2.1. Direct numerical simulations

Direct numerical simulations (DNS) of smooth- and rough-wall channel flow (figure 1) were performed by solving the incompressible Navier–Stokes equations in dimensional form in Nek5000 (Fischer et al. Reference Fischer, Lottes and Kerkemeier2008) as

(2.1) \begin{align} \frac {\partial u_i}{\partial x_i} &= 0, \\[-12pt] \nonumber \end{align}

(2.2) \begin{align} \frac {\partial u_i}{\partial t} + u_{\kern-1pt j} \frac {\partial u_i}{\partial x_{\kern-1pt j}} &= -\frac {1}{\rho }\frac {\partial P}{\partial x_i} + \nu \frac {\partial ^2 u_i}{\partial x_{\kern-1pt j}^2} + f_i, \\[10pt] \nonumber \end{align}

where $u_i$ denotes the velocity components. A constant streamwise body force $f_i=(f_x,0,0)$ is applied, equivalent to a uniform pressure gradient $f_x=-(1/\rho )\,\partial P/\partial x$ . For the smooth-wall case, the domain size is $L_x\times L_y\times L_z=12h\times 2h\times 6h$ , and periodic boundary conditions are imposed in the streamwise ( $x$ ) and spanwise ( $z$ ) directions, with no-slip conditions on all solid surfaces. The equations are solved using a semi-implicit scheme in which the nonlinear terms are advanced by third-order extrapolation (EXT3), and the viscous terms by third-order backward differentiation (BDF3). Spatial discretisation employs curvilinear hexahedral spectral elements with an inf–sup stable $P_N{-}P_{N-2}$ approximation, where degree-7 polynomials are used for velocity, and degree-5 for pressure. Grid spacings are summarised in table 1, and the first off-wall point has $\Delta y^+ \approx 0.11$ for both the smooth- and rough-wall cases.

Table 1.

Summary of simulation parameters, where $N_e$ denotes the number of high-order elements. Domain sizes are normalised by $h$ , and grid spacings are in wall units.

Table 1. Long description

Cases P24, P12, and P06 have Re_tau = 395, k/h = 0.1, domain 12 by 2.2 by 6 in h units, and p/k = 24, 12, and 6. Their element counts are 141372, 158760, and 183204; Delta U+ values are 7.83, 9.61, and 9.32. SM has Re_tau = 395, domain 12 by 2 by 6, 129500 elements, and no roughness. P24H has Re_tau = 590, p/k = 24, k/h = 0.1, domain 12 by 2.2 by 6, and 620240 elements. Streamwise, wall-normal, and spanwise wall-unit spacings are also reported for each case.

Figure 1.

Illustration of the rough-wall configuration with case P12: (a) 3-D view; (b) streamwise-wall-normal ( $x$ – $y$ ) view; (c) local roughness unit with $k =b=0.1h$ and pitch $p =1.2h$ (bar width included).

Figure 1. Long description

Panel (a) shows the channel domain above a lower wall patterned with evenly spaced spanwise-uniform bars. Panel (b) is an x-y section with streamwise length Lx = 12h, total height Ly = 2.2h, and channel half-height h. Panel (c) enlarges one bar: height k = 0.1h, width b = 0.1h, and streamwise pitch p = 1.2h.

2.2. Roughness configuration

The simulation campaign comprises four rough-wall cases and one smooth-wall reference. Three cases, P06, P12 and P24, at $ \textit{Re}_{\tau }=395$ correspond to pitch-to-height ratios $p/k=6$ , $12$ and $24$ , respectively, and are designed to isolate the effect of roughness geometry at fixed Reynolds number. A fourth case, P24H, repeats the P24 geometry at $ \textit{Re}_{\tau }=590$ to examine the sensitivity of the QC statistics to Reynolds number. The rough-wall channel comprises spanwise-continuous, 2-D transverse bars of height and width $k=b=0.1 h$ in a domain of size $L_x\times L_y\times L_z=12h\times 2.2h\times 6h$ . Accordingly, the number of streamwise pitch repetitions is $N_p=L_x/p$ , with $N_p=20$ (P06), $N_p=10$ (P12) and $N_p=5$ (P24, P24H). To enhance resolution near the roughness, the streamwise and spanwise mesh spacings were refined locally. In the streamwise direction, two high-order elements are used across one roughness unit (one pitch), yielding 16 Gauss–Lobatto–Legendre velocity points per unit.

4.1. The QC identification in the instantaneous field

The UMZs are identified following Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) by constructing the PDF of the instantaneous streamwise velocity normalised by the centreline velocity $u/U_{\textit{CL}}$ , within a limited 3-D spatial domain (de Silva et al. Reference de Silva, Hutchins and Marusic2016; Chen et al. Reference Chen, Chung and Wan2020). Peaks of the PDF correspond to separate zones with modal velocities $u_m$ , and the local minimum between adjacent peaks defines the interfacial threshold $u_k$ (de Silva et al. Reference de Silva, Hutchins and Marusic2016, Reference de Silva, Philip, Hutchins and Marusic2017). The PDF method is sensitive to the bin width and peak-recognising schemes (Chen et al. Reference Chen, Chung and Wan2021), and for robustness, a KDE of the PDF is employed following Fan et al. (Reference Fan, Xu, Yao and Hickey2019). The details on the UMZ identification method are provided in Appendix A.

Figure 5 illustrates the procedure using a snapshot from the smooth-wall case. The PDF of $u$ is computed from a limited 3-D domain with window size $l_x/h=2$ in the streamwise direction, and $l_z/h=0.5$ in the spanwise direction. In figure 5(a), two modal velocities (peaks) are identified from the KDE, which correspond to the highest-speed QC (UMZ with the highest modal velocity) with $u_{m,1}\approx 0.96\,U_{\textit{CL}}$ , and the adjacent lower-speed zone with $u_{m,2}\approx 0.84\,U_{\textit{CL}}$ . The interfacial threshold is given by the valley between peaks, $u_k\approx 0.88\,U_{\textit{CL}}$ . Figure 5(b) displays a 2-D slice of the windowed domain used in figure 5(a), in which the contour line encloses the QC at the given $u_k$ .

Figure 5.

The UMZ identification for a representative SM snapshot using a 3-D window of $l_x/h=2.0$ and $l_z/h=0.5$ . (a) The KDE-based PDF of $u/U_{\textit{CL}}$ ; symbols mark modal velocities ( $u_m$ ) and the interfacial threshold ( $u_k$ ). (b) A 2-D slice of the instantaneous field; the black contour denotes $u_k\approx 0.88\,U_{\textit{CL}}$ .

