3.1. Drag

First, we present the total drag force experienced by each of the samples and the reference smooth case at $\textit{Re}_{L} = 12\,200$ , $18\,500$ and $24\,200$ in figure 4. Here, using a three-tiered measurement approach (Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023), we linearly decompose the total drag measured via the dynamometer into three components; $D_{f}$ , the frictional drag force, $D_{p}$ , the pressure drag and $D_{{others}}$ , which is all the three-dimensional and other effects that are not captured by the previous two components ( $D = D_{f} + D_{p} + D_{{others}}$ ). All the drag components are normalised by $(1/2)\rho U_{\infty }^{2} 2Lb$ , where $b$ is the width of the sample in the spanwise direction, and presented in the form of drag coefficients, $C_D^f$ , $C_D^p$ and $C_D^{{others}}$ . On the first tier, the friction drag, due to the shear stress distribution, is found using the integral of the $\langle \tau _{n=0} \rangle$ distribution on both sides of the sample and discussed more in §§ 3.2 and 3.3. On the second tier, we find the pressure drag cumulatively using a control volume analysis, described in more detail by Fu & Raayai-Ardakani (Reference Fu and Raayai-Ardakani2023) and Suchandra & Raayai-Ardakani (Reference Suchandra and Raayai-Ardakani2024), and discussed later in § 3.5. On the last level, the $D_{{others}}$ then is found by subtracting the pressure and fiction drags from the dynamometer measurements.

Figure 4. Decomposition of the total drag force, in terms of the drag coefficient, into friction, $C_D^{f}$ , pressure, $C_D^{p}$ , and $C_D^{{Others}}$ components for experiments performed at global Reynolds numbers (a) $\textit{Re}_{L} = 12\,200$ , (b) $\textit{Re}_{L} = 18\,500$ and (c) $\textit{Re}_{L} = 24\,200$ for all samples. The error bars on the $C_D^f$ are calculated using the upper and lower bounds of the integral of the shear stress discussed in more detail in § SI.2. of the supplementary material. The rest of the error bars correspond to the $95\, \%$ confidence intervals of the profile and total drag measurements. Values on each barare the percentage of the contribution of each component with respect to the total drag of the smooth reference sample. For example, for the [1.5,1.5] sample, at $\textit{Re}_{L} = 18\,500$ , frictional drag is $42.95\, \%$ of the total drag of the smooth reference. In this manner, the sum of the values for smooth samples comes to $100\, \%$ . Average $\overline {\lambda ^{+}}$ and the percentage of total drag reduction (sum of the values on each horizontal line minus 100) for all the samples are listed in the table on the right.

As seen in figure 4, within families of ${\mathcal {R}} = 0.5$ and ${\mathcal {R}} = 1.0$ , concave textures with $\kappa = {\mathcal {R}}$ exhibit the lowest drag force compared with the rest of the samples and convex ones with $\kappa = -{\mathcal {R}}$ the largest drag. Triangular textures with $\kappa = 0$ show drag values in between that of the $\kappa = \pm {\mathcal {R}}$ cases, sometimes close to one or the other depending on the case. However, going to the sharper textures of ${\mathcal {R}} = 1.5$ , at all Reynolds numbers, sample [1.5, 0.0] (triangular) retains the lowest drag, with the concave [1.5, 1.5] riblet in second place and convex [1.5, –1.5] seeing the largest drag for $\textit{Re}_{L} = 12\,200$ and $18\,500$ . At $\textit{Re}_{L} = 24\,200$ the concave and convex samples, [1.5, 1.5] and [1.5, –1.5], experience nearly similar values of drag.

While different from BL past foils, this trend seen in the drag as a function of the $\kappa _2$ is similar to the pattern previously reported in the Couette flow and early Taylor vortex regime in a Taylor–Couette flow (Raayai-Ardakani Reference Raayai-Ardakani2022). For shallow textures with ${\mathcal {R}} \leqslant 1$ , torque of the riblet-covered rotors decreases as $\kappa _2$ is increased (at a constant ${\mathcal {R}}$ ). Going to sharper textures such as $\kappa _2 = 1.5$ , the torque does not monotonically follow the $\kappa _2$ parameter and the lowest recorded torque is around $0 \leqslant \kappa _2 \leqslant 0.5$ for riblets with a spacing of 1 mm (spacing normalised by the gap of the Taylor–Couette cell of 0.13). In addition, previous experiments by Walsh & Lindemann (Reference Walsh and Lindemann1984) show larger drag reductions for concave-up semi-circular riblets compared with triangular textures and nearly no drag reduction for convex-up semi-circular riblets. They also show that an increase in the radius of curvature at the trough of triangular textures can reduce the drag while an increase in the radius of curvature at the peak of the grooves slowly increases the drag experienced by the riblets.

In comparison to the smooth reference, the majority of the cases tested here show some level of drag reduction. For the ${\mathcal {R}} = 0.5$ family, only the [0.5, 0.5] and [0.5, 0.0] samples were drag-reducing in all cases. In the ${\mathcal {R}} = 1.0$ family, the [1.0, 1.0] sample experiences the largest drag reduction among the rest of the family, with [1.0, 0.0] seeing slightly lower reductions, and the [1.0, –1.0] samples experiencing no reduction at $\textit{Re}_{L} = 24\,200$ or reductions of slightly larger than $1\,\%$ for the lower Reynolds numbers. For the ${\mathcal {R}} = 1.5$ family, the [1.5, 0.0] sample is able to stay drag-reducing for all the tested Reynolds numbers, while for $\textit{Re}_{L} = 12\,200$ and $18\,500$ , the [1.5, –1.5]sample is fully drag-increasing and [1.5, 1.5] stays nearly neutral. At the largest Reynolds number of 24 200, all ${\mathcal {R}} = 1.5$ samples become drag-reducing.

Breaking down the measured drag into components, we see that the friction and the pressure drags do not follow the same trend as that of the total drag. Except for a few of the cases reported, the total frictional component of the drag force only experiences marginal changes due to the presence of the textures as discussed more in § 3.2. However, we record substantial levels of reductions in the pressure drag for quite a number of the samples that help in reducing the drag even when the frictional drag is unchanged. While the effect of riblets in changing the pressure distribution and pressure drag has largely been unexplored, recent numerical simulations of riblet-covered bodies (Mele et al. Reference Mele, Tognaccini and Catalano2016, Reference Mele, Tognaccini, Catalano and de Rosa2020; Cacciatori et al. Reference Cacciatori, Brignoli, Mele, Gattere, Monti and Quadrio2022), as well as other skin-friction reduction techniques (Banchetti et al. Reference Banchetti, Luchini and Quadrio2020; Nguyen et al. Reference Nguyen, Ricco and Pironti2021; Quadrio et al. Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022) have also confirmed the ability of these techniques to impact the pressure drag as discussed later in § 3.5.

At the lowest Reynolds number, $\textit{Re}_{L} = 12\,200$ , and among the family of ${\mathcal {R}} = 0.5$ , only the $D_{f}$ of the [0.5, 0.5] sample experiences a reduction of $2.2\, \%$ compared with the smooth reference, contributing $1\, \%$ to the total of $4.1\, \%$ reduction in the total drag. (Note that the values on each of the bars in figure 4 corresponds to the percentage of the contribution of the specific component with respect to the smooth reference sample at the given Reynolds number.) Among the ${\mathcal {R}} = 1.0$ family, both the [1.0, –1.0] and [1.0, 1.0] samples experience $1\, \%$ and $3.2\, \%$ reduction in the $D_{f}$ , contributing to $0.5\, \%$ (out of $1.48\, \%$ ) and $1.6\, \%$ (out of the $3.82\, \%$ ) to the total drag reductions of these samples, respectively. The [1.0, 0.0] sample on the other hand experiences a $4.6\, \%$ increase in the frictional drag, requiring the pressure drag to contribute a $3.6\, \%$ reduction to overcome the added $2.27\, \%$ for the sample to experience a $1.33\, \%$ reduction in the total drag. Similar trends can be extracted for the samples at larger Reynolds numbers (see figure 4).

