arXiv physics/05121022005-12-14physics.flu-dynastro-phnlin.CD

强Turbulence中的概率密度

Probability Densities in Strong Turbulence

Victor Yakhot

备注：: 25 pages and 9 figures
发表信息：: Physica D215 (2006) 166-174
DOI：: 10.1016/j.physd.2006.01.012

摘要：

根据湍流理论的现代发展，“耗散”尺度（u. v.截止值）$η$形成与速度增量相关的随机场$δ_ηu$。本文利用Mellin变换结合Gaussain大尺度边界条件，计算了速度增量的概率密度（PDF）$P(δ_{r}u,r)$和耗散尺度的概率密度（PDF）$Q(η, Re)$，其中$Re$为大尺度雷诺数。由此产生的表达式强烈偏离对数正态PDF $P_{L}(δ_{r}u,r)$经常引用的文献。它示出的小尺度速度波动的概率密度包括有关的大（积分）尺度动力学，这是负责从$P_{L}(δ_{r}u,r)$ $P(δ_{r}u,r)$偏差的信息。开发了一个框架，用于评估涉及空间导数的各种湍流特性的PDF。在Frisch等人（J. Fluid Mech. { \bf 542}，97（2005））中最近讨论的不含伪对数的速度增量的多重分形概率密度的精确关系，不是基于积分的最陡下降估计，并且计算的函数$D(h)$接近于实验数据。一个新的推导（Polyakov，2005），一个众所周知的结果的多重分形理论[弗里施，“湍流。{ \it Legacy of A.N.Kolmogorov}"，Cambridge University Press，1995））的方法。

英文摘要：

According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $η$ form a random field related to velocity increments $δ_ηu$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P(δ_{r}u,r)$ and the PDF of the dissipation scales $Q(η, Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_{L}(δ_{r}u,r)$ 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 $P(δ_{r}u,r)$ from $P_{L}(δ_{r}u,r)$. 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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