3.1 Instantaneous zones

In order to identify the instantaneous UMZs and the layers that separate them, we follow the methodology used by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) and de Silva et al. (Reference de Silva, Hutchins and Marusic2016), extended to exploit the time-resolved information captured in the present experiment. The instantaneous method is based on detecting the local peaks of the PDF of the streamwise velocity component for every velocity snapshot. However, as suggested by de Silva et al. (Reference de Silva, Hutchins and Marusic2016), only the vectors below the turbulent/non-turbulent interface (TNTI) are included, so that the significantly higher outer peak of the non-turbulent region would not overshadow the ones of the UMZs. For the detection of the TNTI, we choose to use a kinetic energy deficit threshold, as described in Chauhan, Philip & Marusic (Reference Chauhan, Philip and Marusic2014a ), Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ) (for more details see appendix A). After the TNTI location is estimated, the vectors belonging to the turbulent region can be identified for every velocity snapshot, and the PDF of the streamwise velocity component can be constructed (see figure 2 a). It has been shown that points of local maxima represent the modal velocities that characterise each UMZ (Adrian et al. Reference Adrian, Meinhart and Tomkins2000; de Silva et al. Reference de Silva, Hutchins and Marusic2016). In order to define the edges of the UMZs in physical space, velocity contours at points of local minima in the PDF can be used. Because the zones and their edges are treated independently, it is not always certain that each peak would be bounded by two edges, so for the peaks that are unbounded, the mid-distance between two peaks is chosen as the edge location (see de Silva et al. Reference de Silva, Hutchins and Marusic2016). A representative example of this procedure can be seen in figure 2. Local peaks in the velocity PDF (triangles in figure 2 a) correspond to each zone’s modal velocity. The velocities at the points between neighbouring peaks (grey dashed lines in figure 2 a) are used to define the contour lines demarcating each UMZ in physical space (solid grey lines in figure 2 b). Repeating this process for consecutive snapshots, the temporal coherence of each detected peak can be analysed (figure 2 c). From a temporal perspective it can be seen that some modal peaks reside within our FOV for very short times (grey symbols in figures 2(a), 2(c) and grey coloured layers in figure 2 b), while others reside for much longer (red symbols in figures 2 a, 2 c and coloured layers in figure 2 b). A detailed discussion on this distinction follows in the next section.

There are some important considerations regarding the instantaneous detection process and the resulting number of peaks ( $N_{UMZ}$ ), that are worth discussing briefly here. Firstly, the streamwise extent of the data and consequently the number of vectors available, were shown by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) to be important factors when detecting the modal velocities. In our case, the available streamwise extent is ${\approx}$ 2600 viscous units ( $0.5\unicode[STIX]{x1D6FF}$ ), including on average 6500 vectors below the TNTI (depending on its vertical location). For the construction of the PDF, the vectors are distributed in 67 bins with size approximately equal to $0.5U_{\unicode[STIX]{x1D70F}}$ , for $U/U_{\infty }\in [0,1]$ . However, it should be noted that, due to spatial resolution effects, only the last 50 bins with $U/U_{\infty }\in [0.26,1]$ are relevant here; no peaks in the PDF are detected for lower velocities. We choose to use the full streamwise extent available for the detection process. However for completeness, we also repeated the process with a reduced streamwise extent of 2000 viscous units; even though there were instantaneous discrepancies in individual snapshots, the resulting statistics were very similar while the trends discussed in the following sections remained unaltered.

Aside from the streamwise extent and number of vectors used, the thresholds used in the peak-detection algorithm itself, are the most important factors in the resulting number of peaks. These include the allowed distance between two peaks $F_{d}$ , (in integer multiples of the bin width), the minimum height for a peak to be considered detectable, $F_{h}$ , and the difference in height of a given peak relative to its neighbouring points (prominence), $F_{p}$ . Depending on these parameters, the resulting average number of detected peaks is found to vary significantly (figure 3). An increase of the minimum required prominence of the peaks with respect to their neighbours results in a gradual decrease of the average detected number of peaks (from 6 to almost 3), as expected. Similarly, when the allowed distance between peaks is significantly increased, no more than 4 peaks on average can be detected. When the peaks are allowed to be next to each other, however, the number of detected peaks can reach up to 7, although in that case differentiating between different UMZs would be almost impossible. Finally, the range of the minimum required peak heights tested here, seemed to have an imperceptible influence on the resulting number of peaks (increasing height leads to slightly lower $N_{UMZ}$ ). It should be noted that the trends discussed in the following sections are robust to the chosen values for these thresholds and are meant to show relative changes rather than represent an absolute value of $N_{UMZ}$ (see also de Silva et al. Reference de Silva, Hutchins and Marusic2016).

Figure 3.

Contours of the average number of peaks detected, with varying minimum peak height, $F_{h}$ , minimum peak distance, $F_{d}$ , and minimum peak prominence from neighbouring points, $F_{p}$ , in the peak-detection algorithm.

In order to facilitate comparison with previous results, we choose a parameter set such that the average number of peaks follows the Reynolds number dependency found by de Silva et al. (Reference de Silva, Hutchins and Marusic2016). These include a minimum peak height of 0.3 with a prominence of at least 0.05 from neighbouring values, and a minimum distance of two bins between detected peaks. We apply this detection process to all the available velocity snapshots independently and, on average ${\approx}4.5$ peaks are detected for $Re_{\unicode[STIX]{x1D70F}}=5300$ , with a distribution that follows the results of de Silva et al. (Reference de Silva, Hutchins and Marusic2016).

