Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-29T01:57:21.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Do extreme events trigger turbulence decay? – a numerical study of turbulence decay time in pipe flows

Published online by Cambridge University Press:  15 February 2021

Takahiro Nemoto Open the ORCID record for Takahiro Nemoto [Opens in a new window]  and
Alexandros Alexakis Open the ORCID record for Alexandros Alexakis [Opens in a new window]
Takahiro Nemoto*
Affiliation:
Philippe Meyer Institute for Theoretical Physics, Physics Department, École Normale Supérieure & PSL Research University, 24 rue Lhomond, Paris Cedex 0575231, France Mathematical Modelling of Infectious Diseases Unit, Institut Pasteur, 25-28 Rue du Docteur Roux, Paris75015, France
Alexandros Alexakis
Affiliation:
Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France
*
†Email address for correspondence: takahiro.nemoto@inserm.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulence locally created in laminar pipe flows shows sudden decay or splitting after a stochastic waiting time. In laboratory experiments, the mean waiting time was observed to increase double-exponentially as the Reynolds number ($Re$) approaches its critical value. To understand the origin of this double-exponential increase, we perform many pipe flow direct numerical simulations of the Navier–Stokes equations, and measure the cumulative histogram of the maximum axial vorticity field over the pipe (turbulence intensity). In the domain where the turbulence intensity is not small, we observe that the histogram is well-approximated by the Gumbel extreme value distribution. The smallest turbulence intensity in this domain roughly corresponds to the transition value between the locally stable turbulence and a metastable (edge) state. Studying the $Re$ dependence of the fitting parameters in this distribution, we derive that the time scale of the transition between these two states increases double-exponentially as $Re$ approaches its critical value. On the contrary, in smaller turbulence intensities below this domain, we observe that the distribution is not sensible to the change of $Re$. This means that the decay time of the metastable state (to the laminar state) is stochastic but $Re$-independent in average. Our observation suggests that the conjecture made by Goldenfeld et al. to derive the double-exponential increase of turbulence decay time is approximately satisfied in the range of $Re$ we studied. We also discuss using another extreme value distribution, Fréchet distribution, instead of the Gumbel distribution to approximate the histogram of the turbulence intensify, which reveals interesting perspectives.

