3.1 Scaling of the attached structures

Figures 4 and 5 represent the distributions of the length ( $l_{x}$ ) and width ( $l_{z}$ ) of the attached structures with respect to $l_{y}$ , respectively. Here, $l_{x}$ and $l_{z}$ are determined based on the bounding box for each structure. There are two distinct growth rates: for the buffer-layer structures ( $l_{y}^{+}<60$ ), $l_{x}$ and $l_{z}$ increase gradually whereas those of the tall structures ( $l_{y}^{+}$ ${>}$ 100) grow rapidly until $l_{y}$ is bounded by $\unicode[STIX]{x1D6FF}$ . For $l_{y}^{+}>100$ , the mean $l_{x}$ and $l_{z}$ (circles) scale with $l_{y}$ , representing that the structures are characterized in terms of $l_{y}$ over a broad range, although there is some dispersion at a given $l_{y}$ . Since the mean $l_{x}$ and $l_{z}$ indicate the sizes of representative structures, this dispersion is associated with attached structures at different stages of stretching (Perry & Chong Reference Perry and Chong1982). Although the mean $l_{x}$ is not linearly proportional to $l_{y}$ , the mean $l_{z}$ especially follows a linear law $l_{z}^{+}\approx 1$ – $1.5l_{y}^{+}$ (orange lines in figures 4 and 5), indicating that the spanwise length scale of the structures is proportional to the distance from the wall, reminiscent of the attached-eddy hypothesis. This point is discussed further below.

Figure 4. Joint PDFs of the logarithms of the length ( $l_{x}$ ) of the attached structures and of their height ( $l_{y}$ ): (a) $u$ , (b) $w$ and (c) $v$ . The inserted dots indicate the mean $l_{x}$ with respect to $l_{y}$ . The orange solid lines are the best fit, $l_{x}^{+}\sim (l_{y}^{+})^{\unicode[STIX]{x1D6FE}}$ of the data for $100<l_{y}^{+}<550$ ; $l_{x}^{+}=17.98(l_{y}^{+})^{0.74}$ in (a), $l_{x}^{+}=12.10(l_{y}^{+})^{0.74}$ in (b) and $l_{x}^{+}=11.34(l_{y}^{+})^{0.69}$ in (c). In (b) and (c), the red contour line represents the contour level of 0.004 in (a) and the red dashed line is consistent with the orange line in (a). The white dashed line ( $l_{x}^{+}=3l_{y}^{+}$ ) indicates the scaling of the vortex and Reynolds shear stress clusters (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). The horizontal dashed line indicates $l_{y}^{+}=\unicode[STIX]{x1D6FF}^{+}$ . The contour levels are logarithmically distributed.

Figure 5. Joint PDFs of the logarithms of the width ( $l_{z}$ ) of the attached structures and of their height ( $l_{y}$ ): (a) $u$ , (b) $w$ and (c) $v$ . The inserted dots indicate the mean $l_{z}$ with respect to $l_{y}$ . The orange solid lines are $l_{z}^{+}=l_{y}^{+}$ in (a), $l_{z}^{+}=1.5l_{y}^{+}$ in (b) and $l_{z}^{+}=l_{y}^{+}$ in (c). In (b) and (c), the red contour line represents the contour level of 0.004 in (a). The red dashed line in (b) is consistent with the orange line in (a). The horizontal dashed line indicates $l_{y}^{+}=\unicode[STIX]{x1D6FF}^{+}$ . The contour levels are logarithmically distributed.

As shown in figure 4, the slope of the mean $l_{x}$ is 0.74 for the wall-parallel components whereas that of the mean $l_{x}$ is 0.69 for the wall-normal component (orange lines) over $100<l_{y}^{+}<550$ . In figure 4(b,c), the red solid line indicates the distribution of the attached $u$ structures and the red dashed line corresponds to the orange line in figure 4(a). As $l_{y}$ increases, the lengths of the attached $u$ and $w$ structures grow at a similar rate beyond that of the attached $v$ structures: for $v$ , the mean $l_{x}$ is only 860 wall units at $l_{y}^{+}=550$ ( $l_{y}/\unicode[STIX]{x1D6FF}\approx 0.55$ ). In addition, the attached $u$ structure is longer than the transverse components. Given that the widths of the attached $u$ structures are comparable with those of the attached $v$ and $w$ (figure 5), the streamwise organization of the $u$ structures contributes to their high volume fraction, 0.67, as indicated in table 1, which is also observed in the instantaneous flow fields (figure 3). To further examine the length distribution, we plot $l_{x}=3l_{y}$ (white dashed line in figure 4) which represents the length distribution of the tall attached clusters ( $l_{y}^{+}>100$ ) of vortex (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006) and of ejection/sweep (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). Note that these structures are closely associated with the velocity clusters identified in the present work. Although these works showed the linear relationship between $l_{x}$ and $l_{y}$ , it should be emphasized that they roughly obtained such a relationship by connecting the ridges of the low contour level. Furthermore, Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) pointed out the nonlinear relationship between $l_{x}$ and $l_{z}$ at a given $l_{y}$ (i.e. $l_{x}l_{y}\propto l_{z}^{2}$ ), as first observed in two-dimensional spectra of the streamwise velocity (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004), even $l_{x}$ and $l_{z}$ vary linearly with respect to $l_{y}$ . However, the attached clusters of $u$ follow $l_{x}l_{y}\propto l_{z}^{1.7}$ , which is approximately quadratic, representing that the power law observed in the attached $u$ structures is suitable for describing the energy-containing motions of the streamwise component. It is also worth highlighting that the distributions of the transverse components are also approximately aligned along $l_{x}l_{y}\propto l_{z}^{2}$ , which is not evident in the corresponding energy spectra (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004). Recently, Hwang (Reference Hwang2015) showed that the energy-containing motions exhibit bimodal behaviour (i.e. the long streaky motions of $u$ and the relatively short vortical structures containing all the velocity components) and that the scaling of these motions satisfies the quadratic distribution of the energy spectra reported in del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004). In this respect, the attached structures of $v$ and $w$ correspond to the short and tall vortical structures described in Hwang (Reference Hwang2015), which are in turn associated with hierarchies of hairpin packets (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000) or tall attached vortex clusters (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006).