Overall, $C_D^{f}$ of ${\mathcal {R}} = 0.5$ and ${\mathcal {R}} = 1.5$ follows a similar trend as that of the total drag force, while for the ${\mathcal {R}} = 1.0$ family, unlike the total drag, the trend in $C_D^{f}$ is non-monotonic with respect to $\kappa _2$ , with the [1.0, 1.0] sample experiencing the lowest and [1.0, 0.0] experiencing the highest $C_D^{f}$ at $\textit{Re}_{L} = 12\,200$ and $18\,500$ , and at the largest $\textit{Re}_{L} = 24\,200$ , the [1.0, 1.0] and [1.0, 0.0] samples have nearly similar values of $C_D^{f}$ , which is larger than that of the [1.0, –1.0] sample. Among the ${\mathcal {R}}= 0.5$ case, the magnitude of the pressure drag also takes a decreasing trend with the $\kappa _2$ parameter, with [0.5, 0.5] experiencing the lowest $C_D^{p}$ and [0.5, –0.5] experiencing the largest one. As a results, [0.5, 0.5] and [0.5, 0.0] also experience a reduction in $C_D^{p}$ compared with the smooth sample while [0.5, −0.5] stays either neutral or $C_D^{p}$ -increasing. Thus, within this family, cumulatively with $C_D^{f}$ , only the [0.5, 0.0] and [0.5, 0.5] samples are drag-reducing. Among the ${\mathcal {R}} = 1.0$ family, at $\textit{Re}_{L} = 12\,200$ and $18\,500$ , the [1.0, 0.0] sample experiences the lowest $C_D^{p}$ , while at $\textit{Re}_{L} = 24\,200$ the [1.0, 1.0] sample has the lowest $C_D^{p}$ and the [1.0, –1.0] sample experiences the largest $C_D^{p}$ among them all. The [1.0, 0.0] sample, which could not offer any frictional reduction, experiences a large enough reduction in the pressure drag to be cumulatively drag-reducing. Similarly, for the [1.0, 1.0] sample, the reduction achievable in the pressure drag enhances the cumulative drag reduction of this sample. For the [1.0, –1.0] sample, the pressure drag is not able to enhance the drag reduction at $\textit{Re}_{L} = 12\,200$ and $18\,500$ , and at $\textit{Re}_{L} = 24\,200$ it even diminishes the drag reduction that was achieved from the frictional component. For the sharpest riblet family, at $\textit{Re}_{L} = 12\,200$ and $18\,500$ , [1.5, 0.0] experiences the lowest pressure drag and [1.5, –1.5] the largest, and only [1.5, 0.0] is able to see a cumulative reduction in the drag force. At $\textit{Re}_{L} = 24\,200$ , the pressure drag experienced by the three samples is nearly similar and all three are able to capture a reduction in the pressure drag, which also leads to them being cumulatively drag-reducing as well.

While the foils used here operate in the laminar regime and have a lower $\textit{Re}_{L}$ compared with the $\textit{Re}_c$ of the previously reported cases with riblets used on airfoils, we use an average wall shear stress based on the total frictional drag force to find an average $\overline {\lambda ^{+}}$ for the entirety of the samples as reported in figure 4. Recognising the difference in the flow fields, we can still attempt to compare the results against the previous reports, especially for the case of the triangular riblets with $\kappa _2 = 0$ . Coustols (Reference Coustols1989) reports a $2.7\, \%$ reduction for $\lambda ^{+}=10$ , while Subashchandar et al. (Reference Subashchandar, Rajeev and Sundaram1995) and Subaschandar et al. (Reference Subaschandar, Kumar and Sundaram1999) report a $15\, \%$ reduction and Caram & Ahmed (Reference Caram and Ahmed1991) report a $13.3\, \%$ reduction. For $10\lt \lambda ^{+}\lt 15$ , Sundaram et al. (Reference Sundaram, Viswanath and Rudrakumar1996) report a $16\, \%$ reduction. For all these cases, the height and spacing of the grooves are the same and, thus, $A^{+} = \lambda ^{+}$ . Chamorro et al. (Reference Chamorro, Arndt and Sotiropoulos2013) report a max imumreduction of $5\, \%$ for $\lambda ^{+} \approx 11{-}14$ . In the cases reported here, the closest situation is the sharpest riblet of ${\mathcal {R}} = 1.5$ where, for $\lambda ^{+} \approx 10$ , the reductions achieved are around $1.88\, \%$ , increasing to $4.5\, \%$ for ${\lambda ^{+}} = 13.36$ and reducing to $4.09\, \%$ for $\lambda ^{+} = 17.27$ . Since the previously reported cases are all for $\textit{Re}_c\gt 10^5$ , one could hypothesise that the larger reductions reported can potentially be related to reductions in the Reynolds shear stresses close to the wall that are absent in the current results.

Lastly, as can be seen from the percentages listed on the bars of figure 4, the contributions attributed to the $D_{{others}}$ , not captured via the PIV measurements, do not show any clear dependence on the textures and have nearly similar values and so are not discussed further. We believe this component is mainly due the three-dimensional edge effects of the low-aspect ratio foils used in the study as well as the impact of a shield (see figure 3) used to reduce the effect of the flow on the strut connecting the sample to the linear variable differential transformer sensor.

3.2. Friction drag

Out of the total drag force exerted on the slender samples, between $40\, \%\!-\!50\,\%$ of the drag is due to the frictional component, $D_{f}$ (as seen from the values written on purple portions of the bar plots in figure 4). This component is comprised of the frictional drag experienced on either side of the sample found using the integral of the spanwise-averaged shear stress distributions as

(3.1) \begin{align} D_{f} = D_{\textit{Front}} + D_{\textit{Back}} = b\left ({\int _0^{L}} \langle {\tau _{n=0}} \rangle _{\textit{Front}}\, {\textrm {d}}x + {\int _0^{L}} \langle {\tau_{n=0}} \rangle _{\textit{Back}}\, {\textrm {d}}x \right). \end{align}

It should be noted that we use the control volume analysis inside the groove as discussed in § SI.2 of the supplementary material to calculate the upper and lower bounds of the difference between the $\langle \tau _{n=0} \rangle$ and $\langle \tau _{{ w}} \rangle$ (where $\langle \tau _{{ w}} \rangle$ is the spanwise-averaged shear stress distribution on the riblet wall) and use them in the integral of (3.1) and present them as the errorbars shown on the $D_f$ in figure 4. Along each side, (3.1) can be divided into the contributions of the LE, LET and Flat region as written in the form of

(3.2) \begin{align} D_{\textit{Front}} = b \left ({\int _0^{x_{\textit{LET}}}} \langle {\tau _{n=0}} \rangle _{\textit{Front}} (x) {\textrm {d}}x + {\int _{x_{\textit{LET}}}^{x_{\textit{Flat}}}} \langle {\tau _{n=0}} \rangle _{\textit{Front}} \,(x) {\textrm {d}}x + {\int _{x_{\textit{Flat}}}^{L}} \langle {\tau _{n=0}} \rangle _{\textit{Front}} \, (x) {\textrm {d}}x \right ) \end{align}

and $D_{Back} = {D_{Back}^{LE}} + {D_{Back}^{LET}} + {D_{Back}^{Flat}}$ . Note that the spanwise-averaged wall shear stress distribution, $\langle {\tau _{n=0}} \rangle$ , is the shear stress component exerted along the surface, or $\hat {s}$ , with $\theta$ the local angle between the $x$ and $\hat {s}$ directions. Thus, with ${\textrm {d}}x = \cos \theta {\textrm {d}}s$ , we have $\langle {\tau _{n=0}} \rangle \cos \theta {\textrm {d}}s = \langle {\tau _{n=0}} \rangle {\textrm {d}}x$ .

In addition to the cumulative form of the $D_{f}$ shown in figure 4 (purple bars), the contributions of each side of the sample, as well as the LE, LET and Flat segments on $D_{f}$ are presented in figure 5. Firstly, due to the small angle of attack, the contribution of the pressure side is larger than the suction side for all the samples, including the smooth reference. In addition, for all the riblet samples except for [0.5, 0.5] at $\textit{Re}_{L} = 18\,500$ and $24\,200$ , and [1.0, –1.0] and [1.5, 0.0] at $\textit{Re}_{L} = 24\,200$ , the pressure sides experience drag increases compared with the smooth reference, and reduction in frictional drag is mostly visible on the suction sides. All members of the ${\mathcal {R}} = 1.5$ family experience a reduction in the frictional drag on the suction side. Among the other families, [0.5, –0.5] becomes $D_{f}$ -increasing at $\textit{Re}_{L} = 18\,500$ and $24\,200$ andthe [1.0, 0.0] sample is either neutral or $D_{f}$ -increasing at $\textit{Re}_{L} = 12\,200$ and $18\,500$ , while the rest of the cases remain $D_{f}$ -decreasing.

