dorsal/arxiv
View SchemaProbability Densities in Strong Turbulence
| Authors | Victor Yakhot |
|---|---|
| Categories | |
| ArXiv ID | physics/0512102 |
| URL | https://arxiv.org/abs/physics/0512102 |
| DOI | 10.1016/j.physd.2006.01.012 |
| Journal | Physica D215 (2006) 166-174 |
Abstract
According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $\eta$ form a random field related to velocity increments $\delta_{\eta}u$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P(\delta_{r}u,r)$ and the PDF of the dissipation scales $Q(\eta, Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_{L}(\delta_{r}u,r)$ often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of $P(\delta_{r}u,r)$ from $P_{L}(\delta_{r}u,r)$. A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function $D(h)$ is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence. {\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on the concepts described in this paper, is also presented.
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"abstract": "According to modern developments in turbulence theory, the \"dissipation\"\nscales (u.v. cut-offs) $\\eta$ form a random field related to velocity\nincrements $\\delta_{\\eta}u$. In this work we, using Mellin\u0027s transform combined\nwith the Gaussain large -scale boundary condition, calculate probability\ndensities (PDFs) of velocity increments $P(\\delta_{r}u,r)$ and the PDF of the\ndissipation scales $Q(\\eta, Re)$, where $Re$ is the large-scale Reynolds\nnumber. The resulting expressions strongly deviate from the Log-normal PDF\n$P_{L}(\\delta_{r}u,r)$ often quoted in the literature. It is shown that the\nprobability density of the small-scale velocity fluctuations includes\ninformation about the large (integral) scale dynamics which is responsible for\ndeviation of $P(\\delta_{r}u,r)$ from $P_{L}(\\delta_{r}u,r)$. A framework for\nevaluation of the PDFs of various turbulence characteristics involving spatial\nderivatives is developed. The exact relation, free of spurious Logarithms\nrecently discussed in Frisch et al (J. Fluid Mech. {\\bf 542}, 97 (2005)), for\nthe multifractal probability density of velocity increments, not based on the\nsteepest descent evaluation of the integrals is obtained and the calculated\nfunction $D(h)$ is close to experimental data. A novel derivation (Polyakov,\n2005), of a well-known result of the multi-fractal theory [Frisch, \"Turbulence.\n{\\it Legacy of A.N.Kolmogorov}\", Cambridge University Press, 1995)), based on\nthe concepts described in this paper, is also presented.",
"arxiv_id": "physics/0512102",
"authors": [
"Victor Yakhot"
],
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"physics.flu-dyn",
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"doi": "10.1016/j.physd.2006.01.012",
"journal_ref": "Physica D215 (2006) 166-174",
"title": "Probability Densities in Strong Turbulence",
"url": "https://arxiv.org/abs/physics/0512102"
},
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