The absence of the linear relationship between $l_{x}$ and $l_{y}$ could be attributed to (i) the preferred azimuthal (or spanwise) inclination of the structures or (ii) the meandering nature of the structures or (iii) the low Reynolds number of the present data ( $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ ). Baltzer, Adrian & Wu (Reference Baltzer, Adrian and Wu2013) suggested that the relatively shorter $u$ structures are aligned with the preferred azimuthal offset in the turbulent pipe flow, which is consistent with the helix angle of roll cells, and form very long structures. Since the long structures possess roll cells (Hutchins & Marusic Reference Hutchins and Marusic2007b ; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016b ; Krug et al. Reference Krug, Yang, de Silva, Ostilla-Mónico, Verzicco, Marusic and Lohse2017), the absence of the linear relationship between $l_{x}$ and $l_{y}$ may arise from the preferred offset of the organized motions. Another possibility is the meandering nature of $u$ regions. In the near-wall region, Jiménez, del Álamo and Flores (Reference Jiménez, del Álamo and Flores2004) showed a power-law relationship between the length and width of the $u$ regions by considering the meandering of the near-wall streaks. Given the meandering of long negative $u$ in the logarithmic region (Hutchins & Marusic Reference Hutchins and Marusic2007a ), the attached $u$ structure beyond the buffer layer may follow a power-law relationship between $l_{x}$ and $l_{z}$ . Finally, the Reynolds number of the present TBL could be insufficient to establish the linear relationship ( $l_{x}\propto l_{y}$ ) in the self-similar range. Recently, Chandra et al. (Reference Chandra, Baidya, Monty and Marusic2017) found the linear relationship in the two-dimensional spectra of the streamwise velocity at $Re_{\unicode[STIX]{x1D70F}}\approx 26\,000$ , whereas the shape of the spectra is aligned along the quadratic form ( $l_{x}l_{y}\propto l_{z}^{2}$ ) at $Re_{\unicode[STIX]{x1D70F}}=2400$ . In the spectra at high Reynolds number, the lower range of large scales follows a quadratic relationship. On the other hand, the larger scales break away from this relationship and are aligned along $l_{x}\propto l_{z}$ . The linear growth of the streamwise length scale was also reported in Baars et al. (Reference Baars, Hutchins and Marusic2017) at $Re_{\unicode[STIX]{x1D70F}}=O(10^{6})$ using a spectral coherence analysis. In figure 4(a), the mean $l_{x}$ of the structures in $550<l_{y}^{+}<750$ follows $l_{x}=3.8l_{y}$ , representing that the larger scales tend towards the linear behaviour. If we assume that these tall structures exist within the upper limit of the logarithmic region ( $y=0.15\unicode[STIX]{x1D6FF}$ ), then the low-Reynolds-number limit for the linear relationship is $Re_{\unicode[STIX]{x1D70F}}=750/0.15=5000$ . Thus, at $Re_{\unicode[STIX]{x1D70F}}>5000$ , we may observe the attached structures of $u$ that follow $l_{x}\propto l_{y}$ in the self-similar range consistent with the distribution of their width. Note that we used $l_{x}$ , $l_{z}$ and $l_{y}$ instead of the wavelengths and the wall-normal location for convenience because the sizes of the identified objects can approximately represent the latter (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006).

