3.1.1 Detecting instantaneous UMZs

The detection of instantaneous UMZs follows the methodology used by Adrian et al. (Reference Adrian, Meinhart and Tompkins2000), de Silva et al. (Reference de Silva, Hutchins and Marusic2016, Reference de Silva, Philip, Hutchins and Marusic2017) and Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018). The instantaneous UMZs are detected from the peaks on the PDFs of the instantaneous streamwise velocity $U$. These peaks correspond to large regions travelling at relatively uniform velocity, namely the zone modal velocity $u_{m}$. Each instantaneous PDF is computed using all the data points in a 3-D snapshot with a streamwise window size of $0.2R$ throughout the $30R$-long pipe and for all the available snapshots over time. The PDFs have a uniform bin size of $1\,\%U/U_{CL}$ with 110 bins covering $U/U_{CL}\in [0,1.1]$. Figure 2(a) shows an example of the instantaneous PDF of $U$ from a 3-D snapshot. Three distinct peaks at three zone modal velocities $u_{m}$ are detected from this snapshot and are marked by ‘▿’. The peak detection scheme used here is similar to de Silva et al. (Reference de Silva, Hutchins and Marusic2016) and Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018). For each instantaneous PDF of $U$, the local maxima are filtered by three constraints, $F_{d}$, $F_{h}$ and $F_{s}$. Here, $F_{d}$ is the minimum distance between each recognised peak, $F_{h}$ is the minimum height of the peak on the PDF and $F_{p}$ is the prominence of the peak compared to the five neighbouring bins on each side of the peak. Peaks are only recognised as $u_{m}$ for $F_{d}>3\,\%U_{CL}$ (3 bins), $F_{h}>0.5$ and $F_{p}>25\,\%$ which requires the recognised peaks to be 25 % higher than the average bin height of its neighbouring $10\,\%U/U_{CL}$. The threshold velocity of the UMZ interface $u_{\unicode[STIX]{x1D705}}$ between each zone is defined at the minimum bin between each two neighbouring peaks of $u_{m}$. In figure 2(a), two $u_{\unicode[STIX]{x1D705}}$ are marked by the dashed lines between each of the two peaks. Figure 2(b) shows the contour of $U$ on a cross-stream plane within the snapshot. The iso-contours of three threshold velocities $u_{\unicode[STIX]{x1D705}}$ separate the flow into three regions, each travelling with small velocity dispersion. The UMZ interface closest to the wall in figure 2(b) corresponds to the lowest $u_{\unicode[STIX]{x1D705}}/U_{CL}\approx 0.8$ in figure 2(a).

Figure 2. An example of the detection of instantaneous UMZs. (a) The PDF of the instantaneous streamwise velocity $U$ computed from the 3-D domain of a snapshot with streamwise window size of $0.2R$. (b) Corresponding contour of $U$ on a cross-stream plane from the same snapshot as in (a). (c) The definition of zone rank $R_{i}$, zone thickness $t|_{R_{i}}$ and zone interface radius $r_{\unicode[STIX]{x1D705}}|_{R_{i}}$ on the same contour as in (b), unwrapped for a quarter of the azimuthal span.

The number of UMZ detected on each PDF, $N_{UMZ}$ depends greatly on the constraints of the peak detection scheme. More strict detection with higher constraints can reduce $N_{UMZ}$ significantly. Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) adjusted their peak detection constraints for a targeted average $N_{UMZ}\approx 4.5$, suggested by the Reynolds number dependency of $N_{UMZ}$ in de Silva et al. (Reference de Silva, Hutchins and Marusic2016) for TBL. In the present study of pipe flow at $Re_{\unicode[STIX]{x1D70F}}=500$, a target $N_{UMZ}\approx 2.5$ estimated from the TBL results is found to be too low. Many non-trivial peaks are seen to be neglected. This could be because, for a pipe and TBL at a similar Reynolds number, there are potentially more UMZs in the pipe, being thinner on average and more refined due to the full circumferential wall confinement. Therefore, in the absence of knowledge on the Reynolds number dependency of $N_{UMZ}$ in a pipe, no targeted $N_{UMZ}$ is pre-defined. The constraints are chosen only for a robust combination that can fairly preserve all distinctive peaks on the PDFs. Figure 3 shows the PDF of $N_{UMZ}$ over all available snapshots. The PDF peaks at $N_{UMZ}=5$ and agrees with the normal distribution found by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) and Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018), indicating that a sufficient number of instantaneous UMZs have been obtained for statistical analysis and UMZ grouping.

Instead of peak detection on the histogram of $U$, Fan et al. (Reference Fan, Xu, Yao and Hickey2019) used the kernel density estimation (KDE) without the requirement of multiple user-defined constraints but a single KDE bandwidth. In the Appendix, part of the statistical characteristics of the UMZs shown in the later sections are reproduced from using the KDE approach for UMZ detection.

Figure 3. PDF of $N_{UMZ}$, the number of UMZs from detected peaks on each instantaneous PDF of $U$ using all vectors in the 3-D snapshot with a streamwise window size of $0.2R$.