Figure 5. Bar plots showing the contributions of LE, LET and Flat regions on either side of the riblet samples to the frictional drag coefficient. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are shown on the top schematic, corresponding to the bounds in the integrals of (3.2).

The LE region captures $14\, \%$ , $10\, \%$ and $7.5\, \%$ of the length of ${\mathcal {R}} = 0.5$ , ${\mathcal {R}} = 1.0$ and ${\mathcal {R}} = 1.5$ families, respectively. The LE region experiences the largest levels of local shear stress as is expected from a developing BL. Without any riblets in this area, the drag experienced by the LE regions (dark magenta bars in figure 5) of all samples of the constant- ${\mathcal {R}}$ family experience nearly the same levels of $D_{f}$ at a constant Reynolds number and, for both sides, the $C_D^{{ LE}}$ are within $0.1\, \%$ of the contributions found for the smooth surface ( $C_{D,0}^{{ LE}}$ ). Thus, the LE regions do not contribute to the drag change and this portion of the frictional drag, $C_D^{{ LE}}$ , is not available for modification by the riblets. With the difference in the starting point of the riblets, i.e. $x_{{ LET}}$ , for the different families, the ${\mathcal {R}} = 0.5$ family experiences the largest contribution from the $C_D^{{ LE}}$ region, with ${\mathcal {R}} = 1.0$ getting a lower contribution, and ${\mathcal {R}} = 1.50$ capturing the lowest contribution from the LE region.

Based on $x_{{ LET}}$ , $11\, \%$ , $15\, \%$ and $17.5\, \%$ of the length of the ${\mathcal {R}} = 0.5$ , ${\mathcal {R}} = 1.0$ and ${\mathcal {R}} = 1.5$ samples is comprised of the growing riblets in the LET region, respectively. Thus, as shown in figure 5, in reverse order compared with the LE regions, the LET portion of the ${\mathcal {R}} = 1.5$ family captures a larger portion of the total $D_{f}$ on either side of the sample, compared with ${\mathcal {R}} = 1.0$ , and ${\mathcal {R}} = 1.0$ has a larger $C_D^{\textit{LET}}$ than the ${\mathcal {R}} = 0.5$ family. With the BL still developing in this region, the magnitudes of the local $\langle \tau _{n=0} \rangle$ are still large in this segment and while only capturing less than $18\, \%$ of the total length, $C_D^{\textit{LET}}$ captures a considerable portion of the total drag force on either side, especially for the sharper riblets of the ${\mathcal {R}} = 1.5$ and ${\mathcal {R}} = 1.0$ families. While the changes in the $C_D^{\textit{LET}}$ is a small portion of the total change the frictional drag experiences, the changes in this region act as a starting point toward drag-reducing or drag-increasing riblets and will be discussed more in § 3.3. The Flat segment, capturing $75\, \%$ of the length, experiences the largest portion of the $C_D^{f}$ .

Cumulatively, the drag on the suction sides of all samples and at all Reynolds numbers follow similar trends with respect to $\kappa _2$ as that of the total $D_{f}$ , as discussed in § 3.1. On the pressure side however, while ${\mathcal {R}} = 1.0$ and $1.5$ follow the same trend as the total $D_{f}$ , the frictional drag of the pressure side ofthe ${\mathcal {R}} = 0.5$ family is non-monotonic with $\kappa _2$ with the [0.5, 0.0] sample experiencing the largest $D_{f}^{\textit{Pressure}}$ . Overall, since the total $D_f$ is the result of the sum of two integrals, the distribution of the $\langle C_{f} \rangle$ on either side of the sample as well as the difference between the $\langle \tau_{n=0} \rangle$ and the $\tau _{0}$ of the smooth reference will be the key parameters in characterising the variations discussed in this section.

3.3. Local shear stress distribution

To explain the trend in the friction drag, and its decomposition in terms of the suction/pressure side, as well as the LE, LET and Flat part of the plate, we focus on the local spanwise-averaged skin-friction coefficient distributions, $\langle C_f \rangle (x)$ . In general, we see fourtypes of $\langle C_f \rangle (x)$ as a function of the local Reynolds number, $\textit{Re}_x = \rho U(x) x/\mu$ , where $U(x)$ is the velocity at the edge of the BL and is nearly always larger than $U_{\infty }$ . Overall, these $\langle C_f \rangle (x)$ distributions show a more pronounced dependence on the total length of the foil and the location along the plate with respect to that ( $x/L$ ). The effect of the global Reynolds number (as dictated by $U_{\infty }$ ) is more visible in the order of magnitude of the values $\langle C_f \rangle (x)$ .

Figure 6. Four types of shear stress distribution observed in flow over riblets and representative examples from the data. (a) Type I forthe [0.5, –0.5] sample, at $\textit{Re}_{L}= 12\,200$ suction side. (b) Type II for the [0.5, 0.0] sample, at ${ Re}_{L}= 18\,500$ pressure side. (c) Type III for the [1.0, 1.0] sample, at ${ Re}_{L}= 12\,200$ suction side. (d) Type IV for the [1.0, 0.0] sample, at ${ Re}_{L}= 18\,500$ suction side. Colours and markers match the colours and markers used in the upcoming plots. Contour plots of (e) $u$ and (f) $v$ for riblet sample [1.0, 1.0] operated at ${ Re}_{L} = 24\,200$ .

In the first type (type I), shown in figure 6(a), the $\langle C_f \rangle (x)$ first follows a fast decreasing trend (as expected) as a function of the ${ Re}_x$ , starting in the LE, continuing in the LET and up to early in the Flat region. Afterward, the decreasing trend continues but at a much lower rate than in the first segment (lower than the ${ Re}_x^{-1/2}$ of the Blasius BL theory (Schlichting & Gersten Reference Schlichting and Gersten2016)), until close to the trailing edge, where the $\langle C_f \rangle (x)$ takes an increasing trend with ${ Re}_x$ . The smooth sample also follows a type I trend for all the Reynolds numbers and on both the suction and pressure sides (Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023). In the second type (type II), shown in figure 6(b), $\langle C_f \rangle (x)$ starts with a decreasing trend throughout LE, LET and early in the Flat region, and then $\langle C_f \rangle (x)$ becomes nearly constant in the Flat region, until close to the trailing edge, $\langle C_f \rangle (x)$ slightly increases. In the third type (type III), shown in figure 6(c), $\langle C_f \rangle (x)$ starts in a similar manner as types I and II, and the difference starts in the second part where after reaching a minimum $\langle C_f \rangle$ in the early portion of the Flat region, $\langle C_f \rangle (x)$ takes an increasing trend continuing to the trailing edge, with the increase at the trailing edge having a slightly faster rate. In the fourth type (type IV), shown in figure 6(d), in the early portion, the same decreasing trend as that of types I, II and III is visible. However, in this case, the decrease continues to a minimum, followed by an increase, leading to a region with near constant $\langle C_f \rangle (x)$ or $\langle C_f \rangle (x)$ -increasing at a very low rate of change, before getting close to the trailing edge where the $\langle C_f \rangle (x)$ increases at a faster rate.

It should be pointed out that in the vicinity of the trailing edge, the $\langle C_f \rangle (x)$ of all samples experiences an increasing trend, different from that usually expected from the BL. This is due to the finite thickness of the sample and how toward the trailing edge while $u$ maintains a familiar behaviour (figure 6 e), the velocity in the $y$ direction, $v$ (figure 6 f), changes sign compared with earlier along the plate, moving toward the body and not away from the surface. In a developing BL, as seen in the earlier portion of the plate, $v$ is always away from the surface (see the colour contours of figure 6 f), but from $x/L \approx 0.7$ , $v$ changes direction toward the plane (change in the colour contours) and, as a result, the velocity profiles are pushed into the surface resulting in a more attached BL and, thus, increase in $\langle C_f \rangle (x)$ for both smooth (Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023) and riblet samples. This trend has been reported in second-order models of a BL over finite-length plates as well(Dennis Reference Dennis1985).

Figure 7. Distribution of $\langle C_f \rangle (x)$ for all the riblet samples of ${\mathcal {R}} = 1.0$ for all the Reynolds numbers. The $C_{f,0}(x)$ of the reference smooth sample on the suction and pressure side are shown with solid and dashed black lines, respectively. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are marked by grey dotted and dash-dotted vertical lines.