Next, we further examine the linear relationship between $l_{z}$ and $l_{y}$ (figure 5). In contrast to the length distributions (figure 4), the attached structures of $u_{i}$ with $l_{y}^{+}$ ${>}$ 100 have a comparable $l_{z}$ in figure 5. Moreover, the mean $l_{z}$ of these structures is linearly proportional to $l_{y}$ up to $l_{y}^{+}=\unicode[STIX]{x1D6FF}^{+}$ (horizontal dashed line). In particular, the mean $l_{z}$ of the streamwise and wall-normal components is aligned along $l_{z}=l_{y}$ and that of the spanwise component behaves in $l_{z}=1.5l_{y}$ , which is slightly wider than the others. The tall attached clusters of the Reynolds shear stress (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012) follow $l_{z}=l_{y}$ in a manner similar to that in figure 5(a,c) since the Reynolds shear stress is the product of $u$ and $v$ . The widths of the tall vortex clusters (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006) are aligned along $l_{z}=1.5l_{y}$ , consistent with those of the attached $w$ , representing that the spanwise scale of large-scale vortical structures is governed by the attached structures of $w$ with a similar $l_{y}$ . In addition, it is worth mentioning that there is a linear ridge, which connects the inner and outer peaks in the spanwise spectra of the streamwise velocity (Hwang Reference Hwang2015). Since the linear ridge represents the self-similarity of the energy-containing motions over a broad range, Hwang (Reference Hwang2015) found that the energy distributions in the spanwise spectra of all the velocities, which resolve turbulent motions at a given spanwise length scale, are characterized by the spanwise length scale. Similar behaviour was also found in Hellström et al. (Reference Hellström, Marusic and Smits2016) by examining the spanwise scale of the energetic modes obtained from a proper orthogonal decomposition. It should be emphasized that the self-similar motions found in the previous studies are statistical structures, which only partially satisfy the concept of self-similar representative eddies originally proposed by Townsend, who tried to describe the correlation statistics of wall turbulence (Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). However, the present result does demonstrate the presence of self-similar structures in instantaneous flow fields. Further evidence for the nature of the identified attached structures in the context of the attached-eddy hypothesis is explored by examining their population density and the associated turbulence intensity profiles in the remaining part of the present work.

Before proceeding with this analysis, we note that the protrusions of the distributions ( $l_{y}\geqslant \unicode[STIX]{x1D6FF}$ ) in figures 4 and 5 were also observed in Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012). Note that we discarded the structures that cross the streamwise domain ( $L_{x}=11.7\unicode[STIX]{x1D6FF}$ ), whereas the objects that cut through the spanwise domain ( $L_{x}=3.2\unicode[STIX]{x1D6FF}$ ) were included. The attached structures of $u$ within the protrusions exhibit $l_{x}=7$ – $11\unicode[STIX]{x1D6FF}$ and $l_{z}=1.5$ – $2\unicode[STIX]{x1D6FF}$ and those of the cross-stream components are $l_{x}=4$ – $6\unicode[STIX]{x1D6FF}$ and $l_{z}=2$ – $3\unicode[STIX]{x1D6FF}$ , i.e. they are shorter but wider than the former. These features are consistent with the bimodal behaviour of the energy-containing motions at the largest spanwise length scale (Hwang Reference Hwang2015). However, it should be stressed that the largest structures in figures 4 and 5 are not aligned along the orange line (i.e. there is scaling failure with respect to $l_{y}$ ), although Hwang (Reference Hwang2015) described that these motions are self-similar based on their spanwise length scale. Interestingly, Perry et al. (Reference Perry, Henbest and Chong1986) conjectured the presence of such large-scale eddies with heights of the order of $\unicode[STIX]{x1D6FF}$ and also noted that these eddies need not be self-similar to the smaller-scale eddies. In this respect, the tallest structures ( $l_{y}\approx \unicode[STIX]{x1D6FF}$ ) within the scaling failure region are the large-scale eddies, which are not geometrically self-similar with $l_{y}$ . Here, Perry et al. (Reference Perry, Henbest and Chong1986) described the tall motions as large scale since large-scale motions such as bulges at the TNTI were defined originally by their heights (Kovasznay, Kibens & Blackwelder Reference Kovasznay, Kibens and Blackwelder1970; Falco Reference Falco1977). Recently, the streamwise length has been used to characterize large-scale structures and in this sense very long attached structures of $u$ are associated with superstructures in TBL (Hutchins & Marusic Reference Hutchins and Marusic2007a ); very-large-scale motions (Kim & Adrian Reference Kim and Adrian1999) or global modes (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004) in internal flows. Hence, a better nomenclature for the attached structures of $u$ with $l_{y}\approx \unicode[STIX]{x1D6FF}$ and with relatively shorter $l_{y}(100<l_{y}^{+}<550)$ could be superstructures (or very-large-scale motions) and large-scale motions, respectively, but we prefer to use the classical description; i.e. we shall describe the structures with $l_{y}\approx \unicode[STIX]{x1D6FF}$ as large-scale eddies based on their heights for our purpose.

