Application of central limit theorems to turbulence problems

  • J. L. Lumley
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 12)


It is shown that (to the extent that the moments involved exist) the existence (≠0) of all (generalized) integral scales is necessary (and sufficient if all moments exist) for integrals over adjacent segments of a stationary process to become asymptotically independent, and sufficient to ensure that existing moments of integrals will become Gaussian. The conditions under which several recent central limit and related theorems for dependent variables have been proven, are shown to be closely related to this requirement. As a consequence of this examination, a slight weakening is suggested of the common condition that the spectrum be non-zero. Several physical problems are described, which may be resolved by the application of such a central limit theorem: longitudinal dispersion in a channel flow (previously treated semi-empirically); the spreading of hot spots, or the expansion of macromolecules; the weak interaction hypothesis (of Kraichnan) for Fourier components. Finally, it is shown that dispersion in homogeneous turbulence is unlikely to be explicable on the basis of a central limit theorem.


Central Limit Theorem Gaussian Process Adjacent Segment Integral Scale Homogeneous Turbulence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Batchelor, G. K. and A. A. Townsend (1956), “Turbulent Diffusion,” Suhveyb in Mechanics, Cambridge, The University Press, 352–399.Google Scholar
  2. Comte-Bellot, G. and S. Corrsin (1966), “The Use of a Contraction to Improve the Isotropy of Grid Generated Turbulence,” J. F.buid Mech., 25, 657–682.Google Scholar
  3. Corrsin, S. (1963), “Estimates of the Relations between Eulerian and Lagrangian Scales in Large Reynolds Number Turbulence,” J. Atnoe. Sci., 20, 115–119.CrossRefGoogle Scholar
  4. Hobson, E. W. (1926), The Theory of Functions of a Real Variaable and the Theory of Fourier's Series, The University Press, Cambridge.Google Scholar
  5. Kolmogorov, A. N. and Yu. A. Rozanov (1960), “On Strong Mixing Conditions for Stationary Gaussian Processes,” Theory of Probability and Its Applicationss, V, 204–208.CrossRefGoogle Scholar
  6. Kraichnan, R. H. (1959), “The Structure of Isotropic Turbulence at Very High Reynolds Numbers,” J. Fluid Mech., 5, 497–543.Google Scholar
  7. Loeve, M. (1955), Probability Theory, Van Nostrand, New York.Google Scholar
  8. Lumley, J. L. (1962), “The Mathematical Nature of the Problem of Relating Eulerian and Lagrangian Statistical Functions in Turbulence,” in Mécanique de Za Turbulence, Edition du CNRS, Paris, 17–26.Google Scholar
  9. Lumley, J. L. (1970), Stochastic Tools in Turbulence, Academic Press, New York.Google Scholar
  10. Lumley, J. L. (1972), “On the Solution of Equations Describing Small-Scale Deformation,” in Symposia Mathematica: Proceedings of Convegno sulla Teoria della Turbolenza al Istituto Nazionale di Alta Matematica, Roma, April 1971. New York, Academic Press.Google Scholar
  11. Maruyama, G. (1949), “The Harmonic Analysis of Stationary Stochastic Processes,” Mem. Fac. Sic. Kyushu Univ. Ser A., 4, 45–106.Google Scholar
  12. Monin, A. S. and A. M. Yaglom (1971), Statistical Fluid Mechanics, Cambridge, M.I.T. Press.Google Scholar
  13. Rosenblatt, M. (1956), “A Central Limit Theorem and a Strong Mixing Condition,” Proc. Natl. Acad. Sci. U.S.A., 42, 43–47.Google Scholar
  14. Rosenblatt, M. (1961a), “Independence and Dependence,” in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistical and Probability (Neyman, ed.), II, Pt. III, U. Cal. Press, Berkeley, 431–443.Google Scholar
  15. Rosenblatt, M. (1961b), “Some Comments on Narrow Band Pass Filters,” Q. Appl. Math., XVIII, 387–393.Google Scholar
  16. Sun, T. C. (1965), “Some Further Results on Central Limit Theorems for Non-linear Functions of a Normal Stationary Process,” J. Math. Mech., 14, 71–85.Google Scholar
  17. Taylor, G. I. (1954), “The Dispersion of Matter in Turbulent Flow through a Pipe,” Proc. Roy. Soc. A, 223, 446–468.Google Scholar
  18. Tennekes, H. and J. L. Lumley (1971), A First Course in Turbulence, M.I.T. Press, Cambridge.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • J. L. Lumley
    • 1
  1. 1.The Pennsylvania State UniversityUniversity ParkPennsylvania

Personalised recommendations