The local distributions of $\langle C_f \rangle (x)$ are presented in figures 7 and SI.4(A-C) of the supplementary material for the suction and pressure sides of all the samples and at all global Reynolds numbers, as well as the accompanying distributions of the difference between the spanwise-averaged shear stress of the riblet samples and the smooth reference, $\unicode{x1D6E5} \langle \tau _{n=0} \rangle$ , normalised locally by that of the smooth reference, $\tau {_0}$ (figure SI.4(D-F)). In addition, a phase map of the type of shear stress distribution as a function of ${\mathcal {R}}$ and $\kappa _2$ for all the samples and the global Reynolds numbers is shown in figure 8. While the $\langle C_f \rangle (x)$ distributions follow only four patterns as driven by the BL development along the samples (more details in the upcoming § 3.4), within each type, we see a variety of patterns when considering the corresponding $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ distribution, as discussed in more detail below.

Figure 8. Phase map summarising the type of distribution of the $\langle C_f \rangle (x)$ as a function of ${\mathcal {R}}$ and $\kappa _2$ for the suction and pressure sides and the global Reynolds numbers. Signs (–/+) on the top right -handside of the markers indicate the drag-reducing/increasing nature of those sides of the samples (frictional component).

For all the cases, prior to the start of the textures, in the LE region the plates experience $\langle C_f \rangle (x)$ nearly identical to that of the smooth reference plate ( $C_{f,0}(x)$ ) as expected. Hence, we record similar values of $C_D^{{ LE}}$ for all the members of each family and the smooth surface as shown in § 3.2. As the riblets start and grow in the LET region, the $\langle C_f \rangle (x)$ starts to deviate from that of the smooth reference. As discussed earlier in § 2.2, the design of the leading edge mimicking the nose of a shark (Lauder et al. Reference Lauder, Wainwright, Domel, Weaver, Wen and Bertoldi2016), with a smooth LE region and gradual growth of the riblets in the LET region, allows the $\langle C_f \rangle (x)$ to incrementally develop with the growth of the textures, and thus avoiding the large levels of shear stress as seen in the leading edge of fullycovered plates (Raayai-Ardakani & McKinley Reference Raayai-Ardakani and McKinley2017). Owing to this development, in the LET we see differences between the $D_{f}^{\textit{LET}}$ contributions recorded for the riblet samples and the smoothsample; on the suction sides, the LET regions of the majority of the samples capture some level of local shear reduction, while on the pressure side, the riblets capture both shear reductions and increases.

The type I shear stress distribution (figures 7, SI.4 and 8) is mainly seen among the members of the shallowest family of ${\mathcal {R}} = 0.5$ , as shown in figure SI.4(Aa–g) and SI.4(Da–g). It dominates the suction sides of ${ Re}_{L} = 12\,200$ and ${ Re}_{L} = 18\,500$ cases for all the members, independent of $\kappa _2$ , and the [0.5, –0.5] sample at ${ Re}_{L} = 24\,200$ . For sharper riblets, this type is only seen to persist on the suction side of the [1.0, 0.0] sample at ${ Re}_{L} = 12\,200$ (figures 7(b) and SI.4(Eb)), and at all tested Reynolds numbers on the suction side of the [1.0, −1.0] sample (figures 7(a,d,g) and SI.4(Ea,d,g)) and pressure side of the [1.5, 0.0] sample (figures SI.4(Cb,e,h) and SI.4(Fb,e,h)).

Among one group of the cases, which occur on the suction sides of all the ${\mathcal {R}} = 0.5$ samples at ${ Re}_{L} = 12\,200$ , and [0.5, 0.0] and [0.5, 0.5] at ${ Re}_{L} = 18\,500$ , as well as the [1.0, –1.0] sample at all the tested Reynolds numbers, in the LET region, $\langle C_f \rangle$ grows to lower values than the $C_{{f},0}$ . After this initial reducing trend in the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ in the LET, samples experience a lower rate of decrease in the $\langle C_f \rangle (x)$ as a function of the ${ Re}_x$ compared with the smooth reference. Thus, they see increasing trends in $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ at times reaching close to the $C_{f,0}(x)$ , until about $x/L \approx 0.8$ where the trailing edge effect sets in and the increasing trend in the $C_{f,0}(x)$ of the smooth reference takes over the increasing trend of $\langle C_f \rangle$ of the riblet samples. This results in both $D_{{ Suction}}^{\textit{LET}}$ and $D_{{ Suction}}^{\textit{Flat}}$ components of these cases experiencing reductions, as shown in figure 5, even with the increasing trend in the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ in the Flat region, and ultimately resulting in overall drag reduction on the suction sides of these samples.

In a second group, as we see on the suction side of [0.5, –0.5] at ${ Re}_{L} = 18\,500$ and ${ Re}_{L} = 24\,200$ , and pressure side of the [1.5, 0.0] sample at ${ Re}_{L}=12\,200$ , $\langle C_f \rangle$ grows to similar or larger values than the $C_{{f},0}$ in the LET, and after the initial $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0 \geqslant 0$ region, with a lower rate of decrease in $\langle C_f \rangle (x)$ compared with the smooth reference along the length, $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ takes an increasing trend in the Flat region prior to decreasing in the trailing edge. Overall, along most of the length, $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0 \geqslant 0$ ; hence, these samples stay $D_f$ -increasing as shown in figure 5.

For the rest of the cases, namely the pressure side of [1.5, 0.0] at ${ Re}_{L} = 18\,500$ and ${ Re}_{L} = 24\,200$ and the suction side of the [1.0, 0.0] sample at ${ Re}_{L} = 18\,500$ , in the LET, $\langle C_f \rangle (x)$ starts at lower values than the smooth reference and later in the Flat region, $\langle C_f \rangle$ crosses over the $C_{f,0}(x)$ , turning shear-increasing, until later in the trailing edge area that some move to slightly shear-reducing. As a result of this, the suction side of the [1.0, 0.0] sample at ${ Re}_{L} = 18\,500$ stays nearly neutral in terms of the $D_f$ changes, and the pressure side of [1.5, 0.0] at ${ Re}_{L} = 18\,500$ is $D_f$ -increasing, while at ${ Re}_{L} = 24\,200$ , the extent of the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \lt 0$ is large enough that the pressure side of [1.5, 0.0] becomes $D_f$ -reducing (one of the few cases where the pressure side is drag-reducing).

Type II shear stress distributions are only seen on the pressure side and they only experience either drag increases or no change compared with the smooth counterpart except for one case of the pressure side of [0.5, 0.5] at ${ Re}_{L} = 18\,500$ , which records a $1.2\, \%$ reduction. This type (see figure 8) is seen on the pressure side of the ${\mathcal {R}} = 0.5$ family at all Reynolds numbers, except for the [0.5, 0.5] sample at ${ Re}_{L} = 24\,200$ (figures SI.4(Aa-Ah) and SI.4(Da-Dh)), as well as the pressure sides of [1.0, 0.0] at ${ Re} = 12\,200$ (figures 7(b) and SI.4(Eb)), [1.0, 1.0] at all tested Reynolds numbers (figures 7(c,f,i) and SI.4(Ec,f,i)) and [1.5, 1.5] at ${ Re} = 12\,200$ (figures SI.4(Cc) and SI.4(Fc)).

In one group of the cases, such as the pressure sides of [0.5, −0.5] and [0.5, 0.0] at ${ Re}_{L} = 12\,200$ and $18\,500$ , [0.5, 0.5] at ${ Re}_{L} = 12\,200$ , as well as [1.0, 0.0] and [1.5, 1.5] at ${ Re}_{L} = 12\,200$ , in the LET region, the $\langle C_f \rangle$ grows to larger than or near equal values to the smooth $C_{f,0}(x)$ and as $\langle C_f \rangle (x)$ becomes constant, $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ keeps increasing, resulting in $D_{\textit{Pressure}}^{\textit{LET}}$ and $D_{\textit{Pressure}}^{\textit{Flat}}$ larger than or near equal to the smooth ones.

In the second group of cases, [0.5, –0.5] and [0.5, 0.0] at ${ Re}_{L} = 24\,200$ , while in the LET, the $\langle C_f \rangle \geqslant C_{{f},0}$ , toward the end of the LET and early Flat region, $\langle C_f \rangle (x)$ moves to lower values than the smooth reference with a small region experiencing $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0\lt 0$ . However, as $\langle C_f \rangle (x)$ becomes constant, it crosses over the smooth reference, ultimately leading to both $D_{\textit{Pressure}}^{\textit{LET}}$ and $D_{\textit{Pressure}}^{\textit{Flat}}$ larger than or near equal to the smooth ones.