Figure 5. Long description

A: The probability density of u/UCL is bimodal. A lower-speed mode occurs near 0.83 and a dominant high-speed mode near 0.97. Symbols mark the modes obtained from the PDF and KDE. The dashed interfacial threshold uk is placed at the intervening valley near 0.88. B: A red high-speed region occupies most of the middle of the channel, while lower-speed yellow, green, and blue regions occur near both walls. The irregular black contour at u approximately 0.88 UCL follows the upper and lower boundaries of the central high-speed region.

4.2. Selection of the identification region

The PDF-based identification of UMZs is sensitive to the spatial window from which the velocity distribution is constructed (de Silva et al. Reference de Silva, Hutchins and Marusic2016). Prior studies often used short streamwise extents (e.g. $l_x\approx 0.25\,h$ ) to resolve UMZs travelling at lower modal velocities, which are typically thinner and thus contribute less prominent PDF peaks (de Silva et al. Reference de Silva, Hutchins and Marusic2016, Reference de Silva, Philip, Hutchins and Marusic2017; Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022). On the other hand, to identify the QC that occupies almost half of the flow domain around the centreline, a statistically converged distribution is required; streamwise windows of order $\mathcal{O}(h)$ (e.g. $l_x\approx 1.2\,h$ in Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014) were found to be robust for planar (particle image velocimetry) measurements. Leveraging 3-D DNS fields, the present study performs the identification in 3-D windows, following the volumetric approach used in internal flows such as pipes (Chen et al. Reference Chen, Chung and Wan2020; Chen et al. Reference Chen, Chung and Wan2021).

In figure 6, a sensitivity analysis is conducted by varying the window extents over $1.0\le l_x/h \le 6.0$ in the streamwise direction, and $0.1\le l_z/h \le 3.0$ in the spanwise direction. This figure presents the distribution of the identified threshold $u_k/U_{\textit{CL}}$ as a scatter plot to visualise how the threshold responds to different combinations of streamwise and spanwise window sizes. A threshold $u_k$ is deemed valid if the KDE distribution exhibits two separate peaks with a distinct intermediate valley. In such a scenario, $u_k$ is defined as the velocity corresponding to the local minimum between the two peaks, which indicates a sharp velocity jump, i.e. a thin shear layer that serves as the boundary demarcating the QC from the exterior zone (de Silva et al. Reference de Silva, Hutchins and Marusic2016; Chen et al. Reference Chen, Chung and Wan2020). By contrast, if the KDE curve is unimodal, which is typically observed amid strong sweep or ejection events, without a discernible valley, then no valid $u_k$ is retrieved under the given sampling window size (see Appendix A). With the histogram-based PDF method, as shown in figure 6(a), a valid $u_k$ is obtained mainly for relatively small windows. Using KDE, as shown in figure 6(b), a valid $u_k$ is recovered over a broader range, and the resulting map of $u_k/U_{\textit{CL}}$ displays a stratified dependence on the window size. Small $(l_x,l_z)$ yields $u_k/U_{\textit{CL}}\to 1$ , increasing $l_z$ drives $u_k/U_{\textit{CL}}\to 0.95$ , and larger $l_x$ further reduces $u_k/U_{\textit{CL}}$ towards $0.9$ .

Figure 6.

Sensitivity of the identified threshold $u_k/U_{\textit{CL}}$ to streamwise and spanwise window sizes for the SM case. Window sizes are varied across $l_x/h \in [1.0, 6.0]$ (streamwise) and $l_z/h \in [0.1, 3.0]$ (spanwise) to determine the interval of window dimensions that enables stable identification of the target threshold $u_k/U_{\textit{CL}}$ . Results from (a) the histogram-based PDF method, and (b) the KDE method.

Figure 6. Long description

Points show uk/UCL as functions of lx/h and lz/h. Most available estimates occur at small spanwise windows, where thresholds cluster around 0.95 to 1.0. The estimates become sparse and more variable as either window dimension increases, illustrating the limited range over which histogram peaks and valleys are identified reliably. B: The KDE method yields thresholds over a much broader set of windows than the histogram method. A stable yellow-green band, corresponding to uk/UCL near 0.93 to 0.97, spans moderate lx/h and small-to-moderate lz/h. Narrow or very large windows show more variation and occasional isolated values.

The spanwise dependence of $u_k$ matches the spatial extent of large-scale motions. Figure 7 shows an instantaneous $z$ – $y$ plane of the streamwise velocity with the contour of $u/U_{\textit{CL}}=0.95$ . The QC boundary exhibits spanwise intermittency with characteristic widths of order $0.5h$ – $1.5h$ , and streamwise extents of order $h$ – $3h$ (Pope Reference Pope2000). Revisiting figure 6(b), $u_k/U_{\textit{CL}}\approx 0.95$ is recovered precisely when the window spans these large-scale motions, indicating that the $0.95\,U_{\textit{CL}}$ threshold is not arbitrary but tied to the geometry of dominant outer-layer motions.

Figure 7.

Instantaneous contour of $u/U_{\textit{CL}}$ in a $z$ – $y$ plane for the SM case. The solid black line shows the QC boundary defined at the threshold $u/U_{\textit{CL}}=0.95$ .

Figure 7. Long description

A red high-speed region extends across the central channel between two irregular black boundaries. Lower-speed yellow, green, and blue patches occur near the lower and upper walls. The core thickness and vertical position vary strongly with spanwise coordinate z/h.

Figure 8.

Convergence of modal velocities $u_m$ and interface threshold velocity $u_k$ using KDE in the SM case: (a) variation with $l_z/h$ at fixed $l_x/h=2$ ; (b) variation with $l_x/h$ at fixed $l_z/h=0.5$ .

Figure 8. Long description

A: The first modal velocity remains close to 0.99 UCL across lz/h. The second mode and threshold are less stable for very narrow windows, then settle near about 0.90 to 0.94 UCL over most moderate and large spanwise windows. B: The first modal velocity remains near 0.98 to 0.99 UCL. The second mode and threshold fluctuate for the smallest lx/h, then stabilize near approximately 0.91 and 0.95 UCL, respectively, over most of the range from lx/h about 1 to 6.

Figure 9.

