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Application of central limit theorems to turbulence problems

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Statistical Models and Turbulence

Part of the book series: Lecture Notes in Physics ((LNP,volume 12))

Abstract

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.

This work supported by the U.S. National Science Foundation, under Grant No. GA18109.

Professor of Aerospace Engineering.

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Authors and Affiliations

  1. The Pennsylvania State University, University Park, Pennsylvania

    J. L. Lumley

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  1. J. L. Lumley
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M. Rosenblatt C. Van Atta

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© 1972 Springer-Verlag

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Lumley, J.L. (1972). Application of central limit theorems to turbulence problems. In: Rosenblatt, M., Van Atta, C. (eds) Statistical Models and Turbulence. Lecture Notes in Physics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-05716-1_1

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  • DOI: https://doi.org/10.1007/3-540-05716-1_1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05716-1

  • Online ISBN: 978-3-540-37093-2

  • eBook Packages: Springer Book Archive

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