2.1. Simulation set-up

A fully-developed turbulent wall flow is simulated in a closed channel configuration using the High Performance Solver for Turbulence and Aeroacoutics Research (HiPSTAR), which is a well-established fourth-order-accurate compressible fluid dynamics code (Sandberg Reference Sandberg2015). Here, we consider a flow Mach number $Ma = 0.2$ to simulate an incompressible turbulent flow over a smooth wall, as well as over riblet surfaces. All these simulations are performed at a matched friction Reynolds number $Re_\tau = hU_\tau /\nu \approx 590$ , where $h$ represents the half-height of the channel along its wall-normal direction, $z$ . The flow is driven by a constant streamwise pressure gradient, which is kept fixed for all the riblet cases. The computational domain, as illustrated in figure 1(a), encompasses dimensions $L_x = 2\pi h$ in the streamwise direction and $L_y = \pi h$ in the spanwise direction. These domain proportions are chosen to effectively capture and resolve the majority of the flow dynamics, aligning with the approach elucidated by García-Mayoral & Jiménez (Reference García-Mayoral and Jimenez2012) concerning suitable simulation domain sizes for riblet studies. A no-slip boundary condition is enforced on the top and bottom walls, while periodic boundaries are implemented at the streamwise and spanwise extremes of the domain.

For the riblet cases, a sawtooth geometry with a 30 $^{\circ }$ tip angle and spanwise tip spacing $s^+ \approx 43$ is considered with riblet height $h_s^+ = 80$ (refer to the dash-dotted red line in figure 1 b), on both the top and bottom walls of the channel (figure 1 b). This geometry is resolved numerically using the second-order-accurate boundary data immersion method described by Schlanderer et al. (Reference Schlanderer, Weymouth and Sandberg2017). In the simulation of riblets, the numbers of mesh grid points in each direction are $N_x\times N_y \times N_z = 384\times 1290\times 793$ , where $z$ is the wall-normal direction. This configuration yields approximately $31$ grid points per riblet period ( $s^+ = 43$ ) based on the spanwise grid spacing $\Delta y^+ = 1.4$ , which is detailed in table 1. These numbers of grid points per riblet period are sufficient for resolving the K–H rollers, as shown previously by the mesh convergence analysis of Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021b), amongst others. Apart from consideration of the conventional riblet geometry with sharp valleys (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011a ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a; Rouhi et al. Reference Rouhi, Endrikat, Modesti, Sandberg, Oda, Tanimoto, Hutchins and Chung2022), simulations are also conducted for a riblet geometry possessing rounded valleys (RR43), with riblet height $h_r^+= 71$ and base diameter $b_r^+ = 6.2$ (figure 1 b). The latter is representative of the practical limitations experienced while fabricating ‘sharp’-valley riblets, with the current dimensions ( $b^+_r$ ) matching exactly with those tested experimentally (discussed in § 3.2). Information regarding all simulations undertaken in this study is included in table 1.

Figure 1. (a) DNS riblet channel domain showing the domain dimensions $L_x, L_y, L_z$ in streamwise, spanwise and wall-normal directions, while the top and bottom walls are covered with sawtooth riblets. Here, $L_z$ is measured from the base of the riblets. (b) Profile view of the rounded-valley riblet design (RR43) and the sharp-valley riblet design (SR43, shown with the dash-dotted red line) showcasing riblet characteristics such as tip spacing $s$ , base width $b_r$ , and height $h_r$ or $h_s$ , where the subscript $r$ or $s$ denotes the height for the rounded- or sharp-valley riblet design, respectively. The green shaded region in (b) corresponds to the cross-sectional area of the groove ( $A_g = l_g^2$ ) for RR43, while the orange shaded region corresponds to the spanwise averaging region, simulating the effect of hot-wire spatial resolution on the K–H detection method discussed in § 4. The dashed vertical line in (b) shows the spanwise centre of the riblet grooves. Throughout this paper, we consider $z = 0$ at the riblet half-height for the corresponding riblet shape (rounded or sharp valley).

Table 1. Geometry and simulation parameters for the fully-developed turbulent channel flow over a smooth wall, rounded-valley riblets (RR43) and sharp-valley riblets (SR43). The parameters include tip spacing $(s^+)$ , riblet height $(h_r^+)$ , ratio of the channel half-height to the riblet height $(h/h_r)$ , base diameter $(b_r^+)$ , square root of the groove cross-section ( $l_g^+$ ), grid spacings in streamwise ( $\Delta x^+$ ), spanwise ( $\Delta y^+$ ) and wall-normal ( $\Delta z^+$ ) directions, domain length $(L_{x}^+)$ and width $(L_{y}^+)$ , the friction Reynolds number ( $Re_\tau \equiv h^+ = L_z^+/2$ ), and the velocity roughness function ( $\Delta U^+$ ). Quantities denoted by a superscript ‘+’ represent viscous scaling, i.e. non-dimensionalised using $U_{\tau }$ and $\nu$ .

2.2. Mean flow properties over the rounded-valley riblets

Figure 2 shows the mean streamwise velocity and Reynolds stress profiles over the rounded-valley riblets (RR43) and compares them against statistics over the smooth wall. Here, for purposes of brevity, we have not shown the profiles for the sharp-valley riblet case (SR43), since they are quantitatively similar to those of the RR43 case. This similarity is expected since the change in $l_g$ between RR43 ( $l_g^+ = 41.3$ ) and SR43 ( $l_g^+ = 41.5$ ) is negligible. Present findings are consistent with previous results reported by Walsh (Reference Walsh1982, Reference Walsh1990) and Walsh & Lindemann (Reference Walsh and Lindemann1984), who found riblet drag reduction performance to be affected on changing the riblet peak geometry but not on changing the valley curvature.

The profiles presented in figure 2 are obtained via time- and ensemble-averaging only along the spanwise centre of the riblet grooves (red dashed line in figure 1 b). The present study considers the origin at the riblet half-height (horizontal black dash-dotted line in figure 1 b), for analysis throughout the paper. This choice, however, does not have any substantive impact on the primary outcomes and findings of this study, which are associated with the detection and characterisation of K–H rollers over drag-increasing riblets. The effects of the choice of origin on drag reduction statistics are discussed in detail and represented with error bars in § 3.4.

In figure 2(a), the mean streamwise velocity profile over the riblet exhibits a noticeable downward shift compared to the present smooth wall profile as well as that from the simulation of Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999), at $Re_\tau = 590$ . Conventionally, this shift has been associated with an increased drag penalty relative to a smooth wall. Here, we quantify this penalty via the roughness function $\Delta U^+$ , which measures the downward shift within the logarithmic region in comparison to a smooth wall profile (Clauser Reference Clauser1954; Hama Reference Hama1954), which is approximately 4.1 for the current riblet case.