3.1.2 Grouping of UMZs

Each instantaneous UMZ has an individual zone modal velocity $u_{m}$ and threshold velocity $u_{\unicode[STIX]{x1D705}}$ at the interface separating it from its adjacent zones. These UMZs detected from all snapshots are classified into a few groups based on their zone modal velocities $u_{m}$. Two grouping methods are used in this study for different purposes. The first grouping method is by the magnitude of $u_{m}$. This is similar to the grouping used in de Silva et al. (Reference de Silva, Hutchins and Marusic2016, Reference de Silva, Philip, Hutchins and Marusic2017) in which UMZs were grouped based on the zone momentum deficit $u_{m}-U_{\infty }$ ($U_{\infty }$ is the TBL free-stream velocity). In the present study, six $u_{m}$ groups denoted as $M_{i}$ with uniform increments of $u_{m}$ are defined by their $u_{m}$ range, as listed in table 1 (left). UMZs in group $M_{6}$ are the closest to the wall with the lowest $u_{m}$ within $M_{1:6}$ whereas group $M_{1}$ with $u_{m}/U_{CL}\in [1.0,1.1)$ is the innermost UMZ of the pipe, travelling above the centreline velocity. Figure 4(a) shows the PDFs of the $N_{UMZ}$ for each $u_{m}$ group of $M_{1:6}$. The distribution of $N_{UMZ}$ in each $u_{m}$ group (bars) follows the time-averaged PDF of $U$ (dashed line). Group $M_{2}$ includes the highest number of UMZ among $M_{1:6}$ and $M_{6}$ has the lowest UMZ count. The count in group $M_{1}$ of UMZs travelling beyond the centreline velocity is relatively low, as expected. The overlaid PDF of UMZ count in the TBL by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) are shown by ‘○’. The PDF distribution is very similar to the time-average PDF of $U$ in a pipe whereas the peak of the PDF of TBL is at a lower $u_{m}/U_{\infty }=0.9$ than a pipe at $U/U_{CL}=0.98$.

The second grouping method follows Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) in which the UMZs are grouped based on the ranking of zone modal velocity $u_{m}$ in each snapshot. For example, in figure 2(a), the zone with the highest $u_{m}$, which is the inner most UMZ in figure 2(b), is ranked as $R_{1}$ and the zone with the lowest $u_{m}$ is ranked the last ($R_{3}$). Figure 2(c) unwraps the first quadrant of the contour in figure 2(b). As illustrated, the three UMZs detected in this snapshot ($N_{UMZ}=3$) are ranked from the centre to the wall as $R_{1}$ to $R_{3}$ with descending $u_{m}$. If a snapshot has $N_{UMZ}=6$, then the six UMZs will be ranked from $R_{1}$ to $R_{6}$ based on their ranking of $u_{m}$. Because of the nature of the wall-bounded flow, the $R_{1}$ zone will always be the innermost zone with the highest $u_{m}$ in the region (see also Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018)). Figure 4(b) shows the PDFs of $N_{UMZ}$ in each ranking group of $R_{1:9}$. The ranks are plotted in reverse order since the lowest rank ($R_{9}$) includes the UMZs travelling at the lowest $u_{m}$ in each snapshot. The count of UMZ continuously decrease from $R_{1}$ to $R_{9}$.

In the following part of this study, UMZ groups based on the range of $u_{m}$ are called $u_{m}$ groups, denoted as $M_{i}$; and UMZ groups based on the ranking of zone $u_{m}$ are called the ranking groups, denoted as $R_{i}$. The symbols and colours for each group in the two grouping methods are consistent throughout this study and are listed in table 1.

Table 1. (Left) The range of modal velocity $u_{m}$, symbols and colours for UMZ groups $M_{i}$ grouped by $u_{m}$. (Right) The symbols and colours used for UMZ groups $R_{i}$ grouped by zone rank on $u_{m}$.

Figure 4. (a) PDF of $N_{UMZ}$, the number of UMZs in each $u_{m}$ group $M_{i}$ based on zone modal velocity $u_{m}$ (bar); the time-average PDF of the streamwise velocity $U$: - - - - -; and the PDF of $u_{m}$ in turbulent boundary layer by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) at $Re_{\unicode[STIX]{x1D70F}}=8000$: ○. (b) PDF of $N_{UMZ}$ in each ranking group $R_{i}$ based on zone rank of $u_{m}$ for $R_{1:9}$.

3.3.1 Wall-normal location of UMZ

The wall-normal location of the UMZs $y_{\unicode[STIX]{x1D705}}$ is averaged for UMZs in each group of the two grouping methods. Figure 6(a) shows $\langle y_{\unicode[STIX]{x1D705}}\rangle$ for $u_{m}$ groups $M_{1:6}$. The angle brackets $\langle \rangle$ indicate conditional average for UMZs in each group. The UMZs in group $M_{6}$ with the lowest $u_{m}/U_{CL}\in [0.5,0.6)$ are very close to the wall with $\langle y_{\unicode[STIX]{x1D705}}\rangle$ almost equal to zero; $y_{\unicode[STIX]{x1D705}}$ gradually increases from $M_{6}$ to $M_{1}$ which has $\langle y_{\unicode[STIX]{x1D705}}\rangle /R\approx 0.75$. The trend follows the mean profile of $U$ plotted as $y(\overline{U})$ (dashed line). The change in $y_{\unicode[STIX]{x1D705}}$ is rather small near the wall for groups $M_{6:4}$ and becomes large for $M_{3:1}$ towards the pipe centre. In figure 6(b) for $\langle y_{\unicode[STIX]{x1D705}}\rangle$ of UMZs in each ranking group $R_{1:7}$, a similar trend is shown. The nonlinear increase of $y_{\unicode[STIX]{x1D705}}$ from $R_{7}$ to $R_{1}$ indicates that the UMZs at lower ranks are more closely distributed nearer the wall while the higher ranked UMZs ($R_{1:3}$) are thicker so that they are further apart from each other towards the pipe centre. The results of $y_{\unicode[STIX]{x1D705}}$ using both grouping methods indicate that the UMZs are not uniformly distributed from the centreline to the wall but in a hierarchical distribution where zones near the wall are more densely populated, similar to TBL (de Silva et al. Reference de Silva, Hutchins and Marusic2016, Reference de Silva, Philip, Hutchins and Marusic2017).

