3. Results and discussion

The reverse transition is achieved using a computational procedure known as the adiabatic protocol (Philip & Manneville Reference Philip and Manneville2011; Ishida et al. Reference Ishida, Brethouwer, Duguet and Tsukahara2017; Tsukahara et al. Reference Tsukahara, Tomioka, Ishida, Duguet and Brethouwer2018; Gokul & Narasimhamurthy Reference Gokul and Narasimhamurthy2022). This procedure begins with a turbulent flow state, from which $\textit{Re}$ is gradually reduced in steps, transitioning the system from a turbulent state to a laminar–turbulent state, and finally to a laminar state. For 3-D k-type roughness, we initiate the adiabatic protocol from $\textit{Re}=400$ , where the flow is turbulent with no significant laminar regions. The $\textit{Re}$ is then reduced in steps of $25$ . After every decrement, the simulation is run until statistical stationarity is confirmed.

Figure 3 shows the outcome of the adiabatic protocol for the 3-D k-type roughness. The contours of streamwise velocity at the midgap ( $z/h=1$ ) are used to visualize the flow structures. The turbulent flow at $\textit{Re}=400$ features high- and low-velocity streaks elongated in the $x$ -direction and alternating in the $y$ -direction. It should be noted that, unlike the 2-D k-type roughness, the elongated streaks in the 3-D k-type roughness are irregularly distributed along the $y$ direction. The streaks in the smooth zones appear broader and more elongated than those in the rough zones. Reducing $\textit{Re}$ to $375$ results in the formation of laminar patches within a turbulent environment. Although laminar spots exist in many areas of the domain, they are yet to self-organize into bands. Gomé et al. (Reference Gomé, Tuckerman and Barkley2023b ) observed that the laminar spots are initially isolated with short lifespans, which later become several and long-lasting, eventually covering a large area and forming laminar–turbulent bands when the $\textit{Re}$ is reduced. A further decrement to $\textit{Re}=350$ yields laminar and turbulent regions coexisting as oblique bands. These bands are also sustained in the flow field at $\textit{Re}=325$ . Further reduction in $\textit{Re}$ leads to relaminarization of the flow at $\textit{Re} = 300$ . Owing to the periodic boundary conditions in the $x$ - and $y$ -directions, the laminar–turbulent bands have wavelengths equal to the corresponding domain dimensions (Gokul & Narasimhamurthy Reference Gokul and Narasimhamurthy2022). As a result, the bands align along the diagonal direction, with the pattern angle approximately equal to $27 ^{\circ }$ . The wavelengths reported in this study do not exactly match those from the experiments by Prigent et al. (Reference Prigent, Grégoire, Chaté and Dauchot2003) but are of the same order of magnitude. Previous DNS studies by Manneville & Rolland (Reference Manneville and Rolland2011) and Ishida et al. (Reference Ishida, Brethouwer, Duguet and Tsukahara2017), etc., have also reported that periodic boundary conditions influence the pattern wavelength. Although the discrepancy in wavelength is a common limitation of DNS studies, it does not undermine the results from this study, as it successfully captures all other features of the pattern. The 3-D k-type roughness sustains laminar–turbulent bands in the transitional Reynolds number range $\textit{Re} \in [325, 350]$ (the transitional range is determined by visualizing the flow in the $x{-}y$ and the $y{-}t$ planes by ensuring the sustained presence of the oblique bands over long simulation times). The smooth PCF sustains these patterns over a broader range, $\textit{Re} \in [325, 400]$ (Ishida et al. Reference Ishida, Brethouwer, Duguet and Tsukahara2017). Compared with smooth PCF, the transition from the turbulent state to the laminar–turbulent pattern stage occurs at a lower Reynolds number for the 3-D k-type roughness. This reduction in the transition Reynolds number can be linked to the vortex-shedding ability of k-type roughness, which can aid streamwise vorticity regeneration and help turbulence persist at lower Reynolds numbers (Gokul & Narasimhamurthy Reference Gokul and Narasimhamurthy2024). On the other hand, the 2-D k-type roughness sustains these patterns within a similar narrow range as the 3-D configuration, but at lower Reynolds numbers, $\textit{Re} \in [300, 325]$ (Gokul & Narasimhamurthy Reference Gokul and Narasimhamurthy2024). The presence of smooth zones within the 3-D roughness configuration reduces its overall effectiveness, making it less capable of sustaining oblique bands at lower Reynolds numbers compared with 2-D k-type roughness. Thus, the transitional range for the 3-D k-type roughness falls between that of the smooth PCF and the 2-D k-type roughness, with some overlap.

