3.1 Properties of the interface

The mean height of the T/NT interface, $\overline{y_{I}}$ , as a function of $Re_{\unicode[STIX]{x1D70F}}$ is shown in figure 5( $a$ ). The boundary-layer thickness $\unicode[STIX]{x1D6FF}$ and the intermittency factor $\unicode[STIX]{x1D6FE}=0.5$ are also overlaid for comparison. The mean position of the interface (figure 5 $a$ ) coincides with the position of $\unicode[STIX]{x1D6FE}=0.5$ . On the other hand, the position of $\unicode[STIX]{x1D6FE}=0.425$ (not shown) is identical to the local 99 % boundary-layer thickness. The agreement between a constant $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}(x)$ is qualitatively similar to the results by Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016), although the $\unicode[STIX]{x1D6FE}$ value is not identical due to the difference in the detector function and threshold. In addition, $\overline{y_{I}}$ grows with $Re_{\unicode[STIX]{x1D70F}}$ at a slightly smaller rate than the boundary-layer thickness $\unicode[STIX]{x1D6FF}(x)$ . The boundary-layer thickness and the T/NT interface height are two flow thicknesses based on the mean streamwise momentum and the turbulence activity, respectively. Hence, the increasing gap between $\overline{y_{I}}$ and $\unicode[STIX]{x1D6FF}$ downstream suggests that the intermittency of the boundary-layer turbulence becomes more significant at higher Reynolds number, and as such, a larger region below the mean boundary-layer edge is encroached by sweeping free-stream laminar flow. The root-mean-square (r.m.s.) of the interface height fluctuation, $y_{I}^{rms}$ , is plotted in figure 5( $b$ ). The excursions of the interface away from its mean location clearly increase in magnitude as the Reynolds number increases downstream. The increase is proportional to the growth of the interface height, or equivalently the boundary-layer thickness, and is most evident when $y_{I}^{rms}$ is normalized by the inlet momentum thickness. The more pronounced excursions of the interface at higher Reynolds numbers imply that larger regions of laminar free-stream fluid will encroach onto the TBL, and vice versa. Another outcome is an increase in the interfacial surface area (figure 5 $c$ ) and entrainment at higher Reynolds numbers, which is consistent with our understanding of high- $Re$ flows (de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013; Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ), although the ratio is smaller than previous studies since we are only concerned with the LSM and envelope.

A spectral analysis of the interface height can effectively characterize its undulation. Figure 6 shows the spanwise spectra of the normalized fluctuation in the T/NT interface height,

Fourier coefficients of the spanwise spectra at both $Re_{\unicode[STIX]{x1D70F}}=680$ and $1000$ peak at spanwise wavenumber $k_{z}=3$ . As the Reynolds number increases, the Fourier coefficients of low- $k_{z}$ modes ( $k_{z}\lesssim 10$ ) become more pronounced, while higher wavenumbers of the normalized fluctuations ( $k_{z}\gtrsim 10$ ) become relatively less important. In other words, the large-scale deformation becomes dominant downstream. A significant aspect of the interface deformation in the span is thus of the order of the boundary-layer thickness. The spectra demonstrate the large-scale nature of the interface morphology and suggests a possible relationship between this morphology and the large-scale turbulent structures inside the boundary layer.

Figure 6.

Spanwise spectra of the normalized interface height fluctuation $Y_{I}^{\prime }$ at ( $a$ )  $Re_{\unicode[STIX]{x1D70F}}=680$ and ( $b$ )  $Re_{\unicode[STIX]{x1D70F}}=1000$ .

Since the interface undulations intensify downstream with the growth of the boundary layer, $y_{i}^{\prime }\sim \unicode[STIX]{x1D6FF}(x)\sin (x)$ , spectra were not computed in that dimension. Instead, the two-point correlation,