The average thickness of the UMZs in each group can be calculated from the radial location $r_{\unicode[STIX]{x1D705}}$ of the zones. The calculation is illustrated in figure 7(a) for the grouping method based on the ranking of $u_{m}$. The illustration is idealised to have three UMZs ranked from $R_{1}$ to $R_{3}$. Note that the thickness of the innermost zone equals its radial location $r_{\unicode[STIX]{x1D705}}$. Figure 7(b) shows the group average thickness $\langle t\rangle$ of the UMZs calculated from $1-\langle y_{\unicode[STIX]{x1D705}}\rangle$. The value of $\langle t\rangle$ from group $R_{7}$ to $R_{1}$ increases monotonically: UMZs are the thinnest nearer the wall and the zone thickness increases when departing from the wall. The slope of $\langle t\rangle$ decreases group by group from the wall ($R_{7}$) toward the pipe centre until the second innermost group ($R_{2}$), then there is a sudden increase in the slope of $\langle t\rangle$ for the innermost zones ($R_{1}$ and $M_{1}$). This supports the previous finding in § 3.2 where the innermost UMZs in the pipe flow were found to be much thicker than the other UMZs in a pipe and the UMZ below the TNTI in a TBL. This is understood as in a pipe and channel, the opposing walls each have a flow similar to a TBL so that the innermost zone in a pipe contains the zones furthest away from the opposing walls interacting at the centre.

Figure 6. Conditionally averaged zone interface wall-normal location $y_{\unicode[STIX]{x1D705}}=1-r_{\unicode[STIX]{x1D705}}$ ($r_{\unicode[STIX]{x1D705}}$ is the radial location) for UMZs grouped by (a) zone modal velocity $u_{m}$, $u_{m}$ groups $M_{1:6}$ and (b) by zone rank of $u_{m}$, ranking groups $R_{1:7}$. The dashed line in (a) is the mean velocity profile of $\overline{U}$ plotted as $y(\overline{U})$.

Figure 7. (a) An illustration of computing the average UMZ thickness of each $u_{m}$ and ranking group, $\langle t\rangle$ from the average UMZ radial location $r_{\unicode[STIX]{x1D705}}$ of each group. (b) The average UMZ thickness $\langle t\rangle$ of each ranking group $R_{i}$.

3.3.2 Zonal means inside and outside the UMZs

In order to investigate how UMZs at different wall-normal locations differ from each other, the zonal mean statistics of the UMZs in different $u_{m}$ groups $M_{1:5}$ are computed. The zonal mean inside a UMZ is averaged in the region between the UMZ lower-bounding interface closer to the wall up to the centreline, in other words, over its radial extent from the pipe centre (see the definition of $r_{\unicode[STIX]{x1D705}}$ in figure 2c); the zonal mean outside a UMZ is averaged over the region outside its lower bounding interface at $r_{\unicode[STIX]{x1D705}}$ towards the wall. Means inside the UMZ are denoted with a hat as $\widehat{U}$ and $\widehat{u^{2}}$; means outside the UMZ are denoted with a tilde as $\widetilde{U}$ and $\widetilde{u^{2}}$, similar to Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014). Figure 8 shows the zonal mean inside ($\widehat{U}$, $\widehat{u^{2}}$) with dot-dashed lines and zonal means outside ($\widetilde{U}$, $\widetilde{u^{2}}$) with dashed lines. The global average profiles of $\overline{U}$ and $\overline{u^{2}}$ with solid lines are shown as references.

In figure 8(a), the zonal mean profiles $U$ are plotted for $u_{m}$ groups $M_{1:5}$. The mean streamwise velocity inside the UMZ ($\widehat{U}$) is always larger than the time mean ($\overline{U}$) whereas the mean outside the UMZ ($\widetilde{U}$) is always smaller than $\overline{U}$. The velocity difference between $\widehat{U}$ and $\widetilde{U}$ for each $u_{m}$ group $M_{i}$ indicates the velocity jump across the UMZ interface (see figure 9). The value of $\widetilde{U}$ for each $u_{m}$ group follows the time mean $\overline{U}$ profile very near the wall and starts to deviate from $\overline{U}$ away from the wall. The $\widehat{U}$ data are available in a large range of wall-normal locations, and can extend very close to the wall at $y/R\approx 0.01$ for group $M_{5}$. This shows the large meandering behaviour of the UMZ. To measure the level of meandering of the UMZs in terms of how far they can extend towards the wall in extreme cases, figure 8(b) shows the minimum wall-normal location $\widehat{y}$ that the UMZ can extend to (where $\widehat{U}$ is last available near the wall). The value of $\widehat{y}$ for group $M_{5:2}$ increases steadily from the wall to the centre from $\widehat{y}/R\approx 0.01$ to $0.05$ whereas the innermost UMZs with $u_{m}>U_{CL}$ in group $M_{1}$ can extend significantly less close to the wall with $\widehat{y}/R\approx 0.22$.