Aggregated PDFs of the first ( $u_{m,1}$ , red) and second ( $u_{m,2}$ , blue) modal velocities for (a) the SM case, and the rough-wall cases (b) P24, (c) P12, and (d) P06. The histograms represent the collected data from all sampled domains, while the solid lines are the corresponding KDE fits. The statistical modal velocities ( $U_{m,1}$ and $U_{m,2}$ ) are marked by triangles at the peaks of the KDE curves. The dashed vertical line indicates the identified threshold velocity $U_k$ .

Figure 9. Long description

A: Red and blue histograms and KDE curves show the first and second modal velocities. The high-speed distribution is narrow and centred at Um,1 approximately 1.00 UCL. The lower-speed distribution is broader and centred at Um,2 approximately 0.93 UCL. The dashed threshold Uk is about 0.95 UCL. B: The red high-speed distribution is narrow with Um,1 approximately 1.00 UCL. The blue low-speed distribution is broader with Um,2 approximately 0.89 UCL. The two distributions overlap around the dashed threshold Uk approximately 0.93 UCL. C: The red high-speed distribution peaks at Um,1 approximately 0.98 UCL. The broad blue low-speed distribution peaks at Um,2 approximately 0.83 UCL and overlaps substantially with the red distribution. The dashed threshold Uk is approximately 0.90 UCL. D: The red high-speed distribution is narrow and centred at Um,1 approximately 0.99 UCL. The blue low-speed distribution is broad and centred at Um,2 approximately 0.82 UCL. The dashed threshold Uk is approximately 0.91 UCL.

To demonstrate convergence of the window size selection, figure 8 shows the variation of $(u_{m,1},u_{m,2},u_k)$ with $l_z/h$ at fixed $l_x/h=2$ , and with $l_x/h$ at fixed $l_z/h=0.5$ (KDE-based identification). For both $l_x$ and $l_z$ , $u_k/U_{\textit{CL}}$ stabilises to $ 0.95$ once the window spans $l_z/h\gtrsim 0.5$ and $l_x/h\gtrsim 2$ . Based on these trends, a standard window of $l_x/h=2$ and $l_z/h=0.5$ is adopted henceforth.

4.3. Determination of a QC boundary

With the standard window, the KDE identification is applied across the domain with spanwise sliding (adjacent windows offset by five grid points). Following Asadi et al. (Reference Asadi, Kamruzzaman and Hearst2022), figure 9 plots the distributions of the highest modal velocity $u_{m,1}$ and the second-highest modal velocity $u_{m,2}$ separately, where both are the KDE-peak velocities as defined in § 4.1, aggregated from all sliding-window analyses across the domain. Table 2 summarises the statistically converged modal velocities ( $U_{m,1}$ for the QC, and $U_{m,2}$ for the adjacent zone, both derived by analysing the probability densities of their individual counterparts $u_{m,1}$ and $u_{m,2}$ from each sampled window), the robust threshold $U_k$ (obtained similarly from the individual $u_k$ across all windows) defined at the valley between modes, and the standard deviation $\sigma _2$ of the second mode.

Table 2.

Summary of the identified modal velocities ( $U_{m,1}$ , $U_{m,2}$ ), the corresponding threshold velocity ( $U_k$ ), and the standard deviation $\sigma _2$ of $u_{m,2}$ for all cases.

Table 2. Long description

Values are listed as Um,1/UCL, Um,2/UCL, Uk/UCL, and sigma_2. P24: 1.00, 0.89, 0.93, 0.10. P12: 0.98, 0.83, 0.90, 0.11. P06: 0.99, 0.82, 0.91, 0.12. SM: 1.00, 0.93, 0.95, 0.06. P24H: 1.00, 0.86, 0.93, 0.11. The rough-wall cases have lower second-mode velocities and larger sigma_2 than SM.

The QC modal velocity $U_{m,1}$ remains nearly invariant (within $\pm 2\,\%$ of $U_{\textit{CL}}$ ) across cases, whereas $U_{m,2}$ is strongly affected by roughness: compared with the SM value $U_{m,2}/U_{\textit{CL}} \approx 0.93$ , $U_{m,2}$ decreases to $0.89\,U_{\textit{CL}}$ (P24), $0.83\,U_{\textit{CL}}$ (P12) and $0.82\,U_{\textit{CL}}$ (P06). The broadening of the $u_{m,2}$ distribution is quantified by $\sigma _2$ . The decrease and broadening of $U_{m,2}$ are consistent with the trend of the drag inferred from $\Delta U^{+}$ , and imply enhanced intermittency of the non-core region over rough walls, which is evaluated in § 5.

As shown in figure 9 and table 2, the QC interface threshold derived from the PDF of $u_m$ yields $U_k/U_{\textit{CL}}\approx 0.93$ (P24), $U_k/U_{\textit{CL}}\approx 0.90$ (P12) and $U_k/U_{\textit{CL}}\approx 0.91$ (P06), slightly lower than the SM value $U_k/U_{\textit{CL}}=0.95$ , since these values are influenced by the widening and displacement of $U_{m,2}$ (see $\sigma _2$ in table 2). For direct comparison, and to target the QC boundary tied to the first mode, a fixed threshold $U_k/U_{\textit{CL}}=0.95$ is adopted across all flow cases.

Furthermore, the larger $\sigma _2$ for rough walls implies that within a given window, the instantaneous distribution becomes less bimodal and exhibits a marked shift of $U_{m,2}$ towards lower modal velocities, typically during strong sweep or ejection events. Figure 10 depicts such dynamics in the instantaneous fields: in figure 10(a) for the P24 case, a sweep-dominated field suppresses the low-velocity peak and precludes a distinct $u_k$ , i.e. no QC boundary can be established, whereas in figure 10(b) for the P12 case, an ejection-dominated field shifts the valley and yields $u_k$ that deviates significantly from the mean $U_k$ . This observed difference in the core topology is consistent with the broader $u_{m,2}$ PDFs and increased intermittency over rough walls (Volino et al. Reference Volino, Schultz and Flack2009, Reference Volino, Schultz and Flack2011).

Figure 10.

Examples of instantaneous events that challenge modal separation. (a) Sweep-dominated field (P24) in which a distinct low-speed mode is absent. (b) Ejection-dominated field (P12) in which the identified $u_k$ deviates from its statistical value.

Figure 10. Long description

A: The x-y contour is dominated by orange and red high streamwise velocity over most of the window. Only small lower-speed patches occur near the upper wall. The absence of a distinct low-speed region makes the two modal populations difficult to separate. B: Large green and blue low-speed patches extend from the upper and lower sides into the channel, splitting the red high-speed region. Irregular black threshold contours surround several disconnected high-speed zones, causing the window-specific threshold to differ from its statistical value.

