Advertisement

A closure hypothesis for the hierarchy of equations for turbulent probability distribution functions

  • T. S. Lundgren
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 12)

Abstract

The hierarchy of equations for turbulent probability distribution functions is closed by relating the three point distribution function to lower order distribution functions. The theory is applied to isotropic, homogeneous turbulence at large wave number giving a nonlinear integral equation for the correlation function at small separation. The Kolmogorov spectrum is found in the inertial range and the Kolmogorov constant is determined.

Keywords

Probability Distribution Function Singular Integral Equation Isotropic Turbulence Dominant Term Nonlinear Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. K. Batchelor, (1960) The Theory of Homogeneous Turbulence, Cambridge University Press.Google Scholar
  2. H. L. Grant, R. W. Stewart and A. Moilliet, (1962) Turbulence spectra from a tidal channel. J. Fluid Mech. 13, 237.Google Scholar
  3. J. O. Hinze, (1959) Turbulence, McGraw-Hill.Google Scholar
  4. E. Hopf, (1952) Statistical hydromechanics and functional calculus, J. Ratl. Mech. Anal. 1, 87.Google Scholar
  5. A. N. Kolmogorov, (1941) Dissipation of energy in locally isotropic turbulence. C. R. Acad. Sci. U.R.S.S. 32, 16.Google Scholar
  6. R. H. Kraichnan, (1965) Preliminary calculation of the Kolmogorov turbulence spectrum, Phys. Fluids 8, 995; (1966) Errata 9, 1884.Google Scholar
  7. M. J. Lighthill, (1960) Fourier Analysis and Generalized Functions, Cambridge University Press.Google Scholar
  8. T. S. Lundgren, (1967) Distribution functions in the statistical theory of turbulence, Phys. Fluids 10, 969.Google Scholar
  9. M. Millionshtchikov, (1941) On the theory of homogeneous isotropic turbulence, C. R. Acad. Sci. U.R.S.S. 32, 619.Google Scholar
  10. A. S. Monin, (1967) Equations of turbulent motion, P.M.M. 31, 1057.Google Scholar
  11. N. Muskhelishvili, (1953) Singular Integral Equations, Noordhoff.Google Scholar
  12. Y. Ogura, (1963) A consequence of the zero fourth order cumulant approximation in the decay of isotropic turbulence, J. Fluid Mech. 16, 33.Google Scholar
  13. I. Proudman and W. H. Reid, (1954) On the decay of a normally distributed and homogeneous turbulent velocity field, Phil Trans. A., 247, 163.Google Scholar
  14. S. A. Rice and P. Gray, (1965) The Statistical Mechanics of Simple Liquids, Interscience.Google Scholar
  15. P. G. Saffman, (1968) Lectures on homogeneous turbulence in Topics in Nonlinear Physics, Ed. N. Zabusky, Springer-Verlag.Google Scholar
  16. T. Tatsumi, (1957) The theory of decay process of incompressible isotropic turbulence, Proc. Roy. Soc. A, 239, 16.Google Scholar
  17. M. VanDyke, (1964) Perturbation Methods in Fluid Mechanics, Academic Press.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • T. S. Lundgren
    • 1
  1. 1.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolis

Personalised recommendations