Figure 8(c,d) shows the zonal mean profiles of $U$ and $u^{2}$ for UMZs in group $M_{2}$ only. This provides a relatively direct comparison between a pipe and channel: UMZs in group $M_{2}$ have $u_{m}/Ucl\in [0.9,1)$ and the channel results are from a single UMZ defined at a fixed $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.95$. In figure 8(c), the pipe profiles of group $M_{2}$ can extend further towards the wall due to the grouping: some of the UMZs in group $M_{2}$ are defined at a lower threshold velocity $u_{\unicode[STIX]{x1D705}}$ than the channel core so that they are naturally thicker and closer to the wall. The differences between the pipe profiles (in red) and the channel profiles (in black) are mainly caused by the difference in $\overline{U}$ due to the Reynolds number difference. Higher Reynolds number of the channel ($Re_{\unicode[STIX]{x1D70F}}=1000$) has $\overline{U}$ developing faster nearer the wall. Otherwise, the trend between $\widehat{U}$, $\widetilde{U}$ and $\overline{U}$ is very similar in a pipe and channel. In figure 8(d), the zonal mean streamwise fluctuation $u^{2}$ is computed relative to zonal means of $U$ following Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014)

(3.1a,b)$$\begin{eqnarray}\widehat{u^{2}}=(U-\widehat{U})^{2},\quad \widetilde{u^{2}}=(U-\widetilde{U})^{2}.\end{eqnarray}$$

The turbulent intensity inside the UMZ, $\widehat{u^{2}}$, is much lower than the turbulent intensity outside the zone, $\widetilde{u^{2}}$. The difference between $\widehat{u^{2}}$ and $\widetilde{u^{2}}$ becomes smaller towards the centreline; $\widehat{u^{2}}$ and $\widetilde{u^{2}}$ cross over at $y/R\approx 0.74$ where $\widehat{u^{2}}$ starts to be larger than $\widetilde{u^{2}}$ until the centreline (see zoomed view inside figure 8d). A similar behaviour of $\widehat{u^{2}}$ and $\widetilde{u^{2}}$ was observed in the channel (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014) and the cross-over point of $\widetilde{u^{2}}>\widehat{u^{2}}$ was also at $y/h\approx 0.74$ ($h$ is the channel half-height). Therefore, despite the Reynolds number effect, the quiescent core of channel and the UMZs ($M_{2}$) of a pipe are very similar.

Figure 8. Zonal mean profiles of the streamwise velocity $U$ for UMZs in $u_{m}$ groups (a) $M_{1:5}$ with colours as in table 1 and (c) $M_{2}$ (in red) overlaid with channel flow results by Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014) (in black). (b) The wall-normal location of the most near-wall data of the zonal mean inside the UMZs of each group in (a), $\widehat{y}$ (symbols as in table 1). (d) The streamwise velocity fluctuation $u^{2}$ for group $M_{2}$ only. The solid lines (——) are the time averages of $U$ and $u^{2}$ in (a,c,d). The dot-dashed lines (— ⋅ —) are the zonal means inside the UMZ, $\widehat{U}$ and $\widehat{u^{2}}$; the dashed lines (- - - -) are the zonal means outside the UMZ, $\widetilde{U}$ and $\widetilde{u^{2}}$.

Figure 9. Conditional averages as functions of the distance from the lower-bounding UMZ interface ($\unicode[STIX]{x1D709}$) for (a) the streamwise velocity $U$ for $u_{m}$ group $M_{2}$, (b) $U$ for groups $M_{1:4}$, (c) velocity gradient $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ for group $M_{2}$, (d) $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ for groups $M_{1:4}$, (e) streamwise velocity fluctuation $u^{\ast 2}$ for group $M_{2}$ and (f) $u^{\ast 2}$ for groups $M_{1:4}$. The legends in (e) and (f) apply to (a,c) and (b,d) respectively. (g) The magnitude of velocity jump in $U$ across the UMZ interface, $\unicode[STIX]{x0394}U$ for groups $M_{1:4}$. (h) The UMZ interface thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ calculated from (3.1) for groups $M_{1:4}$. (g,h) Are overlaid with TBL results by de Silva et al. (Reference de Silva, Philip, Hutchins and Marusic2017) at $Re_{\unicode[STIX]{x1D70F}}=14\,500$ ($\times$), $Re_{\unicode[STIX]{x1D70F}}=8000$ (○), $Re_{\unicode[STIX]{x1D70F}}=2800$ ($+$) and $Re_{\unicode[STIX]{x1D70F}}=1200$ (▵).