Figure 3. Instantaneous streamwise velocity in the midgap ( $z=h$ ) plane for the 3-D k-type roughness at (a) Re = 400, (b) Re = 375, (c) Re = 350 and (d) Re = 325. The regions corresponding to the rough zones are highlighted using rectangular boxes.

Figure 4 shows the spatiotemporal diagram at a Reynolds number just below the upper threshold, $\textit{Re}_t$ (the Reynolds number above which flow is featureless), for the 3-D k-type roughness alongside that of the 2-D k-type roughness for comparison. These diagrams are obtained by recording streamwise velocities along a spanwise line defined at $x/h = 68$ and $z/h = 1$ (located at the centre of the domain) over a total duration of $5500h/U_{w}$ . Interestingly, the alternate laminar and turbulent regions appear obliquely, similar to those in the $x{-}y$ plane. The obliqueness of the laminar–turbulent coexistence in the $y{-}t$ plane is due to the propagation of the pattern in the streamwise direction. The flow is initially fully turbulent for a brief period, after which the laminar spots emerge within the turbulent surroundings. These spots appear predominantly in the smooth zones, as the rough zones are more effective in sustaining turbulence. The laminar spots later combine and form the alternate laminar–turbulent bands. In both rough cases, competing patterns emerge with opposite angles. However, competition ceases, and a pattern oriented along one of the domain diagonals emerges (both orientations are equally likely) in the case of 2-D k-type roughness. Similar competition in smooth PCF has been reported by Prigent et al. (Reference Prigent, Grégoire, Chaté and Dauchot2003), which eventually gives way to a regular pattern in the flow field. In contrast, for the 3-D k-type roughness, the competition between opposite angles persists indefinitely. In addition to the spatiotemporal diagrams, the temporal variation of velocity signals provides valuable insights. We present velocity signals from a point in the rough zone and another in the smooth zone, both located on the same spanwise line specified by $x/h=68$ and $z/h=1$ , where we previously recorded streamwise velocity data for the spatiotemporal diagrams. Figure 5 shows the streamwise velocity at $y/h = 17$ in the rough zone and $y/h = 34$ in the smooth zone for the 3-D k-type roughness (the projections of these locations on the rough wall are shown in figure 1 c). For comparison, the corresponding signal at $y/h=34$ is also shown for the 2-D k-type roughness. In the signals, the flat segments represent laminar regions, while the oscillatory segments indicate turbulent regions. The flat segments are initially very few, as the flow is predominantly turbulent. They appear mainly in the velocity signal at $y/h = 34$ (smooth zone), reinforcing the earlier observation that the laminar spots predominantly emerge in the smooth zones. However, after pattern formation, the signal at $y/h = 34$ (smooth zone) becomes unexpectedly dominated by oscillatory segments. In contrast, the signal at $y/h = 17$ (rough zone) exhibits alternating flat and oscillatory segments, corresponding to laminar and turbulent bands, respectively, a behaviour also observed in the 2-D k-type roughness. The unexpected dominance of oscillatory segments in the velocity signal from the smooth zone could indicate that it is more strongly influenced by the competition between patterns of opposite orientations compared with the rough zone.

Figure 4. Instantaneous streamwise velocity in a spatiotemporal plane for (a) 3-D k-type roughness and (b) 2-D k-type roughness at a Reynolds number slightly below their respective upper threshold ( $\textit{Re}_{t}$ ). The regions corresponding to the rough zones are highlighted using rectangular boxes.

