3.1 Magnitude cumulant analysis: a measure of intermittency

Spatial or temporal series of velocity fluctuations in turbulent flows are typically decomposed into Fourier modes to compute the power spectral density that describes the energetic contribution of eddies of different sizes, or, in statistical terms, identifies scale-dependent variations of the second-order moment, or variance (Singh, Porté-Agel & Foufoula-Georgiou Reference Singh, Porté-Agel and Foufoula-Georgiou2010; Singh et al. Reference Singh, Howard and Guala2014). In addition, the power spectrum provides all the necessary information for re-creating a representative turbulent velocity time series in the case when the velocity increments are Gaussian-distributed over a range of scales (Singh et al. Reference Singh, Lanzoni, Wilcock and Foufoula-Georgiou2011, Reference Singh, Foufoula-Georgiou, Porté-Agel and Wilcock2012); the velocity increments are defined here as ${\rm\Delta}u(t,a)=u(t+a)-u(t)$ as a function of the separation time scale $a$ . However, the distribution of the velocity increments in a turbulent signal usually shows significant deviation from the Gaussian distribution at smaller scales, i.e. for values of $a$ approaching the Kolmogorov time scale ${\it\tau}_{k}$ (see e.g. Malecot et al. Reference Malecot, Auriault, Kahalerras, Gagne, Chanal, Chabaud and Castaing2000; Ruiz-Chavarria et al. Reference Ruiz-Chavarria, Ciliberto, Baudet and Lévêqueb2000). This can also be observed in the probability distribution function (p.d.f.) of velocity increments for WT (figure 3 a) and SLT (figure 3 b), suggesting (i) a multiscaling type of behaviour of turbulent velocity fluctuations, and (ii) that higher order statistical moments are required for reconstructing the original velocity signal. Note that the Kolmogorov time scale is defined here as ${\it\tau}_{k}={\it\eta}/U$ , where ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ is the Kolmogorov length scale, ${\it\epsilon}$ is the average rate of dissipation of turbulent kinetic energy, ${\it\nu}$ is the kinematic viscosity and $U$ is the average convective velocity. This is an advective form of the Kolmogorov time scale, which is different from the Kolmogorov time scale ${\it\tau}=({\it\nu}/{\it\epsilon})^{1/2}$ used in homogeneous turbulence and inadequate when the convective velocity is much larger than the Kolmogorov velocity. The $ITS$ is estimated using the integral of the autocorrelation function of the streamwise velocity.

Figure 3.

Semilogarithmic p.d.f.s of the normalized form of the velocity increments ${\rm\Delta}u(t,a)=u(t+a)-u(t)$ ) for the scales $a={\it\tau}_{k}$ and $a=ITS$ for both WT (a) and SLT (b) datasets. The solid curve denotes the Gaussian distribution. P.d.f.s are shown for $z_{WT}^{+}=1587$ (a) and $z_{SLT}^{+}=1464$ (b).

While a large amount of work has been devoted to high-order structure function analysis (Menevau & Sreenivasan Reference Menevau and Sreenivasan1991; Toschi, Leveque & Ruiz-Chavarria Reference Toschi, Leveque and Ruiz-Chavarria2000; Venugopal et al. Reference Venugopal, Roux, Foufoula-Georgiou and Arnéodo2006; Basu et al. Reference Basu, Foufoula-Georgiou, Lashermes and Arnéodo2007; Kholmyansky, Moriconi & Tsinober Reference Kholmyansky, Moriconi and Tsinober2007; Singh et al. Reference Singh, Fienberg, Jerolmack, Marr and Foufoula-Georgiou2009, Reference Singh, Lanzoni, Wilcock and Foufoula-Georgiou2011; Guala et al. Reference Guala, Metzger and McKeon2010), an alternative method has been proposed in the form of the magnitude cumulant analysis, introduced by Delour, Muzy & Arnéodo (Reference Delour, Muzy and Arnéodo2001) – see also Malecot et al. (Reference Malecot, Auriault, Kahalerras, Gagne, Chanal, Chabaud and Castaing2000) and Chevillard et al. (Reference Chevillard, Roux, Leveque, Mordant, Pinton and Arneodo2005). This approach requires only the second-order magnitude cumulant to unravel the underlying intermittency in the turbulence structure of the velocity signal, under the assumption of log-normality in the increment distribution (see e.g. Delour et al. Reference Delour, Muzy and Arnéodo2001; Chevillard et al. Reference Chevillard, Roux, Leveque, Mordant, Pinton and Arneodo2005; Basu et al. Reference Basu, Foufoula-Georgiou, Lashermes and Arnéodo2007; Singh et al. Reference Singh, Howard and Guala2014). In general, for a random variable $X$ having a p.d.f. $F(x)$ , the moment generating function (also defined as the characteristic function) can be expressed as