3.3.3 Conditional average across the UMZ interface

Conditional averages are computed for flow properties across the UMZ interface as functions of the distance from the interface, $\unicode[STIX]{x1D709}$. Figure 9(a) shows the group average streamwise velocity $\langle U\rangle$ against $\unicode[STIX]{x1D709}$ for a single $u_{m}$ group $M_{2}$, overlaid with results from the channel (Yang et al. Reference Yang, Hwang and Sung2016). The angle brackets $\langle \rangle$ in this section indicate conditional averaging across the interface for UMZs in each $u_{m}$ group; $\unicode[STIX]{x1D709}=0$ indicates the UMZ interface and $\unicode[STIX]{x1D709}>0$ represents the inside of the UMZ. In figure 9(a), the abrupt jumps in $U$ across the interface are similar between UMZs in $M_{2}$ and the quiescent core of the channel: before entering the zone ($\unicode[STIX]{x1D709}<0$), $U$ develops fairly slowly until the near vicinity of the interface where it experiences a sharp change of the rate. The velocity gradient $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ of group $M_{2}$ is shown in figure 9(c), overlaid with the results of the quiescent cores in the channel (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014; Yang et al. Reference Yang, Hwang and Sung2016). A local maximum of velocity gradient corresponding to the abrupt jump in $U$ at the interface is similar here in a pipe. The streamwise fluctuation $u^{2}$ across the UMZ interface is shown in figure 9(e,f). In figure 9(e), for UMZs in group $M_{2}$ of pipe and the quiescent core of channel, $u^{2}$ behaves very similarly in a pipe and channel in which the streamwise fluctuation decreases dramatically across the interface with a local minimum at $\unicode[STIX]{x1D709}\approx 0$, and then remains very low inside the UMZ. In figure 9(f), the $u^{2}$ profiles for groups $M_{3,4}$ are different from groups $M_{1,2}$ because the UMZs in groups $M_{3,4}$ travelling at lower $u_{m}$ are very close to the wall, close enough to capture the rapid increase in $\overline{u^{2}}$ in the near-wall region (see figure 8d for $y/R<0.05$).

Figure 9(b) shows $\langle U\rangle$ against $\unicode[STIX]{x1D709}$ for $u_{m}$ groups $M_{1:4}$. The increase in $U$ across the UMZ interface is larger (higher magnitude) and more abrupt (higher velocity gradient) for interfaces closer to the wall, as found in TBL (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). Figure 9(d) corresponds to figure 9(b), showing the increase in maximum velocity gradient at $\unicode[STIX]{x1D709}=0$ for UMZs nearer the wall travelling at lower $u_{m}$. The magnitude of the velocity jump, $\unicode[STIX]{x0394}U$ is defined similar to Yang et al. (Reference Yang, Hwang and Sung2016), as illustrated in figure 9(a). The value of $\unicode[STIX]{x0394}U$ for UMZs in group $M_{1:4}$ is shown in figure 9(g) with the TBL results (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017) at $Re_{\unicode[STIX]{x1D70F}}=8000$ and $14\,500$. The values of $\unicode[STIX]{x0394}U$ in both pipe and TBL are found to be larger for UMZs travelling at lower $u_{m}$ nearer the wall. The overall magnitude of $\unicode[STIX]{x0394}U$ in the pipe is significantly larger than the TBL. This is partly caused by the lower Reynolds number in the pipe: the number of UMZs is generally lower at lower Reynolds number so that $U$ jumps across fewer interfaces with larger $\unicode[STIX]{x0394}U$ to achieve the centreline velocity. This increase in $\unicode[STIX]{x0394}U$ with decreasing Reynolds number is shown by the TBL results where the $Re_{\unicode[STIX]{x1D70F}}=8000$ case (‘○’) has higher $\unicode[STIX]{x0394}U$ than $Re_{\unicode[STIX]{x1D70F}}=14\,500$ (‘$\times$’). However, the increase in $\unicode[STIX]{x0394}U$ with lowering Reynolds number is not as significant as the difference in $\unicode[STIX]{x0394}U$ between the pipe and TBL. This suggest that the velocity jump of $U$ across the UMZ interface is, on average, stronger in a pipe than TBL.

In figure 9(a–d), the velocity jumps and the peaked velocity gradients at the UMZ interface take place in a narrow distance over $\unicode[STIX]{x1D709}$ as the UMZ interfaces themselves have a thickness (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). The thickness of the interfaces is estimated by

(3.2)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}=\frac{\unicode[STIX]{x0394}\langle U\rangle }{\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}y|_{max}},\end{eqnarray}$$

following Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014) and de Silva et al. (Reference de Silva, Philip, Hutchins and Marusic2017). Figure 9(h) shows the UMZ interface thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ for groups $M_{1:4}$. The value of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ is lower for UMZs closer to the wall as found in the TBL, so that the thinner UMZs nearer the wall also have thinner interfaces accompanied by a sharper and larger jump in $U$. The hierarchical distribution of the UMZ and UMZ interface and the magnitude of $\unicode[STIX]{x0394}U$ together suggest a discrete step-like model of the velocity profile in wall turbulence which has been used as the initial velocity profile in the turbulence modelling by Bautista et al. (Reference Bautista, Ebadi, White, Chini and Klewicki2019). The overlaid TBL results of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ at $Re_{\unicode[STIX]{x1D70F}}=1200{-}14\,500$ are roughly in the range of $u_{m}$ in group $M_{3}$ of the pipe. The interface thickness in the TBL increases when $Re_{\unicode[STIX]{x1D70F}}$ decreases, as expected, since there are less UMZs at larger scales at lower Reynolds number. The UMZ interface in a pipe at a much lower $Re_{\unicode[STIX]{x1D70F}}=500$ has $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ in between the $Re_{\unicode[STIX]{x1D70F}}=2800$ and $1200$ cases of the TBL, indicating that the UMZ interfaces are, on average, thinner in a pipe than TBL. This supports the previous finding where the velocity jump in $U$ was found to be stronger in a pipe so that the velocity jumps in a pipe are more abrupt over thinner interfaces in a pipe than TBL. That the UMZ interfaces are found to be thinner in the DNS data of a pipe than the experimental data of a TBL may also be due to the limited spatial resolution in the experiments.