Figure 2. Viscous scaled profiles over smooth wall (black solid line) and spanwise centre of the riblet grooves (blue dash-dotted line) for (a) mean streamwise velocity $U^+$ and (b) Reynolds shear stress $\langle u^{\prime }w^{\prime }\rangle ^+$ , versus the viscous scaled wall-normal height. The riblet crest location, $z = h_r/2$ is shown with blue dots. The red dashed lines in (a) and (b) are from the channel DNS of Moser et al. (Reference Moser, Kim and Mansour1999) at $Re_\tau = 590$ . The arrow in (a) indicates the downward shift of the riblet velocity profile compared to the smooth wall profile, which is quantified by the velocity roughness function $\Delta U^+ = 4.1$ .

Figure 2(b) shows Reynolds shear stress $\langle u^{\prime }w^{\prime }\rangle ^+$ profiles along the spanwise centre of riblet grooves. The angle brackets ${\langle }{\;}{\rangle }$ here denote parameters averaged in time. The effect of the current drag-increasing riblets on the Reynolds shear stress observed in figure 2(b) is consistent with findings of previous studies (e.g. Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a). It is worth re-stating here that the Reynolds shear stress and mean velocity profiles estimated over the conventionally studied sharp-valley riblets (not presented here) are consistent with those of the rounded-valley riblets, albeit within the range of uncertainty associated with the estimation of the virtual origin. This supports our claim that the rounded valley of the current riblet geometry has negligible effect on the mean momentum field.

2.3. Assessment of the presence of K–H rollers

Figure 3. Two-dimensional pre-multiplied spectral energy densities of (a) $u^{\prime 2}$ , (b) $v^{\prime 2}$ and (c) $w^{\prime 2}$ for smooth surfaces (black solid contour lines) and riblet surfaces (blue shading contours encapsulated in blue contour lines). Contours for both walls are computed at $z^+ \approx 40$ , which corresponds to five viscous units above the riblet crest. The horizontal dashed line is at $\lambda _y^+ = s^+$ .

In the current study, we are simulating $30^\circ$ sawtooth riblets with rounded valleys, which also have viscous scaled spacing ( $s^+ \approx 43$ ) relatively higher than the majority of past investigations (based on sharp valleys). Hence before we can use this database to propose a new K–H detection methodology, it is necessary to investigate whether the rounded-valley shape adversely influences the presence of K–H rollers. In the literature, one of the most common methodologies/metrics adopted to confirm the presence of K–H rollers has been the analysis of the two-dimensional (2-D) spectra of $u$ , $w$ or the Reynolds shear stress co-spectra. For instance, García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011a ) analysed the 2-D energy co-spectra to show that these roller structures are concentrated in the buffer layer, just above the plane of the riblet tips, and extend across the spanwise direction. Their spectra indicated distinct streamwise spacing $\lambda _x^+ \approx 150$ for the K–H rollers, with the energy distributed across a wide spanwise wavelength range $\lambda _y^+ \gtrsim 300$ , consistent with the existence of spanwise rollers. Similar structures were confirmed over $30^\circ$ sawtooth riblets with $s^+ = 30$ by Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a), based on the same metrics.

Inspired by these past investigations, we also compute the pre-multiplied 2-D spectral energy densities ( $k_xk_y\phi _{u_i^\prime u_i^\prime }^+$ ) of all three velocity components, at a wall-normal position 5 wall units above the riblet tips. Here, $k_x$ and $k_y$ are the wavenumbers in the streamwise and spanwise directions. Figure 3 depicts these pre-multiplied 2-D spectra for the RR43 case, where the data from the reference smooth wall at the same wall-normal height is also included for comparison as solid black contours. Here, the energy contours of the $u$ - and $v$ -spectra (figure 3a,b) exhibit nominally similar characteristics for both the riblet and smooth wall cases. However, the $w^\prime$ -spectra in figure 3(c) reveal a distinct energy accumulation in a spectral region near $\lambda _x^+ \approx 280$ , which extends for spanwise wavelengths across $100 \lesssim {\lambda }^+_y \lesssim 1000$ . This indicates the presence of wide structures in the spanwise direction, and relatively shorter structures in the streamwise direction, which is a distinguishing characteristic of the K–H rollers. We consider this as confirmation of the existence of K–H rollers over the present sawtooth riblets with rounded valleys (Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a). It is also worth noting that these K–H rollers exhibit slightly higher streamwise wavelengths, $\lambda _x^+ \approx 280$ , compared to those observed for blade riblets and $30^\circ$ sawtooth riblets with smaller viscous scaled spacing $s^+ = 15{-}30$ ( $\lambda _x^+ \approx 170$ ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a). This suggests that the streamwise wavelength of the K–H rollers is likely dependent on the riblet shape and/or viscous scaled size, which will be investigated further in the present study. However, we noticed that the minor change of the valley between RR43 and SR43 (not shown here) did not change the streamwise wavelength of the K–H rollers.

Figure 4. Instantaneous streamwise wall-shear stress ( $\tau _{w, {\textit{Ins}}}$ ) on the riblet surface, normalised by the average wall-shear stress ( $\langle \tau _w\rangle$ ) on the entire surface. The vertical dashed lines indicate the streamwise distance between two negative events of the streamwise wall-shear stress to be representative of the nominal streamwise distance between subsequent K–H rollers.