3.2 Time-coherent zones

Following the process described above, we can successfully identify UMZs and their edges in all available snapshots, however tracking them in time becomes increasingly difficult for several reasons. From a physical perspective, even when identified correctly, not all modal velocities and zones that appear instantaneously necessarily last longer than a single snapshot in time. Furthermore, while identifying local maxima in independent distributions can be successful most of the time, following the same procedure in a time series does not always result in a smooth evolution of the detected peaks. For these reasons, to identify the peaks and edges in the time-resolved data available, we still use an instantaneous detection for each velocity snapshot, but we further set a temporal threshold ( $\unicode[STIX]{x1D70F}^{+}\approx 20$ ) to remove peaks that remain within the measurement plane for short times (referred to as residence times, $\unicode[STIX]{x1D70F}$ ). Since the characteristic velocity (modal velocity) of each peak is the basis of its instantaneous detection, we also use it to evaluate its temporal coherence. More specifically, peaks that are detected in successive snapshots and vary their modal velocity within a band of $\pm 0.5U_{\unicode[STIX]{x1D70F}}$ (corresponding to a shift of a single bin in the velocity PDF in either direction) from one snapshot to the other, are considered coherent between these snapshots. The effect of this procedure can be seen by contrasting figures 2 and 4; short-lived peaks have been removed in the latter. A video representing the evolution of both short-lived and time-coherent peaks can be found in the supplementary material (movie 1) available at https://doi.org/10.1017/jfm.2018.126. Short-lived peaks are denoted with grey colours while the time-coherent peaks are coloured according to their modal velocity.

Figure 4.

(a) The PDF of $U/U_{\infty }$ , (b) corresponding contour plot of each detected zone and (c) time evolution of the PDF of $U/U_{\infty }$ for the same velocity frames as in figure 2, when all short-lived peaks are removed. Coloured symbols and contours in (a) and (b) respectively, represent the modal velocity of each UMZ. The individual temporal evolution of each UMZ is separately coloured in (c) with contour lines ranging from 0.3 to 7.5 with a spacing of 0.8.

The chosen temporal threshold, taking into account the initial temporal filter applied (resulting in an effective time step of $\text{d}t^{+}=5.1$ ), ensures that a UMZ is present in at least 4 consecutive snapshots. It is important to note that, a spatial filter has already been used for the detection of the UMZs, in the form of the instantaneous PDFs, which include all the vectors of each snapshot spanning $0.5\unicode[STIX]{x1D6FF}$ in the streamwise direction. Therefore, results presented here refer to structures spanning at least $0.5\unicode[STIX]{x1D6FF}$ in length ( ${\approx}$ 2600 viscous units). The removal of short-lived peaks leads to a significant decrease in the total number of peaks, without, however, altering the overall evolution of the number of zones. This can be seen in the PDF of $N_{UMZ}$ after removal of short-lived peaks (figure 5, grey symbols), which is centred around an average of 3.2 peaks.

Figure 5.

The PDF of $N_{UMZ}$ , for all detected peaks (black symbols) and considering time-coherent peaks only (grey symbols).

In the following sections, the geometric characteristics of time-coherent UMZs and associated flow statistics are discussed in detail. It should be noted that this analysis was also performed for snapshot UMZs (i.e. before applying temporal threshold); while we feel the temporal filter more accurately represents the physics, the snapshot approach acts as a benchmark for comparison to earlier studies without time-resolved data. Appendix B discusses the results of the snapshot approach where they differ from the time-coherent approach.

4.1 Modal velocity

The mean value and distribution of $U_{m}$ and their dependence on $N_{UMZ}$ are presented here, for all groups of the two classification systems described above.

Figure 7.

Modal velocities for the two classification systems and their dependence on $N_{UMZ}$ . (a,b) Classification based on momentum deficit, $U_{MD}^{+}$ . Modal velocities, ${U_{m}|}_{MD_{i}}$ for three momentum deficit groups: $U_{MD_{1}}^{+}\in [0,6)$ (◀ ), $U_{MD_{2}}^{+}\in (6,12]$ (▪), $U_{MD_{3}}^{+}\in (12,18]$ ( ▶). Symbols denote mean values and solid lines the pdf extent between the 25th and 75th percentile (grey dashed lines on subfigure b). Grey coloured symbols denote the mean and extent of the example pdf shown in (b). UMZs with low momentum deficit (◀ ) show little variation with increasing $N_{UMZ}$ , while the group of the largest deficit ( ▶) moves to lower modal velocities as new zones appear close to the wall. (c,d) Classification based on zone rank. Modal velocities, ${U_{m}|}_{R}$ , for six zone ranks. Symbols as in figure 6. All ranks move to higher modal velocities as $N_{UMZ}$ increases.

For the first classification system (figure 7 a,b), based on the momentum deficit level of each zone, three different groups are defined, as was mentioned above. The first group with $U_{MD_{1}}^{+}\in [0,6)$ , includes the zones closest to the free stream. The average modal velocity, ${\overline{U}_{m}|}_{MD_{1}}$ (◀ in figure 7 a) of zones belonging to that group, and the spread around it (solid lines in figure 7 a), show virtually no change as $N_{UMZ}$ varies. This suggests that as $N_{UMZ}$ increases and more zones appear, the group of UMZs with low $U_{MD}^{+}$ remains largely unaltered. An example PDF of $U_{m}$ for zones belonging to this group when $N_{UMZ}=5$ , can be seen in figure 7(b) (some slight pixel-locking effects can still be seen in this figure that amplify the prominent modal velocities). The groups of intermediate ( $U_{MD_{2}}^{+}\in [6,12)$ , squares in figure 7 a) and large ( $U_{MD_{3}}^{+}\in [12,18)$ , ▶ in figure 7 a) momentum deficit levels show a slight decrease in the average modal velocity as $N_{UMZ}$ increases, while the distribution around that mean becomes wider. This behaviour indicates that, when $N_{UMZ}$ increases, the newest zones appear near the wall and therefore contribute to the classes of intermediate or large momentum deficit, while the class of zones closest to the free stream remains unaltered. In snapshots where only one zone is present ( $N_{UMZ}=1$ ), the momentum deficit of that single UMZ is still classified to either of the aforementioned groups. As can be seen in figure 7(a), all three groups are represented, suggesting that single UMZs could have $U_{MD}^{+}$ values ranging from 0 to 18. What is, however, not apparent is that, statistically, single UMZs tend to have low momentum deficit levels and mostly contribute to the first group ( $U_{MD_{1}}^{+}\in [0,6)$ ). As $N_{UMZ}$ increases and new zones appear at the wall, the other two groups become increasingly populated.

