4.1 Distribution of UMZ edges in turbulent boundary layers

To examine the geometric properties of the detected UMZ edges as a function of wall-normal height, we first require a quantitative measure of its wall-normal location. To this end, since each UMZ edge has a spatially varying wall-normal location, we employ the streamwise momentum deficit defined as

as a surrogate for wall-normal position. Here, $\widetilde{U}_{i}$ corresponds to the streamwise velocity and is constant along each interface, therefore, $\widetilde{U}_{mom}^{+}$ is also constant along each detected interface. To illustrate that $\widetilde{U}_{mom}^{+}$ is a suitable candidate as a surrogate for wall-normal position, figure 5(a) shows a joint p.d.f. between $(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}$ and $z$ , computed at each grid point along all detected UMZ edges. The results exhibit a clear reduction in wall-normal location with increasing momentum deficit, providing quantitative evidence that $\widetilde{U}_{mom}^{+}$ can be directly related to the wall-normal location. Therefore, for the subsequent analysis, we separate all detected edges of UMZs based on their associated $\widetilde{U}_{mom}^{+}$ into momentum deficit ranges. Here, for simplicity, we have chosen equispaced momentum deficit ranges (indicated by the dashed lines in figure 5 a), which will remain constant for all experimental databases used.

Figure 5(b) shows p.d.f.s of the wall-normal locations computed at the momentum deficit ranges chosen previously in figure 5(a). The ▪, ●, ▴ and ◂ symbols correspond to UMZ edges within the turbulent region at different momentum deficit ranges, while the ▾ symbols correspond to statistics computed exclusively at the TNTI. The results show that statistics computed from UMZ edges that are scarcely present near the wall (▾ and ▪ symbols) exhibit a symmetric distribution. Conversely, interfaces residing closer to the wall (●, ▴ and ◂ symbols) exhibit a positive skewness due to presence of the wall, which acts as a lower bound ( $z/\unicode[STIX]{x1D6FF}>0$ ). We also note that at the TNTI, the p.d.f. exhibits a near Gaussian distribution with a mean wall-normal location of approximately $2\unicode[STIX]{x1D6FF}/3$ , which is in good agreement with prior findings (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ).

Figure 5.

(a) Joint p.d.f. of the streamwise momentum deficit, $(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}$ , and the wall-normal location of UMZ edges within the turbulent region (internal interfaces). Results are presented at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . (b) The p.d.f. of wall-normal locations of all UMZ edges. Results at internal interfaces are binned into momentum deficit ranges, which are indicated by the vertical dashed lines in (a) and also represented by different black symbols. The (green) ▾ symbols correspond to results computed exclusively at the TNTI.

4.2 Geometric properties of UMZ edges

Visual inspection of the UMZ edges shown in figure 3 suggests that they are highly contorted by small-scale wrinkles which ride along large-scale indentations (bulges and valleys). Previously, Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ) showed that the unit length per streamwise distance $\langle l_{s}\rangle /L_{x}$ is greater than one at the TNTI, where $l_{s}$ denotes the interface length and $L_{x}$ represents the corresponding streamwise extent. Here we present an equivalent analysis on the internal interfaces that separate UMZs within the turbulent region. We note for the present study, the notation $L_{x}$ (the streamwise field of view) is limited to the constant streamwise extent used to detect the presence of modal velocities ${\mathcal{L}}_{x}$ (see § 3). Therefore, we are bounded by the upper limit $L_{x}={\mathcal{L}}_{x}$ , which is fixed at approximately $2000$ viscous units, even though our streamwise PIV field of view is larger than this upper bound for certain datasets.

Figure 6.

(a) The average length per unit streamwise length, $\langle l_{s}\rangle /L_{x}$ , of all UMZ edges at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . Symbols are defined on the lower right. (b) Number of boxes $N_{box}$ needed to cover UMZ edges as a function of box size. The solid lines correspond to a power-law least-squares fit in range indicated by the vertical dashed lines. (c) Slope computed from the power-law fits in (b). The grey symbols in (a,c) correspond to results computed over a smaller momentum deficit range in steps of $(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}=0.5$ .

Figure 6(a) shows $\langle l_{s}\rangle /L_{x}$ of all UMZ edges at varying momentum deficit ranges ( ${\sim}$ wall-normal position). The results are presented for $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ and $l_{s}$ is computed along a coordinate system attached to the interface. The black symbols correspond to results computed at internal interfaces in the momentum deficit ranges indicated by the vertical dashed lines, and the green ▾ symbol again represent results computed exclusively at the TNTI. The lowest momentum deficit range of $1.5<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<4.5$ shows that on average, the length of interfaces within this range is approximately 2.5 times the streamwise domain length that it lies on. Thereafter, a reduction of this ratio is observed with increasing momentum deficit (closer proximity to the wall). The higher length of interfaces away from the wall may be caused by the increased presence of large-scale motions residing in this region, which in turn is likely to add large-scale bulges/valleys, increasing the ratio $\langle l_{s}\rangle /L_{x}$ . Such a scenario is consistent with the notion of larger length scales residing further away from the wall (e.g. Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982; Adrian et al. Reference Adrian, Meinhart and Tomkins2000).