3.2 Population density of the attached structures

The population density of the attached structures ( $n_{s}$ ) versus $l_{y}$ is examined to determine whether the attached structures are associated with the hierarchy length scales (Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986). Here, $n_{s,i}$ ( $l_{y}$ ) is defined as

(3.1) $$\begin{eqnarray}n_{s,i}(l_{y})=\frac{N_{i}(l_{y})}{mL_{x}L_{z}},\end{eqnarray}$$

where $N_{i}(l_{y})$ is the number of the attached structure at a given $l_{y}$ , and $m$ is the number of instantaneous flow fields. Note that $n_{s}$ can be obtained by integrating the joint PDFs in figures 4 or 5 along the abscissa. Hence, $n_{s}$ is equivalent to the PDF of the hierarchy scales (Perry & Chong Reference Perry and Chong1982). In figure 6, the distributions decay with $l_{y}$ beyond the buffer layer. In particular, the distribution of the attached $u$ structure (blue) is inversely proportional to $l_{y}$ while that of the cross-stream components follows $n_{s}\propto l_{y}^{-1.3}$ for $290<l_{y}^{+}<550$ . Given the inverse power-law PDF of the hierarchy scales (Perry & Chong Reference Perry and Chong1982), the structures of $u$ in this region are hierarchies of self-similar eddies in the context of the attached-eddy hypothesis (see further discussion in § 5). The onset of the self-similar region for the attached $u$ structures is $l_{y}^{+}\approx 100$ in figures 4 and 5 which is lower than that of the inverse power-law distribution, representing that not all the self-similar structures follow $n_{s}\propto l_{y}^{-1}$ . In addition, the lower and upper limits of the self-similar range ( $100<l_{y}^{+}<550$ ) in figure 4 are consistent with the onset of the logarithmic region (see figure 8) and with the end of the inverse power-law distribution in figure 6, respectively. The population densities of $v$ and $w$ are larger than that of $u$ over a broad range, which is consistent with the numbers of the attached clusters in table 1. The cross-stream components have the same slope and decrease rapidly compared to $u$ , indicating that the attached structures of $u$ exist with larger clusters in instantaneous flow fields. Given that the attached structures of $v$ and $w$ are associated with the tall vortical structures discussed in § 3.1, this result also reveals that the attached structures of the cross-stream components and of $u$ have different characteristics. Furthermore, we can observe a peak at $l_{y}\approx 0.7$ – $0.8\unicode[STIX]{x1D6FF}$ for $u$ and at $l_{y}\approx 0.9\unicode[STIX]{x1D6FF}$ for $v$ and $w$ , indicating the additional weighting for the large-scale structures. As discussed in § 3.1, these large-scale structures are not geometrically self-similar, as is evident in the protrusions around $l_{y}\approx \unicode[STIX]{x1D6FF}$ in figures 4 and 5. This behaviour is consistent with the modified PDF of hierarchy scales (Perry et al. Reference Perry, Henbest and Chong1986), which was proposed to enable the more accurate prediction of the mean velocity defect and the low-wavenumber peak in energy spectra.

Figure 6. Population density of the attached clusters ( $n_{s}$ ) with respect to their height $l_{y}$ . The dashed line is $n_{s}\propto l_{y}^{-1}$ for $u$ and $n_{s}\propto l_{y}^{-1.3}$ for $v$ and $w$ . The inset shows a magnified view of the region in $290<l_{y}^{+}<550$ .

5.1 Uniform momentum zones in the attached structures

As conjectured by Perry & Chong (Reference Perry and Chong1982), the attached eddies are in the form of hierarchies. Each hierarchy is composed of the eddies whose height grows from their initial roll-up height $(O(\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}))$ to the height of the hierarchy ( $l$ ). Here, the height of the hierarchy corresponds to the height of the highest eddy within the hierarchy. In other words, there are several eddies whose heights are less than $l$ at a given hierarchy height $l$ and consequently the number of eddies or hierarchies increases with increasing $l$ . In addition, if we assume that the height of the highest hierarchy is the boundary layer thickness $\unicode[STIX]{x1D6FF}$ , the friction Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ ) indicates the ratio of the highest hierarchy to the initial roll-up height. As a result, the number of eddies or hierarchies can be expected to increase with increasing $Re_{\unicode[STIX]{x1D70F}}$ . In this regard, De Silva, Hutchins & Marusic (Reference de Silva, Hutchins and Marusic2016) argued that a log–linear increase in the number of uniform momentum zones (UMZs) with increasing $Re_{\unicode[STIX]{x1D70F}}$ indicates a hierarchical length scale distribution. This interpretation was further supported by synthesizing the flow field using the attached-eddy model of Marusic (Reference Marusic2001). UMZs that contain roughly uniform streamwise velocity were first observed in the $x$ – $y$ plane of instantaneous flow fields by Meinhart & Adrian (Reference Meinhart and Adrian1995). In addition, UMZs commonly exist in multiple zones along the wall-normal direction (Adrian et al. Reference Adrian, Meinhart and Tomkins2000), implying the hierarchical nature. To further examine the hierarchical characteristics of the attached $u$ structures, we examine the instantaneous streamwise velocity ( $U$ ) within these objects.

