Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-01T12:44:16.526Z Has data issue: false hasContentIssue false
  • English
  • Français

On the strength of the nonlinearity in isotropic turbulence

Published online by Cambridge University Press:  19 September 2013

W. J. T. Bos  and
R. Rubinstein
W. J. T. Bos*
Affiliation:
LMFA-CNRS, Université de Lyon, Ecole Centrale de Lyon, 69134 Ecully, France
R. Rubinstein
Affiliation:
Newport News, VA, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulence governed by the Navier–Stokes equations shows a tendency to evolve towards a state in which the nonlinearity is diminished. In fully developed turbulence, this tendency can be measured by comparing the variance of the nonlinear term to the variance of the same quantity measured in a Gaussian field with the same energy distribution. In order to study this phenomenon at high Reynolds numbers, a version of the direct interaction approximation is used to obtain a closed expression for the statistical average of the mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear term is evaluated and its scaling in the inertial range is investigated as a function of the Reynolds number. Its scaling is dominated by the sweeping by the energetic scales, but this sweeping is weaker than predicted by a random sweeping estimate. At inertial range scales, the depletion of nonlinearity as a function of the wavenumber is observed to be constant. At large scales it is observed that the mean-square nonlinearity is larger than its Gaussian estimate, which is shown to be related to the non-Gaussianity of the Reynolds-stress fluctuations at these scales.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 158 - 170
Copyright
©2013 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

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Bos, W. & Bertoglio, J.-P. 2013 Lagrangian Markovianized field approximation for turbulence. J. Turbul. 14, 99–120.CrossRefGoogle Scholar
Bos, W. J. T., Rubinstein, R. & Fang, L. 2012 Reduction of mean-square advection in turbulent passive scalar mixing. Phys. Fluids 24, 075104.Google Scholar
Chen, H., Herring, J. R., Kerr, R. M. & Kraichnan, R. H. 1989 Non-Gaussian statistics in isotropic turbulence. Phys. Fluids A 1, 1844–1854.CrossRefGoogle Scholar
Chen, S. & Kraichnan, R. H. 1989 Sweeping decorrelation in isotropic turbulence. Phys. Fluids A 1, 2019–2024.CrossRefGoogle Scholar
Gotoh, T. & Fukayama, D. 2001 Pressure spectrum in homogeneous turbulence. Phys. Rev. Lett. 86, 3775–3778.Google Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2003 Spectra of energy dissipation, enstrophy and pressure by high-resolution direct numerical simulations of turbulence in a periodic box. J. Phys. Soc. Japan 72, 983–986.CrossRefGoogle Scholar
Kaneda, Y. 1981 Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107, 131–145.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543.Google Scholar
Kraichnan, R. H. 1964 Kolmogorov’s hypotheses and Eulerian turbulence theory. Phys. Fluids 7, 1723–1734.CrossRefGoogle Scholar
Kraichnan, R. H. 1965 Lagrangian-history closure approximation for turbulence. Phys. Fluids 8, 575–598.Google Scholar
Kraichnan, R. H. 1970 Convergents to turbulence functions. J. Fluid Mech. 41, 189–217.CrossRefGoogle Scholar
Kraichnan, R. H. 1971 An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513–524.Google Scholar
Kraichnan, R. H. & Panda, R. 1988 Depression of nonlinearity in decaying isotropic turbulence. Phys. Fluids 31, 2395–2397.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161.Google Scholar
Nelkin, M. & Tabor, M. 1990 Time correlations and random sweeping in isotropic turbulence. Phys. Fluids A 2, 81–83.CrossRefGoogle Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.CrossRefGoogle Scholar
Servidio, S., Matthaeus, W. H. & Dmitruk, P. 2008 Depression of nonlinearity in decaying isotropic MHD turbulence. Phys. Rev. Lett. 100, 095005.CrossRefGoogle ScholarPubMed
Tennekes, H. 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67, 561–567.CrossRefGoogle Scholar