3.1 Eigenmodes of the leading-order POD modes

The eigenmodes $\unicode[STIX]{x1D6F7}_{i}(x,y)$ of the first six POD modes in the $Re_{\unicode[STIX]{x1D70F}}=377$ and 1093 cases are compared in figures 4 and 5. Topological similarity of $\unicode[STIX]{x1D6F7}_{i}$ can be observed in the first four modes. Such a similarity is also found in other cases, implying a topological invariant of the first four modes, at least in the presently studied $Re_{\unicode[STIX]{x1D70F}}$ range.

Quadrant analysis (Lu & Willmarth Reference Lu and Willmarth1973) is then conducted to categorize the characteristic topology of the leading-order modes. As indicated in figure 4, $\unicode[STIX]{x1D6F7}_{1}$ of the first mode presents one large-scale Q4 event ( $\unicode[STIX]{x1D6F7}^{u}>0$ and $\unicode[STIX]{x1D6F7}^{v}<0$ , with the superscripts $(\cdot )^{u}$ and $(\cdot )^{v}$ indicating the streamwise and wall-normal components, respectively) spanning the whole boundary layer thickness. One pair of Q2 ( $\unicode[STIX]{x1D6F7}^{u}<0$ and $\unicode[STIX]{x1D6F7}^{v}>0$ ) and Q4 events appears in $\unicode[STIX]{x1D6F7}_{2}$ of the second mode. They converge to each other, with an interface being formed in the middle (indicated by solid lines in figure 4 b,e). The inclination angle of this interface is approximately $14^{\circ }$ , consistent with the values of the inclined imprints of the hairpin packets (Adrian et al. Reference Adrian, Meinhart and Tomkins2000; Christensen & Adrian Reference Christensen and Adrian2001; Wu Reference Wu2014).

For the third mode, $\unicode[STIX]{x1D6F7}_{3}$ is characterized as a clockwise-rotating motion with a length scale that is considerably larger than those of typical prograde vortices (Natrajan, Wu & Christensen Reference Natrajan, Wu and Christensen2007; Herpin et al. Reference Herpin, Stanislas, Foucaut and Coudert2013). Such a swirling pattern is partly visualized in $\unicode[STIX]{x1D6F7}_{2}$ (downstream of the Q2/Q4 interface at about $x/\unicode[STIX]{x1D6FF}=1.7$ , $y/\unicode[STIX]{x1D6FF}=0.4$ in figure 4 b,e), and seems to be formed by the Q1 and Q3 events that separate the upstream Q2 and downstream Q4 events in $\unicode[STIX]{x1D6F7}_{3}$ . Note that the flow pattern in the upstream (at about $x/\unicode[STIX]{x1D6FF}=0.4$ , $y/\unicode[STIX]{x1D6FF}=0.4$ in figure 4 c and 4 f) also resembles the Q2/Q4 interface in $\unicode[STIX]{x1D6F7}_{2}$ . Therefore, it is conjectured that $\unicode[STIX]{x1D6F7}_{2}$ and $\unicode[STIX]{x1D6F7}_{3}$ only differ from each other by a phase shift and jointly depict the signature of a large-scale structure convecting downstream. Owing to the limited FOV, a complete picture of this structure is not available yet; however, it deserves to be studied in a wider spatial domain in the future. Moreover, it is stressed that there is a slight difference in the topological pattern of $\unicode[STIX]{x1D6F7}_{3}$ between figure 4(c,f), suggesting the existence of a mild $Re$ effect.

Figure 4.

Vector plot of the eigenmode $\unicode[STIX]{x1D6F7}_{i}(x,y)$ of the leading-order $i$ th POD mode with (a–c)  $Re_{\unicode[STIX]{x1D70F}}=377$ and (d–f)  $Re_{\unicode[STIX]{x1D70F}}=1093$ : (a,d)  $i=1$ ; (b,e)  $i=2$ ; and (c,f)  $i=3$ .

Figure 5.

Vector plot of the eigenmode $\unicode[STIX]{x1D6F7}_{i}(x,y)$ of the leading-order $i$ th POD mode with (a–c)  $Re_{\unicode[STIX]{x1D70F}}=377$ and (d–f)  $Re_{\unicode[STIX]{x1D70F}}=1093$ : (a,d)  $i=4$ ; (b,e)  $i=5$ ; and (c,f)  $i=6$ .

In the fourth mode, the converging Q2/Q4 event remains visible in the downstream part of $\unicode[STIX]{x1D6F7}_{4}$ (after $x/\unicode[STIX]{x1D6FF}=1$ in figure 5 a,d). An additional Q2 event appears upstream of and above the Q4 event, forming another diverging Q4/Q2 pair. The interface clapped in the middle (indicated by dashed lines in figure 5 a,d) has an inclination angle of approximately $45^{\circ }$ . Such a diverging Q4/Q2 pattern resembles the conditional-averaged velocity field associated with a strong backscatter event as revealed by Natrajan & Christensen (Reference Natrajan and Christensen2006), who regarded it as the boundary between two successive hairpin packets.

For higher-order POD modes, the eigenmodes become hierarchically complicated, together with a distinct $Re$ dependence; see figure 5 for example. It is impracticable to describe them by a simple combination of several large-scale quadrant events.

3.2 Coupling of the leading-order POD modes

To explore the dynamic correlation between the leading-order modes, their time coefficients $a_{i}$ are further analysed. Figure 6 shows that, for the $i$ th mode until $i=50$ , the probability density function (p.d.f.) of the normalized $a_{i}/\unicode[STIX]{x1D70E}_{a_{i}}$ (with $\unicode[STIX]{x1D70E}_{a_{i}}$ being the root-mean-square (r.m.s.) value of $a_{i}$ ) always follows a normal distribution that is independent of $Re$ . This seems to suggest that the spatial coherence embodied by each mode appears in time randomly.

Figure 6.

P.d.f. of the time coefficient $a_{i}$ of the $i$ th POD mode. In (a) and (b), $Re_{\unicode[STIX]{x1D70F}}=377$ and 1093, respectively: ○,  $i=1$ ; $+$ ,  $i=2$ ; ▫,  $i=3$ ; $\times$ ,  $i=4$ ; ♢,  $i=5$ ; *,  $i=10$ ; ▿,  $i=50$ . In (c), $i=1$ : ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ;  $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ . Solid lines represent the normal distribution; and $a_{i}$ is normalized by its r.m.s. value  $\unicode[STIX]{x1D70E}_{a_{i}}$ .