In the last group, on the pressure side of the [0.5, 0.5] at ${ Re}_{L} = 18\,500$ , [0.5, 0.0] at ${ Re}_{L} = 24\,200$ and [1.0, 1.0] at all Reynolds numbers, initially, $\langle C_f \rangle$ grows to lower values than $C_{f,0}(x)$ of the smooth reference. But as the $\langle C_f \rangle (x)$ becomes constant, it crosses over the smooth becoming shear-increasing. While $D_{\textit{Pressure}}^{\textit{LET}}$ is smaller than the smooth reference, the shear-increasing portion of the Flat region results in $D_{\textit{Pressure}}^{\textit{Flat}}$ going to larger values than the smooth one, becoming cumulatively $D_{f}$ -increasing (except for the [0.5, 0.5] case at ${ Re}_{L} = 18\,500$ ).

The type III distribution is the least common trend, mainly observed on the suction sides of concave textures [1.0, 1.0] at ${ Re} = 12\,200$ and $18\,500$ (figures 7(c,f) and SI.4(Ec,f)) and [1.5, 1.5] at ${ Re} = 12\,200$ (figures SI.4(Cc) and SI.4(Fc)), and on the pressure side of the convex sample [1.0, –1.0] for all the tested Reynolds numbers (figures 7(a,d,g) and SI.4(Ea,d,g)). On the suction sides, all these cases are $D_{f}$ -reducing, while on the pressure side, all, besides one are $D_{f}$ -increasing.

In the type III cases on the suction sides, from the LET region, the samples start in a shear-reducing pattern that continues to the point where $\langle C_f \rangle$ reaches its minimum and afterward as $\langle C_f \rangle (x)$ starts to increase, $\unicode{x1D6E5} \langle \tau _{n=0} \rangle$ also takes an increasing trend either getting marginally close to the smooth reference or crossing over the $C_{{f},0}$ for a slight bit, before moving to shear-reducing at the trailing edge. With a considerable extent of the LET and Flat region staying in shear-reducing conditions, the $D_{{ Suction}}^{\textit{LET}}$ and $D_{{ Suction}}^{\textit{Flat}}$ stay at lower values than the smooth reference.

On the pressure side, the [1.0, –1.0] sample at ${ Re}_{L} = 12\,200$ and $18\,500$ stays consistently shear-increasing, with the increasing trend of the $\langle C_f \rangle (x)$ along the length past the minimum $\langle C_f \rangle$ resulting in a consistent increase in the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle$ in the Flat region, thus keeping both these cases as $D_{f}$ -increasing. This is while [1.0, –1.0] at ${ Re}_{L} = 24\,200$ starts in a shear-reducing state in the LET, but in the middle of the Flat region, $\langle C_f \rangle (x)$ crosses over the $C_{f,0}(x)$ of the smooth reference, becoming shear-increasing in the latter part. However, the extent of the shear-reducing region is able to maintain a $D_{f}$ -reducing behaviour (one of the few observed on the pressure side).

Type IV distribution is mostly dominant on the suction sides of nearly all samples at ${ Re}_{L} = 24\,200$ except for [0.5, –0.5] and [1.0, –1.0] (figures SI.4(Ah-i), 7(h–i), SI.4(Cg-i), SI.4(Dh-i), SI.4(Eh-i) and SI.4(Fg-i)). Independent of the Reynolds numbers, type IV is dominant on the suction sides of the members of the ${\mathcal {R}} = 1.5$ family except the [1.5, 1.5] sample at ${ Re}_{L} = 12\,200$ and $18\,500$ (figures SI.4(Ca,b,d,e,f) and SI.4(Fa,b,d,e,f)). Besides these two larger groups, the suction side of [1.0, 0.0] at ${ Re}_{L} = 18\,500$ (figures 7(e) and SI.4(Ee)) has a type IV distribution. On the pressure side, [0.5, 0.5] at ${ Re}_{L} = 24\,200$ (figures SI.4(Ai) and SI.4(Di)), [1.0, 0.0] and [1.5, 1.5] at ${ Re}_{L} = 18\,500$ and $24\,200$ (figures 7(e,h), SI.4(Ee,h), SI.4(Cf,i) and SI.4(Ff,i)) and [1.5, –1.5] at all tested Reynolds numbers (figures SI.4(Ca,d,g) and SI.4(Fa,d,g)) experience a type IV distribution. Overall, the samples that experience any of the type I--III distributions at the lowest Reynolds number of $12\,200$ , either sustain the same distribution type at the larger global Reynolds numbers ( $18\,500$ and $24\,200$ ) or ultimately transition to a type IV as the global Reynolds number is increased. Samples [1.5, –1.5] on both sides and [1.5, 0.0] on the suction side capture a type IV distribution for all the tested Reynolds numbers.

Among all type IV distributions on the suction sides, we see the $\langle C_f \rangle (x)$ in the LET region growing to levels lower than the $C_{f,0}(x)$ of the smooth surface, offering shear reduction throughout the LET and resulting in $D_{\textit{Front}}^{\textit{LET}}$ to be lower than that of the smooth reference. Then in the early portion of the Flat region, $\langle C_f \rangle (x)$ keeps its fast rate of decrease along the length of the plate (while the rate of decrease of $C_{{f},0}$ of the smooth sample has started to decline), thus increasing the distance between the $\langle C_f \rangle (x)$ and $C_{{f},0}(x)$ , recording the largest local shear reduction as much as $20\, \%$ for the suction sides of the [0.5, 0.0] and [0.5, 0.5] samples, $30\, \%$ for the suction sides of the [1.5, 0.0] and [1.5, 1.5] cases at ${ Re}_{L} = 24\,200$ , $12\, \%$ for suction side of the [1.0, 0.0] sample at ${ Re}_{L} = 18\,500$ , and about $15\, \%$ for the suction sides of the [1.0, 0.0] and [1.0, 1.0] samples at ${ Re}_{L} = 24\,200$ . Local shear stress reductions ranging from $20\, \%$ to $50\, \%$ have been previously reported by Furuya et al. (Reference Furuya, Nakamura, Miyata and Yama1977), Hooshmand et al. (Reference Hooshmand, Youngs, Wallace and Balint1983) and Gallagher & Thomas (Reference Gallagher and Thomas1984). After reaching its global minimum, the $\langle C_f \rangle (x)$ then takes on an increasing trend that is deterministic of how much $D_{f}$ reduction can be possible for these samples.

In cases such as the suction side of the [1.5, 0.0] sample at all Reynolds numbers, along the increasing and then constant path of the $\langle C_f \rangle (x)$ in the $x$ direction, the $\langle C_f \rangle (x)$ comes marginally close to the $C_{{f},0}(x)$ . But it always stays at $\langle C_f \rangle (x) \lt C_{f,0}(x)$ and never crosses over. Thus, even though non-monotonic and at times not optimal, it maintains a consistent $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \leqslant 0$ all along the length. These cases stay drag-reducing in both $D_{\textit{Suction}}^{\textit{LET}}$ and $D_{\textit{Suction}}^{\textit{Flat}}$ components and cumulatively along the length.

In other cases, such as the suction side of the [1.5, 1.5] and [1.5, –1.5] samples at ${ Re}_{L} = 18\,500$ and $24\,200$ , along the increasing and then constant path of the $\langle C_f \rangle (x)$ , the $\langle C_f \rangle (x)$ is able to cross over the $C_{{f},0}$ , with portions of the sample experiencing $\langle C_f \rangle (x) \gt C_{{f},0}(x)$ . As a result, along the length, the sample starts at $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \leqslant 0$ in the LET and first half of the Flat region, then crossing over to $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \gt 0$ in parts of the second half of the Flat region, before a small region in the trailing edge with $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \leqslant 0$ . For these cases, with the positive and negative $\unicode{x1D6E5} \langle \tau _{n=0} \rangle$ crossing each other out, it is not easy to determine the drag-reducing/increasing nature of the samples without calculation of the integrals of (3.2). Thus, all the type IV distributions on the suction side lead to drag reductions besides the one case of the [1.0, 0.0] sample at ${ Re} = 18\,500$ that experiences a $\langle C_f \rangle (x)$ crossing over the smooth reference and a substantial shearincrease, as shown in figure SI.4(Ee).

