 (2008).
  22. P. D. Mininni, D. Rosenberg, R. Reddy, and A. Pouquet, A hybrid MPI-OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence, Parallel Comput. 37, 316 (2011).
  23. C. Nore, M. Abid, and M. E. Brachet, Decaying Kolmogorov turbulence in a model of superflow, Phys. Fluids 9, 2644 (1997).
  24. E. W. Weisstein, Chi-Squared Distribution (MathWorld—A Wolfram Web Resource, 2017), http://mathworld.wolfram.com/Chi-SquaredDistribution.html.
  25. P. J. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, Global measurement of water waves by Fourier transform profilometry, Exp. Fluids 46, 1037 (2009).
  26. B. Miquel, A. Alexakis, and N. Mordant, Role of dissipation in flexural wave turbulence: From experimental spectrum to Kolmogorov-Zakharov spectrum, Phys. Rev. E 89, 062925 (2014).
  27. E. W. Weisstein, Wiener-Khinchin Theorem (MathWorld—A Wolfram Web Resource, 2017), http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html.
  28. T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167, 353 (1986).
  29. A. Cameron and A. Alexakis, Fate of Alpha Dynamos at Large Rm, Phys. Rev. Lett. 117, 205101 (2016).
  30. G. Michel, F. Petrelis, and S. Fauve, Observation of Thermal Equilibrium in Capillary Wave Turbulence, Phys. Rev. Lett. 118, 144502 (2017).
  31. E. W. Weisstein, Apodization Function (MathWorld—A Wolfram Web Resource, 2017), http://mathworld.wolfram.com/ApodizationFunction.html.

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Phys. Rev. Fluids 2, 114602– Published 6 November, 2017
  • Received 15 May 2017

©2017 American Physical Society

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