Figure 5. Time traces of streamwise velocity at ( $x = 68h$ , $z = h$ ) for (a) 3-D k-type roughness, with data taken at $y/h=17$ (rough zone) and $y/h=34$ (smooth zone), and (b) 2-D k-type roughness at $y/h=34$ , at a Reynolds number slightly below their respective upper threshold ( $\textit{Re}_{t}$ ). To prevent overlap, the data points for the 3-D k-type roughness at $y/h=34$ are vertically shifted by one unit.

Does the competition between patterns of opposite orientations in the 3-D k-type roughness diminishes upon lowering the Reynolds number? The spatiotemporal diagram at a Reynolds number just above the lower threshold, $\textit{Re}_{g}$ (the Reynolds number below which the flow is laminar), indicates that the competition persists (see figure 6 a). Although the competition occasionally pauses for brief intervals in between, it invariably resumes after some time. In contrast, for 2-D k-type roughness, the stable pattern that initially forms remains intact and does not revert to a state of competition (see figure 6 b). The velocity signal for the 3-D k-type roughness at $y/h = 34$ (smooth zone, figure 7 a) is still dominated by the oscillatory segments; however, the number of flat portions is noticeably higher than those observed in figure 5(a) for a Reynolds number just below $\textit{Re}_{t}$ . At $y/h = 17$ (rough zone, figure 7 a), the signal still shows alternating flat and oscillatory segments, indicating laminar and turbulent bands, respectively. The velocity signal for the 2-D k-type roughness (see figure 7 b) shows a similar behaviour but with more consistent flat segments due to a stable pattern.

Figure 6. Instantaneous streamwise velocity in a spatiotemporal plane for (a) 3-D k-type roughness and (b) 2-D k-type roughness at a Reynolds number slightly above their respective lower threshold ( $\textit{Re}_{g}$ ).

Figure 7. Time traces of streamwise velocity at ( $x = 68h$ , $z = h$ ) for (a) 3-D k-type roughness, with data taken at $y/h=17$ (rough zone) and $y/h=34$ (smooth zone), and (b) 2-D k-type roughness at $y/h=34$ , at a Reynolds number slightly above their respective lower threshold ( $\textit{Re}_{g}$ ). To prevent overlap, the data points for the 3-D k-type roughness at $y/h=34$ are vertically shifted by one unit.

Below the lower threshold $\textit{Re}_{g}$ , the turbulent regions disintegrate as laminar patches appear in the turbulent bands (see figure 8). These laminar patches first appear in the smooth zones of the 3-D k-type roughness (see figure 8 a). As a result, the velocity signal at $y/h=34$ (smooth zone, figure 9 a), previously dominated by the oscillatory segments at Reynolds numbers just below $\textit{Re}_{t}$ and above $\textit{Re}_{g}$ (see figures 5 a and 7 a for the velocity signals just below $\textit{Re}_{t}$ and above $\textit{Re}_{g}$ ), now exhibits significant flat segments alongside the oscillatory segments. Similarly, the signal at $y/h=17$ (rough zone, figure 9 a) shows more flat segments than those at higher transitional Reynolds numbers (see figures 5 a and 7 a for velocity signals at higher transitional Reynolds numbers), which indicates the impending laminarization. Notably, the competition between patterns of opposite angles persists in the 3-D k-type roughness, even during the disintegration of laminar–turbulent bands, whereas no such competition is observed in the 2-D configuration. Eventually, all signals become entirely flat, indicating relaminarization.

Figure 8. Instantaneous streamwise velocity in a spatiotemporal plane for (a) 3-D k-type roughness and (b) 2-D k-type roughness at a Reynolds number below their respective lower threshold ( $\textit{Re}_{g}$ ).

Figure 9. Time traces of streamwise velocity at ( $x = 68h$ , $z = h$ ) for (a) 3-D k-type roughness, with data taken at $y/h=17$ (rough zone) and $y/h=34$ (smooth zone), and (b) 2-D k-type roughness at $y/h=34$ , at a Reynolds number below their respective lower threshold ( $\textit{Re}_{g})$ . To prevent overlap, the data points for the 3-D k-type roughness at $y/h=34$ are vertically shifted by one unit.