JFM classification

Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 912 , 10 April 2021 , A38
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767, 1–101.CrossRefGoogle Scholar
Allen, R.J., Warren, P.B. & ten Wolde, P.R. 2005 Sampling rare switching events in biochemical networks. Phys. Rev. Lett. 94, 018104.CrossRefGoogle ScholarPubMed
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.CrossRefGoogle ScholarPubMed
Avila, M., Willis, A.P. & Hof, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127–136.CrossRefGoogle Scholar
Barkley, D. 2011 Simplifying the complexity of pipe flow. Phys. Rev. E 84, 016309.CrossRefGoogle ScholarPubMed
Benavides, S.J. & Alexakis, A. 2017 Critical transitions in thin layer turbulence. J. Fluid Mech. 822, 364–385.CrossRefGoogle Scholar
Bouchet, F., Rolland, J. & Simonnet, E. 2019 Rare event algorithm links transitions in turbulent flows with activated nucleations. Phys. Rev. Lett. 122, 074502.CrossRefGoogle ScholarPubMed
Box, G.E.P. & Tiao, G.C. 2011 Bayesian Inference in Statistical Analysis, vol. 40. John Wiley & Sons.Google Scholar
Celani, A., Musacchio, S. & Vincenzi, D. 2010 Turbulence in more than two and less than three dimensions. Phys. Rev. Lett. 104, 184506.CrossRefGoogle ScholarPubMed
Cérou, F. & Guyader, A. 2007 Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25 (2), 417–443.CrossRefGoogle Scholar
Charras-Garrido, M. & Lezaud, P. 2013 Extreme value analysis: an introduction. J. Soc. Fr. Statistique 154, 66–97.Google Scholar
Chernykh, A.I. & Stepanov, M.G. 2001 Large negative velocity gradients in burgers turbulence. Phys. Rev. E 64, 026306.CrossRefGoogle ScholarPubMed
Eckhardt, B. 2009 Introduction, turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds’ paper. Phil. Trans. R. Soc. Lond. A 367 (1888), 449–455.Google ScholarPubMed
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.CrossRefGoogle Scholar
Fisher, R.A. & Tippett, L.H.C. 1928 Limiting forms of the frequency distribution of the largest or smallest member of a sample. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 24, pp. 180–190. Cambridge University Press.CrossRefGoogle Scholar
Giardinà, C., Kurchan, J. & Peliti, L. 2006 Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603.CrossRefGoogle ScholarPubMed
Goldenfeld, N., Guttenberg, N. & Gioia, G. 2010 Extreme fluctuations and the finite lifetime of the turbulent state. Phys. Rev. E 81, 035304(R).CrossRefGoogle ScholarPubMed
Goldenfeld, N. & Shih, H.-Y. 2017 Turbulence as a problem in non-equilibrium statistical mechanics. J. Stat. Phys. 167 (3), 575–594.CrossRefGoogle Scholar
Grafke, T., Grauer, R. & Schäfer, T. 2015 a The instanton method and its numerical implementation in fluid mechanics. J. Phys. A 48 (33), 333001.CrossRefGoogle Scholar
Grafke, T., Grauer, R. & Schäfer, T. & Vanden-Eijnden, E. 2015 b Relevance of instantons in burgers turbulence. Europhys. Lett. 109 (3), 34003.CrossRefGoogle Scholar
Grigorio, L.S., Bouchet, F, Pereira, R.M. & Chevillard, L. 2017 Instantons in a lagrangian model of turbulence. J. Phys. A 50 (5), 055501.CrossRefGoogle Scholar
Gumbel, E.J. 1935 Les valeurs extrêmes des distributions statistiques. Ann. Inst. Henri Poincaré 5 (2), 115–158.Google Scholar
Heymann, M. & Vanden-Eijnden, E. 2008 Pathways of maximum likelihood for rare events in nonequilibrium systems: application to nucleation in the presence of shear. Phys. Rev. Lett. 100, 140601.CrossRefGoogle ScholarPubMed
Hof, B. de Lozar, A. Kuik, D.J. & Westerweel, J. 2008 Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101, 214501.CrossRefGoogle ScholarPubMed
Hof, B., Westerweel, J., Schneider, T.M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.CrossRefGoogle ScholarPubMed
van Kan, A., Nemoto, T. & Alexakis, A. 2019 Rare transitions to thin-layer turbulent condensates. J. Fluid Mech. 878, 356–369.CrossRefGoogle Scholar
Kuik, D.J., Poelma, C. & Westerweel, J. 2010 Quantitative measurement of the lifetime of localized turbulence in pipe flow. J. Fluid Mech. 645, 529–539.CrossRefGoogle Scholar
Lestang, T., Ragone, F., Bréhier, C.-E., Herbert, C. & Bouchet, F. 2018 Computing return times or return periods with rare event algorithms. J. Stat. Mech. 2018 (4), 043213.CrossRefGoogle Scholar
de Lozar, A. & Hof, B. 2009 An experimental study of the decay of turbulent puffs in pipe flow. Phil. Trans. R. Soc. Lond. A 367 (1888), 589–599.Google ScholarPubMed
de Lozar, A., Mellibovsky, F., Avila, M. & Hof, B. 2012 Edge state in pipe flow experiments. Phys. Rev. Lett. 108, 214502.CrossRefGoogle Scholar
Musacchio, S. & Boffetta, G. 2017 Split energy cascade in turbulent thin fluid layers. Phys. Fluids 29 (11), 111106.CrossRefGoogle Scholar
Nemoto, T. & Alexakis, A. 2018 Method to measure efficiently rare fluctuations of turbulence intensity for turbulent-laminar transitions in pipe flows. Phys. Rev. E 97, 022207.CrossRefGoogle ScholarPubMed
Nemoto, T., Bouchet, F., Jack, R.L. & Lecomte, V. 2016 Population-dynamics method with a multicanonical feedback control. Phys. Rev. E 93, 062123.CrossRefGoogle ScholarPubMed
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Proc. R. Soc. Lond. 35 (224–226), 84–99.Google Scholar
Shimizu, M., Kanazawa, T. & Kawahara, G. 2019 Exponential growth of lifetime of localized turbulence with its extent in channel flow. Fluid Dyn. Res. 51 (1), 011404.CrossRefGoogle Scholar
Smith, L.M., Chasnov, J.R. & Waleffe, F. 1996 Crossover from two-to three-dimensional turbulence. Phys. Rev. Lett. 77 (12), 2467–2470.CrossRefGoogle ScholarPubMed
Tailleur, J. & Kurchan, J. 2007 Probing rare physical trajectories with Lyapunov weighted dynamics. Nat. Phys. 3 (3), 203.CrossRefGoogle Scholar
Teo, I., Mayne, C.G., Schulten, K. & Lelièvre, T. 2016 Adaptive multilevel splitting method for molecular dynamics calculation of benzamidine-trypsin dissociation time. J. Chem. Theory Comput. 12 (6), 2983–2989.CrossRefGoogle ScholarPubMed
Willis, A.P. 2017 The openpipeflow Navier–Stokes solver. SoftwareX 6, 124–127.CrossRefGoogle Scholar
Wygnanski, I.J. & Champagne, F.H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59 (2), 281–335.CrossRefGoogle Scholar