Our examination thus far exhibits a sharp rise in length at the TNTI (▾ symbols) relative to the edges of UMZs within the turbulent region. This trend can also be observed by visually inspecting instantaneous velocity fields (see figure 3 a), where a distinct increase in ‘wrinkles’ is present at the TNTI. We postulate that these differences between internal and external UMZ edges in boundary layers may be attributed to the different fluid ‘states’ present on either side of the edge. Across the TNTI, the fluid is turbulent on one side and non-turbulent on the other side, whereas turbulent fluid can be found either side of the internal UMZ boundaries/edges, albeit with differing intensities. Further, we speculate that entrainment of non-turbulent fluid via small-scale ‘nibbling’, may explain a sharp increase in $l_{s}$ due to an increase in surface area. To investigate this behaviour, the grey symbols show $l_{s}$ computed over smaller ranges of momentum deficit in steps of $(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}=0.5$ . Again the results exhibit a distinct increase at the TNTI, albeit less converged, reaffirming our previous observations.

The presence of both large-scale bulges and finer wrinkles along the detected edges of UMZs motivates us to examine the multiscale characteristics of the interfaces. This would allow us to discern any degree of self-similarity across the range of scales present. Previously, studies have highlighted the geometric fractal behaviour of the TNTI (e.g. Sreenivasan & Meneveau Reference Sreenivasan and Meneveau1986; Siebesma & Jonker Reference Siebesma and Jonker2000), meanwhile, other studies have disputed these observations (e.g. Miller & Dimotakis Reference Miller and Dimotakis1991). However, recent works at high Reynolds numbers in turbulent boundary layers by de Silva et al. (Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013) and also in jets by Mistry et al. (Reference Mistry, Philip, Dawson and Marusic2016) have confirmed a fractal dimension of ${\approx}2.3{-}2.4$ at the TNTI. Due to the comparably high $Re$ available in the present study, we examine whether similar fractal characteristics are present at the edges of UMZs within the turbulent region. We begin with a brief discussion on how the fractal dimension is computed using a box counting analysis (Prasad & Sreenivasan Reference Prasad and Sreenivasan1989, and others). First, each velocity field is divided into square boxes of a certain size, $b$ . Thereafter, we determine the number of boxes, $N_{box}$ , an interface lies on, which is then repeated for a range of box sizes across all available velocity fields. Finally, averaged statistics are computed over all velocity fields to determine if they exhibit a fractal behaviour following $N_{box}\sim b^{-D}$ , where $D$ represents the fractal dimension.

Figure 6(b) shows results for $N_{box}$ as a function of box size $b$ at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . For clarity, $N_{box}$ is pre-multiplied by the box size $b$ to enable us to better discern any deviation from a log–log trend ( $N_{box}\times b\sim b^{-D+1}$ ). Further, we note that the largest box size is restricted by the fixed streamwise extent used to detect the presence of UMZs ( ${\mathcal{L}}_{x}$ ), however minimal influence on the computed fractal dimension was observed at different streamwise extents. The ▾ symbols correspond to the TNTI where a clear linear region is present with a slope of approximately $-0.3$ (solid line), or $D\approx 1.3$ . Consequently, one would obtain $D_{f}=D+1\approx 2.3$ for the surface embedded in three dimensions following the additive rule of co-dimensions (Mandelbrot Reference Mandelbrot1982). These results are in good quantitative agreement with previous findings at the TNTI (Sreenivasan & Meneveau Reference Sreenivasan and Meneveau1986; Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1990; de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013), across a wide range of detection schemes for the TNTI. Further, the magnitude of $D$ is also in agreement with recent findings of Borrell & Jiménez (Reference Borrell and Jiménez2016) from numerical simulations in turbulent boundary layers, who also examined the magnitude of $D$ across a wide range of vorticity thresholds used to detect the TNTI. We note the grey vertical dashed lines in figure 6(b) indicate conservative upper and lower bounds for the self-similar regions observed, so their precise magnitudes should not be taken as definitive estimates. Nevertheless, we observe a lower bound in figure 6(b) that is approximately below the Taylor microscale, $b<\unicode[STIX]{x1D706}_{T}$ . On the other hand at larger scales we again see a deviation, however, it is important to note that this upper limit is likely to be restricted by the streamwise extent of the databases and therefore the self-similar region may extend to even larger scales.