Figure 10 shows the frequency spectra of the streamwise velocity for the rough cases, obtained through fast Fourier transform – an algorithm that transforms a signal from the time domain to the frequency domain, thereby revealing the various frequency components and their associated energy. The velocity signals from the locations $y/h=17$ in the rough zone and $y/h=34$ in the smooth zone for the 3-D k-type roughness, along with the signal from $y/h=34$ for the 2-D k-type roughness are included in the analysis. We focus on the signals at the Reynolds number just above $\textit{Re}_{g}$ , where the laminar–turbulent patterns remain sustained throughout the simulation. The portion of the signal between $1500h/U_{w}$ and $5500h/U{w}$ is used for the analysis, with the initial part excluded to avoid transients. The frequency spectra of the signal at $y/h=17$ (rough zone) reveal a dominant frequency of 0.0032 (see figure 10 a) – a value also observed in the 2-D k-type roughness (see figure 10 c), which corresponds to its pattern frequency. No other peaks in these spectra exhibit energies comparable to those of this dominant frequency. In contrast, the most energetic frequency corresponding to the signal at $y/h=34$ (smooth zone) is 0.0036 (see figure 10 b), along with at least one additional peak whose energy is comparable to that of the dominant frequency. The difference in peak frequencies between the rough and smooth zones can likely give rise to distinct pattern speeds in each region. Moreover, the multiple energetic peaks in the smooth zone, combined with the lack of flat segments in the signal (discussed in conjunction with figure 7 a), can be an indication of a set of complex events in this region that likely correspond to the ongoing competition between patterns of opposite angles.

Figure 10. The frequency spectra for (a–b) 3-D k-type roughness (at $y/h=17$ in the rough zone and $y/h=34$ in the smooth zone) and (c) 2-D k-type roughness (at $y/h=34$ ) at a Reynolds number slightly above their respective lower threshold ( $\textit{Re}_{g}$ ).

In the transitional regime, the small scales of streaks and the large scales corresponding to the long-wavelength patterns have a significant scale separation (Tsukahara et al. Reference Tsukahara, Tomioka, Ishida, Duguet and Brethouwer2018; Gomé et al. Reference Gomé, Tuckerman and Barkley2023a ). This separation justifies the use of a low-pass filter to isolate the large-scale flow associated with the pattern stage (Duguet & Schlatter Reference Duguet and Schlatter2013). Figure 11 shows the wall-normal averaged large-scale flow vectors, superimposed on the streamwise velocity fluctuations at different time instances for $\textit{Re}=325$ , which lies just above the lower threshold $\textit{Re}_{g}$ for the 3-D k-type roughness. The angle $\beta$ denotes the orientation of the large-scale flow vectors relative to the streamwise direction. The angle $\theta$ , which corresponds to the peak of the probability density function of $\beta$ , is shown in figure 12. This angle indicates the dominant direction of the large-scale flow vectors. At $t=3000h/U_{w}$ (see figure 11 a), the oblique bands are oriented along one of the domain diagonals, with no apparent competing fronts of opposite orientation. At this instant, the large-scale flow vectors near the laminar–turbulent interfaces are not perfectly parallel to the diagonal direction. Therefore, the angle with the peak probability ( $\theta =12.63 ^{\circ }$ ) does not match the pattern angle ( $27 ^{\circ }$ ). By $t=4000h/U_{w}$ (figure 11 b), the oblique bands align with the opposite domain diagonal, exhibiting no competing fronts. The large-scale flow vectors near the interfaces are again not perfectly aligned with the diagonal direction, reflected in the deviation of the angle corresponding to the peak probability ( $\theta =15.22 ^{\circ }$ ) away from the pattern angle. At $t=5000h/U_{w}$ and $t=6000h/U_{w}$ (figures 11 c and 11 d, respectively), the patterns of opposite angles coexist, and their competition results in a pronounced deviation of the angles corresponding to the peak probability ( $\theta =2.77 ^{\circ }$ and $\theta =6.39 ^{\circ }$ , respectively) from the pattern angle. A comparison of the large-scale flow vectors and the corresponding probability density functions between the 3-D k-type roughness and its 2-D counterpart is shown in Appendix A (see figures 18 and 19). In the case of 2-D k-type roughness, the large-scale flow vectors near the interfaces are parallel to the diagonal direction (see figure 18 b), and the angle corresponding to the peak probability matches well with the pattern angle (see figure 19). However, in the 3-D k-type roughness, the large-scale flow vectors are misaligned with the diagonal direction (see figure 18 a). Interestingly, even in the absence of competing fronts, large-scale flow vectors are not perfectly parallel to the diagonal direction. Consequently, the peak of the probability density function shifts away from the pattern angle.