6.1. Interfacial dynamics of the QC boundary

For wall-normal conditional averaging of in-core and out-of-core profiles, zonal mean statistics are employed. At a given wall-normal position $y$ , the conditional mean streamwise velocities are designated as $\tilde {u}$ (for locations inside the QC) and $\hat {u}$ (for locations outside the core), corresponding to the zonal averages within and outside the core, respectively. The overall mean streamwise velocity $U$ is derived as a weighted combination of $\tilde {u}$ and $\hat {u}$ , with the weights determined by the intermittency factor $\gamma$ :

(6.1) \begin{align} U = \gamma \hat {u} + (1 - \gamma ) \tilde {u} \end{align}

Additionally, the zonal streamwise turbulent intensity profiles are defined to quantify fluctuations within each zone:

(6.2)

where $u$ denotes the instantaneous streamwise velocity, and $U$ represents the unconditional mean streamwise velocity across the entire flow field.

Figure 16 shows the conditional means of the streamwise velocity and its fluctuation intensities. Across all cases, the in-core mean $\tilde {u}$ is nearly uniform and close to $U_{\textit{CL}}$ , indicating that the QC remains a high-speed, low-shear zone even with rough walls. The near-wall, out-of-core mean velocity profile $\hat {u}$ follows the global mean profile of $U$ and is governed by the roughness sublayer, and typically a smaller pitch (e.g. P06) yields a lower near-wall $\hat {u}$ . Approaching the centreline, $\hat {u}$ values from different rough-wall cases tend to converge. For P12, away from the centreline, $\tilde {u}$ exceeds that of the other rough cases, suggesting more frequent downwash of high-speed fluid from the QC towards the wall under stronger meandering, whereas near the centreline, $\hat {u}$ is the smallest among the rough cases, indicating that the QC more often vacates the centreline, and the exterior flow is less accelerated.

Figure 16.

Conditional statistics of mean velocity and turbulence intensity profiles: (a,c) smooth-wall cases, (b,d) rough-wall cases, for (a,b) streamwise mean velocity profiles $U, \tilde {u}, \hat {u}$ , (c,d) streamwise turbulent intensity profiles $\overline {u'^2}, \tilde {u'}^2, \hat {u'}^2$ . For all plots, dashed lines correspond to the conditional average inside the core ( $\tilde {\vphantom{l}{\cdot }}$ ) and dotted lines correspond to the conditional average outside the core ( $\hat {\vphantom{l}{\cdot }}$ ). In (a,c), solid lines denote the unconditional (global) mean ( $\overline {\vphantom{l}{\cdot }}$ ).

Figure 16. Long description

The global mean rises from about 0.4 UCL at y+ = 10 to one at the centreline. The in-core conditional mean remains near 0.94 to 1.0 UCL. The out-of-core mean follows below the global curve through most of the channel and approaches about 0.9 UCL near the centre. B: Dashed in-core profiles for P24, P12, and P06 remain between about 0.96 and 1.0 UCL. Dotted out-of-core profiles start much lower and increase with y+. P06 is lowest at small y+, P12 is intermediate, and P24 is highest; the differences narrow toward the channel centre. C: The global intensity has a strong near-wall peak above six and then decreases. The in-core conditional intensity stays close to zero over most of the range. The out-of-core intensity is slightly above the global curve near its peak, then falls more rapidly and approaches zero toward the centreline. D: For P24, P12, and P06, dotted out-of-core intensities begin between about 3.6 and 4.5 and decrease toward the centreline. Dashed in-core intensities are below about 0.5 throughout. P24 has the largest out-of-core value closest to the wall, while P06 becomes largest over part of the intermediate range.

The in- and out-of-core conditionally averaged streamwise turbulent intensities in figure 16 show that the QC interior flow has a comparatively low disturbance level, with fluctuations concentrated at the tortuous QC boundary. Near the centreline, $\hat {u}'^2$ falls below $\tilde {u}'^2$ , in line with Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014), indicating that out-of-core structures reaching the centreline have relatively mild turbulent levels. Over rough walls, the near-wall $\hat {u}'^2$ is enhanced by the roughness sublayer, whereas at the centreline, $\hat {u}'^2$ is similar across cases. In this context, P12 exhibits larger $\tilde {u}'^2$ inside the core, indicating stronger in-core streamwise fluctuations associated with the swaying mode. For P06, the near-wall $\hat {u}'^2$ is comparatively higher, reflecting the more direct influence of the roughness sublayer.

Figure 17 shows the interfacial averages in an interface-relative coordinate. Let $y_I$ denotes the instantaneous boundary height ( $y_u$ or $y_l$ ), and define

(6.3) \begin{align} \xi = y - y_I, \quad \xi \gt 0 \text{ points into the QC}, \end{align}

with interfacial averaging along the instantaneous boundary:

(6.4) \begin{align} \big \langle q \,\big |\, \xi \big \rangle _{\varSigma } \;=\; \frac {1}{\mathcal{L}_{\varSigma }}\int _{\varSigma } q(y_I+\xi )\,\textrm{d}s, \end{align}

where $\varSigma$ is the boundary curve in the $x$ – $z$ plane, $\textrm{d}s$ is the line element, and $\mathcal{L}_{\varSigma }$ is its arc length. Here, $q$ denotes any flow quantity of interest (e.g. $u$ , $\varOmega _{z}$ or Reynolds shear stress $u'v'$ ). Equation (6.4) defines the interfacial average of $q$ evaluated at a fixed normal distance $\xi$ from the instantaneous QC boundary $y_{I}$ , integrated along the boundary $\varSigma$ , and normalised by the boundary length $\mathcal{L}_{\varSigma }$ . This operation aligns profiles from different realisations to a common interface-relative origin ( $\xi = 0$ ), so that sharp gradients across the boundary are not smeared by the meandering nature of the interface. The spanwise vorticity is

(6.5) \begin{align} \varOmega _z = \frac {\partial v}{\partial x} - \frac {\partial u}{\partial y}. \end{align}

The velocity profiles in figure 17(a) display a sharp jump at $\xi =0$ , accompanied by a peak of $\varOmega _z$ , which together mark a narrow region of concentrated shear that has been observed in the smooth-wall flows (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014; Chen et al. Reference Chen, Chung and Wan2020). All rough-wall cases show larger velocity jumps and stronger vorticity peaks across the QC boundary around $\xi \approx 0$ . This may be attributed to the roughness sublayer lowering $\hat {u}$ immediately outside the boundary, while a thinner QC places the boundary at an interior location with higher speed, which contributes to the observed increase in the velocity difference. Among the rough-wall cases, P12 exhibits the largest velocity jump and the strongest vorticity. This could be linked to the stronger meandering, which more frequently brings the QC boundary close to the wall.