This process is clearer when the zones are classified based on their rank (figure 7 c,d). Here, the first group ( ▲ in figure 7 c), includes UMZs that have the largest modal velocity in each snapshot. When $N_{UMZ}=1$ , all single UMZs contribute only to that first group and it is now apparent that most of them have modal velocities close to the free stream, albeit with some significant variation (solid lines in figure 7 c), as was mentioned above. More generally for low $N_{UMZ}$ , all groups have modal velocities close to $U_{\infty }$ , with wide distributions, mostly skewed towards the free stream (a representative example can be seen in figure 7 d). As $N_{UMZ}$ increases, the mean modal velocities of all ranks are shown to move to higher values, while the spread around them becomes progressively narrower. Zones of higher rank, that are closer to the wall, show the largest modal velocity increase with increasing $N_{UMZ}$ , while the ones close to the free stream exhibit only a marginal rise. This supports the observations from the first classification scheme, where new zones were shown to appear near the wall and influence the group of the largest momentum deficit. It should also be noted here that, the appearance of newer zones of larger momentum deficit, close to the wall, is in agreement with the attached eddy model, which assumes that the origin of the vortical structures populating the boundary layer is at the wall (see Head & Bandyopadhyay Reference Head and Bandyopadhyay1981 and Perry & Chong Reference Perry and Chong1982).

4.2 Wall-normal location and thickness

Apart from the modal velocity, the UMZs of each group are characterised in physical space by their average wall-normal location, $\overline{Y}$ and thickness, $\overline{t}$ (see figure 6).

When the zones are classified based on their momentum deficit level (figures 8 a,b), the first group, with the lowest momentum deficit (◀ in figure 8 a), is at the largest distance from the wall and shows the largest variation in wall-normal location with $N_{UMZ}$ . More specifically, as $N_{UMZ}$ increases, the zones belonging to that group move further towards the free stream, to accommodate the new ones that appear near the wall. On the other hand, the group of zones with the largest momentum deficit ( ▶ in figure 8 a), is the one closest to the wall, and as $N_{UMZ}$ increases and the group becomes more populated, its average wall-normal location decreases further. This means that, although existing zones will tend to move outwards to accommodate the new ones appearing at the wall, the dominant factor in this group, as $N_{UMZ}$ increases, is the lower wall-normal location of the new zones that appear closer to the wall. It should also be noted here that, as was mentioned above, when $N_{UMZ}=1$ , most of the single UMZs belong to the first group, with the lowest momentum deficit. This however, leads to convergence issues for the statistics of the other two groups, reflected in the large deviations observed in $\overline{Y}|_{MD_{2}}$ and $\overline{Y}|_{MD_{3}}$ (figure 8 a), when $N_{UMZ}=1$ .

Figure 8.

Wall-normal locations for the two classification systems and their dependence on $N_{UMZ}$ . Colours and symbols as in figure 7. (a,b) Wall-normal locations, $Y|_{MD_{i}}$ , for a classification based on momentum deficit. For an increase in $N_{UMZ}$ , the zones with the lowest momentum deficit (◀ ) move to higher wall-normal locations, to account for the newest ones that appear near the wall. (c,d) Wall-normal locations, $Y|_{R_{j}}$ , for a classification based on zone rank. Only the first three ranks are shown for clarity, however the trend is similar for all following ranks. All ranks move upwards while new zones of higher rank appear below them.

When the zones are classified based on their rank (figure 8 c,d), there is a monotonic increase in wall-normal location for zones of all ranks. This supports the observation that as $N_{UMZ}$ increases and zones of higher rank appear closer to the wall, existing UMZs of lower rank with higher modal velocities will move further away from the wall to accommodate for the new ones.

The mean thickness of the different UMZs (see figure 6 b for a schematic definition) is also presented for the two different classification systems (figure 9). When the zones are classified according to their momentum deficit level and in line with the observations from de Silva et al. (Reference de Silva, Hutchins and Marusic2016), the average thickness is higher for groups of zones with low momentum deficit levels and the distribution around that average is wider (figure 9 a,b). The mean thickness of all the groups and the fluctuations around it show a decreasing trend with increasing number of zones. Here again, the large values and wide distributions for $\overline{t}|_{MD_{2}}$ and $\overline{t}|_{MD_{3}}$ when $N_{UMZ}=1$ (figure 9 a), are due to poor convergence for these two groups when only one UMZ is present in our FOV.

A very similar behaviour of decreasing zone thickness when $N_{UMZ}$ increases, can also be seen when the zones are classified according to their rank (figure 9 c,d). An increasing number of zones, will compress the thickness of all the zones, although their wall-normal position will increase (figure 8 c,d).

Figure 9.

Zone thickness for the two classification systems and its dependence on $N_{UMZ}$ . Colours and symbols as in figure 7. (a,b) Zone thickness, $t|_{MD}$ , for a classification based on momentum deficit. (c,d) Zone thickness, $t|_{R}$ , for a classification based on zone rank. Only the first three ranks are shown for clarity, however the trend is similar for all following ranks. Both systems show similar trends of decreasing thickness with increasing $N_{UMZ}$ . Zones closer to the wall have shorter wall-normal extent on average and a much narrower distribution around it.