We can reproduce the same statistics at the edges of UMZs within the turbulent region, which are shown by the ▪, ●, ▴ and ◂ symbols in figure 6(b). Each symbol corresponds to statistics computed at different momentum deficit ranges ( ${\sim}$ wall-normal locations) within the turbulent region, and they all appear to exhibit a linear behaviour of $\ln (N_{box}\times b^{+})$ as a function of $\ln (b^{+})$ . Therefore, similar to the TNTI we hypothesise that a fractal or self-similarity across a range of scales is prevalent at the internal interfaces that demarcate UMZs within the turbulent region. Upon closer inspection, it is evident that the fractal dimension, computed at UMZ edges within the turbulent region and the TNTI differs in magnitude. To better understand this behaviour, figure 6(c) presents the local slopes, computed from the profiles shown in figure 6(b). Here we observe a peculiar trend, where the slope (and subsequently fractal dimension $D$ ) exhibits a nominally constant magnitude at UMZ edges within the turbulent region of $D\approx 1.2$ . However, at the TNTI we obtain a sharp change to $D\approx 1.3$ . We can, therefore, infer that both the TNTI and internal interfaces exhibit self-similarity across a range of scales, however, a lower fractal dimension for the internal interfaces is indicative of a smaller degree of contortions (both large-scale bulges/valleys and small-scale wrinkles). This is also evident from the sharp increase in interface length $l_{s}$ at the TNTI (see figure 6 a).

Figure 7.

(a) Schematic on computing conditional statistics at UMZ edges overlaid on colour contours of streamwise velocity $\widetilde{U}$ . The solid black contours represent the original un-enveloped boundary, and the solid blue line corresponds to the ‘outer’ envelope of the boundary. The ● (red) symbols represent the coordinates $z_{i}$ along the interface and the ● symbols illustrate grid points collected in the near vicinity of each interface. (b) Averaged conditional statistics for the streamwise velocity, $\widetilde{U}$ , at the TNTI. $\unicode[STIX]{x0394}\widetilde{U}$ , $\unicode[STIX]{x1D6FF}_{w}$ and $\unicode[STIX]{x2202}\langle \widetilde{U}\rangle /\unicode[STIX]{x2202}z|_{max}$ correspond to the average jump in streamwise velocity, the width of the jump and the maximum local gradient of the profile, respectively.

4.3 Conditionally averaged statistics

So far we have examined geometric properties at the detected UMZ edges. In this section, we examine conditionally averaged statistical properties in the near vicinity of these UMZ edges using the same techniques commonly used to analyse interfacial layers in boundary layers (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). To perform this, a frame of reference attached to each interface at $z=z_{i}$ is used, as illustrated in figure 7(a). The subscript $i$ denotes a quantity computed at or along the interfaces, and the ● (red) symbols on figure 7 correspond to a few representative locations of the interface, $z_{i}$ . Thereafter, profiles of $\widetilde{U}$ , $\widetilde{W}$ and $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}z$ are collected in the near vicinity of $z_{i}$ (● symbols on figure 7 a), which are averaged along the interface to obtain average conditional statistics (denoted by $\langle \rangle$ brackets). We note that most interfaces are heavily contorted and tend to fall back on themselves, therefore the ‘outer’ envelope of each interface is computed first, providing only a single possible wall-normal location at each streamwise grid point (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2005; Kwon et al. Reference Kwon, Philip, de Silva, Monty and Hutchins2014; Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ). This is illustrated in figure 7(a), where the black solid line (—) represents the original un-enveloped boundary, and the solid blue line (—) corresponds to the ‘outer’ envelope of the boundary. A careful examination of enveloping the interface to compute conditional statistics is described in Kwon et al. (Reference Kwon, Philip, de Silva, Monty and Hutchins2014), where negligible differences were found between the ‘inner’ or ‘outer’ envelope.

Figure 7(b) shows a conditionally averaged profile for $\widetilde{U}$ . The results are shown at the topmost UMZ edge (or TNTI), which reveals a sharp increase in the streamwise velocity of approximately 4 % of $U_{\infty }$ on average (denoted as $\unicode[STIX]{x0394}U$ ), in agreement with previous findings at the TNTI (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). We note that $\unicode[STIX]{x0394}U$ is computed by projecting linear fits to the regions above and below the sharp velocity change on the conditional wall-normal profile of $\widetilde{U}$ (see figure 7 b), following prior work on interfacial layers in boundary layers (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). The shaded region corresponds to the spatial extent across which this sharp change in $\widetilde{U}$ occurs, and is denoted by $\unicode[STIX]{x1D6FF}_{w}$ (to be detailed further). Next, we examine these parameters at the edges of UMZs within the turbulent region.