Figure 9(a) shows an attached structure of negative $u$ (that enclosed by the black box in figure 3 a) and a streamwise slice at $z/\unicode[STIX]{x1D6FF}=2.86$ is illustrated in figure 9(b). This structure extends from the wall to $y\approx 0.8\unicode[STIX]{x1D6FF}$ and in particular, the profile of $U$ at $x/\unicode[STIX]{x1D6FF}=9.77$ (see the inset in figure 9 b) shows several jumps in velocity across the structure (as indicated by the dotted horizontal lines), separating zones of roughly uniform $U$ ; low $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y$ (orange line) also appears within the UMZs. Given that the UMZs produce the local maxima in the histogram of $U$ (Adrian et al. Reference Adrian, Meinhart and Tomkins2000), we plot the histogram of $U$ in the cross-stream plane of the identified structures at $x/\unicode[STIX]{x1D6FF}=9.77$ in figure 9(c). Although there are several local maxima, the two at $U/U_{\infty }\approx 0.5$ and 0.6, which are the consequence of UMZs, are preserved when the data are accumulated over the entire structure (grey line). To further examine the number of UMZs in the attached structures, the joint PDF of the height ( $l_{y}$ ) and the number of local maxima ( $N_{p}$ ) is shown in the inset of figure 9(c). Here, the inserted dots indicate the mean $N_{p}$ at a given $l_{y}$ . The number of UMZs in each structure increases with increasing $l_{y}$ , which confirms the hierarchical nature of the identified structures (De Silva et al. Reference de Silva, Hutchins and Marusic2016). In other words, $l_{y}$ could be consistent with the hierarchy length scale in connection with the results in § 3.

Figure 9. (a) A sample attached structure of negative $u$ (that enclosed by the black box in figure 3 a). (b) Contour of the instantaneous streamwise velocity ( $U$ ) in the $x$ – $y$ plane at $z/\unicode[STIX]{x1D6FF}=2.86$ . The blue line is a slice of the object shown in (a). The black line indicates the instantaneous TNTI. The inset shows a comparison of the instantaneous (orange line) and mean (dashed line) streamwise velocity profiles at $x/\unicode[STIX]{x1D6FF}=9.77$ (indicated by the vertical dashed line in the contour). The grey line represents $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y$ , except the region for $y^{+}<50$ due to the strong shear close to the wall. The horizontal dotted lines indicate step-like jumps of $U$ across the structure, which separate the zones of roughly constant $U$ . (c) Histogram of $U$ in the cross-stream plane of the identified object at $x/\unicode[STIX]{x1D6FF}=9.77$ , which contains the two distinct local maxima (at $U/U_{\infty }\approx 0.5$ and 0.6) that are associated with possible uniform momentum zones (UMZs). The grey line shows the PDF of $U$ within the identified object and the orange circles denote the local maxima. The inset shows the joint PDF of the height ( $l_{y}$ ) and the number of UMZs ( $N_{p}$ ). The inserted dots indicate the mean $N_{p}$ with respect to $l_{y}$ . The contour levels are logarithmically distributed.

Figure 10. (a) Variation of the near peak magnitude of $\overline{u_{a}^{2}}^{+}(\overline{u_{a,max}^{2}}^{+})$ with respect to $l_{y}^{+}$ . The orange line shows $\overline{u_{a,\max }^{2}}^{+}\sim 0.66\ln (l_{y}^{+})$ over $100<l_{y}^{+}<550$ . (b) $\overline{u_{a}^{2}}^{+}-\{-3.010\ln (y/\unicode[STIX]{x1D6FF})\}$ for the attached $u$ structures within $100<l_{y}^{+}<750$ . $l_{y}$ are logarithmically distributed. The solid lines indicate the structures with $l_{y}^{+}=150$ , 190, 240, 300 and 400 and the vertical dashed lines represent the logarithmic region ( $100<y^{+}<0.18\unicode[STIX]{x1D6FF}^{+}$ ).

In this work, we analyse the number of UMZs within the attached structures of $u$ . At a single Reynolds number (i.e. $Re_{\unicode[STIX]{x1D70F}}=980$ ), the number of UMZs increases logarithmically with increasing $l_{y}$ (figure 9), which directly indicates the relationship between the multiple UMZ phenomena (Meinhart & Adrian Reference Meinhart and Adrian1995; Adrian et al. Reference Adrian, Meinhart and Tomkins2000) and the hierarchical distribution of the attached structures of $u$ . Hence, one attached structure of $u$ might be consistent with multiple hairpin packets of various ages and sizes since small packets can be covered by larger packets that convect faster than the smaller ones (Adrian et al. Reference Adrian, Meinhart and Tomkins2000); note that the inclination angle of the $u$ structures ranges from $8.8^{\circ }$ to $16^{\circ }$ , which is similar to that of hairpin packets. In addition, given that the wake of attached vortex clusters identified in del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) is associated with multiple UMZs, the attached structures of $u$ might be shrouded in the vortex clusters. However, we need to perform a more extensive examination of the three-dimensional vortices surrounding the $u$ structures.