was evaluated to examine the characteristics of the interface. In the above expression, $\unicode[STIX]{x1D70E}_{Y_{I}^{\prime }}$ is the standard deviation of the normalized fluctuation. For comparison, the auto-correlations of $u^{\prime }$ and $v^{\prime }$ at the wall-normal reference position $y_{ref}=\overline{y_{I}}$ were also computed. The results are plotted in figure 7 at two downstream locations. The fluctuating interface height shows stronger correlation in the streamwise direction in comparison with the span. The shape of $R_{Y_{I}^{\prime }Y_{I}^{\prime }}$ (figure 7 $a$ ) resembles the two-point correlation of the streamwise velocity fluctuations $R_{u^{\prime }u^{\prime }}$ (figure 7 $b$ ), although the aspect ratio of $R_{Y_{I}^{\prime }Y_{I}^{\prime }}$ is smaller. Based on the contour of $R=0.1$ (dashed line), the streamwise extents of both $R_{Y_{I}^{\prime }Y_{I}^{\prime }}$ and $R_{u^{\prime }u^{\prime }}$ are approximately $2\unicode[STIX]{x1D6FF}$ , and their streamwise dimensions normalized by the reference length scale, $\unicode[STIX]{x1D703}_{in}$ , increase slightly with Reynolds number. The streamwise extent of the interface ridge (or groove), therefore, is as long as the large-scale turbulent motions. In contrast, the correlation of the wall-normal fluctuations $R_{v^{\prime }v^{\prime }}$ (figure 7 $c$ ) is shorter in streamwise extent, which is consistent with Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005) and Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014). The shorter $v^{\prime }$ -coherence is dominated by the spanwise vortices in the intermittent region, which are clustered around large-scale $u^{\prime }$ -structures (Lozano-Durán, Flores & Jiménez Reference Lozano-Durán, Flores and Jiménez2012). The similarity between the streamwise size of the interface deformation and of the large-scale $u^{\prime }$ -coherence motivates study of the T/NT interface characteristics conditioned on the LSMs.

Figure 7.

Two-point correlations in the wall-parallel plane. ( $a$ ) The interface height fluctuations $Y_{I}^{\prime }$ , ( $b$ ) the streamwise velocity fluctuations $u^{\prime }$ and ( $c$ ) the wall-normal velocity fluctuations $v^{\prime }$ . The reference streamwise positions are located at (i)  $Re_{\unicode[STIX]{x1D70F}}=680$ and (ii)  $Re_{\unicode[STIX]{x1D70F}}=1000$ . In (b,c), the reference wall-normal position is at $y=\overline{y_{I}}$ . The dashed contour marks $R=0.1$ .

3.2 Connection between interface deformation and LSM

The results in § 3.1 suggest a connection between the interface and the LSMs. Flow visualizations, such as in figure 3, which shows the LSMs and the deformed T/NT interface, also support the same view. For example, the interface is convex above the low-speed structure (marked ‘A’ in figure 3 a,b), and concave above high-speed structures (marked ‘B’). To clarify and generalize such observations, conditional sampling is used to evaluate the effects of the low- and the high-speed structures on the interface.

The wall-normal positions of the T/NT interface and of the core of the LSM are denoted $y_{I}$ and $y^{s\pm }$ , respectively. The latter can be further classified into the cores of low- and high-speed structures, $y^{s-}$ and $y^{s+}$ , according to (2.4) and (2.5). As shown schematically in figure 8( $a$ ), the core represents a certain position within the structure in the cross-flow plane, while the full structure occupies a large wall-normal and spanwise area around that position. In order to evaluate the correlation between $y_{I}$ and $y^{s\pm }$ , the p.d.f. of each quantity alone (figure 8 b,c) and the joint p.d.f. are computed and visualized in figure 8( $d$ ).

Figure 8.

( $a$ ) Schematic of the LSM and the detected core in the cross-flow plane. Probability density functions of the interface height $y_{I}$ and the vertical position of the detected cores $y^{s\pm }$ at $Re_{\unicode[STIX]{x1D70F}}=1000$ : ( $b$ ) p.d.f. of $y^{s\pm }$ ; ( $c$ ) p.d.f. of $y_{I}$ ; ( $d$ ) joint p.d.f. of $y^{s\pm }$ and $y_{I}$ . (Dashed) The low-speed LSM core $y^{s-}$ and (solid) the high-speed LSM core $y^{s+}$ . Bin sizes are $0.025\unicode[STIX]{x1D6FF}$ in $y_{I}$ and $\unicode[STIX]{x0394}y$ in local $y^{s\pm }$ .

