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VOLUME 59 | ISSUE 8 | PAGE 577
Scaling of correlation functions of velocity gradients in hydrodynamic turbulence
As is demonstrated in Refs. 2 and 3 in the limit of infinitely large Reynolds numbers, the correlation functions of the velocity predicted by Kolmogorov's 1941 theory (K41) are actually solutions of diagrammatic equations. Here we demonstrate that correlation functions of the velocity derivatives, Vaup, should possess scaling exponents which have no relation to the K41 dimensional estimates. This phenomenon is referred to as anomalous scaling. This result is proved in diagrammatic terms: We have extracted a series of logarithmically diverging diagrams, whose summation leads to the renormalization of the normal K41 dimensions. For a description of the scaling of various functions of Vavp, an infinite set of primary fields 0„ with independent scaling exponents Δ„ can be introduced. Symmetry reasons enable us to predict relations between the scaling of different correlation functions. We also formulate restrictions imposed on the structure of the correlation functions due to the incompressibility condition. We also propose some tests which make it possible to check experimentally the conformal symmetry of the turbulent correlation functions. Further, we demonstrate that the anomalous scaling behavior should reveal itself in the asymptotic behavior of the correlation functions of the velocity differences. We propose a method to obtain the anomalous exponents from the experiment.




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