which can be expanded in a Taylor’s series as the following and leading to

where $M_{n}$ is the $n$ th-order moment of $X$ , represented as

Analogously, $M_{n}$ can be estimated by the $n$ th-order derivative of ${\it\phi}_{p}(k)$ at $k=0$ . The cumulant generating function of $X$ can be accordingly defined as

while the cumulant $C_{n}$ of $X$ can be obtained from the $n$ th derivative of ${\it\psi}_{p}(k)$ . Moments and cumulants can be related as

and so on.

If we substitute the variable $X$ with the velocity increment ${\rm\Delta}u$ separated by a given lag $a$ , the magnitude cumulant $C_{n}$ can be expressed as

where

Cumulants $C_{1}$ and $C_{2}$ are scale-dependent functions of the mean $\langle \ln |{\rm\Delta}u|\rangle$ and the variance $\langle (\ln |{\rm\Delta}u|)^{2}\rangle$ (for more details, see Malecot et al. Reference Malecot, Auriault, Kahalerras, Gagne, Chanal, Chabaud and Castaing2000; Delour et al. Reference Delour, Muzy and Arnéodo2001). Figure 4(a) shows the second-order cumulant $C_{2}$ as a function of scale $a$ normalized by the $ITS$ for both WT and SLT (see also Singh et al. (Reference Singh, Howard and Guala2014) for the plot of $C_{1}(a)$ , whose slope from log-linear regime represents the Hölder exponent $\langle h\rangle$ , for the WT dataset). The slope of the log-linear regime obtained from second-order cumulant $C_{2}(a)$ within the range of scales of interest (from Kolmogorov scale to integral time scale, ${\it\tau}_{k}<a<ITS$ ), is used to define the intermittency coefficient $c_{2}$ ; $c_{2}$ represents a compact, scale-independent, parameter describing the structure of each streamwise velocity increment time series, by providing a measure of the inhomogeneous temporal arrangement of its local abrupt fluctuations (Singh et al. Reference Singh, Lanzoni, Wilcock and Foufoula-Georgiou2011, Reference Singh, Foufoula-Georgiou, Porté-Agel and Wilcock2012) and is not to be confused with the above-defined second-order cumulant $C_{2}$ notation.

Figure 4(b) shows the computed intermittency coefficient $c_{2}$ profiles for the WT and SLT cases, for the range of wall-normal distances $z^{+}$ explored here. It is noted that the estimated intermittency coefficients $c_{2}$ obtained from traditional high-order structure function analysis (not shown here for brevity) resulted in similar values as obtained from the magnitude cumulant analysis presented here. As can be seen from figure 4(b), the $c_{2}$ profiles for both WT and SLT are very similar and, in fact, overlap for a range of $z^{+}$ , particularly in the range $z^{+}\sim 70{-}10^{3}$ . For the lower $z^{+}$ , SLTEST data show an increase in $c_{2}$ with decreasing $z^{+}$ , suggesting an increase in the frequency and magnitude of the intense velocity gradients that may have resulted from increasing mean shear as $z^{+}$ decreases (see also Antonia, Orlandi & Romano Reference Antonia, Orlandi and Romano1998; Onorato, Camussi & Iuso Reference Onorato, Camussi and Iuso2000). Note that, in the case of a monofractal signal, where the shape of the increment distributions of the signal does not change with the scale $a$ , the intermittency coefficient $c_{2}$ is precisely $0$ . Also, note that the $c_{2}$ values in our study for both WT and SLT datasets are consistent with the $c_{2}$ observed in Basu et al. (Reference Basu, Foufoula-Georgiou, Lashermes and Arnéodo2007) for a similar dataset in the atmospheric surface layer.