Figure 7.

Time series of the time coefficients of the first six POD modes normalized by $\unicode[STIX]{x1D6FF}/u_{\unicode[STIX]{x1D70F}}$ : (a,c)  $a_{j}(t)$ ; (b,d)  $a_{j}(t+\unicode[STIX]{x0394}t_{2j}^{\,p})$ , where $\unicode[STIX]{x0394}t_{2j}^{\,p}$ is the ‘optimal’ time lag between $a_{j}$ and $a_{2}$ ; (a,b)  $Re_{\unicode[STIX]{x1D70F}}=377$ ; (c,d)  $Re_{\unicode[STIX]{x1D70F}}=1093$ . Dashed vertical lines indicate the phase alignment in (b) and (d).

Wu (Reference Wu2014) identified the so-called ‘dominant turbulent structures’ from instantaneous snapshots presenting extremely large values of $a_{i}$ (with $i=1$ or 2). These structures were regarded as the major contributors to the first two POD modes and were believed to be consistent with the hairpin-packet imprints. To keep the topological similarity between $\unicode[STIX]{x1D6F7}_{1,2}$ and one snapshot with large $a_{1,2}$ , $a_{i}$ of the rest modes of this snapshot should either be close to zero or jointly cancel each other. Referring to the time series $a_{i}(t)$ of the first six modes in the $Re_{\unicode[STIX]{x1D70F}}=377$ and 1093 cases, as shown in figure 7(a,c), this condition is seldom satisfied. In our view, there might be another scenario, i.e. the leading-order modes dynamically acting as a whole, contributing to the large-scale imprints of the hairpin packets.

The joint p.d.f. $P(a_{i}/\unicode[STIX]{x1D70E}_{a_{i}},a_{j}/\unicode[STIX]{x1D70E}_{a_{j}})$ is then inspected. In all the $Re$ cases, any two normalized $a_{i}$ and $a_{j}$ (with $i,j\leqslant 50$ ) follow bivariate normal distributions without any discernible phase preference (see figure 8 a–c for an illustration). Nevertheless, a careful examination of figure 7(a,c) suggests that $a_{i}(t)$ of the leading-order modes do present phase-shift coherence with each other. Following Hellström et al. (Reference Hellström, Ganapathisubramani and Smits2015), the time-lag correlation between $a_{i}$ and $a_{j}$ is calculated to quantify such coherence:

(3.2) $$\begin{eqnarray}R_{ij}(\unicode[STIX]{x0394}t)=\frac{\overline{a_{i}(t)a_{j}(t+\unicode[STIX]{x0394}t)}}{\unicode[STIX]{x1D70E}_{a_{i}}\unicode[STIX]{x1D70E}_{a_{j}}}.\end{eqnarray}$$

In (3.2), $\overline{(\cdot )}$ is the average over time, $\unicode[STIX]{x1D70E}_{a_{i,j}}$ is the standard deviation of $a_{i,j}$ , and $\unicode[STIX]{x0394}t$ is the time lag. Note that figure 7 shows that the present sampling frequency ( $f_{s}=7.5~\text{Hz}$ ) is sufficient for the time-lag correlation analysis of the leading-order modes even at the highest  $Re$ .

Figure 8.

Joint p.d.f. between $a_{2}(t)$ and $a_{j}(t)$ in (a–c) and between $a_{2}(t)$ and $a_{j}(t+\unicode[STIX]{x0394}t_{2j}^{\,p})$ in (d–f), for $Re_{\unicode[STIX]{x1D70F}}=377$ : (a,d)  $j=1$ ; (b,e)  $j=3$ ; and (c,f)  $j=4$ . The contour level ranges from $0.3\,\%$ to $1.2\,\%$ with a step of $0.3\,\%$ . Here $a_{2}$ and $a_{j}$ are normalized by their r.m.s. values $\unicode[STIX]{x1D70E}_{a_{2}}$ and $\unicode[STIX]{x1D70E}_{a_{j}}$ , respectively.

Figure 9.

The time-lag correlation coefficient $R_{2j}$ between $a_{2}(t)$ and $a_{j}(t+\unicode[STIX]{x0394}t)$ , for (a)  $j=1$ , (b)  $j=3$ and (c)  $j=4$ : ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ; $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ . Here $\unicode[STIX]{x0394}t$ is normalized by $\unicode[STIX]{x1D6FF}/u_{\unicode[STIX]{x1D70F}}$ . The inset in (b) illustrates the definition of the uncertainty of both the correlation peak $|R_{2j}^{p}|$ and the corresponding ‘optimal’ time lag $\unicode[STIX]{x0394}t_{2i}^{p}$ . The red solid curve is a localized Gaussian fit around ( $|R_{2j}^{p}|,\unicode[STIX]{x0394}t_{2i}^{p}$ ).

Figure 9 plots $R_{2j}$ between $a_{2}$ and $a_{j}$ (with $j=1$ , 3 and 4) as a function of $\unicode[STIX]{x0394}t$ . In all the studied cases, a correlation peak $|R_{2j}^{p}|$ with a rather high magnitude ( $|R_{2j}^{p}|>0.3$ , as shown in figure 10 a) can always be detected at an ‘optimal’ time lag $\unicode[STIX]{x0394}t_{2j}^{\,p}$ . As illustrated in the inset of figure 9(b), the uncertainty of $|R_{2j}^{p}|$ is estimated by a local Gaussian fit around the peak, while that of $\unicode[STIX]{x0394}t_{2j}^{\,p}$ is defined as the bin spanning the 95 % range of $R_{2j}^{p}$ ; all uncertainties are summarized in figure 10 as error bars. Figure 7(b,d) and figure 8(d–f) show that the shifted $a_{j}(t+\unicode[STIX]{x0394}t_{2j}^{\,p})$ now present remarkable phase alignment with $a_{2}(t)$ . More interestingly, all the $R_{2j}\sim \unicode[STIX]{x0394}t$ curves in figure 9 with various $Re$ collapse with each other when normalizing $\unicode[STIX]{x0394}t$ by the eddy turnover time $\unicode[STIX]{x1D6FF}/u_{\unicode[STIX]{x1D70F}}$ . The corresponding $\unicode[STIX]{x0394}t_{2j}^{\,p}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6FF}$ (shown in figure 10 b) presents a weak $Re$ dependence when taking the estimation uncertainty into account, in distinct contrast to that in an inner scaling by $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}$ . This implies the large-scale nature of the temporal characteristics of the first four leading-order modes.