Figure 11. Wall-normal averaged large-scale flow associated with the laminar–turbulent bands in the 3-D k-type roughness for $\textit{Re}=325$ at $t$ = (a) $3000 h/U_{w}$ , (b) $4000 h/U_{w}$ , (c) $5000 h/U_{w}$ and (d) $6000 h/U_{w}$ . The streamwise velocity fluctuations in the midgap plane ( $z=h$ ) are utilized to depict the laminar–turbulent bands. The regions corresponding to the rough zones are highlighted using rectangular boxes.

Figure 12. The peak ( $\theta$ ) of the probability density function of the large-scale flow angles at time instances $3000 h/U_{w}$ , $4000 h/U_{w}$ , $5000 h/U_{w}$ and $6000 h/U_{w}$ .

We now look closer at the evolution of bands between $t=3000h/U_{w}$ and $t=4000h/U_{w}$ , during which the pattern reorients itself from one domain diagonal to the opposite. Figure 13 shows the contours of instantaneous streamwise velocity at $z/h=0.1$ at different time instances between $t=3000h/U_{w}$ and $t=4000h/U_{w}$ for $\textit{Re}=325$ . The choice of the $z/h=0.1$ plane for visualization is to clearly distinguish the flow structures over the rough and smooth zones of the 3-D k-type roughness. At $3000h/U{w}$ , the pattern is aligned along one of the domain diagonals. However, competing fronts with the opposite orientation periodically originate from the smooth zone (at $t=3250h/U_{w}$ and $t=3550h/U_{w}$ ). The smooth zone appears to be more prone to the formation of competing fronts, which can potentially destabilize the established pattern. This tendency may explain the oscillatory nature of the velocity signal and the presence of multiple high-energy peaks in the corresponding spectra, as discussed earlier. Once initiated, competing fronts in the smooth zone may trigger the formation of similarly oriented fronts within the rough zones (see $t=3700h/U_{w}$ and $t=3850h/U_{w}$ ). The turbulent regions in the rough zones gradually adjust, forming a part of the newly evolving pattern that spans the rough and smooth zones. By $t=4000h/U_{w}$ , the new pattern with the opposite orientation fully emerges, replacing the initial pattern.

Figure 13. Instantaneous streamwise velocity in an $x{-}y$ plane at $z/h=0.1$ for 3-D k-type roughness at $\textit{Re}=325$ , highlighting the evolution of laminar–turbulent bands as they shift orientation from one diagonal to the opposite between the time instances $3000 h/U_{w}$ and $4000 h/U_{w}$ .