In order to provide a clear visual representation of the presence of K–H rollers over the present riblet geometry, we employ another methodology proposed by Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a), of plotting the instantaneous streamwise wall-shear stress on the riblet surface. Previous works have adopted a similar methodology to highlight the extension of the K–H rollers (e.g. within the grooves of riblets; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a), and to reveal their influence on the near-wall region, despite their origin at the crest of the riblets. Figure 4 shows the distribution of the instantaneous streamwise wall-shear stress on the present RR43 riblet surface, revealing quasi-streamwise periodic negative wall-shear stress events (i.e. reversed flow) existing simultaneously and coherently across multiple spanwise adjacent grooves. These events are representative of the spanwise coherent structures (i.e. K–H-like instability) in the overlying flow, at average streamwise spacing approximately 200–400 viscous units ( $\lambda _x^+ \approx 200{-}400$ ), thereby confirming the footprint of the K–H rollers below the riblet crest. The spanwise extent of these structures, however, appears to vary considerably, extending up to the spanwise size of the simulation domain $L_y$ ( $\approx 1800$ viscous units) in certain regions, while being limited to sizes of the order of 100 viscous units in other regions. The ‘intermittent’ nature of the K–H rollers is consistent with observations from previous simulations over riblets (García-Mayoral & Jiménez Reference García-Mayoral and Jimenez2012), reaffirming the idea that these K–H rollers do not always extend across an infinite span in an instantaneous sense. The focus of this study is to explore, through experimental means, whether the spanwise extent or coherence of the K–H rollers changes with variations in the riblet tip spacing (denoted as $s^+$ ), corresponding to changes in friction drag. It is motivated by the Reynolds shear stress co-spectra analysed by Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a) in their figure 8(h,i), in which the signature of the K–H rollers over $30^\circ$ sawtooth riblets was shown to broaden spectrally (i.e. no longer narrowband at one fixed $x$ wavelength), and also shift to a higher wavelength with increasing $l_g^+$ . A similar observation was also noted in the modelling effort by Chavarin & Luhar (Reference Chavarin and Luhar2020), where the streamwise wavelength of spanwise-constant structures (corresponding to the K–H rollers) was found to increase with riblet size. This change in the signature of the K–H rollers will be investigated here comprehensively, by conducting experiments across a significantly extended range of $l_g^+$ (or $s^+$ ) values.

2.4. Near-wall signature of K–H rollers in the instantaneous streamwise velocity

The preceding subsection establishes the presence of K–H rollers for the present DNS of rounded-valley sawtooth riblets with $30 ^\circ$ tip angles (RR43), using methodologies adopted by past simulation studies. Employing the same methodologies for an experimental investigation, however, is challenging owing to the difficulties associated with synchronously measuring multiple velocity components in the close vicinity of the riblet crests (to obtain the Reynolds shear stresses or $w^2$ ), or measuring the wall-shear stress within the grooves. The former is possible only via conducting particle image velocimetry measurements in the streamwise/wall-normal plane close to the riblet crests, or through the use of multi-wire hot-wire anemometry, which face challenges associated with laser reflection from the wall and spatial resolution issues, respectively. These limitations can be bypassed by using conventional single-wire anemometry, which would, however, require the capability of detecting K–H rollers based solely on the streamwise velocity signatures. Here, we explore the possibility of such an approach by using the present numerical simulation as a test case.

Figure 5. (a) Absolute value of the instantaneous streamwise velocity $| u^+|$ within multiple riblet grooves, shown up to only $(x^+ \times z^+) = (1000 \times 200)$ for clarity. (b–d) Samples of $| u^+|$ across the streamwise extent of the DNS domain $0 \lesssim x^+ \lesssim L_x^+$ annotated (b), (c) and (d) in (a). The dashed black line is at the riblet crest $z^+ = h_r/2$ . (e) Plot of $| u^+|$ in an $xy$ -plane located at the riblet mid-height, shown with a magenta dash-dotted line in (a). The $y^+$ range of (a–d) is in the lower part of (e).

The fact that the K–H rollers extend into the valleys and influence the streamwise wall-shear stress simultaneously across multiple adjacent grooves (figure 4) suggests that there is merit in probing simultaneously acquired, instantaneous streamwise velocity signatures across various $z^+$ and along the span. An example of this is depicted in figure 5, which uses the same time instant shown previously in the instantaneous streamwise wall-shear stress in figure 4. To enhance clarity, negative events (i.e. recirculation) are excluded by taking the ‘absolute’ value of the velocity signal, to simulate the scenario of a hot-wire measurement (which is insensitive to flow reversals), and the colour bar is chosen in a way that would reveal the near-wall low-magnitude events. After making these adjustments, the contour plot clearly reveals the presence of recurring high-fluctuation events (relative to the low mean speed near the wall) along the $x$ direction, which are consistent with the negative wall-shear stress events observed in figure 4. These events are spaced at intervals of approximately $200 \lesssim \Delta x^+\lesssim 400$ , which closely matches the wavelength of the K–H rollers noted from spectra in figure 3. It has been verified that the same events can also be observed by simply plotting $u^+$ , and hence are not an artefact of taking the absolute value (not shown here).

Next, to assess the spanwise extent of these extreme fluctuation events, figure 5(b–d) has been selected to depict $\left |u^+\right |$ in the adjacent riblet grooves in the wall-normal plane (see figure 5 a). They indicate that the high-fluctuation events span multiple riblet spacings along the span (here, approximately three riblets, with spanwise spacing approximately 86 wall units), thus confirming their connection with the characteristic overlying K–H rollers and their manifestation on the wall (depicted in figure 4). In figure 5(e), we also display $\left |u^+\right |$ within an $xy$ -plane located at the riblet mid-height. This specific location matches the lowest wall-normal height achievable by the hot-wire sensor (as outlined in § 3). Interestingly, the spanwise extension of the K–H rollers at this particular wall-normal height seems to be comparatively narrower than the extension inferred from the streamwise wall-shear stress shown in figure 4. This difference in spanwise extent is likely owing to the increased disruption of the rollers closer to the crest by enhanced turbulence/mixing activity expected in the region. This is related to increased transpiration across the riblet crest plane, which likely happens for large $l_g^+$ riblets (García-Mayoral & Jiménez Reference García-Mayoral and Jimenez2011b ).

The aforementioned findings thus make it clear that the three-dimensional (3-D) coherence of the K–H rollers can be detected/quantified based solely on streamwise velocity measurements within the grooves. To further support this, the clear signature of the rollers from the one-dimensional pre-multiplied spectra of streamwise velocity fluctuations will also be displayed in § 4 for the comparison between the simulation and experimental results. The present study quantifies the 3-D coherence of the K–H rollers by deploying multiple single normal hot-wire probes, to simultaneously acquire streamwise velocity signals from adjacent riblets along the span, or along the wall-normal direction within the same riblet groove. Cross-correlation analysis is conducted on this dataset to investigate the spatial coherence of the K–H rollers and their sensitivity to  $s^+$ . The next section provides comprehensive details about the measurement methodology employed.