Figure 10(a) shows the conditional average of azimuthal vorticity $\langle \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}\rangle$ against $\unicode[STIX]{x1D709}$ where $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x2202}V_{r}/\unicode[STIX]{x2202}x-\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}r$; $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ peaks in the proximity of the interface similar to $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ in figure 9(d). The initial $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ away from the interface at $\unicode[STIX]{x1D709}/R=-0.2$ is significantly higher for UMZ groups closer to the wall since vorticities are stronger nearer the wall. The local maximum of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ at $\unicode[STIX]{x1D709}=0$ in figure 10(a) and $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ in figure 9(d) are plotted in figure 10(b). The peak azimuthal vorticity and streamwise velocity gradient have very similar trends: $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ both decrease almost log-linearly from the wall ($M_{4}$) towards the centre ($M_{2}$), apart from the innermost UMZs in group $M_{1}$, which shows a lowered rate of decrease. The differences in conditional averages of the innermost UMZs from the other zones are consistent with the findings in the previous §§ 3.3.1 and 3.2.

Figure 10. (a) Conditional-average azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ for $u_{m}$ groups $M_{1:4}$ as a function of the distance from the lower-bounding UMZ interface. (b) The local maximum $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ (in a) and $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ (in figure 8d) at the UMZ interface ($\unicode[STIX]{x1D709}\approx 0$).

3.4.1 Dynamics of the attachment

The local maximum azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ at the UMZ interface in figure 10(a) suggests a strong correlation between the UMZ interfaces and the vortical structures. Figure 11 shows an instantaneous UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$ in (a) and $0.9$ in (b) for the same snapshot in the streamwise plane, superpositioned on the contour of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ (with the prograde vortices in red and the retrograde vortices in blue). Different colour scales are used in figure 11(a,b) to improve the clarity of the contours of the surrounding vortical structures at different strengths. On these streamwise planes, the interface continuously follows the azimuthal vortices with similar strength, so that the $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$ interface in figure 11(a) is generally much closer to the wall, being attached to stronger vortices nearer the wall. This is consistent with the observations by Adrian et al. (Reference Adrian, Meinhart and Tompkins2000) where patches of spanwise vortices were found aligned along the strong shear layers inside a TBL. In figure 11, the UMZ interface threads through the azimuthal vortices and folds into bulges and valleys to form a persistent vortex–interface attachment.

The dynamics of this vortex–interface attachment is further investigated. Supplementary movies of the attachment between vortices and UMZ interfaces over time are available in both fixed and moving frames at https://doi.org/10.1017/jfm.2019.947 (movies 1–4). Figure 12 shows three snapshots from the moving-frame video with a constant time gap. In these snapshots, the interface contorts to follow the inclined trains of eddies (marked with dashed lines) from the wall towards the pipe centre. The interface initially attached to vortex $A$ in figure 12(a) meanders as vortex $A$ deforms and moves with the flow in figure 12(b). In figure 12(c), when vortex A is weakened and merged with weaker eddies, the local interface changes significantly to reattach to a nearby vortex at $A^{\prime }$ to maintain the desirable vortical strength. The weakened vortex $A$ then attaches to an iso-surface defined at a lower velocity threshold.

Figure 11. Contours of azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ in the streamwise plane, super-positioned with instantaneous UMZ interfaces defined at (a) $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$ and (b) $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.9$ for the same snapshot. The background contours of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ in (a,b) only differ in the colour axis.

Figure 12. Contours of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ and the UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.9$ plotted on a streamwise moving frame approximately at the flow speed downstream with time from (a) to (c) with a uniform time gap between each two of the three snapshots. Label $A$ in (a,b) marks a prograde vortex weakened and joined the nearby vortices at $A^{\prime }$ in (c). Label $B$ marks a strong retrograde vortex the interface is attached to. (The arrows show the vector of streamwise and wall-normal velocity fluctuations, $u$ and $v$.)

Figure 13. (a) Contour of azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ in a cross-stream plane, superpositioned with an instantaneous UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.9$. (b) A 3-D instantaneous UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$ with colour based on the azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ on the interface surface. The UMZ interface is shown for $1/6$ of the streamwise extent, unwrapped in the azimuthal direction.

