The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions

Abstract

It is shown that the idea that scaling behavior in turbulence is limited by one outer length $L$ and one inner length $\eta$ is untenable. Every n'th order correlation function of velocity differences $\bbox{\cal F}_n(\B.R_1,\B.R_2,\dots)$ exhibits its own cross-over length ${\eta }_n$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. One result of this Letter is that when all these separations are of the same order $R$ this length scales like ${\eta }_n(R)\sim \eta (R/L{)}^{x_n}$ with $x_n=({\zeta }_n-{\zeta }_{n+1}+{\zeta }_3-{\zeta }_2)/(2-{\zeta }_2)$, with ${\zeta }_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of scaling relations including the ``bridge relation" for the scaling exponent of dissipation fluctuations $\mu =2-{\zeta }_6$.

PRL, Submitted. REVTeX, 4 pages, I fig. (not included) PS Source of the paper with figure avalable at http://lvov.weizmann.ac.il/onlinelist.html

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• Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling
• Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency

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