Acta Mechanica Sinica

, Volume 26, Issue 2, pp 151–157 | Cite as

Corrections to the scaling of the second-order structure function in isotropic turbulence

  • Le Fang
  • Wouter J. T. Bos
  • Xiaozhou Zhou
  • Liang Shao
  • Jean-Pierre Bertoglio
Research Paper


The approach of Obukhov assuming a constant skewness was used to obtain analytical corrections to the scaling of the second order structure function, starting from Kolmogorov’s 4/5 law. These corrections can be used in model applications in which explicit expressions, rather than numerical solutions are needed. The comparison with an interpolation formula proposed by Batchelor, showed that the latter gives surprisingly precise results. The modification of the same method to obtain analytical corrections to the scaling law, taking into account the possible corrections induced by intermittency, is also proposed.


Scaling law Structure function Isotropic turbulence 


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  1. 1.
    Kolmogorov A.N.: Dissipation of energy in locally isotropic turbulence. Doklady Akad. Nauk SSSR. 32, 19 (1941)Google Scholar
  2. 2.
    Kolmogorov A.N.: The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Doklady Akad. Nauk SSSR. 30, 301 (1941)Google Scholar
  3. 3.
    Kolmogorov A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82 (1962)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Anselmet F., Gagne Y., Hopfinger E., Antonia R.: High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63 (1984)CrossRefGoogle Scholar
  5. 5.
    L’vov V., Procaccia I.: “Intermittency” in hydrodynamic turbulence as intermediate asymptotics to kolmogorov scaling. Phys. Rev. Lett. 74, 2690 (1995)CrossRefGoogle Scholar
  6. 6.
    Qian J.: Normal and anomalous scaling of turbulence. Phys. Rev. E 58(6), 7325–7329 (1998)CrossRefGoogle Scholar
  7. 7.
    Obukhov A.M.: The local structure of atmospheric turbulence. Doklady Akad. Nauk SSSR. 67, 643 (1949)MathSciNetGoogle Scholar
  8. 8.
    Monin A.S., Yaglom A.M.: Statistical Fluid Mechanics, vol II: Mechanics of Turbulence. MIT Press, Cambridge (1975)Google Scholar
  9. 9.
    Tatarskii V.I.: Use of the 4/5 kolmogorov equation for describing some characteristics of fully developed turbulence. Phys. Fluids 17, 035110 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Cui G., Zhou H., Zhang Z., Shao L.: A new dynamic subgrid eddy viscosity model with application to turbulent channel flow. Phys. Fluids 16(8), 2835–2842 (2004)CrossRefGoogle Scholar
  11. 11.
    Fang L., Shao L., Bertoglio J.P., Cui G., Xu C., Zhang Z.: An improved velocity increment model based on Kolmogorov equation of filtered velocity. Phys. Fluids 21(6), 065108 (2009)CrossRefGoogle Scholar
  12. 12.
    Batchelor G.K.: Pressure fluctuations in isotropic turbulence. Proc. Cambridge Philos. Soc. 47, 359 (1951)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lohse D., Muller-Groeling A.: Bottleneck effects in turbulence: scaling phenomena in r versus p space. Phys. Rev. Lett. 74(10), 1747 (1995)CrossRefGoogle Scholar
  14. 14.
    Chevillard L., Castaing B., Lvque E., Arneodo A.: Unified multifractal description of velocity increments statistics in turbulence: intermittency and skewness. Physica D 218, 77 (2006)MATHCrossRefGoogle Scholar
  15. 15.
    Meneveau C.: Transition between viscous and inertialrange scaling of turbulence structure functions. Phys. Rev. E 54(4), 3657 (1996)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Barenblatt G., Chorin A., Prostokishin V.: Comment on the paper on the scaling of three-dimensional homogeneous and isotropic turbulence by Benzi et al. Phys. D 127, 105–110 (1999)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kraichnan R.: The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543 (1959)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Batchelor G.K.: The Theory of Homogeneous Turbulence. Cambridge University Press, Cambridge (1953)MATHGoogle Scholar
  19. 19.
    van Atta C.W., Antonia R.A.: Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives. Phys. Fluids 23(2), 252–257 (1980)CrossRefGoogle Scholar
  20. 20.
    She Z.S., Leveque E.: Universal scaling law in fully developed turbulence. Phys. Rev. Lett. 72, 336 (1994)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009

Authors and Affiliations

  • Le Fang
    • 1
  • Wouter J. T. Bos
    • 1
  • Xiaozhou Zhou
    • 2
  • Liang Shao
    • 1
  • Jean-Pierre Bertoglio
    • 1
  1. 1.Laboratoire de Mécanique des Fluides et d’AcoustiqueUniversité de Lyon École centrale de LyonEcullyFrance
  2. 2.Department of Mechanical Engineering, Center for Biomedical Engineering ResearchUniversity of DelawareNewarkUSA

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