Figure 14 shows the visualization of vortical structures through the isosurfaces of $-\lambda _{2}$ . The quantity $\lambda _{2}$ is the second largest eigenvalue of the tensor $S^{2}+\varOmega ^{2}$ , where $S$ and $\varOmega$ are the symmetric and antisymmetric components of the velocity gradient tensor, respectively (Jeong & Hussain Reference Jeong and Hussain1995). Isosurfaces of $-\lambda _{2}$ can capture coherent vortical structures within the turbulent regions, providing a clear distinction between turbulent and laminar bands in 3-D space. The vortical structures visualized here correspond to the dominant flow features, such as streamwise elongated streaks. The streamwise-oriented structures over the smooth zones are more elongated than those in the rough zones (see figure 14 a). These structures are less elongated in the rough zones due to the obstruction caused by the rib roughness. Additionally, rough zones exhibit spanwise-oriented structures near the roughness elements due to the roll-up of shear layers emanating from the ribs (Javanappa & Narasimhamurthy Reference Javanappa and Narasimhamurthy2021). These differences between the rough and smooth zones result in a non-uniform streamwise bandwidth (see figure 14 a) in the 3-D k-type roughness, with the width varying along the spanwise direction. Earlier, we observed that the large-scale flow vectors were misaligned with the diagonal direction even without apparent competing fronts (see figures 11 a and 11 b). The non-uniform bandwidth in 3-D k-type roughness could also contribute to this misalignment, as varying bandwidth can lead to laminar–turbulent interfaces that are not precisely parallel to the diagonal. As the large-scale flow vectors tend to align locally with the misaligned interfaces, their general orientation deviates from the diagonal direction in which the pattern aligns. In contrast, the turbulent bands in 2-D k-type roughness are more uniform in width (see figure 14 b). Furthermore, the 2-D k-type roughness exhibits a higher density of vortical structures, reflecting stronger turbulent activity. The increased turbulence likely accounts for the ability of 2-D k-type roughness to sustain laminar–turbulent bands at lower Reynolds numbers than 3-D k-type roughness.

Figure 14. Isocontours of instantaneous $\lambda _{2}$ normalized by $( {U_{w}^{2}}/{\nu })^{2}$ for (a) 3-D k-type roughness ( $\textit{Re}=325$ ) and (b) 2-D k-type roughness ( $\textit{Re}=300$ ). The surfaces are coloured with wall-normal distance. The isosurfaces correspond to $\lambda _{2} = -1\times 10^{-8}$ . Subpanels (a i) and (b i) display magnified views of a small region, as highlighted by the arrows.

We now analyse the mean flow characteristics of the 3-D k-type roughness. The mean quantities are calculated by averaging 5000 samples collected at intervals of $0.2h/U_{w}$ between $t=5000h/U_{w}$ and $t=6000h/U_{w}$ . It is important to note that the mean quantities include contributions from both turbulent and laminar bands when interpreting their profiles. In the rough zone, the values are further averaged over streamwise pitches and presented at two locations: the midcavity (I) and midrib (II), both at $y/h = 17$ . In the smooth zone, streamwise averaging is performed in addition to time averaging, and the profile is shown at $y/h = 34$ . Figure 15(a) shows the wall-normal variation of the mean streamwise velocity $\bar {u}$ . The variation of $\bar {u}$ at $y/h = 34$ (smooth zone) has a S-shape similar to that in smooth PCF. The corresponding variations at the midcavity (I) and midrib (II) locations in the rough zone (both at $y/h=17$ ) differ from the profile in the smooth zone, showing comparatively lower values due to the resistance imposed by the roughness elements in the rough zone. This deviation is maximum near the rib-roughened stationary wall and reduces towards the smooth-moving wall. The profiles at the midcavity (I) and midrib (II) locations take different paths near the stationary wall, where the influence of the ribs is the strongest, and coincide beyond $z/h=0.5$ . The profile of the mean velocity gradient ( $\partial \bar{u}/\partial z$ ) at $y/h=34$ (smooth zone, figure 15 b) is nearly symmetric about the midgap plane ( $z/h=1$ ), with maximum values at the walls and a minimum at the channel centre. The corresponding profiles at the midcavity (I) and midrib (II) locations in the rough zone (both at $y/h=17$ ) deviate from those in the smooth zone, particularly near the rib-roughened stationary wall. In the rough zone, the values corresponding to the points below the plane of rib crests ( $z/h\lt 0.2$ ) are significantly lower than those in the smooth zone. However, beyond $z/h=0.2$ , the values consistently exceed those in the smooth zone. At the midcavity location (I), the gradient begins with a minimum at the stationary wall, increases to a maximum near $z/h = 0.2$ , decreases to a minimum near the midgap plane ( $z/h = 1$ ), and then rises again as it approaches the moving wall. At the midrib location (II), the gradient is zero below the plane of the rib crests ( $z/h\lt 0.2$ ), rises to a maximum near $z/h=0.2$ and gradually converges with the curve at the midcavity location (I) beyond $z/h=0.5$ .