Figure 17.

Conditionally averaged profiles across the core boundary ( $\xi =0$ ): (a) streamwise velocity $U/U_{\textit{CL}}$ , and (b) spanwise vorticity $\varOmega _z$ , plotted as functions of the interface-relative coordinate $\xi$ . Profiles are shown for the smooth-wall (SM) and rough-wall (P24, P12, P06) cases.

Figure 17. Long description

A: Velocity rises monotonically from the out-of-core side at negative xi to the in-core side at positive xi. The smooth-wall curve increases from about 0.81 to 1.00 UCL. Rough-wall curves begin lower and pass through a steeper transition, reaching slightly above UCL; P12 shows the largest jump. B: Omega_z rises from a moderate value at negative xi to a narrow peak slightly below xi = 0, then falls rapidly and becomes near zero or weakly negative inside the core. P12 reaches the largest peak, followed by P06 and P24, while the smooth-wall peak is lowest.

6.2. Momentum transfer on the QC boundary and its exterior

Linear stochastic estimation (LSE) (Adrian Reference Adrian1996; Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) is used to relate quadrant events to 3-D coherent structures at a near-wall height ( $y^+\approx 47$ ) and on the QC boundary. The LSE method aims to obtain the best linear approximation of the conditional average of the fluctuating velocity field at any spatial position $x'$ , given a predefined velocity event vector at a specified location $x$ . The best linear estimate of the 3-D fluctuating velocity field $ \check {u}_i'(x')$ is expressed as

(6.6) \begin{align} \check {u}_i'(x') = \sum _{j=1}^3 L_{ij}(x', x) \times u_{\kern-1pt j}^*(x), \end{align}

where $ L_{ij}(x', x)$ denotes the linear estimation coefficient matrix. Here, $ u_{\kern-1pt j}^*(x)$ is the velocity event vector at $ x$ , chosen as the second-quadrant (Q2) ejection event. To minimise the mean square error between the linear estimate $ \check {u}_i'(x')$ and the conditional average $ \langle u_i'(x') \mid u_{\kern-1pt j}^*(x) \rangle$ , the coefficients $ L_{ij}(x', x)$ are solved via the Yule–Walker equation:

(6.7) \begin{align} \sum _{j=1}^3 \langle u_k^*(x) \,u_{\kern-1pt j}^*(x) \rangle \,L_{ij}(x', x) = \langle u_i'(x') \,u_k^*(x) \rangle . \end{align}

Here, $ \langle {\cdot }\rangle$ represents the global ensemble average. The term $ \langle u_k^*(x) \,u_{\kern-1pt j}^*(x) \rangle$ is the autocorrelation of the event vector components, and $ \langle u_i'(x') \,u_k^*(x) \rangle$ is the cross-correlation between the fluctuating velocity field and the event vector. The total velocity field, combining the mean velocity and the LSE fluctuating velocity, is given by the best linear estimate of the total flow field:

(6.8) \begin{align} {u}_{i,\textit{LSE}}(x') = U_i + \check {u}_i'(x'). \end{align}

Figure 18.

Three-dimensional schematic of the LSE-conditioned $\textrm{Q}2$ structure (a) at $y^+\approx 47$ and (b) on the instantaneous QC boundary for the smooth wall (grey) and the P12 rough wall (red). Semi-transparent isosurfaces depict the conditional structures, and thin contours show intersections with orthogonal $x$ – $y$ and $y$ – $z$ planes. Spanwise-aligned rectangular blocks indicate the roughness elements.

Figure 18. Long description

Semi-transparent grey and red vortical isosurfaces represent the smooth-wall and P12 conditional structures. The structures rise downstream from the near-wall region and intersect x-y and y-z planes in elongated loops. Black spanwise bars beneath the red structure indicate the P12 roughness elements. B: The conditional structures form compact, overlapping three-dimensional lobes centred near the core-boundary height. Grey denotes the smooth-wall structure and red denotes P12. Outlines on orthogonal x-y and y-z planes show their elongated cross-sections and relative extent.

Figure 19.

The LSE-conditioned structures in 2-D $x$ – $y$ planes at $ z^+ = 157.5$ for smooth and rough walls. (a) The $\textrm{Q}2$ sections at $y^+\approx 47$ and on the instantaneous QC boundary for SM and the rough-wall configurations P06, P12 and P24. (b) The $\textrm{Q}4$ sections at $y^+\approx 47$ and on the instantaneous QC boundary for the same set of cases. The shaded region indicates the roughness height range $0 \leq y^{+} \leq k^{+}$ .

Figure 19. Long description

A: At fixed height near the wall, the coloured case outlines cluster around an inclined elongated structure; rough-wall cases extend farther vertically than SM. At the QC interface, a second group of compact inclined outlines appears near the dotted interface line. A grey band marks the roughness-height range. B: Near the wall, the Q4 outlines form inclined structures with case-dependent streamwise extent, including additional elongated rough-wall contours close to the roughness band. Near the dotted QC-interface line, the outlines are compact and irregular, with P24 and P06 extending farther than SM and P12 in different directions.

Vortices in the reconstructed fields are visualised using the swirling strength criterion $\lambda _{ci}$ (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999; Christensen & Adrian Reference Christensen and Adrian2001). The swirling strength is defined as the imaginary part of the complex eigenvalue pair of the local velocity gradient tensor $\boldsymbol{\nabla }{u}$ . A vortex core is identified, where $\boldsymbol{\nabla }{u}$ possesses a pair of complex conjugate eigenvalues, and $\lambda _{ci}$ is taken as the magnitude of their imaginary part. Regions where $\lambda _{ci} \gt 0$ indicate local swirling motion. The squared swirling strength $\lambda _{ci}^{2}$ is used as the isosurface variable, following the convention of Christensen & Adrian (Reference Christensen and Adrian2001). The conditioning event is taken in the $\textrm{Q}2$ quadrant at the red cross location in figure 20 below, identified from the covariance integrand $u'v'\,P(u',v')$ . To enable consistent cross-case comparisons, isosurfaces are rendered at a fixed percentage of the per-case maximum $\lambda _{ci}^{2}$ rather than at a uniform absolute threshold. Values of $5\,\%$ of the case maximum are used for the fixed-height conditioning cases, and $2\,\%$ for the QC-boundary conditioning cases, with the resulting structural extents $L_x^+$ and $L_y^+$ summarised in table 4.