The p.d.f. of $y^{s+}$ (figure 8 $b$ , solid line) is finite in the range $y^{s+}\lesssim 0.8\unicode[STIX]{x1D6FF}$ only. The decay of the p.d.f. towards the edge of the boundary layer is symptomatic of the relatively small population of the high-speed LSMs away from the wall (Lee et al. Reference Lee, Ahn and Sung2015). In contrast, the cores of the low-speed motions are more frequently observed near the boundary-layer edge, and thus their p.d.f. is finite throughout the range $y^{s-}\lesssim \unicode[STIX]{x1D6FF}$ (dashed line). In a previous study, Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) evaluated the average perturbation fields conditional on the LSM. Their observation that sweeps are located away from the wall relative to ejections is consistent with the p.d.f. of $y^{s\pm }$ in the current study. Figure 8( $c$ ) reports the p.d.f. of $y_{I}$ , conditioned on a finite value of $y^{s\pm }$ , or equivalently on the presence of a large-scale structure. The maximum probabilities of $y_{I}$ are recorded at $1.03\unicode[STIX]{x1D6FF}$ and $0.893\unicode[STIX]{x1D6FF}$ for the low- and high-speed structures, respectively. These values should be contrasted to the mean $\overline{y_{I}}=0.966\unicode[STIX]{x1D6FF}$ at the same streamwise position (see figure 5 $a$ ).

The joint p.d.f. of $y_{I}$ and $y^{s\pm }$ (figure 8 $d$ ) shows modulation of the interface height by the LSMs. When the detected core is located in the log region or more generally below $y^{s\pm }=0.2\unicode[STIX]{x1D6FF}$ (grey region in the figure), the median of the p.d.f. of the interface height is approximately equal to $\overline{y_{I}}$ for both low- and high-speed motions. In other words, there is no clear difference in the height of the T/NT interface when the underlying high- and low-speed structures are in the log region. However, the detected cores in the wake region ( $y^{s\pm }>0.2\unicode[STIX]{x1D6FF}$ ) have a noticeable impact on the interface, which is captured by the bifurcating dashed line in figure 8( $d$ ). The interface height decreases, when the high-speed fluid is detected in the wake region, and increases in presence of low-speed structures. As a result, the vertical spacing between the interface and the cores of these large-scale structures is smaller for the high-speed ones than those at low speed. The difference is clear when contrasting the gap between the dotted line, $y^{s\pm }=y_{I}$ , and the two sets of contours in figure 8( $d$ ). This result confirms that the wall-normal elevation of the T/NT interface is modulated by the underlying low- and high-speed LSMs in the wake region.

Figure 9.

Conditional average of the normalized interface fluctuation based on the core of ( $a$ ) low- and ( $b$ ) high-speed LSMs. Reference position is located at $Re_{\unicode[STIX]{x1D70F}}=1000$ ( $x_{ref}=963\unicode[STIX]{x1D703}_{in}$ ) and $y_{ref}=0.5\unicode[STIX]{x1D6FF}$ , marked as a vertical line.

A statistical representation of the interface shape in the vicinity of positive and negative LSMs can be obtained by computing the conditional average,

Figure 9 shows the isosurface of the result normalized by the mean interface height, $\langle Y_{I}^{\prime }\rangle ^{s\pm }\equiv (\langle y_{I}\rangle ^{s\pm }-\overline{y_{I}})/\overline{y_{I}}$ . The reference position is located at $x_{ref}=963\unicode[STIX]{x1D703}_{in}$ (or $Re_{\unicode[STIX]{x1D70F}}=1000$ ) and $y_{ref}=0.5\unicode[STIX]{x1D6FF}$ . This wall-normal location is within the wake region, and therefore the probabilities of observing low- and high-speed streaks differ (figure 8 $d$ ). Above the low-speed structure (figure 9 $a$ ), the interface is displaced outward towards the free stream. Conversely, a depression in the interface height is observed above the high-speed structure at the reference position (figure 9 $b$ ). The streamwise dimension of the conditionally averaged interface elevation is approximately $2\unicode[STIX]{x1D6FF}$ which is consistent with that of $R_{Y_{I}^{\prime }Y_{I}^{\prime }}$ (figure 7 $a$ ii) and $R_{u^{\prime }u^{\prime }}$ (figure 7 $b$ ii). The displacements are not symmetric: the height of the crest above the low-speed structure is smaller than the depth of the trough associated with the high-speed motion. The figure also shows that the conditional interface shape is elongated in the streamwise direction and is narrow in the span. Similar to the two-point correlation $R_{Y_{I}^{\prime }Y_{I}^{\prime }}$ (figure 7 $a$ ii), the spanwise dimension of the interface deformation, which is identified based on the location where $\langle Y_{I}^{\prime }\rangle ^{s\pm }=0$ , is approximately $1\unicode[STIX]{x1D6FF}$ . Beyond that width, an anti-correlated pattern is observed. Since the low-speed structure is flanked by high-speed structures and vice versa (e.g. Chung & McKeon Reference Chung and McKeon2010; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016), the alternating appearance of crests and troughs in the span is consistent with the arrangement and proposed influence of LSMs.