3.1. Wind tunnel facility

The experiments were conducted in an open-return boundary layer wind tunnel facility housed within the Walter Basset Aerodynamics Laboratory at the University of Melbourne. The tunnel working section has a rectangular cross-section with dimensions $0.94\times 0.38\,\textrm{m}^2$ in the $y$ and $z$ directions, respectively. The streamwise ( $x$ ) length of the working section measures 6.70 m from the boundary layer trip at the inlet. All the measurements in this investigation are conducted at streamwise location $x \approx 3.80$ m from the trip. The tunnel floor consists of eleven interchangeable main plates, each with dimensions $0.72 \times 0.50\,\textrm{m}^2$ ( $y\times x$ ), as well as eight aluminium side rails. Although the tunnel floor is equipped with a heating capability, the floor temperature remained unadjusted for the specific study under consideration. Openings are present in the upper wall of the tunnel at intervals of 0.5 m in the $x$ direction, facilitating the passage of traversing rods for hot-wire anemometer measurements. The schematics of the tunnel facility and detailed information on its characterisation can be found in Monty, Harun & Marusic (Reference Monty, Harun and Marusic2011) and Abu Rowin et al. (Reference Abu Rowin, Xia, Wang and Hutchins2024).

For the present experiments, the wind tunnel is operated within a freestream velocity range $U_\infty = 5{-}20\,\textrm{m}\,\text{s}^{-1}$ . For the smooth wall case, this corresponds to a friction Reynolds number range $850\lesssim Re_{\tau }\lesssim2500$ at the measurement location. Here, the friction Reynolds number is defined as $Re_\tau = \delta U_\tau /\nu$ , where $\delta$ represents the boundary layer thickness (corresponding to a height where the mean velocity $U$ reaches 99 % of the freestream velocity $U_\infty$ ), $U_\tau$ is the friction velocity obtained directly via dedicated drag balance measurements (discussed in § 3.4), and $\nu$ denotes the kinematic viscosity. It is important to highlight that our approach to varying the viscous scaled riblet spacing ( $s^+$ or $l_g^+$ ) involves manipulating the tunnel freestream velocity, thereby influencing $Re_\tau$ . While this methodology may introduce interdependencies between changing $s^+$ and altering $Re_\tau$ , it remains a valid strategy for our investigation. This methodology is also consistent with previous studies, e.g. Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020), Endrikat et al. (Reference Endrikat, Newton, Modesti, García-Mayoral, Hutchins and Chung2022) and Von Deyn et al. (2022). Moreover, this approach mirrors real-world applications, including airflow over aircraft surfaces, where both $Re_\tau$ and $s^+$ naturally vary concurrently across different cruising speeds and phases of flight. In our study, we have taken measures to disentangle the impact of $Re_\tau$ on the obtained results by comparing matched cases of $s^+$ between experiment RR43E and simulation RR43, with distinct $Re_\tau$ values $\sim 1500$ for RR43E and $\sim 600$ for RR43. This careful approach enables us to discern and attribute observed changes to either the variation in $s^+$ or alterations in $Re_\tau$ .

3.2. In-house fabrication of riblets

The riblets employed in this investigation were fabricated with a desired sawtooth cross-section, having characteristic tip angle $30^\circ$ . These riblets had peak-to-peak spacing $s = 2\,\textrm{mm}$ and riblet depth $h_r = 3.7$ mm (refer to figure 1 b for definitions), which was designed to achieve matched $s^+$ and $h^+_r$ (with the DNS) within tunnel freestream range $7\lesssim U_{\infty } \lesssim 10\,\textrm{m}\,\textrm{s}^{-1}$ . The riblets were manufactured in-house, within the Walter Basset Aerodynamics Laboratory, using a MultiCam three-axis CNC router. A 6061 aluminium alloy was chosen as the fabrication material in order to obtain riblets with high thermal conductivity, to be used for subsequent heated floor experiments. Obtaining deep cuts in aluminium using carbide drill bits, however, can be challenging since the drill bits are prone to wear or breakage during machining. This issue was overcome by dividing the machining task across multiple stages, which involved using multiple drill bits of varying sizes to obtain the desired riblet geometry. The final cutting procedure involved the successive passes of 2.0 mm, 1.5 mm and 1.0 mm ball nose cutters, which incrementally removed material at depths 0.70 mm, 0.75 mm and 0.73 mm, respectively. Subsequently, a 0.3 mm tapered ball nose cutter with side angle $15^\circ$ was employed to eliminate an additional 1.1 mm of material. This cutting strategy required approximately five days to complete the machining process for each $720 \times 500\,\textrm{mm}^2$ tile. The resulting shape of the machined riblet tile is presented in figure 6(a) as an example. For the specific tile depicted in figure 6(b), the dimensions of the manufactured riblets were $s = 2\,\textrm{mm}$ and $h_r = 3.3\,\textrm{mm}$ , with base width $b_r \approx 0.3\,\textrm{mm}$ . It is important to note that this rounded-valley riblet shape results in a height difference of approximately 0.4 mm compared to the ideal sharp-valley riblets, for which $h_r = 3.7\,\textrm{mm}$ . However, as mentioned in § 2.2 based on simulations of sharp- and rounded-valley riblets, the comparison of bulk properties such as $\Delta U^+$ or the mean velocity profile revealed no discernible differences between the two cases. Consequently, the present riblet shape with rounded valleys is considered to exhibit a behaviour similar to the conventionally studied sawtooth riblets, i.e. with sharp valleys.

Figure 6. (a) A photograph depicting the overall machined riblet tile, with a finger included for scale. (b) A side view image showcasing the riblet details, including peak-to-peak spacing $s = 2.0\,\textrm{mm}$ , riblet depth $h_r = 3.3\,\textrm{mm}$ , tip radius $\lesssim 30$ µm and base width $b_r \approx 0.3\,\textrm{mm}$ . The riblet shape is delineated by the red solid line.

To ensure a fully developed turbulent boundary layer associated with the specific riblet shape under investigation, it was necessary to cover the entire floor of the wind tunnel (up to $x = 3.8\,\textrm{m}$ ) with riblet surfaces, which required 11 riblet tiles in total. However, owing to constraints on time and resources, only two tiles (with combined streamwise length $\sim1\,{\textrm{m}}$ ) were manufactured using the aforementioned ‘staged machining’ procedure, to yield a riblet geometry consistent with that depicted in figure 6 (i.e. the main riblet geometry). Fabrication of the remaining nine tiles using the same staged machining procedure would have entailed approximately two more months of machining time, exceeding the available resources for this study. Consequently, a more pragmatic approach was employed, wherein $30^\circ$ sawtooth riblets with base width 0.5 mm were fabricated for the remaining 9 tiles (instead of 0.3 mm base width). Independently conducted drag balance measurements revealed that such a slight modification resulted in a deviation in skin-friction drag of less than 2.5 % for the $b_r = 0.5\,\textrm{mm}$ riblets, when compared to the $b_r = 0.3\,\textrm{mm}$ riblets. To ensure consistent experimental conditions, the two machined tiles with base width $b_r = 0.3\,\textrm{mm}$ were placed at, and immediately upstream of, the measurement location. Such an arrangement ensured an upstream fetch of $\approx 0.8\,\textrm{m}$ of the main riblet geometry, which corresponded to approximately 13 boundary layer thicknesses ( $13\delta$ ). By following this approach, it was expected that the boundary layer would be in a fully developed, i.e. near-equilibrium, state associated with that of the more refined ( $b_r \approx 0.3\,\textrm{mm}$ ) riblet geometry.