One could question our method for the detection of UMZs within the $u$ structures: are the local maxima observed in the histogram (figure 9 c) due to the conditional sampling of $U$ in a selective volume? As first observed by Meinhart & Adrian (Reference Meinhart and Adrian1995), the phenomenon of roughly constant uniform momentum in the $x$ – $y$ plane of instantaneous flow fields is a local event that is random and time-varying. In order to quantify the UMZs, Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) analysed the histogram of $U$ with a streamwise domain length of 2000 wall units. De Silva et al. (Reference de Silva, Hutchins and Marusic2016) also used the similar length to detect the UMZs. In other words, these previous studies sampled $U$ selectively in the short streamwise domain at a certain spanwise location. As discussed in Adrian et al. (Reference Adrian, Meinhart and Tomkins2000), the local maxima in the histogram depend on the time averaging and the streamwise length of the data. This behaviour was discussed in Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014), who examined the variation in the local maxima as a function of the streamwise domain length; when the streamwise domain increases up to $6h$ (where $h$ is the channel half-height), several peaks disappear and only one peak survives. To overcome this limitation, the edges of the UMZs must be carefully identified. In the present work, we defined the boundary of the UMZs by examining the three-dimensional $u$ structures. As a result, the histogram of $U$ obtained as a function of the height of each $u$ structure shows several maxima even when the streamwise length of the structures increases (figure 9 c). The negative- $u$ regions in the two-dimensional plane represent the UMZs induced by surrounding vortices (Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Tomkins & Adrian Reference Tomkins and Adrian2003; Hutchins, Hambleton & Marusic Reference Hutchins, Hambleton and Marusic2005). Retrograde vortices can exist on the shear layer of positive- $u$ structures (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012), which may induce the roughly uniform momentum greater than the mean velocity. In this respect, sampling the streamwise velocity within the $u$ structures is an appropriate method for the detection of UMZs in three-dimensional flow fields.

It is worth highlighting the importance of UMZs in the context of the attached-eddy hypothesis. The work of Meinhart & Adrian (Reference Meinhart and Adrian1995) not only demonstrated the general existence of UMZs but also contained that UMZs are connections between the discrete system and the continuous system (Perry & Chong Reference Perry and Chong1982). Meinhart & Adrian (Reference Meinhart and Adrian1995) used the term ‘zones’ to emphasize that the step-like jumps in $U$ are instantaneous phenomena, in contrast to ‘layers’ such as the buffer layer and logarithmic layer, which are defined in terms of the mean quantities (Adrian et al. Reference Adrian, Meinhart and Tomkins2000, § 5.3), and concluded that ‘The zonal structure reported here offers some new ways of conceptualizing the structure of wall turbulence. The conventional decomposition of the boundary layer into a wall layer, a logarithmic layer and a wake region is based on time-averaged properties of regions of fixed vertical dimension. The zones, in contrast, are instantaneous, time-evolving entities, whose characteristics contribute to the mean properties of each of the convection layers.’ In this regard, UMZs are strong evidence for describing the continuous system (Perry & Chong Reference Perry and Chong1982), which is a result of the randomness in the discrete system. Perry & Chong (Reference Perry and Chong1982) assumed that quantum-jump phenomena in the discrete system are smoothed by the randomness and jitter of turbulence, which leads to the inverse power-law PDF of the hierarchy length scale in the continuous system. The present findings show directly that the attached $u$ structures composed of multiple UMZs follow the hierarchical length scale distribution (figure 9) and contribute to the formation of the logarithmic region (figure 8).