Figure 10.

Conditionally averaged velocity fields based on the core of (i) low- and (ii) high-speed LSMs at $Re_{\unicode[STIX]{x1D70F}}=1000$ ( $x_{ref}=963\unicode[STIX]{x1D703}_{in}$ ) and $y_{ref}=0.5\unicode[STIX]{x1D6FF}$ . The figure is visualized in ( $a$ ) the cross-flow plane at $r_{x}=0$ and ( $b$ ) the side view at $r_{z}=0$ . Flood contour and velocity vectors are $\langle u^{\prime }\rangle ^{s\pm }$ and the in-plane velocity fluctuation vectors, respectively. (Solid) the position of the averaged T/NT interface; (dashed) $\langle u^{\prime }\rangle ^{s\pm }=\pm 0.02U_{\infty }$ .

The local change of the T/NT interface height due to the underlying low- and high-speed motions is also examined by computing the conditional average,

The reference position was again placed at $x_{ref}=963\unicode[STIX]{x1D703}_{in}$ (or $Re_{\unicode[STIX]{x1D70F}}=1000$ ) and $y_{ref}=0.5\unicode[STIX]{x1D6FF}$ , and the results are shown in figure 10. The dashed line in the figure marks $\langle u^{\prime }\rangle ^{s\pm }=\pm 0.02U_{\infty }$ , and demonstrates that the shape of the LSM is comparable for both positive and negative velocity perturbations. The conditionally averaged velocity vectors confirm that the ejection event, $\langle u^{\prime }\rangle ^{s-}<0$ , is capped by a crest in the interface (figure 10 $a$ i). In contrast, a sweep event, $\langle u^{\prime }\rangle ^{s+}>0$ , is clearly accompanied by a trough in the interface shape (figure 10 $a$ ii). The subsidence of the interface is accompanied with the negative $\langle v^{\prime }\rangle ^{s+}$ . Also, although sweep events contribute favourably to the Reynolds shear stress and turbulence production, the associated T/NT interface is displaced towards the wall. As a result, the perturbation field associated with high-speed motion does not contribute to turbulence activity near the mean location of the boundary-layer edge.

In the side view (figure 10 $b$ ), the pitch angles of the interface and of the LSM are visible. The estimated angles of the latter (dash-dot line) are approximately $14^{\circ }$ for both the low- and high-speed structures. On the other hand, the inclination angle of the T/NT interface differs in the two configurations. Above the low-speed structures, the angle is positive and shallower, measuring $5.3^{\circ }$ (red solid line), which is in line with our understanding of wall-detached structures nearer to the edge of the boundary layer. Above the high-speed structure, the interface angle is negative, and is equal to $-4.9^{\circ }$ at the upstream position $-\unicode[STIX]{x1D6FF}\lesssim r_{x}\lesssim 0$ (figure 10 $b$ ii).

