Advertisement

Phenomenology and Scaling Theories

  • Emily S. C. ChingEmail author
Chapter
  • 765 Downloads
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

The statistics of the velocity and temperature differences, between measurements taken at two points separated by a distance \(l\), can reveal the structure of turbulence. These structure functions often exhibit power laws or scaling laws in \(l\). We introduce the important concept of energy cascade in turbulent flows and the different theories for the scaling behavior of the velocity and temperature fluctuations. We start with the scaling theory for non-buoyant turbulent flows and then discuss how the presence of buoyancy would affect and modify the scaling behavior. A crossover between the two types of scaling behavior is expected to occur at a length scale, the Bolgiano length, above which buoyancy is significant. Furthermore, there are corrections to these scaling theories due to the intermittent nature of turbulent fluctuations, and we discuss the idea of refined similarity hypothesis used to account for these corrections.

Keywords

Energy cascade Kolmogorov scaling Four-fifth law Obukhov–Corrsin scaling Bolgiano–Obukhov scaling Bolgiano length 

References

  1. 1.
    L.F. Richardson, Weather Prediction by Numerical Process (Cambridge University Press, Cambridge, 2007)Google Scholar
  2. 2.
    A.N. Kolmogorov, The local structure of turbulence in imcompressible viscous fluid for very large Reynolds numbers. C. R. (Dokl.) Acad. Sci. SSSR 30, 301–305 (1941). Reprinted: (1991) Proc. R. Soc. Lond. Ser. A 434, 9–13Google Scholar
  3. 3.
    A.N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence. C. R. (Dokl.) Acad. Sci. SSSR 32:16–18 (1941). Reprinted: (1991) Proc. R. Soc. Lond. Ser. A 434, 15–17Google Scholar
  4. 4.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1987)zbMATHGoogle Scholar
  5. 5.
    U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995)zbMATHGoogle Scholar
  6. 6.
    V. Yakhot, 4/5 Kolmogorov law for statistically stationary turbulence: application to High-Rayleigh-Number Bénard convection. Phys. Rev. Lett. 69, 769–771 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A.M. Obukhov, The structure of the temperature field in a turbulent flow. Izv. Akad. Nauk. SSSR. Ser. Geogr. Geophys. 13, 58–69 (1949)Google Scholar
  8. 8.
    S. Corrsin, On the spectrum of isotropic temperature fluctuations in isotropic turbulence. J. Appl. Phys. 22, 469–473 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Z. Warhaft, Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    I. Procaccia, R. Zeitak, Scaling exponents in nonisotropic convective turbulence. Phys. Rev. Lett. 62, 2128–2131 (1989)Google Scholar
  11. 11.
    I. Procaccia, R. Zeitak, Scaling exponents in thermally driven turbulence. Phys. Rev. A 42, 821–830 (1990)Google Scholar
  12. 12.
    V.S. L’vov, Spectra of velocity and temperature fluctuations with constant entropy flux of fully developed free-convective turbulence. Phys. Rev. Lett. 67, 687–690 (1991)CrossRefGoogle Scholar
  13. 13.
    S. Grossmann, V.S. L’vov, Crossover of spectral scaling in thermal turbulence. Phys. Rev. E 47, 4161–4168 (1993)CrossRefGoogle Scholar
  14. 14.
    R. Bolgiano, Turbulent spectra in a stably stratified atmosphere. J. Geophys. Res. 64, 2226–2229 (1959)CrossRefGoogle Scholar
  15. 15.
    A.M. Obukhov, The influence of Archimedean forces on the structure of the temperature field in a turbulent flow. Dokl. Akad. Nauk. SSR 125, 1246–1248 (1959)Google Scholar
  16. 16.
    A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT Press, Cambridge, 1975)Google Scholar
  17. 17.
    E. Calzavarini, F. Toschi, R. Tripiccione, Evidences of Bolgiano-Obhukhov scaling in three-dimensional Rayleigh-Bénard convection. Phys. Rev. E 66, 016304 (2002)CrossRefGoogle Scholar
  18. 18.
    A.N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A.M. Obukhov, J. Fluid Mech. 13, 77 (1962)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Praskovsky, E. Praskovskaya, T. Horst, Further experimental support for the Kolmogorov refined similarity hypothesis. Phys. Fluids 9, 2465–2467 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    L.-P. Wang, S. Chen, J.G. Brasseur, J.C. Wyngaard, Examination of hypothesis in the Kolmogorov refined turbulence theory through high-resolution simulations. Part 1. Velocity field. J. Fluid Mech. 309, 113–156 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    G. Stolovitzky, P. Kailasnath, K.R. Sreenivasan, Refined similarity hypotheses for passive scalars mixed by turbulence. J. Fluid Mech. 297, 275–291 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Y. Zhu, R.A. Antonia, I. Hosokawa, Refined similarity hypothesis for turbulent velocity and temperature fields. Phys. Fluids 7, 1637–1648 (1995)CrossRefGoogle Scholar
  24. 24.
    E.S.C. Ching, K.L. Chau, Conditional statistics of temperature fluctuations in turbulent convection. Phys. Rev. E 63, 047303 (2001)CrossRefGoogle Scholar
  25. 25.
    E.S.C. Ching, W.C. Cheng, Anomalous scaling and refined similarity of an active scalar in a shell model of homogeneous turbulent convection. Phys. Rev. E 77, 015303(R) (2008)Google Scholar
  26. 26.
    G. Ruiz-Chavarria, C. Baudet, S. Ciliberto, Scaling laws and dissipation scale of a passive scalar in fully developed turbulence. Phys. D 99, 369–380 (1996)zbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of PhysicsThe Chinese University of Hong KongHong KongHong Kong SAR

Personalised recommendations