Conditional wall-normal profiles of $\widetilde{U}$ , $\widetilde{W}$ and $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}z$ of all UMZ edges are presented in figure 8(a–c), respectively. Statistics are collected and averaged at different momentum deficit ranges, and the subscript $i$ denotes statistics computed at $z_{i}$ , for example $\widetilde{U_{i}}$ and $\widetilde{W_{i}}$ corresponds to the streamwise and wall-normal velocity, respectively at the interface. All conditional statistics are computed with respect to the interface location $z_{i}$ as the wall-normal reference on the ordinate, and the mean interface velocities $\widetilde{U_{i}}$ and $\widetilde{W_{i}}$ are used as the abscissa reference in (a) and (b), respectively.

Figure 8.

Conditionally averaged statistics of UMZ edges at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ with respect to the interface location $z_{i}$ . (a–c) correspond to $\langle \widetilde{U}\rangle$ , $\langle \widetilde{W}\rangle$ and the wall-normal shear $\unicode[STIX]{x2202}\langle \widetilde{U}\rangle /\unicode[STIX]{x2202}z$ , respectively. Results at UMZ edges within the turbulent region (internal interfaces) are binned by momentum deficit, $(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}$ , and represented by different black symbols which are defined in figure 6. The green ▾ symbols correspond to results computed exclusively at the TNTI. The inset in (a) shows one standard deviation of the instantaneous $\widetilde{U}$ profiles for the ▴ symbols.

The results for the conditionally averaged streamwise velocity, $\langle \widetilde{U}\rangle$ , reveals a sharp increase in $\widetilde{U}$ when crossing the edge of UMZs (at $z-z_{i}=0$ ). This observation is in agreement with the notion that a large proportion of the shear present in boundary layers is concentrated over a small spatial extent, surrounded by relatively large regions of uniform streamwise momentum (see figure 4). Upon closer inspection, the profiles also reveal a higher $\unicode[STIX]{x0394}\widetilde{U}$ closer to the wall (higher momentum deficit), which is quantified in figure 9(a). These results are in agreement to those reported by Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015) where an increase in velocity jump magnitude at interfacial layers of high shear were observed closer to the wall, with a comparable jump magnitude as a function of $z$ . Collectively, these findings emphasise the significance of the sharp changes in $\widetilde{U}$ (or $\unicode[STIX]{x0394}\widetilde{U}$ ) computed at UMZ edges within the turbulent region, which has not received as much attention as the TNTI to date. The increase in $\unicode[STIX]{x0394}\widetilde{U}$ at larger momentum deficit (closer to the wall) is consistent with the mean flow characteristics of a turbulent boundary layer due to the higher mean velocity gradient in the near-wall region. Consequently, we postulate that the instantaneous $\widetilde{U}$ profile as a function of $z$ in turbulent boundary layers is predominantly composed of a step-like profile (zonal-like arrangement) with larger steps (or jumps in $\widetilde{U}$ ) present on average in the near-wall region. Qualitative evidence of these postulations can be observed in arbitrarily chosen instantaneous velocity profiles (see figure 1), or by visually inspecting $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}z$ contours (see figure 3 b), where the magnitude of $\unicode[STIX]{x2202}\widetilde{U}/\unicode[STIX]{x2202}z$ is higher closer to the wall.

So far the conditional profiles of $\widetilde{U}$ presented in figure 8(a) quantifies the average behaviour, however, instantaneously $\widetilde{U}$ fluctuates significantly near UMZ edges (see figure 1). In an attempt to quantify this, the dashed lines in the inset of figure 8(a) shows one standard deviation of the $\widetilde{U}$ profiles for the ▴ symbols. Although not reproduced here, the other momentum deficit ranges ( ${\sim}$ wall-normal locations) exhibit similar behaviour. Figure 8(b) reproduces the same conditional statistics for the wall-normal velocity component near UMZ edges. The results reveal that the sharp increase in the streamwise velocity at the edge of UMZs for increasing $(z-z_{i})$ is flanked by a decrease in the wall-normal velocity, albeit more gradual. Such a scenario is consistent with the signature of $Q2$ ( $\widetilde{U}<0$ , $\widetilde{W}>0$ ) events below UMZ edges and $Q4$ ( $\widetilde{U}>0$ , $\widetilde{W}<0$ ) events above UMZ edges, relative to the streamwise ( $\widetilde{U_{i}}$ ) and wall-normal ( $\widetilde{W_{i}}$ ) velocity at the interface i.e. consistent with the VITA event, of a stagnation point residing upstream of the head of a hairpin vortex (Adrian et al. Reference Adrian, Meinhart and Tomkins2000).

Figure 9.

