turbulence的Levy-Kolmogorov标度
Levy-Kolmogorov scaling of turbulence
摘要：
柯尔莫哥洛夫双折射标度律被认为是上个世纪最重要的理论突破。它是分析湍流数据的基本方法，这些数据存在于气象、物理、化学、生物和机械现象中。它的一个非常基本的假设是，湍流是一个小尺度的随机高斯过程5。然而，在有限雷诺数的实验数据观察到一个明显的偏离高斯。在这项研究中，通过取代的Navier-Stokes（NS）方程中的标准拉普拉斯表示的耗散与分数拉普拉斯，我们得到分数NS方程的Levy稳定分布，表现出非高斯重尾和分数频率幂律耗散。对该方程进行量纲分析，得到了一种新的线性标度，称为Levy-Kolmogorov标度，其标度指数范围为-3 ~-5/3，对应于不同的Levy过程，并简化为高斯过程下的极限Kolmogorov标度-5/3。本文还讨论了边界效应引起的截断Levy过程和多尺度问题。最后，我们进一步扩展我们的模型，以反映历史依赖的分数布朗运动。
英文摘要：
The Kolmogorov scaling law of turbulences has been considered the most important theoretical breakthrough in the last century. It is an essential approach to analyze turbulence data present in meteorological, physical, chemical, biological and mechanical phenomena. One of its very fundamental assumptions is that turbulence is a stochastic Gaussian process in small scales5. However, experiment data at finite Reynolds numbers have observed a clear departure from the Gaussian. In this study, by replacing the standard Laplacian representation of dissipation in the Navier-Stokes (NS) equation with the fractional Laplacian, we obtain the fractional NS equation underlying the Levy stable distribution which exhibits a non-Gaussian heavy trail and fractional frequency power law dissipation. The dimensional analysis of this equation turns out a new scaling of turbulences, called the Levy-Kolmogorov scaling, whose scaling exponent ranges from -3 to -5/3 corresponding to different Levy processes and reduces to the limiting Kolmogorov scaling -5/3 underlying a Gaussian process. The truncated Levy process and multi-scaling due to the boundary effect is also discussed. Finally, we further extend our model to reflecting the history-dependent fractional Brownian motion.
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