Figure 10.

(a) The $Re$ dependence of the correlation peak $|R_{2j}^{p}|$ and (b) the ‘optimal’ time lag $\unicode[STIX]{x0394}t_{2j}^{\,p}$ : ●,  $j=1$ ; ▴,  $j=3$ ; ▪,  $j=4$ . In (b), $\unicode[STIX]{x0394}t_{2j}^{\,p}$ is normalized by either $\unicode[STIX]{x1D6FF}/u_{\unicode[STIX]{x1D70F}}$ (solid symbols with vertical axis on a) or $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}$ (hollow symbols with vertical axis on b). The error bars show the uncertainty of both $|R_{2j}^{p}|$ and $\unicode[STIX]{x0394}t_{2j}^{\,p}$ estimated by the method illustrated in figure 9 b.

Figure 11.

Matrix plot of the correlation peak $\left|R_{ij}^{p}\right|$ between $a_{i}(t)$ and $a_{j}(t+\unicode[STIX]{x0394}t_{2j}^{\,p})$ : (a)  $Re_{\unicode[STIX]{x1D70F}}=377$ ; (b)  $Re_{\unicode[STIX]{x1D70F}}=525$ ; (c)  $Re_{\unicode[STIX]{x1D70F}}=758$ ; (d)  $Re_{\unicode[STIX]{x1D70F}}=941$ ; (e)  $Re_{\unicode[STIX]{x1D70F}}=1093$ . Only the cell with $|R_{ij}^{p}|>0.3$ is visualized.

For a systematic depiction, $|R_{ij}^{p}|$ of the first 15 modes are presented in a matrix form in figure 11, in which a cutoff threshold of $|R_{ij}^{p}|=0.3$ is applied to mask those $(a_{i},a_{j})$ pairs being less correlated. In general, strong correlations only exist between two modes whose rank gaps $\left|i-j\right|$ are smaller than 5 (as indicated by the dashed lines in figure 11). Lower $Re$ results in a higher rank of the modes that remain correlated with their neighbours. This might be attributed to the insufficient scale separation in the low- $Re$ case. As $Re_{\unicode[STIX]{x1D70F}}$ increases beyond 1000, the number of correlated modes quickly decreases. Nevertheless, $a_{i}$ of the first four modes are persistently correlated with each other throughout the presently studied $Re$ range.

Rempfer & Fasel (Reference Rempfer and Fasel1994) showed that, in a transitional boundary layer, two neighbouring leading-order POD modes with equivalent eigenvalues can be grouped into one conjugated pair to depict the convective motion of the same structure. For a fully developed turbulent boundary layer with abundant structures and complicated inter-scale interactions, finding pairs of conjugated modes is impracticable. In fact, Hellström et al. (Reference Hellström, Ganapathisubramani and Smits2015) identified the existence of low-energy intermediate modes responsible for the transition between energy-containing large-scale modes. The present temporal correlation analysis suggests that, despite the energy hierarchy, the first four modes, which present strong spatial similarity, also exhibit dynamical coherence with each other. They can thus be grouped as a whole to jointly characterize the spatio-temporal evolution of the same type of large-scale structures.

4.1 Indicator of the interfaces of the hairpin packets

According to previous studies (Adrian Reference Adrian2007; Wu Reference Wu2014), the converging Q2/Q4 events and the associated large-scale interfaces in the low-order fields can be attributed to the interaction between the incoming flow and the collective backward induction of hierarchically aligned hairpin heads in a packet. In this scenario, a confluence line with zero speed will always be generated in the middle. The line of $u^{L}=0$ in the $\boldsymbol{V}^{L}$ field, in our view, will serve as a suitable indicator of the large-scale interfaces. Note that the diverging Q4/Q2 events will also produce zero- $u^{L}$ lines, which can be regarded as the dividing boundaries of two neighbouring hairpin packets (Natrajan & Christensen Reference Natrajan and Christensen2006). This case is excluded here by considering only zero- $u^{L}$ lines where the low-pass-filtered spanwise vorticities are negative, i.e. $\unicode[STIX]{x1D714}_{z}^{L}<0$ .

Typical zero- $u^{L}$ lines, which are obtained by connecting the zero-crossing points in the $u^{L}(y)$ profiles at each $x$ station, are illustrated in figures 12 and 13 as bold solid lines. In all the studied cases, more than 50 % of these zero- $u^{L}$ lines have a streamwise length $L_{x}$ longer than $\unicode[STIX]{x1D6FF}$ , as suggested by the cumulative distribution function (c.d.f.) of $L_{x}$ in figure 14. Owing to the truncation of the streamwise FOV, the right end of the $\text{c.d.f.}\sim L_{x}$ curves in figure 14 are underestimated; instead, one can obtain an impression on the possible extension of $L_{x}$ from figure 13. Visual inspection confirms that most zero- $u^{L}$ lines with positive inclination angles are in accordance with the inclined interfaces of the converging Q2/Q4 events. The justification is that no other typical eddies with equivalent length scales are likely to leave such large-scale imprints in the low-order fields.

Figure 14.

The c.d.f. of the streamwise length $L_{x}$ of the zero- $u^{L}$ lines: ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ : $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; and ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ .

Figure 15 shows the wall-normal variation of the p.d.f. of the inclination angle $\unicode[STIX]{x1D703}_{u_{0}}$ of zero- $u^{L}$ lines, which are calculated by the local slope of the detected zero- $u^{L}$ lines at specified $y$ . Note that the negative $\unicode[STIX]{x1D703}_{u_{0}}$ events, whose occurrence probabilities are one-fifth lower than that of the positive $\unicode[STIX]{x1D703}_{u_{0}}$ events, are inconsistent with the inclined pattern of the interfaces and are not counted here. In each $Re$ case, a p.d.f. peak around $\unicode[STIX]{x1D703}_{u_{0}}=10^{\circ }{-}14^{\circ }$ is always presented at $y/\unicode[STIX]{x1D6FF}\approx 0.15$ , i.e. the upper bound of the log layer (Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). Moreover, $P(\unicode[STIX]{x1D703}_{u_{0}})$ is right skewed in the log layer and below but gradually relaxes towards a symmetric distribution in the wake region and above. A smaller $Re$ results in a lower flow layer whereby such relaxation begins. The most probable value of $\unicode[STIX]{x1D703}_{u_{0}}$ monotonically increases with $y/\unicode[STIX]{x1D6FF}$ but decreases with $Re$ . In addition, the magnitude varies between $10^{\circ }$ and $25^{\circ }$ , falling in the range of the averaged inclination angle of the hairpin packets reported by previous studies ( $15^{\circ }{-}20^{\circ }$ in Head & Bandyopadhyay (Reference Head and Bandyopadhyay1981); $12^{\circ }{-}20^{\circ }$ in Adrian et al. (Reference Adrian, Meinhart and Tomkins2000); and $20^{\circ }$ in Natrajan & Christensen (Reference Natrajan and Christensen2006)) via different detection methods. This provides evidence for the idea that the zero- $u^{L}$ lines capture the inclined geometric features of the hairpin packets.