5.2 Near-wall peaks of the streamwise turbulence intensity

The logarithmic variation of the streamwise turbulence intensity (1.2) implies that the magnitude of $\overline{u^{2}}^{+}$ at a fixed $y^{+}$ is proportional to $\ln (Re_{\unicode[STIX]{x1D70F}})$ (Jiménez & Hoyas Reference Jiménez and Hoyas2008; Jiménez Reference Jiménez2012; Marusic et al. Reference Marusic, Baars and Hutchins2017); the near-wall peak $\overline{u_{max}^{2}}^{+}$ , which appears at $y^{+}\approx 15$ , exhibits $\overline{u_{max}^{2}}^{+}\sim \ln (Re_{\unicode[STIX]{x1D70F}})$ . This behaviour was first reported experimentally in DeGraaff & Eaton (Reference DeGraaff and Eaton2000) and they suggested a mixed scaling (i.e. $u_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ ) for $\overline{u_{max}^{2}}^{+}$ in the context of the attached-eddy hypothesis. Subsequently, several experimental and numerical works reported the logarithmic increase of $\overline{u_{max}^{2}}^{+}$ (Marusic & Kunkel Reference Marusic and Kunkel2003; Hoyas and Jiménez Reference Hoyas and Jiménez2006; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013; Lee & Moser Reference Lee and Moser2015; Ahn et al. Reference Ahn, Lee, Lee, Kang and Sung2015, Örlü et al. Reference Örlü, Fiorini, Segalini, Bellani, Talamelli and Alfredsson2017). At extremely high Reynolds numbers ( $Re_{\unicode[STIX]{x1D70F}}>30\,000$ ), some studies (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015) found that the growth of inner peak is absent, but these results might be due to spatial and temporal resolution issues (Sillero et al. Reference Sillero, Jiménez and Moser2013; Marusic et al. Reference Marusic, Baars and Hutchins2017). Given that the behaviour $\overline{u_{max}^{2}}^{+}\sim \ln (Re_{\unicode[STIX]{x1D70F}})$ can be explained with Townsend’s attached-eddy hypothesis, we further examine the variation in the near-wall peak of $\overline{u_{a}^{2}}^{+}$ with $l_{y}$ (figure 7) because $l_{y}$ is associated with the hierarchy length scale.

As pointed out in figure 7, the profiles of $\overline{u_{a}^{2}}^{+}$ have a near-wall peak and its magnitude increases with increasing $l_{y}$ . Figure 10(a) shows the variations of the peak $\overline{u_{a}^{2}}^{+}$ according to $l_{y}^{+}$ . As seen, there is a logarithmic increase in the magnitude of the peak $\overline{u_{a}^{2}}^{+}$ with increasing $l_{y}^{+}$ in $100<l_{y}^{+}<550$ with a slope of 0.66. This result is in good agreement with the results for the slope of the increase in the magnitude of the peak $\overline{u^{2}}$ versus $Re_{\unicode[STIX]{x1D70F}}$ obtained in recent DNSs 0.65 (Sillero et al. Reference Sillero, Jiménez and Moser2013) and in experiments 0.63 (Marusic et al. Reference Marusic, Baars and Hutchins2017); for the DNS dataset of turbulent channel flows up to $Re_{\unicode[STIX]{x1D70F}}=5200$ , the slope is 0.642 (Lee & Moser Reference Lee and Moser2015). Sillero et al. (Reference Sillero, Jiménez and Moser2013) and Marusic et al. (Reference Marusic, Baars and Hutchins2017) noted that this value is approximately half of $A_{1}$ (i.e. the slope of the logarithmic term in 1.2a) since the lower bound of the logarithmic region scales with $Re_{\unicode[STIX]{x1D70F}}^{0.5}$ in contrast to the classical scaling (e.g. $O(100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}})$ ) in Perry & Chong (Reference Perry and Chong1982). As discussed in § 4, the profile of $\overline{u_{a}^{2}}^{+}$ is the result of the conditional averaging of the intense $u$ within the identified structures ( $|u|>\unicode[STIX]{x1D6FC}u_{rms}$ ), leading to the difference in $A_{1}$ ; the magnitude of $A_{1}$ for $\overline{u_{a}^{2}}^{+}$ is larger than that for $\overline{u^{2}}$ ( $A_{1}=1.26$ in Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013. However, the slope of the variation in the magnitude of the peak $\overline{u_{a}^{2}}^{+}$ is similar to that of the variation in the total turbulence intensity, representing that the attached structures of $u$ could play a dominant role in the increase of the near-wall peak for the total turbulence intensity. At the present Reynolds number, the attached structures contribute 20–35 % of $\overline{u^{2}}$ at $y<0.3\unicode[STIX]{x1D6FF}$ , while the contribution of the other components (i.e. detached structures and weak turbulence ( $|u|<\unicode[STIX]{x1D6FC}u_{rms}$ )) is 75–80 % leading to the absence of the logarithmic behaviour in $\overline{u^{2}}$ over $100<y^{+}<0.18\unicode[STIX]{x1D6FF}^{+}$ . In addition, it would be instructive to examine the Reynolds-number dependence of the lower bound of the logarithmic region for $\overline{u_{a}^{2}}$ , which is consistent with the onset of the self-similarity in figures 4 and 5 (i.e. $l_{y}^{+}=100\approx 3Re_{\unicode[STIX]{x1D70F}}^{0.5}$ in the present data), but this task is beyond the scope of the present work.