4.3 Total number of zones

Based on the observations of the previous section, the total number of zones present in a given snapshot is a critical factor influencing UMZ characteristics. This influence could possibly indicate further structural differences of the flow, so here we present some basic statistical properties of the flow when conditioned on $N_{UMZ}$ . As mentioned above, when the total number of zones increases, the thickness of all zones decreases, while their average wall-normal position increases. The combination of these two trends results in an almost constant TNTI, with only a slight decrease for an increasing $N_{UMZ}$ (figure 10). This is also the most notable difference we found between instantaneously detected and time-coherent peaks. More specifically, when all modal peaks are considered, and no temporal threshold is applied, the thickness of all zones still follows a decreasing trend for an increasing $N_{UMZ}$ however, their wall-normal position decreases as well. That leads to significantly lower TNTI location for a high number of zones (see appendix B for a more detailed discussion).

Figure 10.

Variation of the mean location of the TNTI, $\overline{Y}_{i}$ , with the number of UMZs detected. Colour gradient from dark to light denotes increasing number of zones. Red dotted line denotes the global average location of the TNTI.

Figure 11.

Conditionally averaged statistics with varying $N_{UMZ}$ . Blue arrows and lighter greyscales indicate increasing number of zones (from 1 to 6). Dashed lines denote the limits of the logarithmic region based on the global time-averaged streamwise velocity (see figure 1 a). (a) Outer-normalised mean streamwise velocity profile. (b) Difference from the global streamwise average, $\overline{U}$ . (c) Outer-normalised mean wall-normal velocity profile. (d) Difference from the global wall-normal average, $\overline{V}$ . (e) Root-mean-squared profile of the Reynolds shear stress. (f) Difference from the global Reynolds shear stress profile, $\overline{u^{\prime }v^{\prime }}$ . Blue dash-dot lines indicate the cross-over point.

Apart from the TNTI location, flow statistics also show a strong dependence on $N_{UMZ}$ . A large number of zones leads to a strong momentum deficit in the region near the wall ( $y<0.25\unicode[STIX]{x1D6FF}$ ) (figure 11 a,b). This is in line with the conclusions from the previous section, where it was shown that an increasing number of zones leads to a decrease of the mean modal velocity of the lower UMZs (figure 7 b), because new zones tend to appear closer to the wall therefore adding to the momentum deficit in that region. At the same time, within that region, the vertical velocity component is shown to be significantly increased compared to the global average, $\overline{V}$ (figure 11 c,d). This combined behaviour of $U$ and $V$ statistics indicates a large-scale $Q2$ event in the log region for a high $N_{UMZ}$ . When $N_{UMZ}$ is low, the opposite trends are observed in both the streamwise and the wall-normal velocity profiles. Close to the wall there is a strong downward motion (figures 11 c,d), pushing high-speed fluid to regions of lower momentum, thus creating a slight increase of $U$ , reflected in a fuller streamwise velocity profile when compared to the global average, $\overline{U}$ (figures 11 a,b). These two trends suggest the presence of a large-scale $Q4$ event in the log region for a low $N_{UMZ}$ . The region closer to the free stream shows no significant variation, in line with the almost constant average TNTI location (figure 10). It should also be noted here that from all profiles presented, the profile representing $N_{UMZ}=3$ (third profile from the right in figure 11), is averaged over the largest fraction of velocity snapshots (see also PDF in figure 5) and as such is the one closest to the global average.

Finally, conditional averages of the Reynolds shear stress profiles show that closer to the wall, there is increased turbulent activity when compared to the global average for low $N_{UMZ}$ , which then rapidly decreases and only slightly re-surges close to the free stream. In contrast, a high $N_{UMZ}$ is coupled with low turbulent activity very close to the wall, which then significantly increases in the logarithmic region and gradually reaches the global average farther away from the wall. This behaviour can potentially be linked with previously documented small-scale activity when conditioned to a large-scale log-layer structure (amplitude modulation). More specifically, Hutchins & Marusic (Reference Hutchins and Marusic2007b ) showed that the presence of large-scale high-speed events (‘superstructures’) in the logarithmic layer (corresponding to an outer spectral peak ${\approx}6\unicode[STIX]{x1D6FF}$ ), creates an increase of small-scale activity very close to the wall, while low-speed log structures have the opposite effect. They further noticed that this behaviour is reversed after a certain distance from the wall (termed the ‘cross-over’ point) and that this point of reversal is Reynolds number dependent. Later, Mathis, Hutchins & Marusic (Reference Mathis, Hutchins and Marusic2009) supported this observation and proposed that the point where the correlation between the large-scale component with the envelope of the small-scale component crosses zero, happens within the logarithmic region, at an estimated $y_{c}^{+}\propto \sqrt{Re_{\unicode[STIX]{x1D70F}}}$ (blue dash-dot line in figure 11(f), located at $0.04\unicode[STIX]{x1D6FF}$ in the present data), an estimation also supported by Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012). Similar results were presented by Dennis & Nickels (Reference Dennis and Nickels2011), who showed that increased Reynolds shear stress regions were correlated with long low-speed structures while the high-speed ones did not exhibit a similar correlation.

In the present case, when the number of zones is low, there is a significant increase of streamwise momentum in the log region that could signify the passage of high-speed structures. Following the amplitude modulation reasoning and noting that the majority of the available vectors are above the cross-over point (blue dash-dot line in figure 11 f), this would lead to a decrease in the amplitude of the small scales, as is indeed observed in the Reynolds shear stress profile (the only notable exception being the case of $N_{UMZ}=1$ for which a decrease in amplitude is observed outside of the log region). Below the cross-over point the situation reverses and the passage of a high-speed structure in the log region is expected to amplify small-scale activity. The current resolution does not provide sufficient data of that region, however it is possible that new zones are created there that are not detected using the available vectors. Conversely, when the number of zones is high, the log region is characterised by a significant momentum deficit, which in turn can be linked to low-speed structures and increased small-scale activity would be expected. Results again support this argument, with the Reynolds shear stress values being significantly increased compared to global average, when a large number of zones is present in the flow (figure 11 f).