Figure 15.

Wall-normal variation of the p.d.f. of the inclination angle $\unicode[STIX]{x1D703}_{u_{0}}$ of the zero- $u^{L}$ lines: (a)  $Re_{\unicode[STIX]{x1D70F}}=377$ ; (b)  $Re_{\unicode[STIX]{x1D70F}}=525$ ; (c)  $Re_{\unicode[STIX]{x1D70F}}=758$ ; (d)  $Re_{\unicode[STIX]{x1D70F}}=941$ ; and (e)  $Re_{\unicode[STIX]{x1D70F}}=1093$ .

4.2 Relationship between large-scale interface and uniform momentum zones

The definition of the zero- $u^{L}$ line implies a sharp jump of the full-order $u$ across it; this is somewhat similar to the feature of the so-called internal boundaries (IBs) of the uniform momentum zones (UMZs) (de Silva, Hutchins & Marusic Reference de Silva, Hutchins and Marusic2016). On considering that there are viewpoints that UMZs are kinematically linked to the signature of packet-like hairpin structures (Meinhart & Adrian Reference Meinhart and Adrian1995; Adrian et al. Reference Adrian, Meinhart and Tomkins2000; de Silva et al. Reference de Silva, Hutchins and Marusic2016), it is interesting to see whether the concept of zero- $u^{L}$ lines can be accommodated in the paradigm of UMZs.

The modal-velocity-based criterion proposed by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) and de Silva et al. (Reference de Silva, Hutchins and Marusic2016) is adopted here to detect UMZs from instantaneous $u(x,y)$ snapshots. Figure 16 shows that both the p.d.f. of the number of UMZs, $N_{UMZ}$ , and the mean value, $\overline{N}_{UMZ}$ , in the present cases are similar to those at the closest $Re_{\unicode[STIX]{x1D70F}}$ ( $Re_{\unicode[STIX]{x1D70F}}\approx 1200$ ) in de Silva et al. (Reference de Silva, Hutchins and Marusic2016).

Note that $\overline{N}_{UMZ}$ is always larger than 2 (as shown in figure 16 b), indicating a pattern of multiple IBs (and UMZs) frequently appearing in instantaneous snapshots. In contrast, multiple zero- $u^{L}$ lines are seldom observed (see figure 13 for illustration). In the scenario of the coexistence of multi-generations of hairpin packets with various sizes and ages (Adrian Reference Adrian2007), such a difference might be attributed to the fact that zero- $u^{L}$ lines are only related to the ‘primary’ hairpin packets, which have the highest TKE contents and are thus mapped onto the leading-order POD modes. On the other hand, the older generations of hairpin packets, although inactive in TKE contribution, still contribute to the formation of UMZs.

Figure 17 superimposes the zero- $u^{L}$ lines onto the instantaneous $u$ fields with UMZs being partitioned by IBs. It is clearly shown that, if one snapshot presents only two layers of UMZs with one IB (as shown in figure 17 a,c), there will be a strong spatial coherence between the zero- $u^{L}$ line and this IB in the log layer and below; otherwise, the spatial coherence is rather weak (see figure 17 b,d for instance). A further conditional correlation analysis is carried out in appendix A to provide statistical support for this observation.

Figure 16.

(a) P.d.f. of the number of UMZs ( $N_{UMZ}$ ), and (b) mean number of UMZs ( $\overline{N}_{UMZ}$ ) as a function of $Re_{\unicode[STIX]{x1D70F}}$ : ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ; $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ ; and ◂,  $Re_{\unicode[STIX]{x1D70F}}\approx 1200$ in de Silva et al. (Reference de Silva, Hutchins and Marusic2016).

Figure 17.

Illustration of the spatial coherence between the zero- $u^{L}$ lines and the IBs of the UMZs: (a,b) two snapshots at $Re_{\unicode[STIX]{x1D70F}}=377$ ; (c,d) two snapshots at $Re_{\unicode[STIX]{x1D70F}}=1093$ . Pseudo-colour maps show the full-order streamwise velocity fields $U/U_{\infty }$ , bold solid lines are zero- $u^{L}$ lines, bold dash-dotted lines are the IBs of the UMZs, and contours indicate regions with concentrated negative $\unicode[STIX]{x1D6EC}_{ci}$ .

4.3 Coherence between large-scale interfaces and prograde vortices

According to the hairpin-packet model, the inclined interface should possess a strong spatial coherence with the hairpin heads. This is qualitatively visualized in figure 12, where a large portion of negative $\unicode[STIX]{x1D6EC}_{ci}$ concentration regions (with clockwise sense of rotation) are observed to ride on the zero- $u^{L}$ lines. Such a picture can be preliminarily examined by studying the p.d.f. of $u^{L}$ satisfying the condition $\unicode[STIX]{x1D6EC}_{ci}<-\unicode[STIX]{x1D6EC}_{ci}^{\text{T}}$ . Similar to Carlier & Stanislas (Reference Carlier and Stanislas2005) and Herpin et al. (Reference Herpin, Stanislas, Foucaut and Coudert2013), the threshold $\unicode[STIX]{x1D6EC}_{ci}^{\text{T}}$ is set as $1.5\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6EC}}$ , with $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6EC}}$ being the r.m.s. value of the local $\unicode[STIX]{x1D6EC}_{ci}$ , to guarantee that only the core regions of prograde vortices with sufficient strength are included. Figure 18 shows $P(u^{L}/\unicode[STIX]{x1D70E}_{u^{L}}\mid \unicode[STIX]{x1D6EC}_{ci}<-\unicode[STIX]{x1D6EC}_{ci}^{\text{T}})$ at $y/\unicode[STIX]{x1D6FF}\approx 0.1$ , 0.3 and 0.5 in all the studied cases, all of which have more compact p.d.f. peaks around $u^{L}=0$ than that in a normal distribution. This implies that most of the prograde swirling motions appear in the vicinity of the zero- $u^{L}$ lines.