Figure 18(a) shows, in three dimensions, that for P12, the upstream legs of the hairpin structure extend into the gaps between roughness elements, consistent with periodic wakes that form behind each element and interact with the hairpin structure. The hairpin head rises further from the wall, increasing the wall-normal extent of the structure. In $x$ – $y$ planes at $y^+\approx 47$ , $z^+=157.5$ shown in figure 19(a), the upper extent of the $\textrm{Q}2$ structure is comparable across the rough-wall cases, while the streamwise extent and the wall-normal penetration vary with pitch, as quantified in table 4. Among them, P24 displays the longest upstream-leg extension ( $L_x^+\approx 250$ ) and the deepest penetration into the cavities, enabled by the larger element spacing that accommodates more persistent wakes, while P06 yields the most compact footprint ( $L_x^+\approx 199$ ). The modulation of $\textrm{Q}4$ at this height is stronger than that of $\textrm{Q}2$ , with a longer upstream tail, and a clearer wall-normal penetration between elements as $p/k$ increases (figure 19 b), with P24 exhibiting the largest streamwise extent ( $L_x^+\approx 350$ ). These patterns indicate that near-wall momentum transfer is sensitive to the element spacing.

Table 4.

The LSE conditioning parameters and reconstructed-structure metrics. Here, $|E|=\alpha (u^{*2}+v^{*2})^{1/2}$ is the event-vector amplitude (wall units, $\alpha =3$ ), $P_0$ is the probability density at the conditioning point, obtained by bilinear interpolation of the joint PDF, and $L_x^+$ and $L_y^+$ are streamwise and wall-normal extents of the primary $\lambda _{ci}^2$ isosurface contour extracted at $z^+= 157.5$ , using a threshold $5\,\%$ of the case-maximum $\lambda _{ci}^2$ for the fixed-height cases, and $2\,\%$ for the QC-boundary cases.

Table 4. Long description

Columns are SM, P24, P12, and P06. For Q2 at y+ about 47, |E| = 9.5, 9.3, 7.6, 9.1; P0 = 0.025, 0.020, 0.026, 0.028; Lx+ = 234.5, 250.4, 232.0, 199.1; Ly+ = 78.7, 130.1, 124.0, 115.9. For Q4 at y+ about 47, |E| = 6.6, 7.6, 6.6, 7.3; P0 = 0.057, 0.038, 0.046, 0.045; Lx+ = 258.0, 349.9, 277.2, 228.6; Ly+ = 99.0, 146.5, 146.6, 143.5. For Q2 at the QC boundary, |E| = 3.7, 2.9, 2.9, 2.7; P0 = 0.169, 0.375, 0.268, 0.338; Lx+ = 359.2, 427.7, 335.8, 330.1; Ly+ = 195.2, 153.9, 122.0, 163.3. For Q4 at the QC boundary, |E| = 5.5, 6.4, 5.5, 5.3; P0 = 0.065, 0.058, 0.060, 0.072; Lx+ = 326.0, 355.5, 189.3, 315.7; Ly+ = 213.1, 118.8, 75.3, 204.2.

The Q2-conditioned vortical structure at the QC boundary is compared for SM and P12 in figure 18(b). Within the QC interior, the structure is predominantly streamwise-elongated with limited wall-normal penetration, reflecting the quiescent, high-speed character of the core. On the exterior side, a pair of inclined counter-rotating quasi-streamwise vortex legs is present, consistent with the organisation of ejection events in smooth-wall flows (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999; Christensen & Adrian Reference Christensen and Adrian2001). The spatial extent of these legs is appreciably larger for the smooth wall, indicating that the ejection-associated structure is more spatially confined in the rough-wall case.

The $x$ – $y$ cross-sections in figure 19 further quantify these differences. For the Q2 event at the QC boundary (figure 19 a), all rough-wall cases exhibit a reduced wall-normal extent relative to SM, with P12 showing the most pronounced reduction, while P24 retains a comparatively extended streamwise reach. For the Q4 event (figure 19 b), P06 most closely approaches the smooth-wall reference in both $L_x^+$ and $L_y^+$ , whereas P12 yields the most compact structure in both metrics, consistent with its most disrupted QC geometry among the rough cases.

Figure 20 presents the covariance integrand $u'v'\,P(u',v')$ sampled along the instantaneous QC boundary. The red cross marks the conditioning event identified from the joint PDF, whose amplitude $\vert {}E\vert {}$ and probability density $P_0$ are reported in table 4. The distributions exhibit a general asymmetry: the positive- $u'$ side extends to larger amplitudes than the negative- $u'$ side, consistent with the systematically larger $\vert {}E\vert {}$ for Q4 than for Q2 across all cases. For the Q2 contributions, the smooth-wall distribution is the most extended in the streamwise direction, whereas the rough-wall cases are contracted in $\vert {}u'\vert {}$ but enhanced in wall-normal amplitude $v'$ , most markedly for P12. The low $P_0$ for P12 reflects a broad redistribution of events towards larger amplitudes, rather than a concentration at the most probable velocity. For the Q4 contributions, the rough-wall distributions are displaced towards larger amplitudes of both $\vert {}u'\vert {}$ and $\vert {}v'\vert {}$ relative to SM, with P12 exhibiting the strongest bias toward elevated $v'$ , and P06 remaining closest to the smooth-wall reference. This case ordering is consistent with the ranking of interfacial velocity jumps in figure 17(a).

Figure 20.

Contours of the covariance integrand $u'v'\,P(u',v')$ on the instantaneous QC boundary. Contour levels are shown with an increment $\varDelta =0.01$ for negative values in the range $[-0.08,-0.02]$ and for positive values in the range $[0.01,0.03]$ . Plots show the covariance integrand for (a) SM, (b) P24, (c) P12, and (d) P06.