(a) The average jump in streamwise velocity, $\unicode[STIX]{x0394}\widetilde{U}^{+}$ , (b) width across which it occurs, $\unicode[STIX]{x1D6FF}_{w}^{+}$ , computed from the conditionally averaged statistics at UMZ edges at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . Detected UMZ edges are binned by momentum deficit ( $(\widetilde{U}_{i}-U_{\infty })/U_{\unicode[STIX]{x1D70F}}$ ) with the range indicated by the dashed lines, and also by different symbols which are defined in figure 6. The green ▾ symbols correspond to results computed exclusively at the TNTI.

Next, we examine the spatial extent across which we observe a sharp change in streamwise velocity, here denoted as $\unicode[STIX]{x1D6FF}_{w}$ . Due to the qualitative similarity between conditional profiles obtained for the TNTI and at the edges of UMZs, we estimate $\unicode[STIX]{x1D6FF}_{w}$ based on prior work at the TNTI. Specifically, $\unicode[STIX]{x1D6FF}_{w}$ is computed following Brown & Roshko (Reference Brown and Roshko1974), Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ) as

where $\unicode[STIX]{x0394}\widetilde{U}$ and $\unicode[STIX]{x2202}\langle \widetilde{U}\rangle /\unicode[STIX]{x2202}z|_{max}$ correspond to the average jump in streamwise velocity, and the maximum local gradient of the profile, respectively (see figure 7 b). Estimates of $\unicode[STIX]{x1D6FF}_{w}$ are presented in figure 9(b), showing a slight reduction of $\unicode[STIX]{x1D6FF}_{w}$ with increasing momentum deficit (decreasing distance from the wall). One possible basis for this trend could be the presence of larger diameter vortices residing further away from the wall. Such a postulation is in agreement with certain physical models for boundary layers and prior experimental evidence (Perry & Chong Reference Perry and Chong1982; Adrian et al. Reference Adrian, Meinhart and Tomkins2000; Herpin et al. Reference Herpin, Stanislas, Foucaut and Coudert2013; Morrill-Winter & Klewicki Reference Morrill-Winter and Klewicki2013, and others). Alternatively, a more probable explanation for the slight increase in $\unicode[STIX]{x1D6FF}_{w}$ we observe as a function of $z$ could be caused by the proportion of high shear and low shear regions present along UMZ edges. Qualitative evidence of this change in the proportion of high/low shear regions along internal interfaces can be observed on the shear fields, such as the one shown in figure 3(b), where the high shear regions (dark shading) increase in sparsity with increasing $z$ . As a consequence, the increased prevalence of low shear regions along the edges of UMZ further away from the wall may attenuate the magnitude of $\unicode[STIX]{x2202}\langle \widetilde{U}\rangle /\unicode[STIX]{x2202}z|_{max}$ (or increase $\unicode[STIX]{x1D6FF}_{w}$ ) with increasing $z$ . It is also worth noting that the spatial resolution of the measurement (the interrogation window size in our case) is likely to influence the velocity gradient across each interface and the subsequent estimate of $\unicode[STIX]{x1D6FF}_{w}$ . Thus, we can consider the computed $\unicode[STIX]{x1D6FF}_{w}$ to be an upper bound, with some uncertainty as to its exact value. We also note that our results are comparable to $\unicode[STIX]{x1D6FF}_{w}$ estimates from interfacial layers of high shear in boundary layers by Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015), who used a database with twice the spatial resolution. As a final note, our results also reveal the largest $\unicode[STIX]{x1D6FF}_{w}$ at the TNTI with a magnitude similar to that previously reported by Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ), albeit using different detection criteria. Therefore, the magnitude of $\unicode[STIX]{x1D6FF}_{w}$ appears to be somewhat robust to different TNTI detection criteria.

Now we focus our attention on the wall-normal velocity at the interface, $\langle \widetilde{W_{i}}\rangle$ . Results are shown in figure 10(a), which exhibits an increasingly positive magnitude for $\langle \widetilde{W_{i}}\rangle$ closer to the wall, conversely at the TNTI $\langle \widetilde{W_{i}}\rangle$ is slightly negative. One possible explanation for this trend may be made with reference to the streamwise evolution of a boundary layer, where we observe increasing momentum deficit close to the wall (with increasing $x$ ). This, in turn, would push mass away from the wall as a turbulent boundary layer develops (positive $\widetilde{W_{i}}$ ) on average. If we turn our attention to the TNTI (top edge of UMZ furthest away from the wall), $\langle \widetilde{W_{i}}\rangle$ is negative, which is in agreement with the notion that at the TNTI mass from the non-turbulent region is entrained into the turbulent region (see also Chauhan, Philip & Marusic (Reference Chauhan, Philip and Marusic2014a ), Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ), Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015), which discusses the conditionally averaged wall-normal velocity at the TNTI). As a consequence, we now have two competing mechanisms (entrainment at TNTI and the mass pushed up from the near near-wall region). Therefore $\langle \widetilde{W_{i}}\rangle$ is likely to undergo a sign change in the wake region, which appears in figure 10(a) and is also qualitatively illustrated in figure 10(b). Based on such a scenario in conjunction with the streamwise growth rate of the boundary layer, we may hypothesise that the structures responsible for the presence of UMZs encapsulated by high shear regions are growing/moving away from the wall on average. This would be in agreement with findings at interfacial layers of high shear in boundary layers which were reported by Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015) to move away from the wall. However to confirm this, temporally resolved databases would be required to track the edges of UMZs, which are unavailable in this study.