Figure 18.

P.d.f. of $u^{L}$ satisfying $\unicode[STIX]{x1D6EC}_{ci}<-\unicode[STIX]{x1D6EC}_{ci}^{\text{T}}$ at (a)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.1$ , (b)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.3$ and (c)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.5$ : ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ; $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; and ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ . Bold solid lines represent the normal distribution. In all the p.d.f. curves, $u^{L}$ are normalized by their r.m.s. values  $\unicode[STIX]{x1D70E}_{u^{L}}$ .

The conditional-averaged $u^{L}$ field associated with a strong $\unicode[STIX]{x1D6EC}_{ci}$ event is further estimated by linear stochastic estimation (Christensen & Adrian Reference Christensen and Adrian2001):

(4.1) $$\begin{eqnarray}\langle u^{L}(r_{x},y)\mid \unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})\rangle \approx L_{u}\unicode[STIX]{x1D6EC}_{ci}(x,y_{ref}),\quad \text{with}~L_{u}=\frac{\langle \unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})u^{L}(x+r_{x},y)\rangle }{\langle \unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})\unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})\rangle }.\end{eqnarray}$$

In (4.1), $y_{ref}$ is the reference height, $\langle \cdot \rangle$ denotes the average over both the whole snapshot ensemble and the $x$ span, while the estimation kernel $L_{u}$ differs from $\langle u^{L}|\unicode[STIX]{x1D6EC}_{ci}\rangle$ by a factor of $\unicode[STIX]{x1D6EC}_{ci}$ . For simplicity, the unconditional two-point correlation $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ is calculated to approximate $L_{u}$ :

(4.2) $$\begin{eqnarray}R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}(r_{x},y)=-1\times \frac{\langle \unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})u^{L}(x+r_{x},y)\rangle }{\sqrt{\langle \unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})\unicode[STIX]{x1D6EC}_{ci}(x,y_{ref})\rangle \langle u^{L}(x,y)u^{L}(x,y)\rangle }}.\end{eqnarray}$$

Note that the minus sign in (4.2) is to make the sign of $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ consistent with the converging Q2/Q4 events. Figure 19 plots $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ at $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.1$ , 0.3 and 0.5 in the $Re_{\unicode[STIX]{x1D70F}}=1093$ case. Other low- $Re$ cases produce similar results and are not presented. A zero- $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ line, which divides the $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ field into the upper positive part and the lower negative part, is always observed above the probing point, providing statistical evidence for the spatial coherence between the large-scale interfaces and the prograde vortices. Moreover, their wall-normal gap increases with $y_{ref}$ . This implies that the prograde vortices at higher layer have larger sizes and stronger backward induction to push the interfaces further away, consistent with the self-similar growth of wall-attached eddies in the attached-eddy hypothesis (Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982; Marusic & Perry Reference Marusic and Perry1995; Perry & Marusic Reference Perry and Marusic1995; Marusic & Kunkel Reference Marusic and Kunkel2003). This also explains the observation in figure 18(c) that the most probable peak of $P(u^{L}\mid \unicode[STIX]{x1D6EC}_{ci}<-\unicode[STIX]{x1D6EC}_{ci}^{\text{T}})$ at $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.5$ is slightly left offset from $u^{L}=0$ .

Figure 19.

Two-point correlation map of $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}(r_{x},y)$ at $Re_{\unicode[STIX]{x1D70F}}=1093$ for (a)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.1$ , (b)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.3$ and (c)  $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.5$ . Solid and light dotted contours represent positive and negative $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ , respectively, bold solid lines are the zero- $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ lines, and cross marks indicate the probing points.

4.4 Conditional correlation analysis of swirling strength

By examining the whole snapshot ensemble, it is found that, in all the studied $Re$ cases, more than $50\,\%$ of instantaneous $\boldsymbol{V}^{L}$ snapshots contain continuous zero- $u^{L}$ lines with a streamwise length $L_{x}>0.8\unicode[STIX]{x1D6FF}$ . Specifically, a large-scale inclined interface, which can be interpreted as the envelope of one hairpin packet, is a frequently observed event. To further explore the vortex organization within a packet, it is natural to focus on the vicinity of these interfaces. Here, a binarized mask $M(x,y)$ is used for this purpose:

(4.3) $$\begin{eqnarray}M(x,y)=\left\{\begin{array}{@{}ll@{}}1, & \forall ~(x,y)\in H,\\[2.0pt] 0, & \forall ~(x,y)\notin H,\end{array}\right.H=\{(x,y)\mid y\in [y_{u^{L}=0}\pm \unicode[STIX]{x1D6FF}_{y}]\}.\end{eqnarray}$$

In (4.3), $H$ is the neighbouring domain of the inclined interface, determined by shifting the identified zero- $u^{L}$ lines along the $y$ direction over $\pm \unicode[STIX]{x1D6FF}_{y}$ , as illustrated in figure 12. Multiplying $\unicode[STIX]{x1D6EC}_{ci}$ by $M$ yields the conditional $\widetilde{\unicode[STIX]{x1D6EC}}_{ci}$ fields, i.e. $\widetilde{\unicode[STIX]{x1D6EC}}_{ci}=M\unicode[STIX]{x1D6EC}_{ci}$ , which mask those swirling motions far away from the inclined interfaces. The two-point correlation of $\widetilde{\unicode[STIX]{x1D6EC}}_{ci}$ is then calculated as follows to construct a statistical picture of the vortex organization around the probing point:

(4.4) $$\begin{eqnarray}R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}(r_{x},y)=\frac{\langle \widetilde{\unicode[STIX]{x1D6EC}}_{ci}(x,y_{ref})\widetilde{\unicode[STIX]{x1D6EC}}_{ci}(x+r_{x},y)\rangle }{\langle \widetilde{\unicode[STIX]{x1D6EC}}_{ci}(x,y_{ref})\widetilde{\unicode[STIX]{x1D6EC}}_{ci}(x,y_{ref})\rangle }.\end{eqnarray}$$

Figure 20.