Figure 4.

(a) Second-order cumulant $C_{2}$ as a function of scale $a$ normalized by the $ITS$ and (b) vertical profile of intermittency coefficients ( $c_{2}$ ) as a function of vertical wall distance $z^{+}$ for the streamwise velocity for both WT (stars) and SLT (circles). Note that the intermittency coefficients $c_{2}$ shown in figure 4(b) are computed from the second-order cumulants $C_{2}$ (figure 4 a) for a range of scales, i.e. Kolmogorov scale ${\it\tau}_{k}$ to $ITS$ .

3.2 Scale-dependent asymmetry of the velocity increments

From figure 3 it was noticed that the p.d.f.s of the velocity increments, for varying time scale $a$ , are neither Gaussian nor symmetric, a combination that is inferred to point at multiplicative cascade processes and non-local scale interactions (Schertzer et al. Reference Schertzer, Lovejoy, Schmitt, Chiguirinskaya and Marsan1997; Warhaft Reference Warhaft2002; Singh et al. Reference Singh, Foufoula-Georgiou, Porté-Agel and Wilcock2012, Reference Singh, Howard and Guala2014). While non-Gaussianity is expressed through the intermittency coefficient $c_{2}$ , the asymmetry in the distribution can be quantified using either the third or the fifth moment function (as in this study) of the velocity increments (see Warhaft Reference Warhaft2002). Results are provided in figure 5 for WT and SLT velocity time series at $z^{+}\sim 1500$ for the smallest scale separation $a={\it\tau}_{k}$ . The qualitative behaviour of the two curves directs attention towards a Reynolds-number-independent feature of the turbulent boundary layer, regarding the distribution and statistical weight of rare but intense dissipative events.

Figure 5.

Fifth moment functions ( $x^{5}f(x)$ , where $x={\rm\Delta}u(t,a)/\langle ({\rm\Delta}u(t,a))^{2}\rangle ^{1/2}$ ) of the streamwise velocity increments for the WT and the SLT. Note that the p.d.f.s of the velocity increments for both WT and SLT were computed for $a={\it\tau}_{k}$ , at $z_{WT}^{+}=1587$ and $z_{SLT}^{+}=1464$ , respectively.

To better quantify the scale-dependent asymmetry of the velocity increment distribution, we define the asymmetry index as a function of the time scale $a$ and vertical height $z$ , namely $A(a,z)=\langle (X-\overline{X})^{3}\rangle /\langle |(X-\overline{X})^{3}|\rangle$ , where $X={\rm\Delta}u(a,z)$ and $\overline{X}$ is the mean of $X$ . The asymmetry index is plotted in figure 6(a) for different scales, at different heights, for both WT and SLT datasets. As outlined in the introduction, a hump is observed in a range of scales smaller than the $ITS$ in both datasets, consistent in terms of frequency and amplitude with the hump reported by Basu et al. (Reference Basu, Foufoula-Georgiou, Lashermes and Arnéodo2007). Here we confirm the location of the hump in the inertial range using an independent dataset from atmospheric surface layer (SLT) and a new wind tunnel experiment (WT).

Figure 6.