On the pressure side, the [0.5, 0.5], [1.0, 0.0] and [1.5, 1.5] samples at ${ Re}_{L} = 24\,200$ follow a very similar trend as that of the suction sides described above. However, both experience a cross-over to $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \gt 0$ , where only [0.5, 0.5] is able to maintain a $D_{f}$ -reducing behaviour and the shear-increasing region of [1.0, 0.0] and [1.5, 1.5] turn both cases to be $D_{f}$ -increasing (see figure 5). The pressure side of the [0.5, 0.5] sample at ${ Re}_{L} = 24\,200$ is the only case of a type IV distribution that is drag-reducing on the pressure side. As for the rest, in one group of the type IV cases on the pressure side such as the cases of [1.5, 1.5] and [1.5, –1.5] at ${ Re}_{L} = 18\,500$ , initially, we have the $\langle C_f \rangle (x)$ grow to larger values than the $C_{f,0}(x)$ of the smooth case, but close to the end of the LET and start of the Flat region, $\langle C_f \rangle (x)$ crosses over to $\unicode{x1D6E5} \langle \tau _{n=0} \rangle \leqslant 0$ , before turning shear-increasing toward the middle of the Flat region. Another group of the type IV cases, [1.0, 0.0] at ${ Re}_{L} = 24\,200$ and [1.5, –1.5] at ${ Re}_{L} = 12\,200$ , stay in their entirety on the shear-increasing side and, thus, are fully $D_{f}$ -increasing in both LET and Flat regions as well as the entirety of the pressure sides of the plates.

3.4. Distribution of the fitting parameters

The distribution of the $\langle C_f \rangle (x)$ , in presence of the riblets, as a function of the ${ Re}_x$ (shown in § 3.3), can be explained further by looking at the distributions of the two fitting parameters $m$ and $n_0$ for all the samples at all the tested global Reynolds numbers. Since the key drag reduction (or generally change) mechanism in this study is via the flow retardation inside the grooves, the patterns of reduction/increase seen is discussed further in terms of the BL development along the wall and via the distributions of the $m$ and $n_0$ along the length of the sample on either side. Overall, the distribution of the two parameters show an intertwined dependence on each other and the cross-sectional area available to the flow inside the grooves. The $\langle C_f \rangle (x)$ already captures the effect of the increase in the wetted surface area of the riblets compared with a smooth surface, as well as the changes in the shear stress distribution inside the grooves. The distributions of $m$ and $n_0$ are also cumulative views of the flow dynamics and the geometric variations imposed by the presence of the riblets of different shapes, and can further explain the shear stress patterns seen in figure 8. The high frequency variations in the data presented in this and subsequent sections has been filtered with a Savitzky–Golay filter (Savitzky & Golay Reference Savitzky and Golay1964; Savitzky Reference Savitzky1989).

Figure 9. Distribution of $m$ for all the riblet samples of ${\mathcal {R}} = 1.0$ and the reference smooth samples on the suction and pressure side for all tested Reynolds numbers. The $m$ of the reference smooth sample on the suction and pressure side are shown with solid and dashed black lines, respectively. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are marked by grey dotted and dash-dotted vertical lines.

3.4.1. Fitting parameter $m$

The fitting parameter $m$ (presented in figures 9 and SI.5 for all cases) plays a significant role in capturing the local effect of multiple physical phenomena in the flow field; (i) the pressure gradient, (ii) the effect of both the limited (i.e. short) length of the plate and the presence of the riblets on the viscous diffusion, as well as (iii) the effect of the spanwise-averaging operation on the nonlinear advection terms. For all the cases, including the smooth samples, and on both sides, the distribution of $m$ is comprised of three distinct regions: first, a $m\gt 0$ region in the leading edge area, then the $m\lt 0$ area in the middle including the location where $m$ reaches a global minimum and, lastly, the $m\gt 0$ for the rest of the length of the plate toward the trailing edge. As is seen in figures 9 and SI.5, $m$ is larger on the pressure side than the suction side and, as a result, the extent of the $m\lt 0$ region varies quite a bit between the sides and samples, and can take small or large portions of the length of the plate.

In general, $m$ takes a decreasing trend in the early portion of the plate prior to the global minimum and then an increasing trend in the second half toward the trailing edge. The initial decreasing trend is not strictly monotonic for many of the cases on the pressure side; in the LE/LET regions of those, there is a local increasing--decreasing trend that sets in as the flow sees the riblets for the first time. Examples of this are those of [0.5, 0.0] (figure SI.5(Ab)), [0.5, 0.5] (figure SI.5(Ac)), [1.0, −1.0] (figure 9 a) and [1.5, 0.0] (figure SI.5(Cb)) at ${ Re}_{L} = 12\,200$ , [1.0, 0.0] (figure 9 b,e) and [1.5, 1.5] (figure SI.5(Cc,f)) at ${ Re}_{L} = 12\,200$ and $18\,500$ , and [1.5, –1.5] (figure SI.5(Ca,d,g)) at all the Reynolds numbers.

In a FS type of BL equation, at a constant Reynolds number, increasing $m$ monotonically increases the wall shear stress, while at a constant $m$ , increasing the Reynolds number monotonically reduces the wall shear stress. Thus, for all the cases, while prior to the minimum $m$ , the increase in the ${ Re}_x$ and decrease in $m$ work together in reducing the $\langle C_f \rangle (x)$ along the length, the evolution of the $m$ past the minimum is the determining factor for the type of shear stress distribution each sample takes. After the minimum $m$ , along the length of the sample, the increase in the local Reynolds number results in the wall shear stress decreasing. However, the increase in the $m$ disrupts this trend and, depending on the rate of increase of $m$ , the trend in the $\langle C_f \rangle (x)$ is set.

In the type I distribution, the effect of the rate of increase in $m$ on the $\langle C_f \rangle (x)$ is not able to overcome the effect of the ${ Re}_x$ in decreasing the $\langle C_f \rangle (x)$ along the length. Thus, overall we see a decreasing trend in the $\langle C_f \rangle (x)$ along the length prior to the vicinity of the trailing edge albeit at a lower rate than the region prior to the minimum $m$ and lower than the ${ Re}_x^{-1/2}$ of the BL theory. However, as seen in figure SI.5(Aa-g) (suction sides) for the ${\mathcal {R}} = 0.5$ family, figure 9(a,b,d,g) (suction sides) for the ${\mathcal {R}} = 1.0$ family, and figure SI.5(Cb,e,h) (pressure sides) for the ${\mathcal {R}} = 1.5$ family, the rate of increase in $m$ is faster for the riblets than the smooth reference. This reduces the rate of decrease of $\langle C_f \rangle (x)$ along the length. After the minimum $m$ , the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ takes an increasing trend until the vicinity of the trailing edge, at around $x/L \approx 0.8$ , where the rate of increase in $m$ increases more, leading to $\langle C_f \rangle (x)$ of both the riblet and smooth samples increasing (discussed earlier).

In the type II distribution, the increase in $m$ nearly balances that of the increase in the ${ Re}_x$ , resulting in almost constant $\langle C_f \rangle (x)$ prior to the trailing edge effect, as seen in the pressure sides of most of the ${\mathcal {R}} = 0.5$ family (figure SI.5(Aa-h)) and a few cases of the ${\mathcal {R}} = 1.0$ family (figure 9 b,c,f,i) and [1.5, 1.5] (figure SI.5(Cc)). The difference between the rate of increase of $m$ in type I and II distributions is clearly visible for the suction and pressure sides of the ${\mathcal {R}} = 0.5$ family at ${ Re}_{L} = 12, 200$ and $18\,500$ (figure SI.5(Aa-f)).

In type III and IV distributions, the effect of the increase in $m$ fully overcomes and surpasses that of the local Reynolds number. This results in an increasing (and/or near constant) trend in the $\langle C_f \rangle (x)$ along the length. Among these two types, the location of the minimum $m$ is very close to the location of the minimum $\langle C_{f} \rangle (x)$ . Specifically, for those samples that experience type III or IV on the suction sides, the initial decreasing trend in $m$ along the length of the sample has a faster rate than that of the smooth sample. This results in a larger separation between the minimum $m$ of the riblet samples and the smooth reference (for example, see the suction sides of the members of the ${\mathcal {R}} = 1.5$ family in figure SI.5(Ca-i)). Thus, we see large levels of shear reduction prior to the minimum $\langle C_f \rangle (x)$ . Even with the counteracting effect of the increase in ${ Re}_x$ and $m$ , after the minimum $m$ , these samples can maintain a $D_{f}$ -reducing trend on the suction sides. Only on the suction side of the [1.0, 0.0] sample is the rate of increase of $m$ substantially high enough to result in a $D_{f}$ -increasing behaviour.

