4.1. Geometrical properties of the structures

The geometrical properties of the structures are characterized by the two length scales, $l_x^+$ and $l_z^+$, which are normalized by viscous units. Figures 4(a) and 4(b) show the joint p.d.f.s of $l_x^+$ and $l_z^+$ for $PF_s$ and $NF_s$, respectively. The joint p.d.f.s cover a wide range of $l_x^+$ and $l_z^+$, which implies that the skin-friction events are multiscale and the result of the collective effect of motions with different length scales (de Giovanetti et al. Reference de Giovanetti, Hwang and Choi2016; Yang & Lozano-Durán Reference Yang and Lozano-Durán2017; Cho et al. Reference Cho, Hwang and Choi2018; Agostini & Leschziner Reference Agostini and Leschziner2019a; Baidya et al. Reference Baidya, Baars, Zimmerman, Samie, Hearst, Dogan, Mascotelli, Zheng, Bellani and Talamelli2019; Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2019). The mode of the joint p.d.f.s for $PF_s$ and $NF_s$ follows approximately

(4.1)\begin{equation} l_x^+=0.64(l_z^{+})^{1.44} \end{equation}

for $l_z^+\geq 100$. For reference, figure 4(c) shows the two-dimensional spectra of $u'$ in the viscous region. The streamwise wavelength ($\lambda _x^+$) and the spanwise wavelength ($\lambda _z^+$) follows $\lambda _x^+ \propto (\lambda _z^+)^2$, which coincides with the power law between $\lambda _x^+$ and $\lambda _z^+$ of the $u'$ spectra in the logarithmic region reported by Del Álamo et al. (Reference Del Álamo, Jiménez, Zandonade and Moser2004). One possible explanation for the nonlinear relationship between $\lambda _x^+$ and $\lambda _z^+$ (or $l_x^+$ and $l_z^+$) is the meandering nature of the near-wall $u'$ structures. At low Reynolds numbers, Jiménez, Del Álamo & Flores (Reference Jiménez, Del Álamo and Flores2004) observed a power law relationship between $\lambda _x^+$ and $\lambda _z^+$ for near-wall streaky structures when taking their meandering nature into account. Some recent studies have reported a linear relationship between $\lambda _x^+$ and $\lambda _z^+$ at very large Reynolds number. Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017) investigated the two-dimensional spectra of $u'$ in a turbulent boundary layer at $Re_\tau =26\,000$, and found a linear relationship of $\lambda _x^+\propto \lambda _z^+$. Therefore, $l_x^+$ and $l_z^+$ of the $\tau _x'$ structures might also follow a linear relationship at sufficiently high Reynolds numbers.

Figure 4.

(a,b) Joint p.d.f.s of $l_z^+$ and $l_x^+$ for $PF_s$ (a) and $NF_s$ (b). The dashed straight lines in panel (a) and panel (b) denote $l_x^+=0.64(l_z^{+})^{1.44}$. All contour levels of the p.d.f.s are logarithmically distributed. (c) Two-dimensional premultiplied spectra of $u'$ in the viscous region. The dashed straight lines in panel (c) denote $\lambda _x^+ \propto (\lambda _z^+)^{2}$. Contour lines are 0.25, 0.5 and 0.75 of the maximum. For all figures, black lines denote $Re550$, red lines denote $Re950$ and blue lines denote $Re2000$.

Under the attached-eddy model framework (Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982) and assuming that wall-attached eddies are space filling, Srinath et al. (Reference Srinath, Vassilicos, Cuvier, Laval, Stanislas and Foucaut2018) and Solak & Laval (Reference Solak and Laval2018) pointed out that the p.d.f. of $l_x^+$ should obey $p \propto (l_x^+)^{-2}$. The marginal p.d.f.s of $l_x^+$ for $\tau _x'$ structures is shown in figure 5. It can be seen that asymmetrical features are exhibited between $PF_s$ and $NF_s$. For $NF_s$, the p.d.f.s of $l_x^+$ approximately follow $p\propto (l_x^+)^{-2}$, which is consistent with the prediction of Srinath et al. (Reference Srinath, Vassilicos, Cuvier, Laval, Stanislas and Foucaut2018) and Solak & Laval (Reference Solak and Laval2018); whereas for $PF_s$, the relation is less accurate. This result suggests that $NF_s$ may be the direct outcome of the space-filling wall-attached eddies, which will be analysed in detail in §4.5.

