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The Kolmogorov–Obukhov Statistical Theory of Turbulence

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Abstract

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

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Acknowledgements

The author would like to acknowledge a large number of colleagues with whom the results in this paper have been discussed and who have provided valuable insights. They include Ole Barndorff-Nilsen and Jurgen Schmiegel in Aarhus, Henry McKean, K.R. Sreenivasan and R. Varadhan in New York, Z.-S. Zhe in Beijing, Ed Waymire in Oregon, E. Bodenschatz and H. Xu in Gottingen, Michael Wilczek in Munster and M. Sørensen in Copenhagen. He also benefitted from conversations with J. Peinke, M. Oberlack, E. Meiburg, B. Eckhardt, S. Childress, L. Biferale, L.-S. Yang, A. Lanotto, K. Demosthenes, M. Nelkin, A. Gylfason, V. L’vov and many others. This research was supported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10, whose support is gratefully acknowledged.

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  1. Center for Complex and Nonlinear Science and Department of Mathematics, University of California, Santa Barbara, USA

    Björn Birnir

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  1. Björn Birnir
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Correspondence to Björn Birnir.

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Communicated by P. Newton.

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Birnir, B. The Kolmogorov–Obukhov Statistical Theory of Turbulence. J Nonlinear Sci 23, 657–688 (2013). https://doi.org/10.1007/s00332-012-9164-z

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