Extremal-point density of scaling processes: From fractional Brownian motion to turbulence in one dimension

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 $\mu$, 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 $\xi \left(H\right)$ is found to be $\xi \left(H\right)=H$, where $H$ is Hurst number, while for MRW $\xi \left(\mu \right)$ 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 ${\text{Re}}_{\lambda }=720$, and a turbulent boundary layer with the momentum thickness $\theta$ based Reynolds number ${\text{Re}}_{\theta }=810$. 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 agrees well with the corresponding sublayer structures inside the turbulent boundary layer.

Figure 1

Synthesized fractional Brownian motion data for various Hurst number $H$ using the same random numbers in the Wood-Chan algorithm with 1024 data points in each realization. Visually, with the increasing of $H$, the fBm curve becomes more and more smooth, i.e., has less extremal points. For display clarity, these curves have been vertically shifted.

Figure 2

Joint pdf $p({\varphi }_1,{\varphi }_2)$ for (a) $H=1/3$ and (b) $H=0.75$. The corresponding normalized conditional pdf $p\left(y\right)$ at various ${\varphi }_1$ and ${\varphi }_2$ for (c) $H=1/3$ and (d) $H=0.75$. For comparison, the normal distribution is illustrated as a solid line. Dashed lines in (a) and (b) denote the calculated ${\varphi }_1$ and ${\varphi }_2$ value.

Figure 3

Comparison of the measured $\mathcal{I}\left(H\right)$ from direct counting, Eq. (1) ($◯$), Eq. (6) ($\square$) via calculating ${\varphi }_i$ and Eq. (4) via calculating $\langle x_t{\left(t_i\right)}^2\rangle$ and $\langle x_{tt}{\left(t_i\right)}^2\rangle$ ($▵$), with theoretical predictions by Eqs. (9) (solid line: $\frac{1}{\pi }arccos(2^{2H-1}-1)$), (13) (dashed-dotted line: $2\sqrt{\frac{2-2H}{4-2H}}$), and (16) (dashed line: $\frac{2-2H}{3-2H}$), respectively. The inset shows the relative errors between Eq. (9) and others.

Figure 4

(a) Relative standard deviation ${\sigma }_H\left(L\right)/{\mathcal{I}}_{\mathrm{T}}\left(H\right)\times 100\%$ for various $H$. (b) The compensated data using fitted parameters to emphasize the power-law behavior. The inset shows the measured scaling exponent $\gamma \left(H\right)$, where the dashed line shows the value $1/2$, expected from the central limit theorem. The error bar is the $95\%$ fitting confidence interval.

Figure 5

Illustration of the filtering effect: (a) raw data with $H=0.5$ and 10,000 data points; (b) $\ell =100$ data points; (c) $\ell =500$ data points. Increasing $\ell$ removes more local extremal points.

Figure 6

(a) Measured $\mathcal{I}(H,\ell )$ with $H=0.2,0.4,0.6$, and 0.8. The solid line is a power-law fit. For display convenience, $\mathcal{I}(H,\ell )$ is normalized by $\mathcal{I}(H,0)$. (b) The measured scaling exponent $\xi \left(H\right)$ versus $H$. The error bar is the standard deviation obtained from 100 realizations. The inset shows the relative error $\mathrm{Er}\left(H\right)$ versus $H$.

Figure 7

Illustration of the discrete MRW cascade process. Each step is associated with a scale ratio of 2. After $p$ steps, the total scale ratio is $2^p$. The synthesized multifractal measure ${\epsilon }_{\mu }\left(t\right)$ has a lognormal statistics. The corresponding intermittency correction is controlled by the parameter $\mu ={\sigma }^2(ln{W}_n)/ln\lambda$.

Figure 8

Illustration of the synthesized MRW $x_{\mu }\left(t\right)$, for one realization with various intermittency parameter $\mu$. For display clarity, the curves have been vertical shifted. The measured $\mathcal{I}\left(\mu \right)\simeq 0.5011$ is independent of $\mu$ since the same random number are used to construct the multifractal measure ${\epsilon }_{\mu }\left(t\right)$.