Figure 5.

Marginal p.d.f.s of $l_x^+$ for $PF_s$ (a) and $NF_s$ (b). The red dashed lines in figures represent $p\propto (l_x^+)^{-2.0}$.

The marginal p.d.f.s of aspect ratio $l_x^+/l_z^+$ are shown in figure 6. The p.d.f.s of $PF_s$ and $NF_s$ differ in shape, but they are insensitive to the Reynolds number. The p.d.f. profiles of $NF_s$ are flatter than those of $PF_s$, suggesting that $l_x^+/l_z^+$ of $NF_s$ is more uniformly distributed. The p.d.f.s reach their maxima at $l_x^+/l_z^+\approx 5$, regardless of the Reynolds number and the sign of the $\tau _x'$ structure.

Figure 6.

Marginal p.d.f.s of $l_x^+/l_z^+$ for the structures.

4.2. Spatial distribution of the structures

We investigate next the spatial distribution of $\tau _x'$ structures. Let us consider two structures, labelled as $i$ and $j$. Figure 7 provides a schematic of the two structures. Their relative separation in the streamwise and spanwise direction are defined as

(4.2a,b)\begin{equation} \delta_x=\frac{x_c^{(j)}-x_c^{(i)}}{d^{(i,j)}}\quad \textrm{and}\quad \delta_z=\frac{z_c^{(j)}-z_c^{(i)}}{d^{(i,j)}}, \end{equation}

respectively, where ($x_c$, $z_c$) is the centre of the bounding rectangle belonging to the structure $i$ or $j$, and $d^{(i,j)}$ is the mean diagonal of the two rectangles. The absolute distance of the two structures is defined as

(4.3)\begin{equation} r_{i,j}=\sqrt{(x_c^{(j)}-x_c^{(i)})^2+(z_c^{(j)}-z_c^{(i)})^2}. \end{equation}

Figure 7.

The schematic of the distance between two $\tau _{x}'$ structures.

Following the works of Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017) and Osawa & Jiménez (Reference Osawa and Jiménez2018), we only consider structures with $1/2\le l_z^{(j)}/l_z^{(i)} \le 2$, i.e. the structures sharing similar spanwise size.

Figure 8 shows the joint p.d.f.s of $\delta _x$ and $\delta _z$ for $Re950$. Similar results are obtained for the other two cases, which are not shown here for brevity. Figure 8 shows that structures tend to be organized side by side along the spanwise direction, while the low probabilities around $\delta _z \thickapprox 0$ indicates that the structures do not overlap along the spanwise direction.

Figure 8.

Joint p.d.f.s of the relative spacings in $Re950$. (a) The closest $PF_s$ relative to a $PF_s$, (b) the closest $NF_s$ relative to a $NF_s$ and (c) the closest $PF_s$ relative to a $NF_s$.

Figure 9 displays the variations of $r_{i,j}^+$ as a function of $l_z^+$. For $l_z^+>100$, $NF_s$ follow the approximately the power law relationship $r_{i,j}^+\approx 5.3(l_z^+)^{0.8}$, regardless of the Reynolds number, whereas a similar trend is absent for $PF_s$.

Figure 9.

Variations of the absolute distance ($r_{i,j}^+$) as a function of $l_z^+$. (a) $PF_s$ and (b) $NF_s$. The red dashed lines in panel (b) denotes $r_{i,j}^+\approx 5.3(l_z^+)^{0.8}$.

