Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T10:27:12.456Z Has data issue: false hasContentIssue false
  • English
  • Français

Superfast amplification and superfast nonlinear saturation of perturbations as a mechanism of turbulence

Published online by Cambridge University Press:  15 October 2020

Y. Charles Li Open the ORCID record for Y. Charles Li [Opens in a new window] ,
Richard D. J. G. Ho ,
Arjun Berera  and
Z. C. Feng
Y. Charles Li*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO65211, USA
Richard D. J. G. Ho
Affiliation:
School of Physics, University of Edinburgh, EdinburghEH9 3JZ, UK
Arjun Berera
Affiliation:
School of Physics, University of Edinburgh, EdinburghEH9 3JZ, UK
Z. C. Feng
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO65211, USA
*
†Email address for correspondence: liyan@missouri.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Ruelle predicted that the maximal amplification of perturbations in homogeneous isotropic turbulence is exponential $\exp ({\sigma \sqrt {Re} \,t})$ (where $\sigma \sqrt {Re}$ is the maximal Lyapunov exponent). In our earlier works, we predicted that the maximal amplification of perturbations in fully developed turbulence is faster than exponential and is given by $\exp ({\sigma \sqrt {Re} \sqrt {t} +\sigma _1 t})$ where $\sigma \sqrt {Re} \sqrt {t}$ is much larger than $\sigma \sqrt {Re} \, t$ for small $t$. That is, we predicted superfast initial amplification of perturbations. Built upon our earlier numerical verification of our prediction, here, we conduct a large numerical verification with resolution up to $2048^3$ and Reynolds number up to $6210$. Our direct numerical simulation here confirms our analytical prediction. Our numerical simulation also demonstrates that such superfast amplification of perturbations leads to superfast nonlinear saturation. We conclude that such superfast amplification and superfast nonlinear saturation of ever existing perturbations suggest a mechanism for the generation, development and persistence of fully developed turbulence.

JFM classification

Turbulent Flows: Homogeneous turbulence Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A27
Check for updates
Copyright
© The Author(s), 2020. 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.)

Footnotes

‡

Present address: Marian Smoluchowski Institute of Theoretical Physics, Jagiellonian University, Lojasiewicza 11, 30-348 Krakow, Poland.

References

REFERENCES

Aurell, E., Boffetta, G., Crisanti, A., Paladin, G. & Vulpiani, A. 1996 Growth of non-infinitesimal perturbations in turbulence. Phys. Rev. Lett. 77, 1262–1265.CrossRefGoogle Scholar
Aurell, E., Boffetta, G., Crisanti, A., Paladin, G. & Vulpiani, A. 1997 Predictability in the large: an extension of the concept of Lyapunov exponent. J. Phys. A 30, 1–26.CrossRefGoogle Scholar
Berera, A. & Ho, R. 2018 Chaotic properties of a turbulent isotropic fluid. Phys. Rev. Lett. 120, 024101.CrossRefGoogle ScholarPubMed
Boffetta, G., Cencini, M., Falcioni, M. & Vulpiani, A. 2002 Predictability: a way to characterize complexity. Phys. Rep. 356, 367–474.CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2017 Chaos and predictability of homogeneous-isotropic turbulence. Phys. Rev. Lett. 119, 054102.CrossRefGoogle ScholarPubMed
Bohr, T., Jensen, M. H., Paladin, G. & Vulpiani, A. 2005 Dynamical Systems Approach to Turbulence, 1st edn. Cambridge University Press.Google Scholar
Crisanti, A., Jensen, M. H., Vulpian, A. & Paladin, G. 1993 Intermittency and predictability in turbulence. Phys. Rev. Lett. 70, 166–169.CrossRefGoogle ScholarPubMed
Feng, Z. & Li, Y. 2020 Short term unpredictability of high reynolds number turbulence – rough dependence on initial data. arXiv:1702.02993, http://faculty.missouri.edu/liyan/Rough-Num.pdf.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Inci, H. 2015 On the regularity of the solution map of the incompressible Euler equation. Dynam. Part. Differ. Eq. 12, 97–113.CrossRefGoogle Scholar
Inci, H. & Li, Y. 2019 Nowhere-differentiability of the solution map of 2D Euler equations on bounded spatial domain. Dynam. Part. Differ. Eq. 16, 383–392.CrossRefGoogle Scholar
Kaneda, Y. & Ishihara, T. 2006 High-resolution direct numerical simulation of turbulence. J. Turbul. 7, N20.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.CrossRefGoogle ScholarPubMed
Li, Y. 2005 Invariant manifolds and their zero-viscosity limits for Navier–Stokes equations. Dynam. Part. Differ. Eq. 2, 159–186.CrossRefGoogle Scholar
Li, Y. 2013 Major open problems in chaos theory and nonlinear dynamics. Dynam. Part. Differ. Eq. 10, 379–392.CrossRefGoogle Scholar
Li, Y. 2014 The distinction of turbulence from chaos – rough dependence on initial data. Electron. J. Differ. Eq. 104, 1–8.Google Scholar
Li, Y. 2017 Rough dependence upon initial data exemplified by explicit solutions and the effect of viscosity. Nonlinearity 30, 1097–1108.Google Scholar
Li, Y. 2018 Linear hydrodynamic stability. Not. Am. Math. Soc. 65, 1255–1259.Google Scholar
Linkmann, M. F. & Morozov, A. 2015 Sudden relaminarization and lifetimes in forced isotropic turbulence. Phys. Rev. Lett. 115, 134502.CrossRefGoogle ScholarPubMed
Lucas, D. & Kerswell, R. 2015 Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27, 045106.CrossRefGoogle Scholar
Machiels, L. 1997 Predictability of small-scale motion in isotropic turbulence. Phys. Rev. Lett. 79, 3411–3414.CrossRefGoogle Scholar
Ruelle, D. 1979 Microscopic fluctuations and turbulence. Phys. Lett. 72A, 81–82.CrossRefGoogle Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.CrossRefGoogle Scholar
van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107, 114501.CrossRefGoogle ScholarPubMed
Viswanath, D. 2007 Recurrent motions within plane couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Yoffe, S. R. 2012 Investigation of the transfer and dissipation of energy in isotropic turbulence. PhD thesis, University of Edinburg, arXiv:1306.3408.Google Scholar