3.4.2 The three-dimensional attachment

Figure 13(a) shows the vortex–interface attachment in the azimuthal direction of the pipe on a cross-stream plane for a UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$ corresponding to figure 11(b). The azimuthal vortices have elongated tube shapes in the azimuthal direction and vortices at similar strengths vary in their wall distance. Similar to the attachment shown in the streamwise direction in figure 11 and also Adrian et al. (Reference Adrian, Meinhart and Tompkins2000), the UMZ interface contorts to attach to individual vortices and switches at the gaps between the azimuthal vortices to attach to the next vortex at desired strength. The 3-D attachment for the UMZ interface is shown in figure 13(b) for a UMZ interface defined at $u_{\unicode[STIX]{x1D705}}/U_{CL}=0.8$. The attachment of the UMZ interface to vortical structures is significantly biased to the prograde vortices (in red) whereas the retrograde vortices (blue) are rarely found on the interface surface. To quantify the attachment, figure 14(a) shows the time-average PDFs of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ on the 3-D interface surface of the UMZs in $u_{m}$ groups $M_{1:4}$. Through group $M_{1}$ to $M_{5}$ from the centre to the wall, there is a continuous increase in the skewness of prograde vortices (defined as positive $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$) denoted as $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$, which is the strongest closest to the wall. The negative ends of the PDFs representing attachment on retrograde vortices ($\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$) show small changes for different UMZ groups. The magnitude of prograde vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ at a fixed PDF of $10^{-4}$ for each $u_{m}$ group (marked by ‘$\times$’ along the dashed line at PDF$=10^{-4}$) is shown in figure 14(b). The increase in $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ against UMZ modal velocity $u_{m}$ shows a marked linearity, even for the fastest travelling UMZs in group $M_{1}$ which have been found to be different from the other groups in the previous results. The linear increase of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ towards the wall against $u_{m}$ has a slope of $\text{d}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}/\text{d}u_{m}\approx 66.4$ with negligible variation (maximum 4.8 %) between different $u_{m}$ groups.

Figure 14. (a) PDF of the azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ on the 3-D UMZ interfaces for $u_{m}$ groups $M_{i=1:5}$. Here, $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ is defined as positive for prograde vortices and negative for retrograde vortices. (b) The magnitude of positive prograde azimuthal vorticity, $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ at a probability density of $10^{-4}$ for groups $M_{1:5}$ (marked by ‘$\times$’ in (a)).

Figure 15. Conditional-average profiles of total azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$, prograde vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ and retrograde vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$ as functions of distance from the UMZ interface ($\unicode[STIX]{x1D709}$) for $u_{m}$ group $M_{2}$. The total azimuthal vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}-\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$. The inset shows the ratio between the prograde and retrograde vorticity $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}/\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$ against $\unicode[STIX]{x1D709}$.

Figure 15 shows the conditional-average vorticity of the prograde ($\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$) and retrograde ($\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$) vortices for UMZs in group $M_{2}$ against $\unicode[STIX]{x1D709}$. At the interface ($\unicode[STIX]{x1D709}=0$) where the total $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ peaks in figure 10, here, it shows that nearly all $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}$ is contributed from $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}$ while $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}$ decreases to a local minimum at the UMZ interface. The distribution of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p,r}$ for group $M_{2}$ is, again, very similar to the quiescent core in the channel (Yang et al. Reference Yang, Hwang and Sung2016). Inside figure 15, the ratio of prograde and retrograde vorticity peaks at the interface, achieving $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{p}/\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D703}}^{r}\approx 25$ for UMZs in group $M_{2}$. Results in this section suggest that the contorted shape of the UMZ interface is a consequence of its attachment to the azimuthal/spanwise vortices which move three-dimensionally with varying wall-normal location. Additionally, the attachment is predominantly to prograde vortices.

3.5.1 The 2-D contortion

The contortion of the UMZ interface surface in the streamwise and azimuthal directions are computed from the 3-D instantaneous velocity field. The 2-D contortion is measured by the UMZ interface length in the streamwise ($r-x$ plane) and azimuthal ($r-\unicode[STIX]{x1D703}$ plane) directions, denoted as $l_{x}$ and $l_{\unicode[STIX]{x1D703}}$ respectively. Here, $l_{x}$ and $l_{\unicode[STIX]{x1D703}}$ are computed by an algorithm that calculates the exact length of the interface using the high resolution 3-D data. The lengths of the islands outside the enclosed region of the main part of the UMZ are preserved because it is found that, in three dimensions, the frequently observed islands on the 2-D planes are relatively rare. A majority of the islands are connected in three dimensions as part of the three-dimensional interface contortion. In figure 16, the projections on the streamwise and cross-stream planes show examples for $l_{x}$ and $l_{\unicode[STIX]{x1D703}}$, respectively.

The lengths of the interface per unit length in both directions are shown in figure 17(a). Here, $l_{x}$ is normalised by the streamwise extent of the pipe $L_{x}=30R$ while $l_{\unicode[STIX]{x1D703}}$ is normalised by the perimeter of a circle with a radius equal to the UMZ interface average radius $r_{\unicode[STIX]{x1D705}}$: $L_{\unicode[STIX]{x1D703}}=2\unicode[STIX]{x03C0}\langle r_{\unicode[STIX]{x1D705}}\rangle$. Because $\langle r_{\unicode[STIX]{x1D705}}\rangle$ varies for UMZs in different $u_{m}$ groups, the unit length in the azimuthal direction $L_{\unicode[STIX]{x1D703}}$ is not constant but is larger nearer the wall. In figure 17, both $l_{x}/L_{x}$ and $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ increase as UMZs move away from the wall towards the centre of the pipe through $M_{1:5}$, so that the 2-D contortion intensifies for interfaces away from the wall in both directions. The UMZ interface in the azimuthal direction is always more contorted than the streamwise direction as $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ is always higher than $l_{x}/L_{x}$ in all $u_{m}$ groups. The same trend was found for the quiescent core of a channel (Yang et al. Reference Yang, Hwang and Sung2016) which has $l_{x}/L_{x}$ overlaid in figure 17(a) as ‘▫’ and $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ by ‘

$M_{1,2}$

$l_{x}/L_{x}$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$

$l_{x}/L_{x}$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$

$l_{x}/L_{x}$

$L_{\unicode[STIX]{x1D703}}$

$r_{\unicode[STIX]{x1D705}}$

$L_{\unicode[STIX]{x1D703}}^{\prime }=2\unicode[STIX]{x03C0}R$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}^{\prime }$