Figure 20. Long description

A: The largest-magnitude contours lie in Q2, where u-prime is negative and v-prime positive, and in Q4, where u-prime is positive and v-prime negative. Smaller positive contours appear in Q1 and Q3. A red cross marks the selected Q2 conditioning event. B: The topology is similar to the smooth-wall case but the Q2 lobe is narrower and concentrated closer to u-prime = 0, while the Q4 lobe is broad and extends to large positive u-prime and negative v-prime. Positive Q1 and Q3 lobes remain smaller. C: A narrow high-gradient Q2 lobe lies just left of u-prime = 0 and above v-prime = 0. The broad Q4 lobe occupies positive u-prime and negative v-prime and has the largest extent. Smaller positive lobes occur in Q1 and Q3. D: The Q2 lobe is narrow and nearly vertical at small negative u-prime, while the Q4 lobe is broad at positive u-prime and negative v-prime. The Q1 and Q3 positive lobes are weaker. The overall arrangement closely resembles P12 with modest changes in contour extent.

The wall-normal evolution of quadrant fractions complements these observations. The fractional contributions $\langle u'v'\rangle _Q/\langle u'v'\rangle$ versus $y^+$ in figure 21 show that P24 and P06 approximately follow the smooth-wall reference, with modest increases of $\textrm{Q}2$ and $\textrm{Q}4$ in the QC exterior. Case P12 departs for $y^+\gtrsim 200$ , and all four quadrant contributions separate from the smooth-wall curves. This indicates that near the QC boundary, stronger interfacial shear modifies the composition of momentum-carrying events, echoing the roughness-sensitive Q2 behaviour noted by Zhang et al. (Reference Zhang, Feng, Huo and Jiang2026) in boundary layers.

Strong-event weighting is assessed using the hyperbolic-hole filter following Lu & Willmarth (Reference Lu and Willmarth1973). For quadrant $Q$ at fixed $y$ ,

(6.9) \begin{align} \frac {\langle u'v'\rangle _Q(H)}{\langle u'v'\rangle } \;=\; \frac {{\displaystyle \iint}_{(u',v')\in Q} \, u'v'\,P(u',v'\mid y)\,I(H)\,\textrm{d}u'\,\textrm{d}v'}{\langle u'v'\rangle }, \\[-28pt] \nonumber \end{align}

(6.10) \begin{align} I(H)=\big (|u'v'|\gt H\,\langle {-u'v'}\rangle \big ). \\[0pt] \nonumber \end{align}

The hyperbolic-hole analysis at the instantaneous QC boundary (figure 22) shows that the dominant stress-carrying events are concentrated in Q1 and Q4, with Q2 contributing only a minor fraction of the total stress. This is consistent with the rightward bias of the covariance integrand in figure 20, reflecting the character of the QC boundary as a high-speed interface across which fast fluid impinges upon the slower exterior. Among the cases, the smooth-wall Q2 contribution shifts towards larger $H$ relative to all rough-wall cases, indicating that ejection stress is more concentrated in strong events for the smooth wall, whereas rough-wall ejections are comparatively diffuse. The case ordering of Q4 contributions follows the amplitude redistribution pattern in figure 20: P12 exhibits the largest Q4 contribution across the full range of $H$ , and P06 remains closest to the smooth-wall reference.

Figure 21.

Fractional quadrant contributions to the Reynolds shear stress plotted versus the wall coordinate $y^+$ . The ordinate is $\langle u'v'\rangle _{Q}/\langle u'v'\rangle$ , with $\langle u'v'\rangle _{Q}=\iint _{(u',v')\in Q} u'v'\,P(u',v'\mid y)\,\textrm{d}u'\,\textrm{d}v'$ . Plots are arranged by quadrant: (a) $\textrm{Q}2$ , (b) $\textrm{Q}1$ , (c) $\textrm{Q}3$ , (d) $\textrm{Q}4$ . Curves correspond to SM and the rough-wall configurations P24, P12 and P06.

Figure 21. Long description

A: The Q2 fraction rises rapidly from the wall to a broad level around 0.7 to 0.9, then increases steeply near the channel centre. P12 is generally highest, followed by P24 and P06, while SM is lowest over much of the profile. B: The Q1 fraction remains weakly negative over most of the wall-normal range and becomes more negative near the channel centre. P12 is consistently the most negative, P24 is next, and SM and P06 have smaller magnitudes. C: Q3 contributions remain near minus 0.1 to minus 0.25 through much of the channel. P12 is generally most negative, with P24 next. All curves turn sharply downward as y+ approaches the centreline. D: The Q4 fraction quickly reaches about 0.55 to 0.7, remains comparatively flat through the middle region, and rises steeply near the centreline. P12 is generally highest and SM lowest, with P24 and P06 between them.

Figure 22.

Hyperbolic-hole filtered quadrant contributions on the instantaneous QC boundary. The ordinate is $\langle u'v'\rangle _Q(H)/\langle u'v'\rangle$ , with the indicator $I(H)=(|u'v'|\gt H\,\langle {-u'v'}\rangle )$ . Plots are for (a) $\textrm{Q}2$ , (b) $\textrm{Q}1$ , (c) $\textrm{Q}3$ , (d) $\textrm{Q}4$ . For display purposes, the abscissa is reversed for $\textrm{Q}2$ and $\textrm{Q}3$ , and the ordinate sign is reversed for $\textrm{Q}1$ and $\textrm{Q}4$ . Curves correspond to SM, P24, P12 and P06.

Figure 22. Long description

A: With the horizontal axis reversed from H = 50 to 0, all curves remain near zero for strong filtering and rise only at low H. At H = 0, the smooth-wall Q2 fraction is largest, while the rough-wall cases are smaller and close together. B: At H = 0, P12 has the largest negative Q1 fraction, followed by P24, P06, and SM. The magnitude decreases approximately exponentially with increasing H, and all curves approach zero by H around 40 to 50. C: With the reversed horizontal axis, the curves stay near zero for H above roughly 10 and become negative only as H approaches zero. SM has the largest magnitude at H = 0, while P24, P12, and P06 are smaller. D: At H = 0, P12 has the largest Q4 contribution, followed by P24, P06, and SM. All curves decrease monotonically with H and are close to zero by H about 40 to 50.

Quadrant fractions across the interface-relative coordinate $\xi$ (negative outside the core, and positive inside) in figure 23 highlight where roughness effects are concentrated. Stresses vary sharply on the exterior side, and become nearly case-independent inside the QC. Immediately outside the interface ( $\xi \approx -0.1$ ), P12 exhibits the largest magnitudes across all quadrants, which indicates that all types of events are intensified at the boundary for this case. This is consistent with the larger velocity jump and the stronger spanwise vorticity peak observed for P12 in figure 17(a). For P06, increases in quadrant contributions occur slightly farther from the interface on the exterior side ( $\xi \approx -0.2$ ), which aligns with the influence of the roughness sublayer – its elevation promotes momentum exchange that extends towards the QC boundary.