Overall, the pressure sides of the samples capture larger values of $m$ than the suction sides. With the nonlinear impact of the rate of change of $m$ on the rate of change in the $C_f$ (keeping all other parameters constant), we see that while the rate of change of $m$ along the length is quite similar between the suction and pressure sides, depending on the sign and magnitude of the $m$ , this translates into different rates of change in $\langle C_f \rangle (x)$ between the two sides of the same sample. Mathematically, one can see that as $m$ is increased the variations in the $C_f$ slowly decreases; for example, increasing $m$ from $-0.02$ to $-0.01$ results in a $\Delta C_f/ C_f(m=0) = 0.066$ (normalised by the $C_f$ of $m=0$ FS solution), while increasing $m$ from 0.01to 0.02 results in $\Delta C_f/ C_f(m=0) = 0.056$ . Thus, comparing the two sides of the samples, with $m_{{ pressure}} \geqslant m_{{ suction}}$ , on the pressure sides, the rate of increase in $\langle C_f \rangle (x)$ due to the increase in $m$ is weaker than on the suction side and, therefore, with the increase in the ${ Re}_x$ , the changes in $\langle C_f \rangle (x)$ on the suction side (as well as the $\unicode{x1D6E5} \langle \tau _{n=0} \rangle /\tau _0$ ) show larger variations along the length of the sample compared with those on the pressure sides. This leads to seeing more type I/IV distributions on the suction side (24/27 riblet cases) and more type II distributions on the pressure side (13/27) and a few type IV cases with smaller variations compared with those of the suction sides (8/27)(see figure 8).

To explore the effect of $m$ further, we perform the spanwise-averaging operation on the Navier--Stokes equation in the $x$ direction for flow past a generic riblet surface and organise and write it in a form resembling that of the BL equation (derivation in § SI.5 of the supplementary material for the Flat region and § SI.6 for the curved LE and LET regions), i.e.

(3.3) \begin{align} \rho \left ( \langle u \rangle \dfrac {\partial \langle u \rangle }{\partial x} + \langle v \rangle \dfrac {\partial \langle u \rangle }{\partial y} \right ) = - \dfrac {\partial \langle P^{*} \rangle }{\partial x} + \mu \dfrac {\partial ^{2} \langle u \rangle }{\partial y^{2}}, \end{align}

where in the Flat region ( $x\gt x_{{ Flat}}$ )

(3.4) \begin{align} -\dfrac {\partial \langle P^{*} \rangle }{\partial x} = - \underbrace {\dfrac {\partial \langle p \rangle }{\partial x}}_{\unicode {x2460}} + \underbrace {\mu \dfrac {\partial ^{2} \langle u \rangle }{\partial x^{2} } + {\mathcal{Z}}}_{\unicode {x2461}} + \underbrace {\rho \left ( \langle u\rangle \dfrac {\partial \langle u \rangle }{\partial x} + \langle v \rangle \dfrac {\partial \langle u \rangle }{\partial y} - \dfrac {\partial \langle u u \rangle }{\partial x} - \dfrac {\partial \langle u v \rangle }{\partial y} \right )} _{\unicode {x2462}}, \end{align}

(3.5) \begin{align} {\mathcal{Z}} = \left \{ \begin{alignedat}{3} & - \dfrac {2\mu }{\lambda } \dfrac {\partial u}{\partial z} \Big \vert _{z=z_{{ w1}}^{+}}, \ \ \ \ \ & y_{\textit{trough}} \leqslant y \leqslant y_{\textit{peak}} = h/2, \\ &0,\ \ \ \ \ &\vert y \vert \gt y_{\textit{peak}}, \end{alignedat} \right . \end{align}

and in the riblet-covered leading edge, LET, ( $x_{{ LET}} \lt x \leqslant x_{{ Flat}}$ )

(3.6) \begin{eqnarray} - \dfrac {\langle P^{*} \rangle }{\partial s} &= & - \underbrace {\dfrac {\partial \langle p \rangle }{\partial s}}_{\unicode {x2460}} + \underbrace {\mu \dfrac {\partial ^{2} \langle u_{s} \rangle }{\partial s^{2}} + {\mathcal{Z}} }_{\unicode {x2461}} \nonumber\\ && \underbrace {\rho \left ( \langle u_{s} \rangle \dfrac {\partial \langle u_{s} \rangle }{\partial s} + \langle v_{n} \rangle \dfrac {\partial \langle u_{s} \rangle }{\partial n} - \dfrac {\partial \langle u_{s} u_{s} \rangle }{\partial s} - \dfrac {\partial \langle u_{s} v_{n} \rangle }{\partial n} \right )}_{\unicode {x2462}} + \underbrace {{\mathcal{K}_2}}_{\unicode {x2463}} , \end{eqnarray}

replacing $y$ with $n$ and $u$ with $u_{s}$ in the definition of the $\mathcal{Z}$ in (3.5), where $u_{s}$ and $v_{n}$ are the tangential and normal velocities along the $s$ and $n$ directions with respect to the wall of the curved leading edge (see figure 1), and $\mathcal{K}_2$ is the contribution from the curvature of the leading edge as shown in § SI.6 of the supplementary material. In this format, all the excess terms compared with the BL equation are captured via an equivalent pressure gradient term, defined as $\partial \langle P^{*} \rangle / \partial x$ (or $\partial \langle P^{*} \rangle / \partial s$ ), so that (3.3) matches the form of the FS family of BL equations.

Here, locally, by fitting the collected data to the FS equations, we capture the $\partial \langle P^{*} \rangle /\partial x$ term via the $m$ parameter that is then written in the form of $-{\partial \langle P^{*} \rangle }/{\partial x} = \rho U(x)^{2} {m}/{x}$ , where the distribution of this term follows the same trend as the $m$ and the $1/x$ acts as a scaling factor that decreases along the length. Note that, as mentioned in § 2.4.2, for simplicity, in the LE and LET, we perform the fitting with $x$ instead of $s$ and let $m$ also capture the effect of this difference and, thus, use $-{\partial \langle P^{*} \rangle }/{\partial s} = \rho U(x)^{2} {m}/{x}$ in the curved leading edge area. Thus, the $m$ parameterrepresents the cumulative effect of $\unicode {x2460}$ the pressure gradient, $\unicode {x2461}$ the non-negligible viscous diffusion term in the $x$ direction and the viscous diffusion term in the $z$ direction due to the three-dimensional effect inside the grooves, as well as $\unicode {x2462}$ the residues from the nonlinear terms left from the averaging operation. In the leading edge area, $m$ also captures the effect of the curvature both through the $\mathcal{K}_2$ term (see § SI.6 of the supplementary material) and the use of $x$ in the fitting process. In the LEarea, (3.3) turns back into (see § SI.6)

(3.7) \begin{align} \rho \left ( u_{s} \dfrac {\partial u_{s} }{\partial s} + v_{n} \dfrac {\partial u_{s} }{\partial n} \right ) = - \dfrac {\partial P^{*} }{\partial s} + \mu \dfrac {\partial ^{2} u_{s} }{\partial n^{2}}; \quad -\dfrac {\partial P^{*} }{\partial s} = - \underbrace {\dfrac { \partial p }{\partial s}}_{\unicode {x2460}} + \underbrace {\mu \dfrac {\partial ^{2} u_{s} }{\partial s^{2} }}_{\unicode {x2461}} + \underbrace { {{\mathcal{K}}_1} }_{\unicode {x2463}} \end{align}

Figure 10. Distribution of ${\mathcal {G}}$ or difference between the $\partial \langle P^{*} \rangle /\partial x$ and $\partial \langle p \rangle /\partial x$ terms in dimensionless form for all the riblet samples of ${\mathcal {R}} = 1.0$ on the suction and pressure side of the riblet samples. The results for the smooth reference for all the tested Reynolds numbers are shown with solid and dashed black lines for the suction and pressure sides, respectively. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are marked by grey dotted and dash-dotted vertical lines.