Figure 15. Profiles of (a) mean streamwise velocity ( $\bar {u}$ ) and (b) its gradient ( $\partial \bar{u}/\partial z$ ) along the wall-normal direction for 3-D k-type roughness at $\textit{Re}=325$ . The profiles are plotted at $y/h=17$ (rough zone) and $y/h=34$ (smooth zone). In the rough zone, the values are shown for the midcavity (I) and midrib (II) locations. The inset shows a roughness pitch with locations I and II highlighted.

Table 2 shows a comparison of viscous drag ( $\overline {V_{d}}$ ), pressure drag ( $\overline {P_{d}}$ ), friction velocity ( $u_{\tau }$ ) and friction Reynolds number ( $\textit{Re}_{\tau }$ ) of the 3-D k-type roughness with those of the 2-D k-type roughness and smooth PCF. For the rough zones of the stationary wall, the values are derived using the methodology established by Mahmoodi-Jezeh & Wang (Reference Mahmoodi-Jezeh and Wang2020), using the expressions: $\overline {V_{d}} = (\nu/\lambda) \int _{0}^{\lambda } ( {\partial \bar {u}}/{\partial z} )_{w} {\rm d}x$ , $\overline {P_{d}} = (1/\rho \lambda) \int _{0}^{k} ( \overline {P}_{\!{up}} - \overline {P}_{{down}} ){\rm d} z$ and $u_{\tau } = \sqrt {\overline {V_{d}} + \overline {P_{d}}}$ . Here, $\overline {P}_{\!{up}}$ and $\overline {P}_{{down}}$ denote the mean pressures on the upstream and downstream faces of a rib, respectively. The 3-D k-type roughness, with its alternating rough and smooth zones on the stationary wall, results in mixed drag behaviour. The smooth zones contribute solely to the viscous drag, whereas the rough zones generate both viscous and pressure drag, with the pressure drag being the dominant (components of drag from the rough and smooth zones are shown in table 3 in Appendix A). As a result, the viscous drag for the 3-D k-type roughness is less than that of the smooth PCF but greater than that of the 2-D k-type roughness. In contrast, the pressure drag for the 3-D k-type roughness is less than that of the 2-D k-type roughness. In the present study, the friction velocity for the rough stationary wall ( $u_{\tau S}$ ) and the smooth moving wall ( $u_{\tau M}$ ), which should be equal in principle, are marginally different for the rough PCF cases due to slight discrepancies introduced by interpolations near the roughness elements. Therefore, we introduce a global friction velocity defined as $u_{\tau }=\sqrt { {u_{\tau S}^{2}+u_{\tau M}^{2}}/{2}}$ , to provide a single reference value for comparison. The friction velocity and corresponding friction Reynolds number for the 3-D k-type roughness are greater than those of the smooth PCF but lower than those of the 2-D k-type roughness. While 3-D k-type roughness generates higher friction than smooth PCF, its alternating rough and smooth zones produce less turbulence than the 2-D k-type roughness, limiting its ability to sustain stable laminar–turbulent bands as effectively as 2-D k-type roughness.

Table 2. Viscous drag ( $\overline {V_{d}}$ ), pressure drag ( $\overline {P_{d}}$ ) and friction velocity ( $u_{\tau }$ ) for the cases studied. Here $S$ and $M$ are subscripts indicating stationary and moving wall, respectively. The expressions for global friction velocity and global friction Reynolds number are given as $u_{\tau }=\sqrt { {u_{\tau S}^{2}+u_{\tau M}^{2}}/{2}}$ and $\textit{Re}_{\tau }=u_{\tau }h/\nu$ , respectively.