4.3. Scale-dependent properties of the structures

In this section we investigate the properties of the $\tau _x'$ structures conditioned on $l_z^+$, including the population density, intensity of $\tau _x'$ and marginal p.d.f.s of $\tau _x'$ at several selected $l_z^+$.

The scale-dependent population density is defined as

(4.4)\begin{equation} n_d(l_z^+)=\frac{N_i(l_z^+)}{N_F A_w^+}, \end{equation}

where $N_i(l_z^+)$ is the number of the structures with a given $l_z^+$, $N_F$ is the number of flow fields used to accumulate statistics and $A_w^+ = 2 L_x^+ L_z^+$. Figure 10 shows the distribution of $n_d$ as a function of $l_z^+$ for all cases. It can be seen that, for $NF_s$, $n_d$ decays logarithmically with $n_d \propto (l_z^+)^{-2.5}$ when $l_z^+> 100$; whereas a constant power-law decaying rate is less defined for $PF_s$. The latter suggests that $PF_s$ and $NF_s$ are distributed in different manners.

Figure 10.

Variations of the population density ($n_d$) as a function of $l_z^+$. (a) $PF_s$ and (b) $NF_s$. The red dashed line in panel (b) denotes $n_d \propto (l_z^+)^{-2.5}$.

The scale-dependent intensity of $\tau _x'$ is expressed as

(4.5)\begin{equation} I_{\tau_{x}'}^+(l_z^+)=\sqrt{\frac{1}{N_{\tau}}\sum_{i=1}^{N_{\tau}}\left[ \frac{\tau_{i,x}'(l_z^+)}{\tau_{x,rms}'}\right]^2 }, \end{equation}

to the structures with a given $l_z^+$, and $\tau _{i,x}'(l_z^+)$ denotes the instantaneous streamwise wall-shear stress at the $i$th grid point. In this manner, $I_{\tau _{x}'}^+(l_z^+)$ represents the intensity of the streamwise wall-shear fluctuations carried by the structures with spanwise size $l_z^+$. Figure 11 shows the variations of $I_{\tau _{x}'}^+$ as a function of $l_z^+$. For both $PF_s$ and $NF_s$, $I_{\tau _{x}'}^+$ collapses well, showing the independence of $I_{\tau _{x}'}^+$ on the friction Reynolds number. For all cases, the expected values of $I_{\tau _{x}'}^+$ are higher for $PF_s$ than for $NF_s$, but the former rarely attain size larger than $l_z^+\approx 300$.

Figure 11.

Variations of $I_{\tau _{x}'}^+$ carried by $PF_s$ and $NF_s$ as a function of $l_z^+$.

This asymmetrical feature suggest that large-scale outer motions affect mainly the negative skin-friction events. The phenomenon is reminiscent of the asymmetric modulation of near-wall streamwise velocity fluctuations by positive and negative outer large-scale motions (Agostini & Leschziner Reference Agostini and Leschziner2016a,Reference Agostini and Leschzinerb, Reference Agostini and Leschziner2018, Reference Agostini and Leschziner2019a). It also implies that the symmetric modelling of positive and negative wall-attached structures by Baidya et al. (Reference Baidya, Baars, Zimmerman, Samie, Hearst, Dogan, Mascotelli, Zheng, Bellani and Talamelli2019) for the $u'$ spectrum might be improved by accounting for the flow asymmetries.

Another interesting feature revealed by figure 11 is that as $l_z^+$ increases, the intensity $I_{\tau _{x}'}^+$ of $NF_s$ converges to a constant value, i.e. $I_{\tau _{x}'}^+\approx 1.03$, which suggests that the intensities of large-scale $NF_s$ are independent of $l_z^+$. The marginal p.d.f.s of $\tau _{x}'(l_z^+)/\tau _{x,rms}'$ for $NF_s$ at several selected $l_z^+$ is shown in figure 12 for case $Re950$. Similar trends are observed in the other two cases (not shown). As $l_z^+$ increases, the p.d.f.s gradually converge to a normal distribution, which is well described by the Gaussian function,