4.4 Large-scale $Q2$ and $Q4$ events in the log region

The above discussion on amplitude modulation is based on the assumption that the observed large-scale $Q2$ ( $Q4$ ) events in the log region indicate the passage of low- (high)-speed structures that have a substantial coherence in space. Since the available velocity snapshots are time resolved, the time for which such events reside within our FOV can be computed and their spatial extent can be subsequently estimated. In order to assess this concept we first classify all available snapshots based on whether they match the resulting profiles on figure 11(b,d). More specifically, for every velocity snapshot, the deviation, in the log layer, of the $\overline{U}|_{x}$ and $\overline{V}|_{x}$ velocity profiles (averaged across the streamwise extent of the FOV) from the global means, $\overline{U}$ and $\overline{V}$ , is estimated. The signed integral between the resulting profiles (see figure 12) and the $y$ -axis is the basis of our definition for the log-layer events. We define as a large-scale $Q2$ event a combination of a negative integral $\overline{U}|_{x}-\overline{U}$ value and a positive integral $\overline{V}|_{x}-\overline{V}$ value, in the log region (a representative example is depicted on figure 12 a,b). The opposite behaviour (figure 12 c,d) is classified as a $Q4$ event. Overall, such events are found to account for $74\,\%$ of all instantaneous fields. This is in line with the results on three-dimensional structures in a turbulent boundary layer by Dennis & Nickels (Reference Dennis and Nickels2011), who documented a high correlation of increased streamwise momentum with a downwards motion and a momentum deficit region with an upwards movement.

It should be noted here that this definition of $Q2$ and $Q4$ events differs from the standard quadrant analysis in that we do not classify the events based on pointwise velocity fluctuations, but rather on Reynolds shear stress values averaged over the size of our FOV. In that sense a large-scale $Q2$ ( $Q4$ ) event as defined here, would indicate an average ejecting (sweeping) motion for an area covering the whole streamwise extent and up to the end of the log region in the wall-normal direction. However, individual vectors within that area could still be classified in either of the four quadrants based on pointwise Reynolds shear stress.

Figure 12.

Streamwise-averaged velocity profiles for two different snapshots representing a large-scale $Q2$ (a,b) and $Q4$ (c,d) event at the log region. ▼ Denote average velocity values in the log region lower than the global average and ▲ average velocity values in the log region higher than the global average.

Figure 13.

The PDF of residence times within the FOV for large-scale $Q2$ and $Q4$ events. Dashed lines denote the average residence times for both event types.

The resulting residence times (figure 13), show that on average, these profiles persist for $0.5\unicode[STIX]{x1D6FF}/U_{\infty }$ within the measurement domain (for both $Q2$ and $Q4$ events, dashed lines in figure 13) and the longest of them reach times of $3\unicode[STIX]{x1D6FF}/U_{\infty }$ (maximum recorded event of $6\unicode[STIX]{x1D6FF}/U_{\infty }$ ). It should be stressed here that the profiles in figure 12 are a result of streamwise averaging and therefore indicate that the events considered are at least $0.5\unicode[STIX]{x1D6FF}$ long. Assuming moreover that the events move with approximately the local mean velocity (temporal mean at a wall-normal location of $y=0.1\unicode[STIX]{x1D6FF}$ for each event), the average time scales found indicate high-speed structures that are at least $0.9\unicode[STIX]{x1D6FF}$ long while low-speed ones are at least $0.8\unicode[STIX]{x1D6FF}$ long. While such an assumption on the convection velocities has long been disputed (see Fisher & Davies Reference Fisher and Davies1964; Zaman & Hussain Reference Zaman and Hussain1981; Kim & Hussain Reference Kim and Hussain1993; Davoust & Jacquin Reference Davoust and Jacquin2011, among others), recently, Geng et al. (Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015) showed that in the logarithmic region of turbulent boundary layers, which is of interest here, the use of the streamwise mean as the convection velocity is a very good approximation. The maximum recorded residence times suggest the passage of structures of up to $6\unicode[STIX]{x1D6FF}$ , in line with reported scales of large-scale motions and superstructures in the logarithmic layer (see Ganapathisubramani et al. Reference Ganapathisubramani, Longmire and Marusic2003; Tomkins & Adrian Reference Tomkins and Adrian2005; Hutchins & Marusic Reference Hutchins and Marusic2007b ; Dennis & Nickels Reference Dennis and Nickels2011, among others). On average however, both $Q2$ and $Q4$ events have similar but significantly shorter spatial extents. The main reason behind this is the much shorter temporal coherence of the vertical velocity $V$ when compared to the streamwise one. The reported long scales in the literature mainly refer to low or high streamwise velocity structures. In the present data, when only a high or low $U$ velocity event is tracked in time, the resulting residence times have a similar distribution to the $Q2$ and $Q4$ events (figure 13), but are on average twice as long ( $1.2\unicode[STIX]{x1D6FF}/U_{\infty }$ ), suggesting spatial extents closer to the documented values, albeit still shorter. Another reason for this is that our measurement domain is a streamwise wall-normal plane and that very long high and low momentum structures have been shown to alternate in the log region while also exhibiting a meandering motion in the spanwise direction (Hutchins & Marusic Reference Hutchins and Marusic2007a ,Reference Hutchins and Marusic b ; Zheng & Longmire Reference Zheng and Longmire2014). Therefore, it can be argued that this presence of $Q2$ ( $Q4$ ) events, observed in the velocity profiles, signifies the passage of low (high) momentum structures through our field of view and their recorded time scale depends on their streamwise alignment with the measurement plane. A similar organisation of side-by-side $Q2$ and $Q4$ events in the logarithmic region of channels was also presented by Lozano-Duran, Flores & Jimenez (Reference Lozano-Duran, Flores and Jimenez2012), who showed that these events form a predominant structure which is completed by a vortex cluster primarily located within the $Q2$ event but extending slightly underneath the $Q4$ as well.