Two-point correlation map $R_{\unicode[STIX]{x1D6EC}_{ci}\unicode[STIX]{x1D6EC}_{ci}}$ of the unfiltered $\unicode[STIX]{x1D6EC}_{ci}$ fields in (a–c) and $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ of the filtered $\widetilde{\unicode[STIX]{x1D6EC}}_{ci}$ fields in (d–f), together with the streamwise variation of $R_{\unicode[STIX]{x1D6EC}_{ci}\unicode[STIX]{x1D6EC}_{ci}}^{max}$ and $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{max}$ in (g–i), which are the maximum values of the wall-normal profiles $R_{\unicode[STIX]{x1D6EC}_{ci}\unicode[STIX]{x1D6EC}_{ci}}(y)$ and $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}(y)$ at each $r_{x}$ station. Here $Re_{\unicode[STIX]{x1D70F}}=1093$ ; $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.1$ in (a,d,g); $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.3$ in (b,e,h); and $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.5$ in (c,f,i). The contour levels in (a–f) are ( $-0.01$ , 0.01, 0.1, 0.15, 0.3, 0.6). In (d–f), the bold dashed lines are the zero- $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ lines shown in figure 19. In (g–i), the dashed lines are the square of the unfiltered $R_{\unicode[STIX]{x1D6EC}_{ci}\unicode[STIX]{x1D6EC}_{ci}}^{max}$ , the solid lines are the filtered $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{max}$ with the mask offset $\unicode[STIX]{x1D6FF}_{y}=0.2\unicode[STIX]{x1D6FF}$ , and the dash-dotted lines are $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{max}$ with $\unicode[STIX]{x1D6FF}_{y}=0.1\unicode[STIX]{x1D6FF}$ . The cross marks indicate the secondary correlation peaks $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{p}$ . The definitions of $\unicode[STIX]{x1D703}_{1,2}$ and $\unicode[STIX]{x1D706}_{x1,x2}$ to be discussed in § 5.1 are given in (f). The inset of (i) illustrates the uncertainty estimation of both $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{p}$ and the related $\unicode[STIX]{x1D706}_{x1,x2}$ .

The magnitude of the mask offset $\unicode[STIX]{x1D6FF}_{y}$ in (4.3) will inevitably affect the resulting $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ in (4.4). In all the studied cases, $\unicode[STIX]{x1D6FF}_{y}$ is selected to be $0.2\unicode[STIX]{x1D6FF}$ , sufficiently larger than the radius of most spanwise-rotated vortices (Herpin et al. Reference Herpin, Stanislas, Foucaut and Coudert2013). The major consideration in selecting such a large value is to apply a loose constraint to include all the swirling events that are possibly related to the hairpin-packet imprints. In the following discussion on the geometric statistics of the vortex organization (in § 5.1), the peak values in the $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}(r_{x},y)$ map and the corresponding spatial positions serve as the key parameters. Appendix B uses a systematic test to show that these parameters are rather robust to $\unicode[STIX]{x1D6FF}_{y}$ in a range of $\unicode[STIX]{x1D6FF}_{y}=(0.10{-}0.25)\unicode[STIX]{x1D6FF}$ .

Figure 20(d–f) gives the map of $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}(r_{x},y)$ in the $Re_{\unicode[STIX]{x1D70F}}=1093$ case at $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.1$ , 0.3 and 0.5. Other cases with smaller $Re$ have similar $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ maps and are not presented. In addition to the round-shaped autocorrelation region, which represents one spanwise-rotated vortex being probed at $r_{x}=0$ and $y=y_{ref}$ , two elongated positive $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ strips (with $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}>0.1$ ) always appear on both flanks of it. They are in spatial accordance with the zero- $R_{\unicode[STIX]{x1D6EC}_{ci}u^{L}}$ lines (red dashed lines in figure 20 d–f) and can be attributed to the scattering distribution of spanwise vortices, whose sense of rotation is the same as that being probed at the autocorrelation peak, along the large-scale Q2/Q4 interfaces. Importantly, secondary peaks $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{p}$ exist in the $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ strips (indicated by cross marks in figure 20 d–f). They are highlighted in figure 20(g–i), which plots the streamwise variation of the maximum value of the $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}(y)$ profile at each $r_{x}$ station. Figure 20(g–i) further shows that, below $y_{ref}/\unicode[STIX]{x1D6FF}=0.3$ , a decrease in the mask offset from $\unicode[STIX]{x1D6FF}_{y}=0.2$ to $\unicode[STIX]{x1D6FF}_{y}=0.1$ will not remarkably affect their magnitudes and spatial positions.

When considering the condition invoked in the conditional correlation scheme, i.e. only the vortices in the vicinity of the large-scale Q2/Q4 interfaces are concerned, the existence of the secondary correlation peaks provides statistical evidence for the hierarchical alignment of multiple prograde vortices within one packet. In contrast, the two-point correlation of the unconditional $\unicode[STIX]{x1D6EC}_{ci}$ fields fails to reveal such a picture. As shown in figure 20(a–c), two negative correlation trails are observed on both flanks of the autocorrelation peak, inclining at an angle of about $\pm 45^{\circ }$ . Natrajan et al. (Reference Natrajan, Wu and Christensen2007) attributed them to pairs of prograde and retrograde vortices related to the $\unicode[STIX]{x03A9}$ -shaped shoulder of hairpin vortices. On the other hand, the positive $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}$ strips now degrade to very weak positive $R_{\unicode[STIX]{x1D6EC}_{ci}\unicode[STIX]{x1D6EC}_{ci}}$ trails (with the magnitude being less than 0.01) without discernible correlation peaks (as shown in figure 20 g–i). Such a significant difference stresses the necessity of excluding the contamination of irrelevant swirling motions when forming a statistical picture of the vortex organization within a packet.

