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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: [email protected]
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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
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

‡

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

References

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