The conditionally averaged statistics presented above (figure 11), have shown that velocity profiles exhibit a large, sweep-like behaviour in the log region for cases of a low number of zones present in the flow, while large-scale ejection-like events dominate for high $N_{UMZ}$ . The TNTI location is shown to be almost constant when conditioned on the number of zones (figure 10) with only a slight decrease with increasing $N_{UMZ}$ , although this trend was much stronger for an instantaneous peak detection (appendix B). In order to further investigate the correlation between $N_{UMZ}$ , $\overline{Y}_{i}$ , and the presence of these high- or low-speed log structures, conditional averaging is used again. This time however, the latter two parameters are used as conditions, while the resulting variations in $N_{UMZ}$ are recorded.

Figure 14.

The PDF of the average number of zones present in the flow, $\overline{N}_{UMZ}$ , for varying TNTI location, $\overline{Y}_{i}$ and magnitude of log region events. Average number of zones (a) and PDF of realisations for large-scale $Q2$ , $Q4$ events in the log region (b). In total, they occur in $74\,\%$ of all cases. Average number of zones (c) and PDF of realisations (d) for large-scale $Q1$ , $Q3$ events in the log region ( $26\,\%$ of all cases).

We follow the procedure discussed above to identity $Q2$ and $Q4$ events in the log region and we subsequently estimate their magnitude together with the instantaneous TNTI location in every snapshot. The joint PDF of these two parameters is computed, showing how many times a certain combination of events takes place (figure 14 b). The average number of zones over these realisations is subsequently estimated (figure 14 a) and the resulting map of $\overline{N}_{UMZ}$ shows that, indeed, a similar correlation as the one shown in figures 10 and 11 exists. More specifically, a $Q2$ event in the log layer results on average in a high number of zones ( $\overline{N}_{UMZ}\geqslant 5$ ), a number which tends to increase further with increasing ejection magnitude. The presence of a $Q4$ event on the other hand results in a low number of zones, in line with the conditional averages in figure 11. The TNTI location is again shown to have a limited effect on the resulting number of zones (see also figure 10). These results (figure 14 a,b), were based on the presence of either a $Q2$ - or $Q4$ -like large-scale event in the log region. Overall such events occur in $74\,\%$ of the total snapshots available and therefore the trends observed in figure 14(a,b) will be the ones dominating in a global sense. However, for completeness, a similar map of $\overline{N}_{UMZ}$ is presented for the other type of events, namely the $Q3$ (inward) and $Q1$ (outward) events (figure 14 c,d). The resulting PDFs show that these events are significantly weaker in magnitude and they mostly represent cases of average $N_{UMZ}$ . Even though there is no clear segregation of extremes like the one in figure 14(a) (instances with very low number of zones are almost non-existent), there is still a strong trend of increasing $\overline{N}_{UMZ}$ for low-speed structures present in the log region ( $Q3$ events). Considering all quadrant events, an increase of the streamwise velocity is shown to be correlated with a decrease in the number of zones ( $Q1$ and $Q4$ events), while a lower than average streamwise velocity has the opposite effect ( $Q2$ and $Q3$ events). When the large-scale high-speed events are combined with a downwards motion there is a further decrease in the number of zones. Conversely, when the low-speed events are combined with an upwards motion ( $Q2$ events), the number of zones increases further.

Finally, a comment should be made regarding the temporal coherence of the large-scale events in the log region and how it could affect the time scales of UMZs. As mentioned above, large-scale $Q2$ and $Q4$ events in the log layer that have average residence times of $0.5\unicode[STIX]{x1D6FF}/U_{\infty }$ , can potentially trigger the organisation of the flow into either low or high $N_{UMZ}$ , respectively. Individual UMZs however, represent flow regions that have an almost constant modal velocity, either higher or lower than the local mean, and as such are generally expected to have shorter time scales than the large-scale log events (which include regions of all flow velocities higher or lower than the local mean).

5.1 Number of zones

Since UMZs with short temporal coherence have been removed, we can estimate the time for which a certain zone-number structuring resides within our FOV, and how often this structuring reappears. This definition is shown schematically in figure 16(a) for an example of 300 consecutive images. The residence time of a certain number of zones or group of zones (the large number of zones group is highlighted here) inside the measurement plane is denoted with $\unicode[STIX]{x1D70F}_{w}^{+}$ , while $\unicode[STIX]{x1D70F}_{f}^{+}$ denotes the time after which the same $N_{UMZ}$ or group of zones reappears.

Figure 16.

(a) Time evolution of $N_{UMZ}$ (time-coherent UMZs) for the same 300 consecutive images as in figure 4(c). Background colours denote the classification of $N_{UMZ}$ into the three groups defined in figure 15. Examples of how long a high number of zones resides within the measurement plane, $\unicode[STIX]{x1D70F}_{w}^{+}$ , and how often it reappears, $\unicode[STIX]{x1D70F}_{f}^{+}$ , denoted with black. (b,c) For how long, on average, the boundary layer is structured into a certain number of zones (b) and the frequency of repetition of this structuring (c). Each number of zones is marked with ◼ and each characteristic group with ▼. Solid lines mark the range of each group. Colours as in figure 15. ♦ Symbols denote the time (b) and frequency (c) for which the boundary layer is structured in an average or high number zones (light grey) or in a low or average number of zones (black), while dashed lines mark the range of each of these cases respectively.