(4.6)\begin{equation} p(\zeta)\big|_{l_z^+}= a^{{-(\zeta-b)^2}/{2c^2}}, \end{equation}

where the fitting constants are $a=1.223$, $b=-0.784$ and $c=0.44$, and $\zeta$ stands for the abscissa in figure 12. The fitting curve is linked to the constant value of $I_{\tau _{x}'}^+$ in figure 11. The definition of $I_{\tau _{x}'}^+$ in (4.5) can be rewritten as

(4.7)\begin{equation} I_{\tau_{x}'}^+(l_z^+)= \sqrt{\int_{-\infty}^{-\alpha} \zeta^2 p(\zeta)\big|_{l_z^+}\,\textrm{d}\zeta}. \end{equation}

Figure 12.

The marginal p.d.f.s of $\tau _{x}'/\tau _{x,rms}'$ for $NF_s$ at several selected $l_z^+$ by using the case of $Re950$. The red curve denotes a fitting curve by Gaussian function. The vertical dashed line denotes $\tau _{x}' = -0.5 \tau _{x,rms}'$.

Substitution of (4.6) into (4.7) yields

(4.8)\begin{equation} I_{\tau_{x}'}^+(l_z^+)\approx 1.027, \end{equation}

which is consistent with the constant value ($I_{\tau _{x}'}^+\approx 1.03$) shown in figure 11.

The constant value of $I_{\tau _{x}'}^+$ for large-scale $NF_s$ can be speculated and explained by the properties of the large-scale structures in the near-wall region. Hwang (Reference Hwang2016) reported that large-scale $u'$ motions should scale with the inner units in the vicinity of the wall due to viscous effects, which is supported by the measurements of the high-Reynolds-number turbulent boundary layers (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). Cho et al. (Reference Cho, Hwang and Choi2018) further pointed out that this inner scaling is related to the balance between the turbulent transport and dissipation at large scale in the near-wall region. The readers are referred to Hwang (Reference Hwang2016) and Cho et al. (Reference Cho, Hwang and Choi2018) for more details.

To further investigate the observation of $I_{\tau _{x}'}^+\approx constant$, we compute two-dimensional structures of $u'$ at a given $y$-location using the clustering methodology. The negative near-wall $u'$ structures in the wall-parallel plane are identified following the same approach described in § 3. We have checked that the threshold within $0.4 \le \alpha \le 0.7$ has no impact on the conclusions below. Similarly to the definition of $I_{\tau _{x}'}^+$ in (4.5), we introduce a scale-dependent streamwise turbulence intensity as

(4.9)\begin{equation} I_{u'}^+(l_z^+,y^+)=\sqrt{\frac{1}{N_u}\sum_{i=1}^{N_u}\left[ \frac{u_i'(l_z^+,y^+)}{u_{\tau}} \right]^2 }, \end{equation}

where $N_u$ is the total number of the grid points belonging to the negative $u'$ structures with a conditional $l_z^+$ at $y^+$, and $u_i'(l_z^+,y^+)$ denotes the negative streamwise velocity fluctuations carried by the $i$th connected grid point. Analogous to $I_{\tau _{x}'}^+$, $I_{u'}^+(l_z^+,y^+)$ characterizes the intensity of the streamwise velocity fluctuations of the negative $u'$ structures for a given $l_z^+$ at $y^+$.