5.1 Wall-normal variation of $\unicode[STIX]{x1D703}_{1,2}$ and $\unicode[STIX]{x1D706}_{x1,x2}$

Figure 21 summarizes $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ as a function of $y_{ref}/\unicode[STIX]{x1D6FF}$ in all the studied cases. A linear wall-normal growth of $\unicode[STIX]{x1D703}_{1}$ is always presented below $y_{ref}/\unicode[STIX]{x1D6FF}\approx 0.17{-}0.19$ , above which a saturation will be gradually achieved. The larger $Re_{\unicode[STIX]{x1D70F}}$ , the lower the flow layer at which the saturation begins. In contrast, $\unicode[STIX]{x1D703}_{2}$ is independent of $y_{ref}/\unicode[STIX]{x1D6FF}$ . Its magnitude is always larger than that of $\unicode[STIX]{x1D703}_{1}$ in the log layer and below, implying a wall-normal steepening of the vortex alignment visualized in figure 20(d). Above the log layer, the magnitudes of $\unicode[STIX]{x1D703}_{2}$ and $\unicode[STIX]{x1D703}_{1}$ are rather close, all falling in the range of $10^{\circ }{-}15^{\circ }$ and being negatively correlated with $Re_{\unicode[STIX]{x1D70F}}$ .

Recall that the hairpin-packet model interprets the large-scale interface as a result of the collective backward induction of multiple hairpin vortices. Such a picture can be evaluated by comparing $\unicode[STIX]{x1D703}_{2}$ to the most probable inclination angle $\unicode[STIX]{x1D703}_{u_{0}}^{mod}$ of the zero- $u^{L}$ lines. As shown in figure 21(c), $\unicode[STIX]{x1D703}_{u_{0}}^{mod}$ is generally larger than $\unicode[STIX]{x1D703}_{2}$ , and the difference between them increases with $y_{ref}/\unicode[STIX]{x1D6FF}$ but decreases with $Re_{\unicode[STIX]{x1D70F}}$ . In the context of the hairpin-packet model, this observation implies that prograde vortices prefer to align at a shallower angle than that of the inclined interfaces; their backward induction strengths are enhanced at higher flow layer but degrade at higher  $Re$ .

Figure 21.

Wall-normal variation of the averaged inclination angle between neighbouring prograde vortices, $\unicode[STIX]{x1D703}_{1}$ in (a) and $\unicode[STIX]{x1D703}_{2}$ in (b), as well as the difference between the most probable inclination angle $\unicode[STIX]{x1D703}_{u_{0}}^{mod}$ of the zero- $u^{L}$ lines and $\unicode[STIX]{x1D703}_{2}$ in (c). Symbols: ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ; $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ . The definitions of $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ are illustrated in figure 20(f).

Figure 22 shows the wall-normal variation of $\unicode[STIX]{x1D706}_{x1}$ and $\unicode[STIX]{x1D706}_{x2}$ in both the outer and inner scaling. Considering the uncertainty bin shown in figure 22(c,d), both $\unicode[STIX]{x1D706}_{x1}$ and $\unicode[STIX]{x1D706}_{x2}$ present quasi-linear growth in the near-wall region, which are followed by either saturation or very slow variation once $y_{ref}$ exceeds a critical height $y_{\unicode[STIX]{x1D706}_{x}}^{s}$ (the superscript $(\cdot )^{s}$ is short for saturation). Unlike $\unicode[STIX]{x1D703}_{1,2}$ , the difference between $\unicode[STIX]{x1D706}_{x1}$ and $\unicode[STIX]{x1D706}_{x2}$ is rather small. In the inner scaling, the magnitudes of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ monotonically increase with $Re_{\unicode[STIX]{x1D70F}}$ , the slopes $\unicode[STIX]{x2202}\unicode[STIX]{x1D706}_{x1,x2}^{+}/\unicode[STIX]{x2202}y_{ref}^{+}$ in the linear growth stage are rather close among all the studied cases, and the critical heights $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ are positively correlated with $Re_{\unicode[STIX]{x1D70F}}$ . In contrast, none of these features are observed in the outer scaling. The inner-scaled nature of $\unicode[STIX]{x1D706}_{x1,x2}$ , in our view, is consistent with the well-known idea that inner-scaled near-wall events are responsible for the generation of packets of vortical structures (Asai et al. Reference Asai, Minagawa and Nishioka2002; Schoppa & Hussain Reference Schoppa and Hussain2002; Brandt et al. Reference Brandt, Schlatter and Henningson2004; Brandt & de Lange Reference Brandt and de Lange2008; Wang, Pan & Zhang Reference Wang, Pan and Zhang2009; Jodai & Elsinga Reference Jodai and Elsinga2016; Wu Reference Wu2017).

Figure 22.

Wall-normal variation of the averaged streamwise spacing of neighbouring prograde vortices within a packet, $\unicode[STIX]{x1D706}_{x1}$ in (a) and (c), and $\unicode[STIX]{x1D706}_{x2}$ in (b) and (d): (a) and (b) are the outer-scaling plots, while (c) and (d) are the inner-scaling plots. Symbols: ○,  $Re_{\unicode[STIX]{x1D70F}}=377$ ; $+$ ,  $Re_{\unicode[STIX]{x1D70F}}=525$ ; ▫,  $Re_{\unicode[STIX]{x1D70F}}=758$ ; $\times$ ,  $Re_{\unicode[STIX]{x1D70F}}=941$ ; ♢,  $Re_{\unicode[STIX]{x1D70F}}=1093$ ; *,  $Re_{\unicode[STIX]{x1D70F}}\approx 1141$ in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004). In (c) and (d), all the inclined dashed lines have the same slope to indicate the quasi-linear growth of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ below $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ . The shaded regions show the uncertainty bins of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ estimated via a $99\,\%$ $R_{\widetilde{\unicode[STIX]{x1D6EC}}_{ci}\widetilde{\unicode[STIX]{x1D6EC}}_{ci}}^{p}$ criterion illustrated in figure 20(i), which indicates a full separation of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ among the four smallest $Re$ cases. The definitions of $\unicode[STIX]{x1D706}_{x1}$ and $\unicode[STIX]{x1D706}_{x2}$ are illustrated in figure 20(f), and the definitions of $\unicode[STIX]{x1D706}_{x}^{s+}$ and $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ are given in figure 22(c).

5.2 Discussion of the $Re$ dependence of $\unicode[STIX]{x1D706}_{x1,x2}$

Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) reported a logarithmic wall-normal growth of the mean streamwise spacing of neighbouring prograde vortices, which were regarded by them as the hairpin heads. This conflicts with the quasi-linear growth of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ and the subsequent saturation observed here. For a comparison, the data of $\unicode[STIX]{x1D706}_{x}^{+}\sim y^{+}$ at $Re_{\unicode[STIX]{x1D70F}}\approx 1141$ in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) are reproduced in figure 22(c,d). Despite the remarkable scattering of $\unicode[STIX]{x1D706}_{x}^{+}$ , which can be attributed to the large uncertainty of the acceleration measurement in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004), a saturation of $\unicode[STIX]{x1D706}_{x}^{+}$ seems to be presented in the outer region if abnormal data are omitted. Meanwhile, the near-wall part of $\unicode[STIX]{x1D706}_{x}^{+}\sim y^{+}$ in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) can be reasonably characterized by a liner correlation.