Figure 17.

The PDF of the number of zones at time $t+\text{d}t$ , based on the number of zones at time $t$ . Green dots denote data points. Linear interpolation was used on a finer mesh for visual clarity.

The average residence time of the different zone-number states is found to be of the order of 10 viscous time units and exhibits only a very slight decreasing trend as $N_{UMZ}$ increases (figure 16 c). The frequency with which each number of zones appears in time reflects the PDF distribution of $N_{UMZ}$ (figure 5); increase in frequency is observed for an average number of zones, while both extremes appear progressively less often (figure 16 d). A change in the relative time scales is expectedly observed when instead of individual zones, the three characteristic zone-number groups are considered. The average zone-number group exhibits residence times almost twice as long as either of the high or low zone-number groups and it is also the most frequent state (as was expected from the PDF distribution of $N_{UMZ}$ ). As can be seen in figure 16(a), the transition from the average state to either extreme is not monotonic. This implies that, starting at the most frequent state of an average $N_{UMZ}$ , there could be a shift towards an extreme (either high or low), a return to an average number of zones, and a shift again towards the same extreme. The time scales of all such transitions were reported above, however it was the presence of one extreme or the other that was shown in the previous section to be linked with large-scale structures in the log layer. Therefore, the residence time of a high or average zone-number state ( ${\sim}0.33\unicode[STIX]{x1D6FF}/U_{\infty }$ ) that does not get into a low zone behaviour (marked with

in figure 16 b), could be indicative of the presence, within our measurement plane, of a large-scale ejection that was shown to be linked with this behaviour. Conversely, the significantly higher residence time ( ${\sim}0.58\unicode[STIX]{x1D6FF}/U_{\infty }$ ) of a low or average number of zones, could indicate the presence of a high-speed structure that was previously shown to be linked with it.

Apart from the time scales, we can further investigate the evolution of this structuring process by finding the most probable number of UMZs at a time $t+\text{d}t$ , given that at time $t$ , the flow is organised into $N_{UMZ}$ . As was discussed, the time step $\text{d}t$ after which a change occurs, has a weak dependence on the number of zones at time $t$ and is of the order of 10 viscous time units. Results show that the division of an average number of zones ( $3\leqslant N_{UMZ}<5$ ) is the most preferable state (figure 17). Given a low $N_{UMZ}$ the most probable change will be towards a higher number, while given a high number of UMZs at time $t$ , the opposite trend is observed. For an average $N_{UMZ}$ , there is almost equal probability of moving towards either a lower or a higher number of zones. It is interesting to note that all zone-number changes are symmetric. For example a change from a structuring of $N_{UMZ}=5$ to one with $N_{UMZ}=3$ will happen approximately as many times as the reverse ( $N_{UMZ}=3$ followed by $N_{UMZ}=5$ ). Due to this symmetry, when estimating the most probable situation that preceded (at time $t-\text{d}t$ ) each realisation at time $t$ , the results are almost identical to the ones presented in figure 17.

5.2 Individual UMZs

The analysis of the previous section was relevant to the time scales and evolution of the structuring process of the boundary layer into a given number of zones. However, the results did not include information on the temporal coherence of individual UMZs. Each zone is detected based on modal velocities that are present in each snapshot. Several of the detected peaks also appear in following snapshots, while others die away. Based on the change of each peak’s modal velocity in consecutive snapshots, individual UMZs can be tracked. As mentioned above, we only consider peaks that persist inside the FOV for times longer than $\unicode[STIX]{x1D70F}^{+}\approx 20$ . For these peaks, the average residence time is ${\approx}0.35\unicode[STIX]{x1D6FF}/U_{\infty }$ and shows that these structures persist within our FOV for time scales shorter than $\unicode[STIX]{x1D6FF}/U_{\infty }$ (figure 18). As was mentioned in the previous section, UMZs are shown to be linked with large-scale structures in the log region, however their temporal coherence is based on the time that a single modal velocity persists within the measurement plane and as such are generally expected to have much shorter time scales than the corresponding log region events. The largest residence times found ( $6-7\unicode[STIX]{x1D6FF}/U_{\infty }$ ), show that there are cases where individual UMZs do indeed reach time scales that, assuming convection velocities that range from $0.5-0.9U_{\infty }$ , may indicate structures with lengths larger than $3\unicode[STIX]{x1D6FF}$ (LSMs) and reaching up to $6\unicode[STIX]{x1D6FF}$ (superstructures).

Figure 18.

Frequency of appearance of time-coherent UMZs (in $\%$ of the total), depending on their residence time, $\unicode[STIX]{x1D70F}$ . Dashed line denotes average residence time.

Large-scale structures and superstructures are also known to meander in the spanwise direction and therefore it is likely for them to appear and disappear from our streamwise–wall-normal measurement plane, effectively breaking their observed time coherence and as such the residence times of UMZs as well, depending on their meandering amplitude and alignment with the FOV. Another explanation for the lack of longer residence times, comes from recent studies suggesting that the observed spectral peaks in the outer region that have been linked to the presence of very large structures, might be due to a sequence of structures that are repeated in time for very long distances, without necessarily an elongated spatial coherence: Kim & Adrian (Reference Kim and Adrian1999) and later Baltzer, Adrian & Wu (Reference Baltzer, Adrian and Wu2013) suggested that the large-scale motions (LSMs) observed in high $Re$ pipe flows align to form VLSMs, an observation later supported by Lee & Sung (Reference Lee and Sung2011) for superstructures in turbulent boundary layers. Following this reasoning, Hellström, Ganapathisubramani & Smits (Reference Hellström, Ganapathisubramani and Smits2015) and Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) using experimental and numerical data of a pipe flow respectively, showed that the observed coherence of the very large structures in the frequency domain does not infer spatial coherence of the same size and that the emerging outer spectral peak reflects an alignment of smaller structures.