Earlier work by Eisma et al. (Reference Eisma, Westerweel, Ooms and Elsinga2015), which did not differentiate the effect of low- and high-speed motion, showed a similarity in the turbulence statistics at the interface and at internal shear layers. In the present context, these layers correspond to the outer edges of the LSMs (Adrian et al. Reference Adrian, Meinhart and Tomkins2000; Christensen & Adrian Reference Christensen and Adrian2001; Adrian Reference Adrian2007). However, the conditionally sampled fields in figure 10 demonstrate the qualitative difference between the T/NT interface and the outer edge of the LSM, in particular for high-speed structures. This point is further emphasized by computing conditional statistics of the turbulence in the vicinity of the interface.

3.3 Conditional statistics in the vicinity of the interface

Previous studies have reported the turbulence statistics as a function of distance to the interface (e.g. Bisset et al. Reference Bisset, Hunt and Rogers2002; Chauhan et al. Reference Chauhan, Philip and Marusic2014a ; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014). Since the low- and high-speed motions contribute differently to the conditional statistics in the bulk (Lee et al. Reference Lee, Ahn and Sung2015), it is anticipated that the near-interface statistics are also dependent on the adjacent LSMs. We therefore examine the low- and high-speed large scales, and evaluate their respective contributions to the near-interface dynamics.

Figure 11.

Joint p.d.f. of the conditional total velocity and the vertical distance of the LSM core from the interface (flood) at $Re_{\unicode[STIX]{x1D70F}}=1000$ . (Solid) The expected value of the p.d.f. of the conditional velocity at the LSM core. ( $a$ ) Streamwise and ( $b$ ) wall-normal velocities. (i) Low- and (ii) high-speed structures. The number of samples is larger than $1000$ at each $y$ -position. Dashed line indicates $\langle u_{i}\rangle _{I}$ .

Figure 11 shows the joint p.d.f. of the conditional total velocity and the vertical separation between the LSM core and the interface. The expected value of the probability distribution is also plotted. Similar to Ishihara et al. (Reference Ishihara, Ogasawara and Hunt2015), a local averaging in the streamwise direction is applied to improve convergence of the conditional mean,

where $\unicode[STIX]{x0394}L_{x}$ is set to $10\unicode[STIX]{x1D703}_{in}$ . Note that the average also includes points across the cores of the large-scale structures in the homogeneous spanwise direction. In addition to the conditional sampling based on the low- and high-speed structures separately, the mean velocity evaluated as a function of the vertical distance from the interface, $\langle u_{i}\rangle _{I}(y-y_{I})$ , is plotted for comparison (dotted line). The p.d.f. of the low-speed events (figure 11 $a$ i) vanishes near the interface ( $y^{s\pm }-y_{I}=0$ ), while that of high-speed ones (figure 11 $a$ ii) is finite. As such the expectation of observing a low-speed event, here defined with the core of LSM, is zero in the near-interface region, even though the structures are associated with ejections and their large volume can reach farther away from the wall than the high-speed counterparts. The p.d.f. of the positive LSM more closely shadows the unconditional mean profile near the T/NT interface.

Previous studies reported the presence of a rapid change in the mean streamwise velocity near the interface (e.g. Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ; Ishihara et al. Reference Ishihara, Ogasawara and Hunt2015). As noted in the discussion of figure 4, such abrupt change is weak, e.g. in the inset of figure 11( $a$ ii), due to the small vorticity threshold $|\unicode[STIX]{x1D714}|^{\ast }$ adopted for the identification of the T/NT interface. The p.d.f. indicates that the jump in $\langle u\rangle _{I}$ coincides with the absence of low-speed events near the interface. High-speed events, on the other hand, do not contribute appreciably to the jump due to the weak shear near the outer edge of the boundary layer.

The p.d.f.s of the conditional vertical velocity are plotted in figure 11( $b$ ). Consistent with the kinematics of ejections and sweeps (Wallace Reference Wallace2016), the low-speed LSM is mainly associated with positive $\langle v\rangle _{I}$ and the high-speed LSMs are due to negative $\langle v\rangle _{I}$ . The distinction here is that the origin of the vertical coordinate axis is attached to the interface, and we report the advection velocity of the boundary-layer, or vortical, flow only. In addition, the figure also shows the entraining motion beneath the interface, where the high-speed event is predominant.