Figure 23.

Quadrant-conditioned fraction of the Reynolds shear stress across the QC interface, plotted against the interface-relative coordinate $\xi$ (positive into the core, and $\xi =0$ at the instantaneous interface, as in figure 15). The ordinate is $\langle u'v'\rangle _Q/\langle u'v'\rangle$ . Plots are for (a) $\textrm{Q}2$ , (b) $\textrm{Q}1$ , (c) $\textrm{Q}3$ , (d) $\textrm{Q}4$ . Curves correspond to SM, P24, P12 and P06.

Figure 23. Long description

A: From xi = −0.2 toward the interface, Q2 fractions decrease to minima slightly below zero. They then rise sharply across xi = 0 and level near 0.9 to 1.1 inside the core. P12 and P06 are highest far outside the core, while the cases become closer inside. B: Q1 fractions are negative on both sides of the interface. Their magnitude increases toward xi = 0, reaches a sharp minimum at or just outside the interface, and then decreases inside the core. P12 is most negative, followed by P24 and P06, with SM least negative. C: The Q3 fractions rise from larger negative values at xi = −0.2 to a broad maximum near xi = −0.05, then decline slightly across the interface. P12 has the largest negative magnitude outside and inside, while SM is generally least negative. D: Q4 fractions increase from the outer side to maxima near xi = −0.02, then fall sharply through the interface and level at positive values. P12 has the largest peak, followed by P24, P06, and SM.

The interfacial stresses discussed above form part of the total stress balance. To establish how each stress component contributes to the momentum transport across roughness configurations, figure 24 presents the double-averaged decomposition

(6.11) \begin{align} \tau _{\textit{total}}^{+} = \tau _\nu ^{+} + \tau _R^{+} + \tau _D^{+}, \end{align}

where $\tau _\nu ^{+}=(\nu \,\textrm{d}{U}/\textrm{d}y)/u_\tau ^2$ is the viscous stress, $\tau _R^{+}=-\langle \overline {u'v'}\rangle /u_\tau ^2$ is the Reynolds shear stress, and $\tau _D^{+}=-\langle \overline {u}''\,\overline {v}''\rangle /u_\tau ^2$ is the dispersive shear stress, with $\overline {u}''=\overline {u}-U$ and $\overline {v}''=\overline {v}-V$ denoting the departures of the time-averaged velocities from their spatial averages. The three terms represent, respectively, the contributions of viscous friction, turbulent fluctuations and time-averaged spatial flow inhomogeneity to the total momentum transport over rough walls.

For the rough-wall cases, $\tau _\nu ^+$ exhibits a positive peak near the roughness crest, and decays rapidly above it, with the peak width narrowing as $p/k$ decreases, while its direct contribution to the outer-layer momentum balance remains limited.

The profile of $\tau _D^+$ differs markedly among the three rough-wall cases. For P06 (figure 24 d), $\tau _D^+$ attains a small negative value near the crest height, and decays rapidly above it, remaining the smallest in magnitude among the rough-wall cases across the full wall-normal extent. The distributions of both $\tau _D^+$ and $\tau _R^+$ remain the closest to the smooth-wall reference among the rough-wall cases, consistent with the near-smooth-wall QC geometry and interfacial statistics established in § 5–§ 6.1.

For P12 (figure 24 c), the negative peak of $\tau _D^+$ near the crest height is the largest in absolute value among all three rough-wall cases, indicating that the spatially non-uniform time-averaged flow transports a greater fraction of momentum upwards against the mean gradient in this configuration. The elevated outer-layer $\tau _R^+$ reflects the intensified wall-normal turbulent motions and enhanced interfacial shear documented in § 5–§ 6.1, both of which are associated with the pronounced wall-ward QC excursions of P12.

For P24 (figure 24 b), the negative region of $\tau _D^+$ is broader in wall-normal extent, and its influence persists further into the outer layer relative to the other cases. The larger element spacing allows the time-averaged recirculation within each cavity to develop more fully, sustaining a spatially heterogeneous mean flow pattern over a greater wall-normal distance.

Figure 24.

Stress decomposition profiles versus $y^{+}-k^{+}$ for (a) SM, (b) P24, (c) P12 and (d) P06. The curves show the viscous stress $\tau _\nu ^{+}$ (solid line), the Reynolds shear stress $\tau _R^{+}$ (dashed), the dispersive shear stress $\tau _D^{+}$ (dot-dashed, rough-wall cases only), and the total stress $\tau _{\textit{total}}^{+}$ (dotted). The grey vertical line at $y^{+}-k^{+}=0$ marks the roughness crest height. The legend shown in figure (b) applies to all plots.

Figure 24. Long description

A: Viscous stress starts near one and rapidly decays to nearly zero. Reynolds shear stress starts near zero, rises to carry most of the stress, and then decreases toward the centreline. Their sum, the dotted total stress, follows an approximately linear decline from the wall to zero. B: Near y+ − k+ = 0, viscous stress has a local positive peak, Reynolds stress rises rapidly, and dispersive stress becomes negative before recovering toward zero. Away from the crest, Reynolds stress dominates and the total stress decreases approximately linearly. C: Immediately around y+ − k+ = 0, dispersive stress reaches a large negative minimum and viscous stress has a sharp positive peak. Reynolds stress rises through the roughness layer and dominates farther from the wall. The total stress remains close to a linear outer decline. D: Viscous stress peaks just above the crest, while dispersive stress shows a modest negative excursion and returns quickly toward zero. Reynolds stress becomes dominant outside the roughness layer, and the total stress decreases almost linearly with wall distance.

Overall, under a strong swaying mode, P12 shows the expected signature at the QC boundary: stronger interfacial shear leads to larger deviations from the mean and therefore to stronger events. Case P06 remains close to the smooth-wall behaviour in both structure and statistics, in line with its QC geometry. Taken together, the covariance integrand, the conditional structures and the quadrant analyses suggest that similarity in QC geometry is linked to similarity in momentum transfer. For the present configuration, momentum transfer near the wall appears to be primarily influenced by the roughness sublayer, while further away from the wall, its behaviour is more aligned with the evolution of the meandering QC.