Figure 13(a) displays the variations of $I_{u'}^+$ as a function of $l_z^+$ at several near-wall locations for $Re950$. Similar trends are observed for the other two cases (not shown here). It can be seen in figure 13(a) that as $l_z^+$ increases, $I_{u'}^+$ tends to be constant with $y^+\le 7$. The constant value of $I_{u'}^+$ (denoted by $(I_{u'}^+)_{constant}$) increases with $y^+$. Figure 13(b) displays the variation of $(I_{u'}^+)_{constant}$ as a function of $y^+$ for all cases. A linear relationship between $(I_{u'}^+)_{constant}$ and $y^+$ holds for the three Reynolds numbers considered,

(4.10)\begin{equation} (I_{u'}^+)_{constant}\approx C y^+, \end{equation}

where $C$ is a constant. The results support the dimensional analysis in Cho et al. (Reference Cho, Hwang and Choi2018); LSM of $u'$ in the vicinity of the wall tend to follow inner scaling. According to (4.10), we can assume that the streamwise velocity within the large-scale structures of negative $u'$ follow the relationship

(4.11)\begin{equation} \frac{u'\big|_{l_z^+}}{u_{\tau}}\approx -C y^+. \end{equation}

Figure 13.

(a) Variations of $I_{u'}^+$ at several selected $y^+$ in $Re950$. (b) The convergent values of $I_{u'}^+$ ($(I_{u'}^+)_{constant}$) with respective to $y^+$ and the black solid line denotes $(I_{u'}^+)_{constant}=0.38y^+$.

Additionally, the streamwise wall-shear fluctuations are given by

(4.12)\begin{equation} \tau_x' =\mu \frac{\partial {u'}}{\partial y}, \end{equation}

and substituting (4.11) into (4.12) yield

(4.13)\begin{equation} \frac{|\tau_x'|_{NF_s}}{\rho u_\tau^2} \approx C, \end{equation}

which indicates that the magnitude of streamwise wall-shear fluctuations carried by large-scale $NF_s$ is independent of $l_z^+$ and tends to be a constant value, consistent with figure 11.

4.4. Correlation between $\tau _x'$ structures and outer motions

It is known that the energy-containing motions populating the logarithmic and outer regions leave a ‘footprint’ on the wall surface (Hutchins et al. Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011; Mathis et al. Reference Mathis, Marusic, Chernyshenko and Hutchins2013; Agostini & Leschziner Reference Agostini and Leschziner2019a). Investigating the fluid patterns in the proximity of the wall-shear events can shed light on the generation mechanism of the $\tau _x'$ structures. To this end, we calculate the correlation between the $\tau _{x}'$ structures and the energy-containing motions populating the logarithmic and outer regions by

(4.14)\begin{equation} R(r_x,y,r_z)\lvert_{l_z}= \frac{E\left[\tau_{x}'(x_0,0,z_0)\lvert_{l_z}u'(x_0+r_x,y,z_0+r_z)\right]}{\sigma_{\tau_{x}'}\sigma_{u'}}, \end{equation}

where $E$ is the expected value, $(x_0,0,z_0)$ is the reference point, $(r_x,y,r_z)$ represents spatial differences in the streamwise, wall-normal and spanwise direction, respectively, and $\sigma _{\tau _{x}'}$ and $\sigma _{u'}$ are the standard deviation of $\tau _{x}'$ and $u'$, respectively.

Figure 14 shows the isosurfaces of $R=0.2$ for $PF_s$ and $NF_s$ with $l_z^+=100$ and 300 for case of $Re950$. It is apparent that well-organized streaky structures emerge in the proximity of $\tau _{x'}$ structures. The streaks associated with $PF_s$ are wider and longer than those associated with $NF_s$. This feature is robust for varying $R$, $l_z^+$ and the Reynolds number. If we set both $r_x^+$ and $r_z^+$ in (4.14) to zero, then $R(0,y^+,0)$ (denoted as $R_a(y^+)$ hereafter) represents the conditional correlation just above the target structures. The variations of $R_a(y^+)$ as a function of $y^+$ and $y/h$ are shown in figure 15. The large-scale $\tau _{x}'$ structures (both $PF_s$ and $NF_s$) are significantly correlated with $u'$ above the wall. For $PF_s$, $R_a$ remains positive across the whole channel height when $l_z^+>50$; whereas for $NF_s$, $R_a$ remains positive until the wall-normal distance $y_c^+$, which increases with $l_z^+$.