Asymmetry index of the streamwise velocity increments as a function of scale for WT (black lines) and SLT (blue lines) at $70<z^{+}<3.3\times 10^{3}$  (a) and at $z^{+}>4\times 10^{4}$  (b). The $x$ -axis represents the time scale $a$ normalized by the $ITS$ .

3.3 Reynolds-number-independent asymmetry in turbulent boundary layers

As the difference in Reynolds number between SLTEST and the SAFL wind tunnel is of the order of two decades, the emerging hump-like feature might be universal to turbulent boundary layer flows. One of the main objectives of this paper is to compare systematically the amplitude and location of the hump at varying wall-normal distances, providing different normalization options for both the vertical coordinate $z$ and the velocity increment time scale $a$ . The main hypothesis put forward is how the asymmetry index evolves in different regions of the turbulent boundary layer, for the two available Reynolds-number experiments. In order to allow a direct comparison between the laboratory-scale (WT) and field-scale (SLT) datasets, we kept a consistent normalization for $a$ , using the integral time scale for normalization, $a/ITS$ .

As discussed earlier in the introduction, scale interactions near the wall are the main source of intermittency and anisotropy observed in high-order statistics of the streamwise velocity (Guala et al. Reference Guala, Metzger and McKeon2010, Reference Guala, Metzger and McKeon2011; Singh et al. Reference Singh, Howard and Guala2014). In the continuous cycle of mutual influence between (inner) near-wall turbulence and (outer) ${\it\delta}$ -scale motions, we argue that a single scaling may not work across the whole range of turbulent scales and wall distances. For this purpose, we investigate here if inner ( $z^{+}=zu_{{\it\tau}}/{\it\nu}$ ) or outer ( $z/{\it\delta}$ ) unit normalization will lead to a collapse of the anisotropy (asymmetry index) curves $A(a,z)$ for a specific range of temporal scales $a$ and vertical locations $z$ . In this respect, we recall that the spacing between near-wall streaks ( $\simeq 100$ viscous wall units) has been proved to be fairly invariant across a wide range of Reynolds number (Metzger & Klewicki Reference Metzger and Klewicki2001). Recognizing the key role of near-wall structures to inner–outer scale interaction processes, we first attempt to use the viscous scale ${\it\nu}/u_{\ast }$ for wall distance normalization. We compare scale-dependent asymmetry curves for the two datasets at the fixed $z^{+}$ location in figure 6(a,b). Here the normalizing time scale for the velocity increment is kept as $ITS$ , which of course varies in the two Reynolds-number flows.

In figure 6(a), we note how the hump maintains its amplitude and temporal location for a wide range of $z^{+}$ , independent of the Reynolds number, unambiguously marking a statistical property of turbulent boundary layer flows. However, for $z^{+}>4000$ (figure 6 b), we start to notice appreciable deviations for large time scales $a$ approaching the $ITS$ , suggesting a change in scale interaction processes, and possibly in wall turbulence organization at the SLT high-Reynolds-number flow. For even larger wall distances, inspected in figure 6(b), the asymmetry peak is lost within ample oscillations that might be the result of the interactions between large- and very-large-scale motions populating the logarithmic layer of the SLT flow; in such a case, we must note that the SLT dataset may not be long enough to ensure statistical convergence of the scale-dependent asymmetry index in the full temporal–vertical domain investigated. This said, figure 6(b) confirms that the flow region where asymmetry statistics overlap is confined in the $O(10^{2}{-}10^{3})$ range of viscous wall units. Note that an independent set of data from a wind tunnel experiment (not shown here) showed similar trends and results to those presented here.

Figure 7.

Asymmetry index of the streamwise velocity increments as a function of scale for WT (black line) and SLT (blue lines) for $z^{+}<10^{2}$ . The $x$ -axis represents the time scale $a$ normalized by the $ITS$ .