Figure 23.

The $Re$ dependence of (a) the saturated $\unicode[STIX]{x1D706}_{x}^{s+}$ and (b) the corresponding detachment height $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ . In (a), symbols are: ○,  $\unicode[STIX]{x1D706}_{x1}^{s+}$ ; $+$ ,  $\unicode[STIX]{x1D706}_{x2}^{s+}$ ; ▫,  $\unicode[STIX]{x1D706}_{x}^{s+}$ in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) with error bar indicating the r.m.s. value in the slow variation stage. In (b), symbols are: ○,  $y_{\unicode[STIX]{x1D706}_{x1}}^{s+}$ ; $+$ ,  $y_{\unicode[STIX]{x1D706}_{x2}}^{s+}$ . The solid straight lines in (a) and (b) indicate a $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling, and the dashed line in (b) shows a $y_{M}^{+}=3.9Re_{\unicode[STIX]{x1D70F}}^{1/2}$ formulation (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009) suggesting the geometric centre of the log layer.

The quasi-linear growth of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ below $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ implies a self-similarity of the streamwise spacing of neighbouring vortices within one packet, which is consistent with the linear growth of the length scales of conditional eddies associated with hairpin vortices (Tomkins & Adrian Reference Tomkins and Adrian2003; Dennis & Nickels Reference Dennis and Nickels2011). Meanwhile, the saturation of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ beyond $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ indicates a detachment of this self-similar growth. As shown in figure 23(a), the saturated $\unicode[STIX]{x1D706}_{x1,x2}^{s+}$ , which are measured by the mean value of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ beyond $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ , follows a $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling. The saturated $\unicode[STIX]{x1D706}_{x}^{s+}$ in the dataset of Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) are estimated in a similar manner: (1) low-pass filtering the $\unicode[STIX]{x1D706}_{x}^{+}\sim y^{+}$ curve, with abnormal points being manually removed in advance; and (2) calculating the average value and the r.m.s. value of the low-pass-filtered $\unicode[STIX]{x1D706}_{x}^{+}$ in the region with slow wall-normal variation as $\unicode[STIX]{x1D706}_{x}^{s+}$ and its uncertainty, respectively. As shown in figure 23(a), the estimated $\unicode[STIX]{x1D706}_{x}^{s+}$ in Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004), despite the rather large uncertainty, shows a reasonable agreement with the present $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling.

Wu & Christensen (Reference Wu and Christensen2006) reported a $Re_{\unicode[STIX]{x1D70F}}^{-1/2}$ scaling of the inner-scaled population density $\unicode[STIX]{x1D6F1}_{p}^{+}$ of prograde vortices. Referring to the definition of $\unicode[STIX]{x1D6F1}_{p}^{+}$ , which measures the average number of detected prograde vortices per unit area at one wall-normal height, the $\unicode[STIX]{x1D6F1}_{p}^{+}\sim Re_{\unicode[STIX]{x1D70F}}^{-1/2}$ correlation implies a $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ dependence of the averaged streamwise gap between two neighbouring prograde vortices, consistent with the present scaling of $\unicode[STIX]{x1D706}_{x1,x2}^{s+}\sim Re_{\unicode[STIX]{x1D70F}}^{1/2}$ .

Furthermore, in the concept of the intermediate meso-layer (Sreenivasan Reference Sreenivasan1989; Klewicki et al. Reference Klewicki, Fife and Wei2009), the location $y_{RSmax}^{+}$ of the maximum Reynolds stress scales with $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ . Morrison et al. (Reference Morrison, McKeon, Jiang and Smits2004) assumed that the magnitude of the maximum RSS scales approximately with $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ in pipe flows. Eyink (Reference Eyink2008) derived a $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ dependence of the location of the maximum RSS by a modified attached-eddy approach. In addition, Alfredsson, Segalini & Orlu (Reference Alfredsson, Segalini and Orlu2011) proposed that the wall-normal height $y_{u,max}^{+}$ of the outer $u_{rms}$ peak scales with $Re_{\unicode[STIX]{x1D70F}}^{0.56}$ by $y_{u,max}^{+}=0.81Re_{\unicode[STIX]{x1D70F}}^{0.56}$ .

In the present case, the near-wall quasi-linear growth of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ and the $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ dependence of $\unicode[STIX]{x1D706}_{x1,x2}^{s+}$ lead to a $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling of $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ . This is evidenced by figure 23(b), in which $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ is estimated as the wall-normal position where $\unicode[STIX]{x2202}\unicode[STIX]{x1D706}_{x1,x2}^{+}/\unicode[STIX]{x2202}y_{ref}^{+}$ presents a sharp drop. The data of Christensen et al. (Reference Christensen, Wu, Adrian and Lai2004) are not included due to the difficulty in precisely estimating $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ in their case. A relation of $y_{M}^{+}=3.9Re_{\unicode[STIX]{x1D70F}}^{1/2}$ , which was proposed by Mathis et al. (Reference Mathis, Hutchins and Marusic2009) to formulate the geometric centre of the log layer, is also provided in figure 23(b) (as dashed line). Clearly, the saturation of $\unicode[STIX]{x1D706}_{x1,x2}^{+}$ occurs above the centre of the log layer, where a secondary $u_{rms}$ peak is expected under high- $Re$ condition (Mathis et al. Reference Mathis, Hutchins and Marusic2009). The locations of $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ are further superimposed onto the $u_{rms}^{+}(y^{+})$ profiles in figure 1(b). This gives a hint that $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ locates at the right end of the ‘shoulder’ of the $u_{rms}^{+}(y^{+})$ profile (due to the low-to-moderate $Re$ , no outer $u_{rms}$ peak is observed here), beyond which $u_{rms}$ presents a quick drop. Such an observation suggests that $y_{\unicode[STIX]{x1D706}_{x}}^{s+}$ is related to the commencement of the intermediate meso-layer, above which the hairpin packets generated by the near-wall events gradually detach from the wall and become less active in contributing to the streamwise turbulent fluctuations.