Figure 10.

(a) Conditionally averaged wall-normal velocity, $\langle \widetilde{W}\rangle$ at UMZ edges. Results are presented at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ and are computed at the interface/edge location, $z_{i}$ . The vertical dashed line corresponds to $\langle \widetilde{W}\rangle =0$ , and the symbols are as defined in figure 6. (b) Schematic of $\langle \widetilde{W}\rangle$ at UMZ edges overlaid on colour contours of streamwise velocity $\widetilde{U}$ . The black and green arrows correspond to UMZ edges within the turbulent region and the TNTI, respectively.

5.1 Velocity and length scales

Previously, figure 8(a) showed conditional statistics computed on a frame of reference attached to each interface, which revealed a persistent sharp rise in streamwise velocity, $\unicode[STIX]{x0394}\widetilde{U}$ across a finite thickness, $\unicode[STIX]{x1D6FF}_{w}$ . The results in figure 11 reproduces the same statistics now computed across all four experimental databases. Again all results are presented on a frame of reference attached to each interface at $z_{i}$ . Each column in figure 11 represents a different momentum deficit range. We note that conditional statistics at the higher deficit ranges are unavailable for the lower $Re$ databases due to the lack of near-wall data. To examine the appropriate velocity scales, the rows of figure 11(a–f) present conditional statistics normalised by $U_{\infty }$ and $U_{\unicode[STIX]{x1D70F}}$ , respectively. The results show good qualitative collapse over the range of Reynolds numbers considered, with no discernible difference between scaling the profiles by $U_{\infty }$ or $U_{\unicode[STIX]{x1D70F}}$ . We note that both $U_{\infty }$ and $U_{\unicode[STIX]{x1D70F}}$ are weak functions of $Re_{\unicode[STIX]{x1D70F}}$ , therefore, an even larger $Re$ range would be necessary to fully examine the appropriate velocity scale. Nevertheless, an attempt to quantify the appropriate velocity scale based on these qualitative observations are presented later in this work (see figure 13 a,b). We also note that a more rational explanation for scaling with $U_{\unicode[STIX]{x1D70F}}$ has been provided by Chauhan et al. (Reference Chauhan, Philip and Marusic2014a ), albeit only for the TNTI.

Next, we turn our attention to the length scales prevalent at the UMZ edges. To this end, figure 12 shows conditional statistics with the length scale on the ordinate normalised by (a) viscous units, (b) the boundary layer thickness and (c) the Taylor microscale. We note results are only shown for a single momentum deficit range ( $1.5<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<4.5$ ), however, other momentum deficit ranges also show similar behaviour and are not reproduced here for brevity. Figure 12(a) presents normalisation of length scales by viscous units, revealing poor collapse for the $Re$ range examined. Similar observations are also present when normalised by the boundary layer thickness, $\unicode[STIX]{x1D6FF}$ (see zoomed region of figure 12 b). On the other hand, the zoomed region of figure 12(c) reveals that normalisation by the Taylor microscale, $\unicode[STIX]{x1D706}_{T}$ , exhibits improved agreement. An attempt to quantify these qualitative observations on the prevalent velocity and length scales will follow next.

Accordingly, figure 13 shows a collection of characteristic properties of the conditional statistics as a function of $Re$ . Specifically, we examine the magnitude of the sharp change in streamwise velocity, $\unicode[STIX]{x0394}\widetilde{U}$ , and thickness across which it occurs, $\unicode[STIX]{x1D6FF}_{w}$ , in (a,b) and (c–e), respectively. This enables us to quantify some of the qualitative observations from figures 11 and 12. We note that the parameters $\unicode[STIX]{x0394}\widetilde{U}$ and $\unicode[STIX]{x1D6FF}_{w}$ are computed based on functional fits to conditional statistics (see figure 7), therefore precise numerical values for these characteristic properties must be taken with some degree of caution. Figure 13(a,b) show $\unicode[STIX]{x0394}\widetilde{U}$ computed across the entire extent of the boundary layer as a function of $Re$ normalised by $U_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ , respectively. Although visual inspection of conditional profiles of $\widetilde{U}$ in the near vicinity of the interface (see figure 11) showed no discernible difference between scaling by $U_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ , the results in figure 13(a) exhibits slightly better agreement is attained for $U_{\unicode[STIX]{x1D70F}}$ , albeit from a functional fit. Nevertheless, this result is in agreement with the fact that $U_{\unicode[STIX]{x1D70F}}$ serves as the appropriate velocity scale for the mean velocity deficit and the turbulent stresses in the outer region (Tennekes & Lumley Reference Tennekes and Lumley1972). Additionally, prior work has reported that the sharp change in $\widetilde{U}$ across the TNTI in turbulent boundary layers also exhibits invariance when scaled with $U_{\unicode[STIX]{x1D70F}}$ (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ). Therefore, our findings extend this behaviour to the edges of UMZs within the turbulent region, allowing us to infer that both the external (TNTI) and internal UMZ edges in boundary layers have turbulent motions with velocities that scale with the friction velocity.

