Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect
Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect
复制标题
湍流传输的精确重正化数学模型，II：分形界面、非高斯统计和扫掠效应
DOI：
10.1007/bf02099212
复制
发表时间：
1992
影响因子：
2.4
通讯作者：
A. Majda
中科院分区：
文献类型：
--
作者：
M. Avellaneda;A. Majda
本文继续研究了作者最近提出和发展的一个湍流输运模型，该模型采用了精确重整化理论。对该模型的三个重要问题进行了严格的数学分析：（1）被动平流标量的重整化高阶统计，如界面的对距离分布和分维;（2）非高斯湍流速度统计对重整化理论的影响;一个特别强调的是放在重整化理论在附近的模拟模型中的柯尔莫哥洛夫谱的值。在作者的早期论文中，它建立了Kolmogorov值是在相变边界的精确重整化理论。在这里发现，定性模型，尽管它的简单性，包含在附近的Kolmogorov值，显着量的定性行为的湍流传输已被发现在最近的实验中，并提出了唯象理论。特别地，当Kolmogorov谱从相变边界的一侧作为极限时，对色散和分维缺陷为2/3的界面的Richardson 4/3定律在模型中严格地作为极限出现;对Richardson定律的替代修正与最近文献中提出的修正形式相同，并且界面具有分形维数缺陷1/3，当从相变的另一侧接近Kolmogorov谱时，在模型中发生。在最近的湍流实验中，被动标量的水平集和界面的分维值约为1/3或2/3的缺陷是普遍存在的。对于非高斯分布，对应于紧支撑的均值为零的团块的归一化积分（B.56）的渐近正态性。后一个事实的证明是以与命题B.3的步骤2相同的方式完成的，使用的事实是，相应的随机过程
$$\tilde V_\delta（s）$$
有有限的依赖域。这就是命题B.4的证明。
AbstractThis paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors. Three important topics are analyzed with complete mathematical rigor for this model: (1) Renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces, (2) the effect of non-Gaussian turbulent velocity statistics on renormalization theory, (3) the “sweeping” effect of additional large scale mean velocities. A special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the Kolmogorov-spectrum in the model. In the authors' earlier paper, it was established that the Kolmogorov value is at a phase transition boundary in the exact renormalization theory. It is found here that the qualitative model, despite its simplicity contains, in the vicinity of the Kolmogorov value, a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories. In particular, the Richardson 4/3-law for pair dispersion and interfaces with fractal dimension defect of 2/3 occur in the model rigorously as limits when the Kolmogorov spectrum is approached as a limit from one side of the phase transition boundary; alternative corrections to the Richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1/3, occur in the model when the Kolmogorov spectrum is approached from the other side of the phase transition. It is very interesting that fractal dimension defects of roughly the value either 1/3 or 2/3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments. As regards non-Gaussian the asymptotic normality of normalized integrals (B.56) corresponding to compactly supported blobs with mean zero. The proof of this latter fact is done in the same way as Step 2, Proposition B.3, using the fact that the corresponding random processes
$$\tilde V_\delta (s)$$
have finite domain of dependence. This concludes the proof of Proposition B.4.
DOI：
--
发表时间：
2006
期刊：
影响因子：
--
作者：
N. Okamoto;K. Yoshimatsu;Y. Kaneda;Y. Kaneda
通讯作者：
Y. Kaneda