dorsal/arxiv
View SchemaAnomalous scaling in homogeneous isotropic turbulence
| Authors | M. J. Giles |
|---|---|
| Categories | |
| ArXiv ID | physics/0012047 |
| URL | https://arxiv.org/abs/physics/0012047 |
| DOI | 10.1088/0305-4470/34/21/302 |
Abstract
The anomalous scaling exponents $\zeta_{n}$ of the longitudinal structure functions $S_{n}$ for homogeneous isotropic turbulence are derived from the Navier-Stokes equations by using field theoretic methods to develop a low energy approximation in which the Kolmogorov theory is shown to act effectively as a mean field theory. The corrections to the Kolmogorov exponents are expressed in terms of the anomalous dimensions of the composite operators which occur in the definition of $S_{n}$. These are calculated from the anomalous scaling of the appropriate class of nonlinear Green's function, using an $uv$ fixed point of the renormalisation group, which thereby establishes the connection with the dynamics of the turbulence. The main result is an algebraic expression for $\zeta_{n}$, which contains no adjustable constants. It is valid at orders $n$ below $% g_{\ast}^{-1}$, where $g_{\ast}$ is the fixed point coupling constant. This expression is used to calculate $\zeta _{n}$ for orders in the range $% n=2$ to 10, and the results are shown to be in good agreement with experimental data, key examples being $\zeta_{2}=0.7$, $\zeta_{3}=1$ and $% \zeta_{6}=1.8$.
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"abstract": "The anomalous scaling exponents $\\zeta_{n}$ of the longitudinal structure\nfunctions $S_{n}$ for homogeneous isotropic turbulence are derived from the\nNavier-Stokes equations by using field theoretic methods to develop a low\nenergy approximation in which the Kolmogorov theory is shown to act effectively\nas a mean field theory. The corrections to the Kolmogorov exponents are\nexpressed in terms of the anomalous dimensions of the composite operators which\noccur in the definition of $S_{n}$. These are calculated from the anomalous\nscaling of the appropriate class of nonlinear Green\u0027s function, using an $uv$\nfixed point of the renormalisation group, which thereby establishes the\nconnection with the dynamics of the turbulence. The main result is an algebraic\nexpression for $\\zeta_{n}$, which contains no adjustable constants.\n It is valid at orders $n$ below $% g_{\\ast}^{-1}$, where $g_{\\ast}$ is the\nfixed point coupling constant. This expression is used to calculate $\\zeta\n_{n}$ for orders in the range $% n=2$ to 10, and the results are shown to be in\ngood agreement with experimental data, key examples being $\\zeta_{2}=0.7$,\n$\\zeta_{3}=1$ and $% \\zeta_{6}=1.8$.",
"arxiv_id": "physics/0012047",
"authors": [
"M. J. Giles"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1088/0305-4470/34/21/302",
"title": "Anomalous scaling in homogeneous isotropic turbulence",
"url": "https://arxiv.org/abs/physics/0012047"
},
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