Figure 14.

(a,b) Isosurfaces of $R=0.2$ for $PF_s$ (a) and $NF_s$ (b) with $l_z^+=100$ in $Re950$. (c,d) Isosurfaces of $R=0.2$ for $PF_s$ (c) and $NF_s$ (d) with $l_z^+=300$ in $Re950$. The colour contours at $r_x^+=0$ exhibit the plane-cut correlations.

Figure 15.

Variations of $R_a(y^+)$ as a function of $y^+$ and $y/h$ for $PF_s$ (a) and $NF_s$ (b) in $Re950$.

The asymmetry between the correlations of $PF_s$ and $NF_s$ can be interpreted from the perspective of the quadrant decomposition (Wallace Reference Wallace2016). The joint p.d.f.s of $u'^+$ and $v'^+$ just above $PF_s$ and $NF_s$ with $l_z^+=310$ are displayed in figure 16. It can be seen that at all wall-normal positions, $PF_s$ are mainly associated with the sweep events ($u'^+>0$, $v'^+<0$), whereas $NF_s$ are strongly correlated with the ejection events ($u'^+<0$, $v'^+>0$) in the near-wall region ($y^+\leq 150$). This observation is in line with the profiles of $R_a(y^+)$ shown in figure 15. In this sense, it is reasonable to hypothesize that $PF_s$ are generated by strong sweeps, while $NF_s$ mainly arise from the ejections, consistent with the asymmetry between sweeps and ejections.

Figure 16.

Joint p.d.f.s of $u'^+$ and $v'^+$ at several $y^+$, conditioned on $PF_s$ (a,c,e,g) and $NF_s$ (b,d,f ,h) with $l_z^+=310$ in $Re950$. $(a,b) \ y^{+} = 20, \ (c,d) \ y^{+} = 80, \ (e,\,f) \ y^{+} = 150 \ {\rm and} \ (g,h) \ y^{+} = 250.$

Next, we pay attention to the signature of Townsend's attached eddies. The logarithmic decay of the second-order moment of $u'$ is considered to be one of the most compelling indicators in favour of the attached-eddy model (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Meneveau & Marusic Reference Meneveau and Marusic2013). Inspired by this observation, the streamwise turbulence intensity conditioned on the $\tau _{x}'$ structures is calculated by

(4.15)\begin{equation} \overline{u_{c}^{'2}}^{+}(y^+,l_z^+) =\frac{1}{N_{\tau}}\sum_{i=1}^{N_{\tau}}\left[ \frac{u_i'(x_0,y^+,z_0)\lvert_{l_z^+}}{u_{\tau}} \right]^2, \end{equation}

where $u_i'(x_0,y^+,z_0)\lvert _{l_z^+}$ is the streamwise velocity fluctuation just above the $\tau _x'$ structures, with a spanwise length of $l_z^+$.

Figure 17 shows the variations of $\overline {u_{c}^{'2}}^{+}$ as a function of $y^+$ and $y/h$ for the structures with several selected $l_z^+$. The streamwise turbulence intensities of the full channel ($\overline {u^{'2}}^{+}$) are also plotted for comparison. Note that the profiles of $\overline {u_{c}^{'2}}^{+}$ are truncated at $y_c^+$. A remarkable finding is that $\overline {u_{c}^{'2}}^{+}$ decays logarithmically for $NF_s$ similarly to $\overline {u^{'2}}^{+}$, but the same trend is not so observed for $PF_s$.

Figure 17.

Wall-normal variations of $\overline {u_{c}^{'2+}}$ for $PF_s$ (a) and $NF_s$ (b) in $Re950$ for selected $l_z^+$. The profile of $\overline {u^{'2+}}$ is included for comparison.