The SLT data provide the opportunity to further explore the scale-dependent asymmetry very close to the wall ( $z^{+}<100$ ; see figure 7). As the surface is approached, the amplification of $A(a,z)$ in the Kolmogorov scale range contributes to increase the peak value while smearing the hump signature. With intermittency and small-scale asymmetry both high at the wall, scale coupling is inferred to occur within a different layer and population of wall turbulent structures, as compared to the case discussed above in the logarithmic layer. Following a structural interpretation, figure 7 suggests a strong effect of large- and very-large-scale motions at the surface, where the mean shear is strongest, literally sweeping the wall and distorting the distribution of the near-wall streamwise velocity at those small scales that were traditionally marked as homogeneous and isotropic. Recently Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013) quantified third-order central moments of velocity fluctuations obtained at high Reynolds number and observed increasing skewness with decreasing wall-normal distance. These observations are in qualitative agreement with our results, suggesting that scale coupling increases as the wall distance decreases. The evolution of the asymmetry peak appears to occur quite sharply in the $(a,z)$ domain, suggesting a marked change in the wall structure organization. This is consistent with the spatial transition between the near-wall streaks and the ramp-like structure organization above $z^{+}\sim 70$ , which marks the lower end of the logarithmic layer. With increasing distance from the wall ( $10<z^{+}<50$ ), the streaks’ asymmetry is reduced at the small time scales, until the hump $z^{+}>70$ clearly emerges in the temporal (inertial range) and wall-normal domain of the attached eddies (see Perry, Henbest & Chong Reference Perry, Henbest and Chong1986).

The $A(a,z)$ vertical distribution suggests that local high shear induced by large-scale structures impinging the wall is felt by near-wall streaks, leading to increased intermittency (see Guala et al. Reference Guala, Metzger and McKeon2010, and also figure 4 b), asymmetry and non-local energy transfer (Warhaft Reference Warhaft2002; Singh et al. Reference Singh, Howard and Guala2014). The asymmetry profiles also indicate that the ramp-like structures inferred in Guala et al. (Reference Guala, Metzger and McKeon2011) to populate, at the atmospheric scale (high Reynolds number), a relatively near-wall region of the boundary layer up to $z^{+}\sim 5\times 10^{3}$ display the asymmetry-hump signature in the same vertical range. The role of ramp-like structures is confirmed by employing the attached eddy time scale $z/u_{\ast }$ as a replacement for the $ITS$ : in figure 8(a), asymmetry curves in the near-wall region are presented as a function of $a/(z/u_{\ast })$ and observed to partially overlap across a range of scales consistent with those reported in figure 6(a).

Figure 8.

(a) Asymmetry index of the streamwise velocity increments as a function of scale for WT (black lines) and SLT (blue lines) at $z^{+}<10^{3}$ . (b) Asymmetry index of the streamwise velocity increments as a function of scale for WT (black lines) and SLT (blue lines) for comparable $z/{\it\delta}$ locations. The $x$ -axis in (a) represents the time scale $a$ normalized by $z/u_{\ast }$ , whereas in (b) it represents the time scale $a$ normalized by the $ITS$ .

The validity of inner, viscous, scaling and its conceptual implications are further evidenced in figure 8(b). When asymmetry index curves are compared for both WT and SLT at different $z^{+}$ but similar $z/{\it\delta}$ locations, no significant data collapse is observed. Near-wall scaling is thus confirmed in a region well above the near-wall spectral peak of Hutchins & Marusic (Reference Hutchins and Marusic2007a ) and well within the logarithmic layer, where outer scaling would have been, in principle, an equally valid option. While we acknowledge that a full comparison should be carried out using measurements in the outer layer at the SLTEST Reynolds number (not available here), we must recognize that strong scale interactions, intermittency and anisotropy are typically observed in high-shear flows, thus naturally residing in the near-wall region. Therefore, it is speculated that outer scale normalization for $A(a,z)$ will hardly apply in large-scale geophysical flows, and that the asymmetry hump will remain a near-wall turbulence signature, with near wall defined in viscous units.