Huang Yongxiang, Wang Lipo, Schmitt F G, Zheng Xiaobo, Jiang Nan, Liu Yulu
State Key Laboratory of Marine Environmental Science, College of Ocean and Earth Sciences, Xiamen University, Xiamen 361102, China.
UM-SJTU Joint Institute, Shanghai JiaoTong University, Shanghai, 200240, China.
Phys Rev E. 2017 Jul;96(1-1):012215. doi: 10.1103/PhysRevE.96.012215. Epub 2017 Jul 17.
In recent years several local extrema-based methodologies have been proposed to investigate either the nonlinear or the nonstationary time series for scaling analysis. In the present work, we study systematically the distribution of the local extrema for both synthesized scaling processes and turbulent velocity data from experiments. The results show that for the fractional Brownian motion (fBm) without intermittency correction the measured extremal-point-density (EPD) agrees well with a theoretical prediction. For a multifractal random walk (MRW) with the lognormal statistics, the measured EPD is independent of the intermittency parameter μ, suggesting that the intermittency correction does not change the distribution of extremal points but changes the amplitude. By introducing a coarse-grained operator, the power-law behavior of these scaling processes is then revealed via the measured EPD for different scales. For fBm the scaling exponent ξ(H) is found to be ξ(H)=H, where H is Hurst number, while for MRW ξ(μ) shows a linear relation with the intermittency parameter μ. Such EPD approach is further applied to the turbulent velocity data obtained from a wind tunnel flow experiment with the Taylor scale λ-based Reynolds number Re_{λ}=720, and a turbulent boundary layer with the momentum thickness θ based Reynolds number Re_{θ}=810. A scaling exponent ξ≃0.37 is retrieved for the former case. For the latter one, the measured EPD shows clearly four regimes, which agrees well with the corresponding sublayer structures inside the turbulent boundary layer.
近年来，已经提出了几种基于局部极值的方法来研究非线性或非平稳时间序列，以进行标度分析。在本工作中，我们系统地研究了合成标度过程和实验得到的湍流速度数据的局部极值分布。结果表明，对于未进行间歇性校正的分数布朗运动（fBm），测量得到的极值点密度（EPD）与理论预测吻合良好。对于具有对数正态统计的多重分形随机游走（MRW），测量得到的EPD与间歇性参数μ无关，这表明间歇性校正不会改变极值点的分布，但会改变其幅度。通过引入一个粗粒化算子，然后通过测量不同尺度下的EPD揭示了这些标度过程的幂律行为。对于fBm，发现标度指数ξ(H)为ξ(H)=H，其中H是赫斯特数，而对于MRW，ξ(μ)与间歇性参数μ呈线性关系。这种EPD方法进一步应用于从雷诺数Re_λ = 720的风洞流动实验获得的湍流速度数据，以及雷诺数Re_θ = 810的湍流边界层。对于前一种情况，得到的标度指数ξ≃0.37。对于后一种情况，测量得到的EPD明显显示出四个区域，这与湍流边界层内相应的子层结构吻合良好。