To further support the upward shift of the profiles in figure 7(a), we subtract the logarithmic function (i.e. $-3.010\ln (y/\unicode[STIX]{x1D6FF})$ ) from (4.1) in figure 10(b). As seen, there is a region where the difference between $\overline{u_{a}^{2}}$ and $-3.010\ln (y/\unicode[STIX]{x1D6FF})$ is constant and the difference increases with increasing $l_{y}$ . This trend also reflects the collective contribution of the attached $u$ structures with heights less than a given $l_{y}$ . The results in figure 10(b) indicate that the slope of each $\overline{u_{a}^{2}}$ is close to $-3.010$ and their logarithmic variations shift upwards with increasing $l_{y}$ , representing that the additive constant in (1.2) would depend on the Reynolds number; similar behaviour was also reported in Jiménez & Hoyas (Reference Jiménez and Hoyas2008) by assessing the turbulence intensity data over a wide range of $Re_{\unicode[STIX]{x1D70F}}$ . Note that the range of the logarithmic behaviour is small in the present work due to the Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}\approx 10^{3}$ ). It would be valuable to examine this trend at high $Re_{\unicode[STIX]{x1D70F}}$ (i.e. increase the range of $l_{y}$ ).

5.3 Nested hierarchies

As discussed earlier, attached $u$ structures with a given $l_{y}$ represent the collective contributions of the structures with heights less than $l_{y}$ . In addition, these structures are composed of multiple uniform momentum zones (UMZs). The combination of these two results demonstrates the concept of nested hierarchies proposed by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000). They suggested that small packets (or UMZs) exist within larger packets due to the difference in their convection velocities. The smaller packet contained within the larger one induces the lower streamwise momentum (i.e. strong retardation of $U$ ). Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) described such flow patterns as a nested hierarchy which leads to the presence multiple UMZs; see Adrian et al. (Reference Adrian, Meinhart and Tomkins2000, figure 22). Interestingly, the regions of different streamwise momentum are depicted in a closed loop even though a certain level of $U$ (i.e. interfaces of UMZs) extends from the left edge to the right edge of the flow field. This schematic representation resembles the flow patterns associated with attached $u$ structures.

Figure 11(a,d) shows a side view of sample attached $u$ structures with $l_{y}=0.33\unicode[STIX]{x1D6FF}$ and $l_{y}=0.12\unicode[STIX]{x1D6FF}$ , respectively. As seen, the streamwise length of the shorter structure in figure 11(d) is $l_{x}=0.8\unicode[STIX]{x1D6FF}$ which is approximately half of the larger structure in figure 11(a). As $\unicode[STIX]{x1D6FC}$ increases, the size of the structure in figure 11(a) shrinks and is split into smaller structures that are attached and detached objects with intense $u$ in figure 11(b,c). The structures in figure 11(c) are nested within the region of $-2.2u_{rms}<u<-1.5u_{rms}$ . In other words, a larger structure (or a higher hierarchy) contains several smaller structures (or shorter hierarchies less than a given $l_{y}$ ) with intense $u$ . In particular, the split structure close to the wall ( $0<l_{x}/\unicode[STIX]{x1D6FF}<0.8$ ) is similar to the shorter structure identified at $\unicode[STIX]{x1D6FC}=1.5$ in figure 11(d). This is evident when we compare the wall-normal variations of $U$ . Figure 11(e,f) shows the wall-normal profiles of $U$ at the positions ( $l_{x}/\unicode[STIX]{x1D6FF}=0.5$ and $l_{z}/\unicode[STIX]{x1D6FF}=0.5$ ) denoted by the vertical dashed lines in (a) and (d). As seen, the profile of the larger structure (figure 11 e) exhibits several velocity jumps (at least more than two UMZs) across the wall-normal direction whereas that of the shorter one (figure 11 f) shows one UMZ. In addition, the velocity deficit is more evident in figure 11(e); the dashed line indicates the wall-normal profiles of the mean streamwise velocity. At $y/\unicode[STIX]{x1D6FF}<0.03$ , in particular, the magnitude of $U$ in figure 11(f) is similar to that of the mean streamwise velocity whereas $U$ in figure 11(e) shows the larger velocity deficit. This behaviour indicates that one attached structure of $u$ with $l_{y}$ is the collective contributions of the shorter structures ( ${<}l_{y}$ ) since intense $u$ regions (figure 11 b,c) belong to the taller one (figure 11 a). Recently, Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) showed that the increase in the number of UMZs results in a strong retardation of $U$ close to the wall because additional UMZs (i.e. additional momentum deficit) are observed in the near-wall region. This is similar to the result in the present work that taller structures of $u$ have strong momentum deficit close to the wall leading to the increase of the near-wall peak of $\overline{u_{a}^{2}}$ with $l_{y}$ . Based on the mean streamwise and wall-normal velocities associated with UMZs, Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) reported that a high number of UMZs is associated with large-scale structures of ejections. Given that the attached structures of negative $u$ can contain the ejection event ( $u<0$ and $v>0$ ), this phenomenon can be explained in the context of nested hierarchies. Note that a low number of UMZs exhibits the sweep event ( $u$ ${>}$ 0 and $v<0$ ) which is associated with the attached structures of positive $u$ in the present work. This may indicate that the number of UMZs is relatively small in the positive- $u$ structures compared to that in the negative- $u$ objects at a given $l_{y}$ . In future efforts, it would be valuable to explore the attached structures of negative and positive $u$ and examine their hierarchical nature related to multiple UMZs.