Figure 13.

Characteristic properties computed from conditional profiles in the near vicinity of the edges of UMZs. (a,b) The average jump in streamwise velocity, $\unicode[STIX]{x0394}\widetilde{U}^{+}$ , normalised by $U_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ , respectively. (c–e) The width across which the jump occurs, $\unicode[STIX]{x1D6FF}_{w}$ , normalised by $U_{\unicode[STIX]{x1D70F}}$ , $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D706}_{T}$ , respectively. Results in (c–e) are computed for UMZ edges in the momentum deficit range $1.5<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<4.5$ .

Next, we attempt to quantify the width, $\unicode[STIX]{x1D6FF}_{w}$ , across which the sharp change in streamwise velocity occurs as a function of $Re$ . To this end, figure 13(c) presents $\unicode[STIX]{x1D6FF}_{w}$ normalised with $\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ as the length scale. The results show an increase in $\unicode[STIX]{x1D6FF}_{w}^{+}$ at low $Re$ which seems to plateau to an asymptotic limit as $Re$ increases. This behaviour at high $Re$ is encouraging as it is in close agreement with prior findings at the TNTI (Chauhan et al. Reference Chauhan, Philip and Marusic2014a ). Therefore, we may infer similarity between both length and velocity scales of turbulent motions at all edges of UMZs (inclusive of the TNTI) in boundary layers. Further similarities to the TNTI are also exhibited for normalisation of $\unicode[STIX]{x1D6FF}_{w}$ as a fraction of boundary layer thickness ( $\unicode[STIX]{x1D6FF}$ ) and the Taylor microscale $\unicode[STIX]{x1D706}_{T}$ , which are shown in figure 13(c,d), respectively. Specifically, normalisation of the width by $\unicode[STIX]{x1D6FF}$ exhibits that the high shear regions present at UMZ edges are thinner relative to $\unicode[STIX]{x1D6FF}$ at higher $Re$ , which is in agreement with visual inspections of the instantaneous shear at different $Re$ (see figure 3). On the other hand, $\unicode[STIX]{x1D6FF}_{w}$ normalisation by $\unicode[STIX]{x1D706}_{T}$ reveals improved invariance with $Re$ which is in agreement with qualitative evidence previously shown in figure 12. As a consequence, the width of the internal interfaces appear to be $O(\unicode[STIX]{x1D706}_{T})$ . Similar observations were also reported by Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015) at a fixed $Re$ across a range of wall-normal locations.

Figure 14.

The average length ( $l_{s}$ ) per unit streamwise length ( ${\mathcal{L}}_{x}$ ) computed from all experimental datasets. (a) Corresponds to the statistics computed exclusively at the TNTI. (b,c) correspond to edges of UMZs within the momentum deficit ranges ( ${\sim}$ wall-normal location) $1.5<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<4.5$ and $4.5<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<7.5$ , respectively. Symbols correspond to different datasets and are defined in figure 11. The solid line in (a) and dashed lines in (b,c) correspond to a power-law fit with a slope of 0.225 and 0.15, respectively.

If we turn our attention to the surrounding regions away from the UMZ edge ( $z\ll z_{i}$ or $z\gg z_{i}$ ), or the UMZs themselves, the results show the best agreement when scaled with $\unicode[STIX]{x1D6FF}$ (see figure 12 b). Therefore, we may postulate that the appropriate length scales for the structures associated with the UMZs are $O(\unicode[STIX]{x1D6FF})$ (see de Silva et al. Reference de Silva, Hutchins and Marusic2016). Collectively, a possible explanation for the lengths scales observed both within UMZs ( $O(\unicode[STIX]{x1D6FF})$ ) and at their associated edges ( $O(\unicode[STIX]{x1D706}_{T})$ ) can be postulated following Sreenivasan, Ramshankar & Meneveau (Reference Sreenivasan, Ramshankar and Meneveau1989) (also see da Silva & Taveira Reference da Silva and Taveira2010), where the time scale, $T$ , over which the large structures act is $(\unicode[STIX]{x1D6FF}/U_{\unicode[STIX]{x1D70F}})$ . Here, $U_{\unicode[STIX]{x1D70F}}$ is the relevant turbulent velocity scale across the turbulent boundary layer. During this time, $T$ , the viscous diffusion will act over a distance

and consequently $l_{vis}\sim \unicode[STIX]{x1D706}$ (e.g. Tennekes & Lumley Reference Tennekes and Lumley1972).

