Extremal-point density of scaling processes from fractal Brownian motion to turbulence in one dimension

Abstract: 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 with the intermittency parameter $\mu$, suggesting that the intermittency correction does not change the distribution of extremal points, but change 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 $\xi(H)$ is found to be $\xi(H)=H$, where $H$ is Hurst number, while for MRW $\xi(H)$ shows a linear relation with the intermittency parameter $\mu$. Such EPD approach is further applied to the turbulent velocity data obtained from a wind tunnel flow experiment with the Taylor scale $\lambda$ based Reynolds number $Re_{\lambda}= 720$, and a turbulent boundary layer with the momentum thickness $\theta$ based Reynolds number $Re_{\theta}= 810$. A scaling exponent $\xi\simeq 0.37$ is retrieved for the former case. For the latter one, the measured EPD shows clearly four regimes, which agree well with the four regimes of the turbulent boundary layer structures.