Figure 7. (a) Schematic of the riblets showing two hot-wire sensors deployed within the riblet grooves at spanwise separation $\Delta y = s$ . (b) A photograph of two hot-wire sensors within the riblet grooves, for spanwise separation $\Delta y = 3s$ .

3.3. Hot-wire anemometry and two-point measurement set-up

Previous discussions (§ 2.4) on the near-wall footprint of the K–H rollers, in the instantaneous total streamwise velocity, have paved the way for detection of K–H rollers based on conventional hot-wire anemometry (i.e. using single-normal hot-wire sensors). The miniature design of these sensors also makes it convenient to deploy the hot-wire probes within the grooves of the riblets, having 2 mm spacing at the peak. In the present study, we deployed a modified version of a Dantec boundary layer type probe (55P15), to facilitate measurements as low as the mid-height of the present riblet geometry (i.e.  $z\approx0$ ; figure 7). The sensor is prepared in-house using a Wollaston wire with a Pt-core of diameter $d\approx2.5$ µm, which is exposed to a length $l\sim0.5\,{\textrm{mm}}$ via etching using nitric acid. This ensures a length-to-diameter ratio ( $l/d$ ) nearly equal to 200, while also keeping the spanwise extent of the sensor within 1.0 mm, which is required to access the riblet mid-height. The length of the stubs, on either side of the Pt-sensor, was approximately 0.1 mm to avoid conduction losses from the prongs while minimising the overall width of the probe.

During experiments, the hot-wire probe was operated in constant temperature mode using a custom-designed Melbourne University constant temperature anemometer at overheat ratio 1.8. Probe calibration was conducted in the freestream of the wind tunnel both before and after each velocity profile measurement, based on fifteen freestream velocities ranging from zero to 1.2 times the desired freestream velocity of the specific measurement. The resulting hot-wire voltage curves are then fitted with a third-order polynomial curve to obtain the calibration coefficients. For each boundary layer profile measurement, the hot-wire probe traversed at least 40 wall-normal locations, spaced logarithmically across $0\lesssim z\lesssim\delta$ . At each of these $z$ locations, the hot-wire signal was acquired at sampling rate 40 kHz for duration $T = 300$ s, ensuring total acquisition time $T{U_{\infty }}/{\delta }\gtrsim20\,000$ . The viscous scaled sampling period, denoted as $t^+$ , varies between 0.09 and 1.57 amongst the various measurements, and is well within the recommendations made by Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009).

Using constant physical dimensions of hot-wire sensors (length $l \approx 0.5\,\textrm{mm}$ and diameter $d \approx 2.5$ µm), across different $Re_{\tau }$ (i.e. $U_{\infty }$ ) cases for the riblets meant that the viscous scaled length of the sensor ( $l^+ = l U_\tau /\nu$ ) varies across $7\lesssim l^+\lesssim27$ (see table 2). Although these $l^+$ values are in close proximity to the recommended threshold $l^+ = 20$ for minimal spatial filtering (Ligrani & Bradshaw Reference Ligrani and Bradshaw1987; Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009), some attenuation of the small-scale energy is inevitable for increasing $l^+$ . This attenuation, however, is not expected to significantly impact the primary conclusions of this study, since the spanwise wavelengths associated with the K–H rollers are significantly larger than the spatial cut-off associated with the hot-wire resolution (refer to figure 3).The same is evident from the detailed analysis presented later in section 5. It is also important to note that the proximity of the hot-wire sensor to the metallic riblet surface, especially when the sensor is inside the groove (figure 7), means that the wire is subjected to additional heat losses. This leads to an increased output voltage from the sensor, which incorrectly corresponds to a higher local time-averaged velocity value (Khoo et al. Reference Khoo, Chew and Teo2000). Although these effects can impact the magnitude of the mean velocity measurements within the riblet grooves, they are anticipated to have minimal impact on the primary conclusions of this study, which are predominantly based on the spectral/Fourier content of the streamwise velocity fluctuations (instead of its physical magnitude). For this reason, as well as the sensor’s inability to identify instantaneous flow reversals (i.e. $U \lt 0$ ), the velocity magnitudes estimated within the valley regions could not be deemed accurate. Further details on the impact of heat losses on the hot-wire measurements, in comparison with simulation results, can be found in § 3.4.

Table 2. Bulk flow and geometry properties of the rounded-valley riblet at $x =3.8\,\textrm{m}$ obtained from velocity measurements across a freestream velocity in the range $U_\infty \approx 5{-}20\,\textrm{m}\,s^{-1}$ . The parameters include the boundary layer thickness ( $\delta$ ), the blockage ratio ( $h_r/\delta$ ), the friction velocity ( $U_\tau$ ), the skin fiction coefficient ( $C_f \equiv 2/(U_\infty ^+)^2$ ), and the hot-wire spanwise spatial resolution ( $l^+$ ). Boundary layer profiles were also acquired over a smooth wall at matched $U_{\infty }$ and $\sim l^+$ as those indicated by $^*$ in the table.