Table 3. Viscous drag ( $\overline {V_{d}}$ ), pressure drag ( $\overline {P_{d}}$ ) and friction velocity ( $u_{\tau }$ ) for the 3-D k-type roughness. For the stationary wall, contributions from rough and smooth zones are shown separately. Subscripts $S$ and $M$ indicate stationary and moving walls, while superscripts $RZ$ and $SZ$ indicate the rough and smooth zones of the stationary wall, respectively. The friction velocity for stationary wall is estimated as $u_{\tau S} = \sqrt {(1/2)(\overline {V}_{{\rm d}S}^{RZ} + \overline {P}_{{\rm d}S}^{RZ} + \overline {V}_{{\rm d}S}^{SZ})}$ . The expressions for global friction velocity and global friction Reynolds number are given as $u_{\tau }=\sqrt { {u_{\tau S}^{2}+u_{\tau M}^{2}}/{2}}$ and $\textit{Re}_{\tau }=u_{\tau }h/\nu$ , respectively.

In the transitional regime, the mean spanwise velocity ( $\bar {v}$ ) is not zero for the 3-D k-type roughness (see figure 16 a). At $y/h=34$ (smooth zone), $\bar {v}$ starts from zero at the stationary wall, increases to a maximum and then decreases, crossing zero at $z/h=1$ . Then it increases in the opposite direction, reaches another peak and decreases to zero on the moving wall. On the other hand, the profiles at the midcavity (I) and midrib (II) locations in the rough zone (both at $y/h=17$ ) exhibit similar oscillatory behaviour but with multiple peaks and sign changes. The variations at the midcavity (I) and midrib (II) locations, which take distinct paths in the lower-half of the domain, merge into a single curve in the upper-half. The mean wall-normal velocity ( $\bar {w}$ ) is nearly zero at $y/h=34$ (smooth zone, figure 16 b). At $y/h=17$ (rough zone), $\bar {w}$ is non-zero, with negative values at the midcavity location (I), driven by the downward flow into the cavities, and positive values at the midrib location (II), resulting from the upward deflection as the fluid moves over the ribs. The curves at the midcavity (I) and midrib (II) locations follow two separate trajectories and converge only near the moving wall. The discussion so far has not addressed the impact of indefinite competition between opposite angles on the mean flow characteristics. To assess the impact of the ongoing competition, which is highly transient in nature, a comparison of mean quantities estimated over two distinct time windows is shown in figure 17 in Appendix A. The profiles of $\bar {u}$ and $\bar {w}$ from these time windows differ only slightly in magnitude. However, $\bar {v}$ shows pronounced quantitative and qualitative variations, indicating a strong sensitivity to the indefinite competition between oppositely oriented patterns. Therefore, analysing $\bar {v}$ and comparing its profiles in rough and smooth zones does not lead to generalized conclusions, as the observed trend only represents a transient rather than a permanent state. In contrast, the drag characteristics are largely unaffected by the ongoing competition. The present study does not address the higher-order statistical quantities in the 3-D k-type roughness due to significant temporal variations caused by continuous competition between bands of opposite angles.

Figure 16. Profiles of (a) mean spanwise velocity ( $\bar {v}$ ) and (b) mean wall-normal velocity ( $\bar {w}$ ) along the wall-normal direction for 3-D k-type roughness at $\textit{Re}=325$ . The profiles are plotted at $y/h=17$ (rough zone) and $y/h=34$ (smooth zone). In the rough zone, the values are shown for the midcavity (I) and midrib (II) locations.

Figure 17. Wall-normal variation of (a) mean streamwise velocity ( $\bar {u}$ ), (b) mean spanwise velocity ( $\bar {v}$ ) and (c) mean wall-normal velocity ( $\bar {w}$ ) estimated over the time windows $t = 4000h/U_{w}-5000h/U_{w}$ (in blue) and $t = 5000h/U_{w}-6000h/U_{w}$ (in red) at $\textit{Re}=325$ . Profiles are plotted at: $y/h = 17$ (rough zone, location I); $y/h=17$ (rough zone, location II); $y/h=34$ (smooth zone).