5.2 Scaling of geometric properties

An examination of the average interface length $l_{s}$ as a function of $Re$ is presented in figure 14(a), which shows results computed exclusively at the TNTI. Figure 14(b,c) reproduces the same statistics for the UMZ edges within the turbulent region at two different momentum deficit ranges ( ${\sim}$ wall-normal locations). At the TNTI we observe an increase in $l_{s}$ with $Re$ , which can be attributed to the increased range of scales present at high $Re$ that leads to a more convoluted TNTI. A similar behaviour is also observed at the edges of the UMZs within the turbulent region shown in figure 14(b,c), albeit with a smaller slope (slower increase in $l_{s}$ with $Re$ for the internal interfaces as compared to the TNTI). This slower increase in $l_{s}$ with $Re$ for the internal interfaces is in agreement with the observed differences in fractal dimension between internal shear layers and the TNTI as reported in § 4.2.

Accordingly, the $Re$ variation of the lengths presented in figure 14 can be estimated based on the self-similar nature of the fractal scaling. At a particular $Re$ the length of the interface scales like $l_{s}\sim b^{-D+1}$ for a given scale $b$ , where $D\approx 1.3$ and $1.2$ for the TNTI and the internal interfaces, respectively (see figure 6). If we assume that there is no variation to the length below the Kolmogorov length scale $\unicode[STIX]{x1D702}$ , then the length reported in figure 14 follows

Additionally, it is evident that the largest cut-off of the power-law scaling is likely to be at a constant fraction of $\unicode[STIX]{x1D6FF}$ (e.g. de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013), suggesting

Now, $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FF}\sim Re_{\unicode[STIX]{x1D70F}}^{-3/4}$ and $l_{s}(\unicode[STIX]{x1D6FF})\sim L_{x}$ , which result in

This suggests,

These two scalings are shown by solid and dashed lines in figure 14(a–c), respectively, and exhibit reasonable agreement with the data over a decade in $Re$ .

Figure 15.

(a) The p.d.f. of the wall-normal location of the TNTI, $z_{i}$ , at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . (b) Velocity jump $\unicode[STIX]{x0394}\widetilde{U}^{+}$ (see § 4.3) in streamwise velocity at TNTI modelled as a step function. (c) Resultant velocity profile from convolving (a,b). (d) Velocity deficit of the modelled profile, $U_{\infty ,m}^{+}-\overline{U}_{m}^{+}$ (solid line) in (c) overlaid on the mean velocity deficit, $U_{\infty }^{+}-\overline{U}^{+}$ (dashed blue line).

Figure 16.

Modelled results in figure 15 reproduced for all UMZ edges at $Re_{\unicode[STIX]{x1D70F}}\approx 14\,500$ . (a) P.d.f. of $z_{i}$ , where the dashed line corresponds to the TNTI (same as figure 15 a), and each solid line corresponds to UMZ edges within the turbulent region. The solid lines correspond to five equispaced momentum deficit ranges within the bounds $1<(U_{\infty }-\widetilde{U}_{i})/U_{\unicode[STIX]{x1D70F}}<10$ . (b) The corresponding velocity jumps, $\unicode[STIX]{x0394}\widetilde{U}^{+}$ , modelled as step functions. (c) Velocity profiles from convolving the corresponding profiles in (a,b). (d) Mean velocity profile, $\overline{U}_{m}^{+}$ , obtained by the superposition of the profiles in (c). (e) Dotted line: modelled mean velocity deficit, $U_{\infty ,m}^{+}-\overline{U}_{m}^{+}$ , dashed line: mean velocity deficit, $U_{\infty }^{+}-\overline{U}^{+}$ , from the same database. (f) Mean velocity profiles as a function of $z^{+}$ corresponding to the deficit profiles in (e).

Figure 17.

Modelled results for the mean flow computed at (a) $Re_{\unicode[STIX]{x1D70F}}\approx 2500$ from a DNS databases of Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013), (b) $Re_{\unicode[STIX]{x1D70F}}\approx 2800$ and (c) $Re_{\unicode[STIX]{x1D70F}}\approx 8000$ . ● symbols: modelled mean velocity profile, $\overline{U}_{m}^{+}$ , dashed line: mean velocity profile, $\overline{U}^{+}$ , from the same database.