During two-point measurements (conducted independently from the boundary layer traverse experiments discussed above), two hot-wire sensors were mounted on a unique multi-axis traversing system (figure 7), to enable reconstruction of two-point correlations of the streamwise velocity fluctuations in the cross-plane, following Deshpande et al. (Reference Deshpande, de Silva, Lee, Monty and Marusic2021). The system permitted independent numerical control of the wall-normal as well as spanwise offsets between the two hot-wire sensors, thereby facilitating quantification of the 3-D spatial coherence of the K–H rollers (with the third dimension being time). Accurate wall-normal positioning of the sensors was critical for the present measurements, and was ensured with the help of a traversable stereoscopic microscope (TS-30HS), having a depth-of-field 30 µm. The wall-normal height of each sensor was measured relative to the riblet peaks by positioning the microscope above each wire, which was itself located above the crest of the riblets. The relative distance between the wire and the crest was determined with the help of a digital micrometer, which measured the distance traversed by the microscope to focus from one to another. Further, to facilitate the visual observation of the sensor as it traversed into the riblet groove, a high-resolution camera system was positioned upstream of the wire location, looking into the riblet groove. This arrangement permitted real-time monitoring of the wire as it was meticulously manoeuvred along the spanwise and wall-normal directions, to position within the riblet groove. It is worth noting here that this process of positioning the wire within the riblet groove can be subject to various sources of errors, which include uncertainties originating from the microscope system (with precision 30 µm) and the transverse encoder resolution (with precision 5 µm). The net uncertainty associated with initial positioning of the wire was estimated to be $\sim 30$ µm (corresponding to within 1.8 viscous units). See Abu Rowin et al. (Reference Abu Rowin, Xia, Wang and Hutchins2024) for more details on the uncertainty analysis. Apart from the uncertainty in hot-wire positioning during wind-off conditions, a small downward movement in the wire position was also noted when the air flow was turned on. This can be attributed to aerodynamic forces that occur in any conventional hot-wire experiment, to a certain extent. It is possible that this shift becomes more pronounced at higher flow velocities.

Figure 8. (a) Viscous scaled mean streamwise velocity $U^+$ as a function of the viscous scaled wall-normal distance $z^+$ from smooth (grey dashed lines) and riblet (filled blue circles). The crest location $z^+ = h_r^+/2$ is shown with a short vertical black line for each case. The law-of-the-wall (solid red line), log-law (red dash-dotted line) and spanwise centre of the riblet grooves (blue dash-dotted line) for the current DNS RR43 of the rounded-valley riblets are also included. (b) Velocity roughness function $\Delta U^+$ of the current riblet measurement (filled blue circles) shown alongside the current DNS RR43 of the rounded-valley riblets (), $90^\circ$ triangular riblets of Bechert et al. (Reference Bechert, Bruse, Hage, Van der Hoeven and Hoppe1997) (), trapezoidal riblets of Von Deyn et al. (Reference Von, Lars, Gatti and Frohnapfel2022) (), and $30^\circ$ sawtooth riblets () and blade riblets () of Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a).

3.4. Validation of the mean statistics from experiments

The mean velocity profiles for smooth and riblet cases are depicted in figure 8(a). Here, $U^+ = U/U_\tau$ represents the mean viscous scaled streamwise velocity, and $z^+ = z U_\tau /\nu$ denotes the viscous scaled wall-normal distance. The freestream velocity is varied within the range $U_\infty = 8{-}20\,\textrm{m}\,\textrm{s}^{-1}$ . The measurement over the smooth wall (shown with dashed grey shaded lines) is also performed for comparison purposes. To determine the friction velocity ( $U_\tau$ ) for both smooth and riblet cases, we utilised an in-house drag balance, as deployed in previous studies on trapezoidal riblets (Endrikat et al. Reference Endrikat, Newton, Modesti, García-Mayoral, Hutchins and Chung2022) and rough walls (Ramani et al. Reference Ramani, Schilt, Nugroho, Busse, Jelly, Monty and Hutchins2024). Details and schematics of the drag balance can be found in Ramani et al. (Reference Ramani, Schilt, Nugroho, Busse, Jelly, Monty and Hutchins2024). A summary of the primary flow and riblet parameters at each $U_\infty$ condition, including $U_\tau$ , is provided in table 2. As expected for the smooth wall data in figure 8(a), regardless of the $Re_\tau$ (i.e. $U_\infty$ ) values, all the velocity profiles overlap in the inner ( $U^+ = z^+$ ) and logarithmic ( $U_{log }^+ = \kappa ^{-1}\log (z^+) + A$ ) regions. In this context, $\kappa = 0.384$ and $A = 4.17$ represent the logarithmic constants.

Riblets have been shown previously to alter the slope in the logarithmic region ( $\kappa$ ) (Endrikat et al. Reference Endrikat, Newton, Modesti, García-Mayoral, Hutchins and Chung2022). Thus forcing the current mean velocity profiles to match the smooth wall $\kappa$ slope by altering the virtual origin was deemed unreliable. Consequently, the wall-normal origin for all riblet $U^+$ profiles in figure 8(a) is assumed to be at the riblet mid-height ( $h_r/2$ ). The hot-wire measured profile $U^+$ (RR43E ) aligns well with the current simulation RR43 (), except within the valley, where the hot-wire signal overestimates $U^+$ due to heat losses to the wall. This overestimation affects the accuracy of the measured velocity magnitude, making it difficult to directly determine the inflection point from the velocity profile, which is often indicative of instability in shear flows (Rayleigh Reference Rayleigh1880). The measured $U^+$ profiles over the riblet surfaces are shifted downwards compared to that of the smooth wall. This downward shift, namely the Hama roughness function $\Delta U^+$ , is shown in figure 8(b). Owing to unavailability of the correct virtual origin for the current riblet study, we vary the origin between the riblet mid-height and crest. Such variations of the virtual origin lead to changes in $\Delta U^+$ , which are presented by error bars in figure 8(b). It is worth nothing here that the uncertainties about the exact virtual origin for each riblet case do not affect the discussion regarding K–H rollers, as all the following results related to the K–H rollers characteristic are primarily reported with reference to the riblet mid-height. Results of the current DNS RR43 of the rounded-valley riblets (), $90^\circ$ triangular riblets of Bechert et al. (Reference Bechert, Bruse, Hage, Van der Hoeven and Hoppe1997) (), trapezoidal riblets of Von Deyn et al. (Reference Von, Lars, Gatti and Frohnapfel2022) (), and $30^\circ$ sawtooth riblets () and blade riblets () of Endrikat et al. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a) are also included. For the current data, $\Delta U^+$ appears to gradually increase with $l_g^+$ , implying increased drag compared to the smooth wall. The small discrepancy between the experimentally estimated $\Delta U^+$ (for $l_g^+ \approx 44$ ) and that of the simulations (with $l_g^+ \approx 42$ ) could be attributed to the different flow geometry (boundary layer for the experiments and channel for the simulation), the mismatched $h_r/\delta$ , the valley rounding and/or the measurement uncertainty, e.g. hot-wire positioning and drag balance uncertainty.