To quantify the logarithmic decaying of $\overline {u_{c}^{'2}}^{+}$, we define the indicator function, $\varXi =y(\partial \overline {u_{c}^{'2}}^{+} /{\partial y})$. Figure 18 shows the values of $\varXi$ as a function of $y^+$ for $NF_s$. The value of $\varXi$ calculated for $\overline {u^{'2}}^{+}$ is also included. Compared with the $\varXi$ for the full channel data, slightly better defined plateaus are observed for $NF_s$ in the range of $100\leq y^+\leq 0.2h^+$, which spans the conventional logarithmic region (Jiménez Reference Jiménez2018; Baars & Marusic Reference Baars and Marusic2020b). The wrinkles in figure 18 may be caused by a lack of DNS samples. The observation is striking, since the logarithmic decay of streamwise turbulence intensity has been only observed at much higher Reynolds numbers (Marusic & Kunkel Reference Marusic and Kunkel2003; Jiménez & Hoyas Reference Jiménez and Hoyas2008; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Lee & Moser Reference Lee and Moser2015; Yamamoto & Tsuji Reference Yamamoto and Tsuji2018). For example, Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) reported a logarithmic decay in turbulent pipe flows at $Re_\tau \geq 20\,000$, and Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018) showed the logarithmic decay in the DNS of turbulent channel flows at $Re_\tau =8000$. Our observations at lower Reynolds numbers underlines the fact that the large-scale $NF_s$ identified in the present study have active connection with the attached eddies populating the logarithmic region. The results are also consistent with Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2020), who showed that Townsend's wall-attached eddies are present in wall turbulence at Reynolds numbers that are traditionally considered low.

Figure 18.

Wall-normal variations of the indicator function $\varXi$ for $NF_s$ in (a) $Re550$, (b) $Re950$ and (c) $Re2000$.

To further assess the flow properties above the $\tau _x'$ structures, the marginal p.d.f.s of $u'$ at $y^+=150$ above $PF_s$ and $NF_s$ with $l_z^+=310$ are compared with the p.d.f.s for the full channel data in figure 19(a). The individual root mean square of $u'$ is used for normalization. It should be noted this wall-normal position is below $y_c^+$ and located in the logarithmic region. The p.d.f. of $u'$ above $NF_s$ is highly symmetric and follows a Gaussian distribution as that for $u'$ for the full channel; whereas the p.d.f. of $u'$ above $PF_s$ is highly skewed. Similar results are found in the other two Reynolds numbers, which are not shown here for brevity. Bearing in mind that $u'$ should exhibit a Gaussian distribution in the logarithmic region according to the attached-eddy model (Meneveau & Marusic Reference Meneveau and Marusic2013; Marusic & Monty Reference Marusic and Monty2019), our results suggest again that $NF_s$ are linked to wall-attached eddies, whereas $PF_s$ are not.

Figure 19.

(a) The marginal p.d.f.s of $u'/u'_{rms}$ at $y^+=150$ conditioned on $PF_s$ and $NF_s$ with $l_z^+=310$ for $Re950$, comparing with the p.d.f. of the full channel data. (b) Variations of the inner peaks $\overline {u_{c,m}^{'2}}^{+}$ as a function of $l_z^+$ for $NF_s$. The solid and dashed lines denote $\overline {u_{c,m}^{'2}}^{+} \propto 1.27 \ln (Re_l)$ and $\overline {u_{c,m}^{'2}}^{+} \propto 5.48 \ln (Re_l)$, respectively.

Another consequence of the attached-eddy model is the increase of the inner peaks of the second-order moment of $u'$ as $\ln (Re_{\tau })$, which has been validated in previous numerical and experimental studies (Marusic & Kunkel Reference Marusic and Kunkel2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Meneveau & Marusic Reference Meneveau and Marusic2013; Lee & Moser Reference Lee and Moser2015). By assuming that the inner peaks of the streamwise turbulence intensities are the outcome of the wall-attached eddies residing in the logarithmic region, Meneveau & Marusic (Reference Meneveau and Marusic2013) found that the magnitude of the peaks can be predicted as