Previous studies examined the sudden change in the turbulence stresses near the interface (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015; Ishihara et al. Reference Ishihara, Ogasawara and Hunt2015), which was interpreted as a viscous effect of the superlayer. Another interpretation is provided here based on the population and the intensity of the LSM. Figure 12( $a$ i– $c$ i) shows the conditional average $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle _{I}^{s\pm }(y^{s\pm }-y_{I})$ , where the perturbations are evaluated relative to $\langle u_{i}\rangle _{I}$ . The spanwise stress is not shown because the spanwise perturbation is negligible at the centre of the LSM which is induced by counter-rotating streamwise roll cells (Hwang et al. Reference Hwang, Lee, Sung and Zaki2016). All components of the conditional stresses at the cores of LSMs are generally stronger than their unconditional counterparts (dashed lines in the figure). The streamwise stress (figure 12 $a$ i) shows stronger turbulence activity at the cores of the low-speed structures in comparison with the high-speed structures. Local averaging samples both low- and high-speed structures, as well as non-turbulent free-stream fluid that is engulfed into the boundary layer. In the present case, only the turbulent events are included. In addition, since the focus is placed on the T/NT interface, statistics are computed relative to the interface location. As a result, only high-speed events contribute appreciably to the stresses in the immediate vicinity of the interface. The wall-normal stress (figure 12 $b$ i) conditioned on a high-speed event shadows the unconditional quantity, with the stress from low-speed events being dominant away from the interface. The unconditional Reynolds shear stress (dotted line in figure 12 $c$ i) has a small negative value near the interface. However, the shear stress conditioned on a high-speed event remains positive value due to the finite strength of the sweep motion.

Figure 12.

Conditional average of the turbulent stress at the LSM core. $Re_{\unicode[STIX]{x1D70F}}=1000$ . Reynolds ( $a$ ) streamwise, ( $b$ ) wall-normal and ( $c$ ) shear stress. (i) Velocity fluctuations relative to $\langle u_{i}\rangle _{I}$ and (ii) relative to $\langle u_{i}\rangle _{I}^{s\pm }$ . (Blue) Low- and (red) high-speed structures. The number of samples is larger than $1000$ at each $y$ -position. Dashed line indicates $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle _{I}$ .

The conditional turbulence stresses are re-evaluated relative to the conditional mean velocity $\langle u_{i}\rangle _{I}^{s\pm }$ in figure 12( $a$ ii– $c$ ii), and exhibit similar trends as those in figure 12( $a$ i– $c$ i). Note that the velocity perturbation in $\langle u_{i}^{\prime \prime }u_{j}^{\prime \prime }\rangle _{I}^{s\pm }(y^{s\pm }-y_{I})$ is defined as

The results demonstrate that $\langle u_{i}^{\prime \prime }u_{j}^{\prime \prime }\rangle _{I}^{s\pm }$ is smaller than $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle _{I}^{s\pm }$ . The reduction in the stress when the perturbation is measured relative to a different mean can be explained using figure 11( $a$ i). The markings ‘A’ and ‘B’ on the figure show an example of the moments relative to $\langle u\rangle _{I}$ and $\langle u\rangle _{I}^{s\pm }$ , at $(y^{s\pm }-y_{I})=0.3\unicode[STIX]{x1D6FF}$ . The latter is clearly smaller, and the more accurate measure which is reported in figure 11( $a$ i). Note that Kwon et al. (Reference Kwon, Hutchins and Monty2016) also remarked the importance of appropriately defining the mean profile in the computation of the turbulence stress in the intermittent region. Although the magnitude of the conditional turbulence stress can be affected by the choice of reference velocity, the current results clearly show that the stress in the low-speed event is stronger than the high-speed counterpart.

In general, the profiles of the Reynolds stresses due to the high-speed motion shadow the unconditional average near the interface. On the other hand, away from the interface, it is the Reynolds stresses due to the low-speed motion that contributes most to the unconditional average. Interpretation of the statistics based on the large scales is analogous to the attached eddy hypothesis (Townsend Reference Townsend1976), which anticipates the turbulence statistics using the intensity function and the probability of the large-scale structures. Here, however, the statistics were examined from the perspective of an observer located at the T/NT interface.

Figure 13.

Large-scale deformation of the T/NT interface (white) and the underlying large-scale turbulent motions. Blue and red represent the low- and high-speed structures, respectively.