5.1. Variation in spectral signature of K–H rollers

Before we investigate the variation in spatial coherence of the K–H rollers as a function of $l_g^+$ , it is important to identify the variation (if any) in the auto-spectral signatures. For this, we begin by analysing ${f}{{\phi }^+_{u^\prime u^\prime }}$ plotted in figure 11, which is estimated from hot-wire data acquired over riblets at various $s^+$ : $27 \lesssim s^+ \lesssim 110$ and $1000 \lesssim Re_{\tau } \lesssim 3500$ (refer to table 2). Here, considering that all the spectra are obtained from $u^{{\prime }+}$ time series, we have opted to plot them as functions of the viscous scaled time/frequency scales, i.e. $T^+={U^2_{\tau }}/{f{\nu }}$ , instead of applying Taylor’s hypothesis (where $f$ indicates frequency). Consistent with figure 9, a distinct spectral signature of the K–H rollers is evident at small time scales ( $T^+\lesssim 200$ ) for $z^+$ below the riblet crest, irrespective of the variation in $s^+$ . Their spectral signature has been marked by black rectangles in figure 11, which clearly reflect a shift in the characteristic viscous scaled time scale of the K–H rollers with increasing $s^+$ . Potentially, this can also be interpreted as an increase in the streamwise periodic offset between the K–H rollers, with increasing $s^+$ . It is worth noting that due to the absence of an estimate of the convection velocity at different tested $l_g^+$ cases, one could argue that the observed increase in the K–H wavelength might be linked to an increase in the convection velocity. Therefore, in future analyses, investigating the convection velocity could provide further insights into this issue. Another noteworthy observation from figure 11 is the reduction in spectral energy associated with the K–H rollers, with increasing $s^+$ . This could be associated with the penetration of large-scale structures ( $T^+ \gtrsim 500$ ) into the riblet valleys with increasing $s^+$ , which is indicated by the relative darkening and broadening of the energy contours below the dashed line.

Figure 11. The viscous scaled one-dimensional pre-multiplied spectra of streamwise velocity fluctuations, denoted as $f\phi _{u^\prime u^\prime }^+$ , obtained from experimental data. The spectra are plotted as functions of the viscous scaled time scale $T^+$ and the wall-normal distance normalised by half the riblet height, represented by $z/(h_r/2)$ . (a–e) Plots corresponding to different values of tip spacing, ranging from $s^+ \approx 27$ to $s^+ \approx 110$ . The horizontal dashed lines indicate the location of the riblet crest ( $z/(h_r/2) = 1$ ). The rectangles highlight the narrowband frequency observed below the riblet crest, which serves as an indication of the presence of K–H rollers.

To better demonstrate the effect of $s^+$ on the existence of K–H rollers and the penetration of large scales into the groove, figure 12(a) showcases ${f}\phi _{u^\prime u^\prime }^+$ at the mid-height of the riblet ( $z/(h_r/2) \approx 1$ ) for various $s^+$ cases plotted against $T^+$ . The red marker $\times$ indicates that the peak of ${f}\phi _{u^\prime u^\prime }^+$ (which is representative of the K–H rollers) is most energetic for the lowest $s^+$ value tested (approximately 27). However, as the riblet tip spacing ( $s^+$ ) increases, this peak generally undergoes attenuation and shifts towards higher $T^+$ values. The attenuation is not monotonic: the peak ${f}\phi _{u^\prime u^\prime }^+$ of the $s^+ = 110$ case is higher than peaks of $s^+ = 66, 88$ . This trend could potentially be attributed to the downward displacement of the wire at higher flow speeds, as discussed in § 3.3. Figure 12(b) depicts the variation in spectral signature ( $T^+$ ) of the K–H rollers (referred to as $T_{KH}^+$ ) against $s^+$ , which demonstrates a quasi-linear relationship between these two variables for $s^+ \lesssim 110$ . These observations reaffirm that the frequency of occurrence of K–H rollers is influenced by the riblet tip spacing ( $s^+$ ) and/or $Re_\tau$ .

In this context, it is relevant to highlight the study by Ho & Huerre (Reference Ho and Huerre1984), who characterised the frequency of K–H rollers by estimating the Strouhal number, defined as $St = \theta / (T_{KH} U_{{KH}})$ . Here, $U_{{KH}}$ denotes the convection velocity of the rollers, while $\theta$ represents the momentum thickness of the mixing layer associated with K–H instability. Ho & Huerre (Reference Ho and Huerre1984) determined a Strouhal number $St \approx 0.032$ for K–H instability, a result later validated by Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002). In the present study, Strouhal numbers were observed within the range 0.023–0.034 across the tested values of $s^+$ , assuming $U_{{KH}} \approx U_{{crest}} \approx 6U_\tau$ (figure 8 a) and $\theta \approx (1/4)\delta _w \approx (1/4) z_{KH}$ , where $\delta _w$ is the vorticity thickness characterising the vertical extent $z_{KH}$ of the roller structures, later measured to be 1.5 to $0.5h_r/2$ (see figure 14 b). These observed values align closely with those reported by Ho & Huerre (Reference Ho and Huerre1984) and Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002), further supporting the identification of the present structures as K–H rollers.

Figure 12. (a) The viscous scaled one-dimensional pre-multiplied spectra of streamwise velocity fluctuations $f\phi _{u^\prime u^\prime }^+$ at the riblet mid-height versus $T^+$ . The darker blue shading illustrates greater $s^+$ . The red ‘ $\times$ ’ marks indicate the peaks of $f\phi _{u^\prime u^\prime }^+$ for K–H rollers at $T_{KH}^+$ . (b) Plot of $T_{KH}^+$ as a function of $s^+$ . The secondary $x$ -axis in (b) shows the corresponding $Re_\tau$ values; it is not linear.

5.2. Variation in spatial coherence of K–H rollers

Here, we employ the cross-spectral analysis demonstrated previously in figure 10, to characterise the spatial (i.e. spanwise and wall-normal) extent of the K–H rollers. Figure 13 shows the cross-spectra of the streamwise velocity fluctuations computed between the $u^{{\prime }+}$ signal from a fixed ‘reference’ sensor (positioned approximately at the mid-height of a riblet groove), and the $u^{{\prime }+}$ signal from a wall-normal traversing probe, for all $s^+$ cases tested. To examine the spanwise coherence of the rollers, the spanwise offset between the two sensors ( $\Delta y$ ) is varied across $0 \lesssim {\Delta }y/s \lesssim 5$ . For the case ${\Delta }y=0$ , both sensors are situated within the same riblet groove, with the fixed sensor located at the riblet mid-height while the other sensor traverses vertically away from the fixed wire, starting from a minimum wall-normal separation of approximately 0.4 mm. The plots in figure 13 are arranged from left to right to reflect the increasing values of $27 \lesssim s^+ \lesssim 110$ , while the vertical arrangement from top to bottom represents the ascending values of $0 \lesssim \Delta y/s \lesssim 5$ . Here, since our primary focus lies on the K–H rollers, the cross-spectra are limited to $T^+ \lesssim 300$ , for clarity and relevance. To isolate the effect of $Re_\tau$ of the K–H rollers, we also included the results from the simulation of RR43 in figure 13(c), next to the matched $s^+$ from the experiment over RR43E in figure 13(b).