(4.16)\begin{equation} \overline{u_m^{'2}}^{+}\sim A_1\ln Re_{\tau}, \end{equation}

where $A_1$ is the constant in the eddy intensity function in (1.2a). Here, we define a scale-based Reynolds number as

(4.17)\begin{equation} Re_{l}=\frac{l_z\overline{u_{\tau,c}}(l_z)}{\nu}, \end{equation}

where $\overline {u_{\tau ,c}}(l_z)$ is the mean friction velocity carried by the structure with a given $l_z$, defined by

(4.18)\begin{equation} \overline{u_{\tau,c}}(l_z)=\frac{1}{N_{\tau}}\sum_{i=1}^{N_{\tau}} u_{\tau,i}\lvert_{l_z}, \end{equation}

where $u_{\tau ,i}\lvert _{l_z}$ denotes the instantaneous friction velocity at the $i$th grid point belonging to the structures with a given $l_z$. In this regard, $Re_l$ can be interpreted as a local Reynolds number.

Figure 19(b) aids the examination of the growth rate of the inner peaks $\overline {u_{c,m}^{'2}}^{\,+}$ for $PF_s$ and $NF_s$. The value of $\overline {u_{c,m}^{'2}}^{+}$ for $NF_s$ grows logarithmically with $\ln (Re_l)$ with a slope of $1.27$, in the regions of $100\le Re_l \le 200$ for $Re550$, $100\le Re_l \le 400$ for $Re950$ and $100\le Re_l \le 700$ for $Re2000$. The growth rate of 1.27 is in agreement with the value of $A_1$ in the eddy intensity function (1.2a), as reported in a collection of studies $A_1\simeq 1.25\text {--}1.26$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Sillero et al. Reference Sillero, Jiménez and Moser2013; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014a; Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). For $PF_s$, the peaks increase with $\ln (Re_l)$ with a growth rate of 5.48 for all cases, which is significantly larger than that of $NF_s$ and far from the usual value for $A_1$ discussed above.

Recently, Hwang (Reference Hwang2015) reported that a single attached eddy is composed of two distinct elements. One is a streak carrying streamwise velocity fluctuations and some spanwise velocity fluctuations, another is the short and tall vortical structure containing all the velocity fluctuating components. The scenario is consistent with a series of previous studies (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Del Álamo et al. Reference Del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). Seen in this context, it is not difficult to find that the footprint of the streak is the near-wall inactive part of the attached eddy as it does not carry the Reynolds stress. In addition, the long streaks are typically represented as the low-momentum regions (Kim Reference Kim1999; Adrian et al. Reference Adrian, Meinhart and Tomkins2000; Hwang Reference Hwang2015; Lee, Sung & Adrian Reference Lee, Sung and Adrian2019). Taking these observations, it would be reasonable to view the $NF_s$ as the footprints of the attached eddies populating the logarithmic region, which has been demonstrated from several perspectives in the present study. Finally, we conjecture that the formation of $PF_s$ is caused by the sweep of vortical structures, probably including the second element of a single attached eddy and the ‘hairpin vortex packet’, a conceptual structure proposed to depict the LSM (Hutchins, Hambleton & Marusic Reference Hutchins, Hambleton and Marusic2005; Adrian Reference Adrian2007; Lee et al. Reference Lee, Sung and Adrian2019). There are three reasons: (i) the correlation of $PF_s$ in figure 14 reaches a higher wall-normal location than that of $NF_s$ with a given $l_z^+$, which is consistent with the spatial image of a single attached-eddy in the study of Hwang (Reference Hwang2015) and the spatial location of LSM; (ii) the correlations of large-scale $PF_s$ in figures 14(c) and 15(a) could be characterized by outer scales, which behave similarly to the scales of LSM (Tomkins & Adrian Reference Tomkins and Adrian2002; Adrian Reference Adrian2007; Baars & Marusic Reference Baars and Marusic2020a); and (iii) figure 16 suggests that $PF_s$ are mainly associated with the sweep events far from the wall.