$l_{x}/L_{x}$

$l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}^{\prime }$

$l_{x}/L_{x}$

Figure 17. (a) UMZ folding intensity in two dimensions: the average length of UMZ interface per unit length in the streamwise direction, $l_{x}/L_{x}$ and the azimuthal direction, $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ for $u_{m}$ groups $M_{1:5}$. The streamwise extent $L_{x}=30R$ is constant and the azimuthal extent $L_{\unicode[STIX]{x1D703}}=2\unicode[STIX]{x03C0}\langle r_{\unicode[STIX]{x1D705}}\rangle$ varies for UMZs in different groups with $\langle r_{\unicode[STIX]{x1D705}}\rangle$ as the average radial location of the UMZ interface of a group. The markers are $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}^{\prime }$ with constant azimuthal extent $L_{\unicode[STIX]{x1D703}}^{\prime }=\unicode[STIX]{x03C0}R$ for all groups. The overlaid results from TBL by de Silva et al. (Reference de Silva, Philip, Hutchins and Marusic2017) at $Re_{\unicode[STIX]{x1D70F}}=14\,500$ ($\times$), $Re_{\unicode[STIX]{x1D70F}}=8000$ (○), $Re_{\unicode[STIX]{x1D70F}}=2800$ ($+$) and $Re_{\unicode[STIX]{x1D70F}}=1200$ (▵) are for $l_{x}/L_{x}$ only. The overlaid channel flow results by Yang et al. (Reference Yang, Hwang and Sung2016) at $Re_{\unicode[STIX]{x1D70F}}=1000$ have $l_{x}/L_{x}$ (▫) and $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ (

). (b) UMZ interface folding intensity in three dimensions: the average surface area of the 3-D UMZ interface $S_{\unicode[STIX]{x1D705}}$ per unit area $L_{x}L_{z}$ for groups $M_{1:6}$.)

The 2-D contortion in the streamwise direction $l_{x}/L_{x}$ from TBL by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) of $Re_{\unicode[STIX]{x1D70F}}=1200{-}14\,500$ is also overlaid on the pipe results in figure 17(a). There is a clear increase in $l_{x}/L_{x}$ when the Reynolds number increases in the TBL. For $u_{m}$ groups $M_{3,4}$, the streamwise contortion of UMZs in the pipe is very close to the TBL at a roughly doubled Reynolds number $Re_{\unicode[STIX]{x1D70F}}=1200$. This supports the previous findings in conditional averages where UMZs in a pipe at a lower Reynolds number are quantitatively more similar to a TBL at higher Reynolds numbers.

3.5.2 The 3-D contortion

The 3-D contortion of the UMZ interface is measured by the surface area of the interface $S_{\unicode[STIX]{x1D705}}$ per unit area $L_{x}L_{\unicode[STIX]{x1D703}}$ where $L_{x}L_{\unicode[STIX]{x1D703}}$ is the area of a smooth surface at the interface average radial location $\langle r_{\unicode[STIX]{x1D705}}\rangle$ with zero contortion. Figure 17(b) shows $S_{\unicode[STIX]{x1D705}}/L_{x}L_{\unicode[STIX]{x1D703}}$ for $u_{m}$ groups $M_{1:6}$; $S_{\unicode[STIX]{x1D705}}/L_{x}L_{\unicode[STIX]{x1D703}}$ shows a very similar trend to $l_{\unicode[STIX]{x1D703}}/L_{\unicode[STIX]{x1D703}}$ in which the increase is rather slow initially in groups $M_{6:4}$ with a UMZ interface area roughly twice the unit area. The 3-D contortion is the lowest for UMZs closest to the wall ($M_{6}$) and achieves the highest $S_{\unicode[STIX]{x1D705}}/L_{x}L_{\unicode[STIX]{x1D703}}\approx 11$ in group $M_{1}$. It is interesting to note that $S_{\unicode[STIX]{x1D705}}$ of the inner most UMZ interfaces ($M_{1}$) is over 10 times the non-contorted reference area. The contortion of the UMZ interfaces behaves similarly as a function of the wall distance, both the 2-D and 3-D contortions of UMZ interfaces intensify when departing from the wall. This may be partly due to the suppression from the wall where interfaces are less free to meander and also because that the streamwise fluctuation is on smaller scales nearer the wall. The increase in UMZ interface contortion with distance from the wall shows the increase in meandering of the LSMs with wall distance (Kevin & Hutchins Reference Kevin and Hutchins2019). The rate of increase of the 3-D contortion is shown in figure 18 as the gradient of $\unicode[STIX]{x2202}S_{\unicode[STIX]{x1D705}}/\unicode[STIX]{x2202}u_{m}$. The slope of the semi-log straight line $\unicode[STIX]{x2202}^{2}S_{\unicode[STIX]{x1D705}}/\unicode[STIX]{x2202}u_{m}^{2}=10^{c}$ has the exponent $c=9.7$ with a maximum 15 % variation between different $u_{m}$ groups.

Figure 18. The gradient of the 2-D and 3-D interface folding intensities $l_{x}$, $l_{\unicode[STIX]{x1D703}}$ and $S_{\unicode[STIX]{x1D705}}$ in figure 17 against UMZ modal velocity $u_{m}$.