Figure 13. The viscous scaled co-spectra of streamwise velocity fluctuations $f\phi _{{u^\prime {u_{\textit{ref}}^\prime }}}^+$ of a fixed signal located at approximately the riblet mid-height and other wall-normal traversing signals for (a–f) $27 \lesssim s^+ \lesssim 110$ and (I–VI) $0 \lesssim \Delta y/s\lesssim 5$ . Plots in (c), as highlighted with a red box, are results obtained from the DNS for RR43, with the exact configuration ( $z$ and $y$ locations, spatial resolution and absolute value of the signal) of the experiments of the other columns. The horizontal solid lines indicate the location of the riblet crest at $z/(h_r/2) = 1$ . The rectangles highlight the positive correlation below the crest as indication of K–H rollers. The greentick and redcross symbols refer to detected and undetected K–H rollers, respectively.

In agreement with the observations made in figure 11, the plots in figure 13(aI $-$ eI) at $\Delta y/s = 0$ reveal a positive correlation below the riblet crest, i.e. ${f}\phi _{{u^\prime {u_{\textit{ref}}^\prime }}}^+ \gt 0$ , at $T^+$ associated with the K–H rollers (indicated by rectangles). Consistent with figure 12, the peak of this correlation shifts to higher $T^+$ values as $s^+$ increases, while the strength of the positive correlation appears to diminish with increasing $s^+$ . Further to that, the wall-normal extent of this positive correlation, which represents the wall-normal coherence of the K–H rollers, initially extends above the riblet tips for $s^+ \approx 27$ , but decreases in height until it is confined near the mid-height of the riblet, at $s^+ \approx 110$ . Hence the first row of plots in figure 13 suggest that the wall-normal coherence as well as the strength of the K–H rollers decreases as $s^+$ increases. Next, to investigate the cross-spectra for various spanwise separations, we just view the columns of co-spectra in figure 13 for each viscous scaled riblet size $s^+$ . For the smallest riblet size ( $s^+ = 27$ , in figure 13 aI–aVI) the distinct signature of the K–H rollers is evident for all spanwise separations ( $\Delta y/s \lesssim 5$ ). For larger $s^+ \gtrsim 43$ cases of the experimental results, however, coherence of these K–H rollers appears to decrease as $\Delta y/s$ increases.

The influence of $Re_\tau$ on the characteristics of K–H rollers becomes evident upon comparing figure 13(b) for RR43E at $Re_\tau \approx 1500$ and figure 13(c) for RR43 at $Re_\tau \approx 600$ . Examining the first row of figure 13(b,c) at $\Delta y/s = 0$ , the coherence of K–H rollers is notably narrower in band $50 \lesssim T^+ \lesssim 300$ for RR43 in comparison to the range $T^+ \gtrsim 50$ for RR43E. This distinction is largely attributed to the deeper penetration of large-scale structures into the riblet valleys, influenced by the substantially higher $Re_\tau$ in the case of RR43E, aligning with the observations of figure 11. Of particular interest is the discernible interruption in the spanwise extension of K–H rollers induced by the elevated $Re_\tau$ in the RR43E case (see figures 13(b,c)I–VI). The coherence of K–H rollers experiences a loss at $\Delta y/s \approx 4$ for RR43E, whereas it remains comparatively robust until $\Delta y/s \approx 5$ for the RR43 case. These findings collectively underscore the substantial influence of increasing $Re_\tau$ in disturbing the K–H rollers. We should also note that between the current simulation (RR43) and experimental (RR43E) cases, the ratios of the riblet height to the boundary layer thickness are substantially different, and could also contribute to the differences observed between the two cases.

Figure 14. (a) Contours of the peaks of $\phi _{{u^\prime {u_{\textit{ref}}^\prime }}}^+$ at $T_{KH}$ below the crest as a function of the two wires viscous scaled spanwise separation $\Delta y^+$ and $s^+$ . The $+$ marks show the measurement points, while the rest of the contour is interpolated. (b) The variation in wall-normal coherence ( $z_{KH}/(h_r/2)$ ) as a function of $s^+$ , shown with the $\times$ marks. The secondary $x$ -axis in (a) and (b) shows the corresponding $Re_\tau$ values; it is not linear.

Overall, figure 13 suggests that both the spanwise and wall-normal extents of the K–H rollers are influenced by $s^+$ or $Re_\tau$ . Notably, for large $s^+ \approx 110$ , the spatial coherence of the K–H rollers is confined to a single valley, indicating a deviation from their typical ‘spatial extension’ along the span. To analyse the spanwise extension of the rollers in viscous scaling (i.e. $\Delta y^+$ ), instead of $\Delta y/s$ as done in figure 13, we plot in figure 14(a) contours of the peak of $f\phi _{{u^\prime {u_{\textit{ref}}^\prime }}}$ (i.e. at $T^+_{KH}$ ) at $z^+ \lesssim h^+_r/2$ , as a function of $\Delta y^+$ and  $s^+$ . Since in this study $s^+$ is varied simultaneously with $Re_\tau$ , the corresponding $Re_\tau$ values for each case are shown on the top axes of figure 14. This highlights the coupled variation and underscores the limitation in isolating the individual effects of $s^+$ and $Re_\tau$ within the present dataset. The strength of the K–H rollers appears to slightly reduce as $s^+$ increases. This, however, should not be confused with the spanwise extent of the near-wall cycle, given that it has a distinctly different $T^+\sim100$ than noted for $T^+_{KH}$ in figure 12(a). Figure 14(b) additionally presents the wall-normal extension of K–H rollers, denoted as $z_{KH}/(h_r/2)$ plotted against $s^+$ for values acquired from the first row ( $\Delta y = 0$ ) of figure 13. Here, $z_{KH}$ is detected at $T_{KH}$ when $f\phi _{{u^\prime {u_{\textit{ref}}^\prime }}}^+ \approx 0.05$ . As elaborated previously, it is evident that increasing $s^+$ (and $Re_\tau$ ) leads to a discernible reduction in the wall-normal extension of K–H rollers. We speculate that the increasing penetration of relatively large scales into the riblet grooves, with increasing $s^+$ or $Re_{\tau }$ , affects the velocity mismatch between the valley and above the riblet crest.