Figure 9

Contour plot of the measured relative error between the structure function scaling exponent $\zeta \left(q\right)$ and the lognormal prediction by Eq. (26). The typical value $0.2\le \mu \le 0.4$ for the Eulerian turbulent velocity is indicated by a horizontal solid line.

Figure 10

(a) Measured EPD $\mathcal{I}\left(\mu \right)$ for MRW. The errorbar is the standard deviation from 100 realizations. Due to the intermittency effect, Eq. (6) overestimates $\mathcal{I}\left(\mu \right)$. (b) The relative error $\mathrm{Er}\left(\mu \right)$ for the measured $\mathcal{I}\left(\mu \right)$ (open symbols) and the standard deviation $\sigma$ (closed symbols), where a theoretical value $\mathcal{I}\left(\mu \right)=1/2$ is considered as the reference case.

Figure 11

Experimental pdfs for the first-order derivative of MRW, $\delta x=x_{\mu }^{\prime }\left(t\right)$, with various $\mu$. For display clarity, the pdfs have been normalized by their respective standard deviation.

Figure 12

Measured $\mathcal{I}(\mu ,\ell )$ versus $\mu$ at various scale $\ell$ from (a) direct counting by Eq. (1), (b) Eq. (6). The solid line is an exponential fitting. Exponential-law is observed for both approaches, but with different trends.

Figure 13

(a) Experimental scaling exponent $\pi \left(\ell \right)$ versus $\ell$. Due to the violation of the joint Gaussian distribution requirement, Eq. (6) fails to measure $\pi \left(\ell \right)$. (b) Contour plot of the measured ratio ${\mathcal{I}}_{\mathrm{T}}(\mu ,\ell )/\mathcal{I}(\mu ,\ell )$ to confirm the overestimation of $\mathcal{I}(\mu ,\ell )$ by Eq. (6).

Figure 14

$\mathcal{I}(\mu ,\ell )$ versus $\ell$ from (a) direct counting and (b) Eq. (6). For display clarity, the curves have been vertical shifted. The corresponding scaling exponent $\xi \left(\mu \right)$ is fitted in the range $100\le \ell \le 1000$.

Figure 15

Scaling exponent $\xi \left(\mu \right)$ versus $\mu$ from (a) direct counting ($◯$), and (b) Eq. (6) ($\square$), where a linear fitting is illustrated by a solid line. The horizontal dashed line indicates the value $1/2$ for Brownian motion.

Figure 16

(a) Measured ${\mathcal{I}}_H\left(\ell \right)$ for the turbulent velocity obtained from a wind tunnel experiment [36] with a Reynolds number ${\text{Re}}_{\lambda }\simeq 720$. For display clarity, the curve for $u-\mathrm{T}$ and $v-\mathrm{T}$ have been vertically shifted by multiplying 1.5 and 0.5, respectively. The solid line is the power-law fit in the range $0.0001\le \tau \le 0.01\mathrm{s}$, corresponding to a frequency in the range $100<f<10000\mathrm{Hz}$. (b) The corresponding curves compensated by the fitting parameters.

Figure 17

Schematic representation of the boundary layer experimental setup [40] with a unit cm. The inflow is $4.5{\mathrm{ms}}^{-1}$ with a Reynolds number ${\text{Re}}_{\theta }\simeq 810$. A trip-wire with a twisted-wire is located at $8\mathrm{cm}$ downstream to accelerate the turbulent boundary layer development. The measurement is performed at $145\mathrm{cm}$ downstream ($\oplus$).

Figure 18

(a) Measured Fourier power spectrum at different height $y^{+}$. (b) The corresponding $\mathcal{I}(y^{+},\ell )$ versus $\tau$.

Figure 19

The measured EPD $\mathcal{I}(y^{+},\tau )$ versus $y^{+}$ reproduced in a log-log view, together with the four regimes of the turbulent boundary layer structure.