Figure 19.

UMZs are classified according to whether they reach a modal velocity, $U_{m}$ belonging to each of the three shown velocity bands, at least once during their residence time $\unicode[STIX]{x1D70F}$ . (a) Frequency of appearance of UMZs within each velocity band (in $\%$ of the total UMZs within that band), depending on their residence time. Each UMZ might contribute to multiple bands, only once to each, according to the different modal velocities it corresponds to during its residence time. Dashed lines denote the average residence time of UMZs within each band. (b) Duration of observation of UMZs within each velocity band (in $\%$ of the total observation time of all UMZs within that band) according to their residence time. Each UMZ might contribute to multiple bands, every time it reaches a modal velocity belonging to any of them. Dashed lines denote the average observation time of UMZs in each group.

Throughout each zone’s residence time within the FOV, both their respective rank as well as the number of zones present at every snapshot vary. The naturally arising question is whether the modal velocity of each zone and the total number of zones present in the flow during each zone’s residence time have any influence on that characteristic time. The time sequence presented in figure 4(c) shows that zones with large modal velocities have larger time scales than the ones that appear closer to the wall. This is further supported from the residence time of structures that, during this time, have a modal velocity belonging to one of the four modal velocity groups shown in figure 19(a). Each zone might contribute to all groups (only once to each though), according to the different modal velocities it corresponds to during its residence time. Although each modal velocity band contains zones that persist within the measurement plane for various times, there is a clear increase of the average residence time (denoted with dashed lines), for zones closer to the free stream. Also, even though low modal velocity bands include some UMZs with long residence times, the majority of their observation time includes UMZs with shorter time scales (see figure 19 b). For example, 50 % of the total observation time of the lowest modal velocity group (denoted with light grey in figure 19) is spent on structures with residence times shorter than $\unicode[STIX]{x1D70F}^{+}=40$ . For the high modal velocity bands, the observation time is almost equally spent between structures of all time scales. This essentially means that for any individual snapshot, it is much more probable for a high modal velocity peak to have a long residence time than it is for a low modal velocity peak one. Furthermore, the mean observation times of each velocity group show that in any individual snapshot there is also a much higher probability of finding a peak that belongs in a high velocity group. That trend supports a hierarchical organisation of structures in the wall-normal direction (new slower zones appearing close to the wall, while older faster ones are located closer to the free stream) and is also in line with the schematic depiction of a hairpin packet organisation by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000).

Figure 20.

UMZs are classified according to whether they encounter any of the three shown zone-number groups, at least once during their residence time, $\unicode[STIX]{x1D70F}$ . (a) Frequency of appearance of UMZs, within each zone-number group (in $\%$ of the total within that group), depending on their residence time. Each UMZ might contribute to multiple groups, only once to each, according to the number of other zones appearing and dying away during its residence time. Dashed lines denote the average residence time of UMZs within each group. (b) Duration of observation of UMZs within each zone-number group (in $\%$ of the total observation time of all UMZs within that group) according to their residence time. Each UMZ might contribute to multiple groups, every time it encounters them. Dashed lines denote the average observation time of UMZs in each group.

When the residence times are grouped according to $N_{UMZ}$ the results show that zones appearing in snapshots where the boundary layer is organised into many layers are less coherent in time (mean residence times of $0.14\unicode[STIX]{x1D6FF}/U_{\infty }$ ) and even the longer ones do not typically exceed residence times of $0.25\unicode[STIX]{x1D6FF}/U_{\infty }$ (figure 20 a). Also, in the high number of zones group, shorter-lived structures are shown to dominate the total time of peak observation (figure 20 b). This shows that for any snapshot belonging to a high zone-number group, there is a much higher possibility that any of the peaks present is a short-lived one. On the other hand, UMZs that exist in snapshots characterised by a low $N_{UMZ}$ , tend to persist much longer (on average almost 4 times longer than the high zone-number group), with maximum residence times exceeding $\unicode[STIX]{x1D6FF}/U_{\infty }$ and they are also shown to spend the majority of these residence times in the low zone-number group (figure 20 b). For example, zones with residence times of $\unicode[STIX]{x1D70F}^{+}=250$ account for 4.5 % of the total observation time of the low zone-number group, while only for 1 % of the total observation time of the high zone-number group. This trend can be mapped into a Reynolds number dependency following the scaling of the number of zones with $Re_{\unicode[STIX]{x1D70F}}$ (de Silva et al. Reference de Silva, Hutchins and Marusic2016). More specifically, a low number of zones ( $N_{UMZ}<3$ ) would correspond to a $Re_{\unicode[STIX]{x1D70F}}\sim 10^{3}$ while a high number of zones to a $Re_{\unicode[STIX]{x1D70F}}\sim 10^{4}$ suggesting that the UMZs present in the flow tend to decrease in size and extent as the Reynolds number increases. A similar behaviour was reported by Vallikivi, Ganapathisubramani & Smits (Reference Vallikivi, Ganapathisubramani and Smits2015), who found that the wavelength of the outer spectral peak increases when normalised with inner variables and decreases in the case of an outer-scale normalisation, suggesting that the scales associated with it decrease in size as the Reynolds number increases. Another explanation for this time scale variation could come from the organisation of the flow in the log region. More specifically, it was shown above that a low number of zones is on average correlated with a large-scale $Q4$ event in the log layer, while a $Q2$ most often leads to a high number of zones. Lozano-Duran et al. (Reference Lozano-Duran, Flores and Jimenez2012) showed that, large-scale $Q4$ events exist in pairs with $Q2$ structures and that low-speed events are often engulfed by high-speed ones and are significantly smaller in length and width. This documented difference in size, could be the reason for the much larger time scales observed when the number of zones is low.