as expected, missing the $\unicode {x2462}$ term compared with the riblet regions with $-{\partial P^{*}}/{\partial s} = \rho U(x)^{2} ({m}/{x})$ . Separate from the curve fitting operation, we use the PIV data and find the pressure gradient following § 2.4.3 and find the dimensionless pressure gradient terms in the form of

(3.8) \begin{align} -\dfrac {L}{({1}/{2}) \rho U(x)^{2}} \dfrac {\partial \langle p \rangle }{\partial x} \end{align}

on a horizontal line parallel to the Flat portion of the samples at a height of $y = \pm 0.6 h$ , which is at a distance of $0.1 h$ or $\lambda /2$ from the Flat part of the surface on either side, where the three-dimensional effects of the riblets have mostly faded away and it is safe to assume that $\mathbf{u}(x,y,z) = \langle \mathbf{u} \rangle (x,y)$ and $p(x,y,z) = \langle p \rangle (x,y)$ . Therefore, we can find the difference between the $\partial \langle p \rangle /\partial x$ and $\partial \langle P^{*} \rangle /\partial x$ , corresponding to terms $\unicode {x2461}$ and $\unicode {x2462}$ (as well as $\unicode {x2463}$ in the LE and LET regions), found via the two separate methods, in a dimensionless form as

(3.9) \begin{align} {\mathcal {G}} = \dfrac {-L}{(1/2) \rho U(x)^{2}} \left ( \dfrac {\partial \langle P^*\rangle }{\partial x} - \dfrac {\partial \langle p \rangle }{\partial x} \right ) = \dfrac {2mL}{x} + \dfrac {L}{({1}/{2}) \rho U(x)^{2}} \dfrac {\partial \langle p \rangle }{\partial x} \end{align}

and present it for all the riblet samples in figures 10 and SI.7, with the results for the smooth sample shown as reference. (In the LE and LET regions the $\partial /\partial s$ has been transformed to $\partial /\partial x$ using the chain rule $\partial /\partial x = \partial s/\partial x \ \partial /\partial s$ .) As seen in this figure, the non-zero difference between the two terms confirms that the $\unicode {x2461}$ , $\unicode {x2462}$ and $\unicode {x2463}$ have limited non-negligible effects for a finite length sample. Firstly, in the absence of the riblets for the smooth reference, in the LE and LET regions, (3.7) holds while in the Flat region, (3.3) and (3.4) are simplified to

(3.10) \begin{align} \rho \left ( u \dfrac {\partial u }{\partial x} + v \dfrac {\partial u }{\partial y} \right ) = - \dfrac {\partial P^{*} }{\partial x} + \mu \dfrac {\partial ^{2} u }{\partial y^{2}} ; \qquad -\dfrac {\partial P^{*} }{\partial x} = - \underbrace {\dfrac { \partial p }{\partial x}}_{\unicode {x2460}} + \underbrace {\mu \dfrac {\partial ^{2} u }{\partial x^{2} }}_{\unicode {x2461}}. \\[-18pt] \nonumber \end{align}

The viscous term in the streamwise direction, $\unicode {x2461}$ , as seen in figures 10 and SI.7is mostly zero or negative along both sides of the plate in the Flat region. In the leading edge area, the effect of both $\unicode {x2461}$ and $\unicode {x2463}$ are present and, hence, ${\mathcal {G}}$ captures its largest deviation from zero and stays mostly negative. Since the smooth surface (of the configuration used here) does not experience any form of flow reversal, we do not expect $\unicode {x2461}$ to become positive for this sample and, thus, the small regions of ${\mathcal {G}}\gt 0$ in the LE and LET are most likely due to the contributions from the curvature terms $\unicode {x2463}$ .

Moving to the riblet-covered samples, the distribution of ${\mathcal {G}}$ follows a very similar trend to that of the smooth surface, and especially for the shallowest riblets of the ${\mathcal {R}} = 0.5$ family (figure SI.7(A)), the distributions are nearly the same with slight differences recorded for the ${ Re}_{L} = 18\,500$ cases in the early portions of the Flat region. Similarly, for all the cases, in the LE region, ${\mathcal {G}}$ is nearly the same as that of the smooth reference and in the LET region slight differences can be seen for members ofthe ${\mathcal {R}} = 1.0$ and $1.5$ families (figures 10 and SI.7(C)), which could be due to any of the $\unicode {x2461}$ , $\unicode {x2462}$ or $\unicode {x2463}$ terms.

The main differences between the ${\mathcal {G}}$ of the smooth and riblet surfaces in the Flat region is seen among the ${\mathcal {R}} = 1.0$ and $1.5$ families. For example, in the early portion of the Flat region of [1.5, 1.5] at all Reynolds numbers (figure SI.7(Cc,f,i)) and [1.5, 0.0], [1.5, -1.5], [1.0, 1.0] and [1.0, 0.0] at Reynolds numbers of $18\,500$ and $24\,200$ (figures SI.7(Cd,e,g,h) and 10(e,f,h,i), ${\mathcal {G}}$ is lower than that of the smooth one. This is due to the effect of both $\unicode {x2462}$ and contributions from the $\mathcal{Z}$ terms. In the latter portion of the Flat region, especially for samples [1.0, 0.0], [1.0, 1.0], [1.5, -1.5] and [1.5, 1.5], unlike the smooth reference, ${\mathcal {G}}$ turns positive for some extent of the plate length. In this region, as the flow inside the grooves has had sufficient distance to develop, it is more likely for the slow down inside the riblets to lead to near stagnant flow inside the grooves with potential for small recirculation regions (Raayai-Ardakani & McKinley Reference Raayai-Ardakani and McKinley2017) turning the viscous terms $\unicode {x2461}$ positive, and giving more weight to the contributions of $\unicode {x2461}$ than $\unicode {x2462}$ here (see more in the upcoming § 3.4.2). The difference tends to become more visible toward the sharpest textures and also toward samples with larger $\kappa _2$ where the available cross-sectional area within the riblets allows for $\unicode {x2461}$ and $\unicode {x2462}$ terms to capture larger variations and push ${\mathcal {G}}$ to deviate from that of the smooth reference. These cases experience type II, III (all) and IV shear distributions.

Overall, the similarity of the $\mathcal{G}$ of the riblet and smooth samples and the order of magnitude of the difference observed between the two gives us confidence that the spanwise-averaging operation is able to capture a large part of the flow dynamics and is a credible method for extracting valuable information in studying the effect of riblets and the three-dimensional nature of the flow inside their grooves.

3.4.2. The effective origin, $n_0$

The distribution of the $n_0$ for the suction and pressure sides of all the riblet samples and for all the tested Reynolds numbers are shown in figures 11 and SI.8. In all the plots, the design location of the trough of the riblets and the measured locations are shown for comparison and the $n_0$ values are normalised with $\lambda /2$ , which is the more consistent dimensions among all the printed samples. For the majority of the cases tested, the distributions of $n_0$ , on both sides of the samples, follow a non-monotonic behaviour where initially, following the growth of the riblet height in the LETregion, the magnitude of the effective origin increases, reaching a maximum in the early portions of the Flat region, before decreasing to a non-zero minimum. Afterward again increasing in the vicinity of the trailing edge of the samples until the end of the body. In a few of the cases, namely the suction side of [0.5, 0.5] at ${ Re}_{L} = 12\,200$ (figure SI.8(Ac)), the pressure side of [1.0, -1.0] at ${ Re}_{L} = 12\,200$ (figure 11 a) and the suction side of [1.0, −1.0] at ${ Re}_{L} = 18\,500$ (figure 11 d), after the initial increase in the magnitude $n_0$ , in the Flat region, the effective origin stays nearly constant for a portion of the length of the sample until close to the trailing edge where $n_0$ increases toward the end of the plate. In all the cases, the magnitudes of the $n_0$ on the suction sides are larger than or similar to the pressure sides.

Figure 11. Distribution of the effective origin, $n_0$ , for all the riblet samples of ${\mathcal {R}} = 1.0$ on the suction and pressure side for all tested Reynolds numbers. The location of the design and measured troughs are also marked on the figures. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are marked by grey dotted and dash-dotted vertical lines.

Generally, within the ${\mathcal {R}} =$ constant families, at a constant Reynolds number, the magnitude of the $n_0$ tends to be larger for $\kappa _2={\mathcal {R}}$ (concave) and $\kappa _2 = 0.0$ (triangular) where there is a larger cross-sectional area available in the riblets for the flow to develop compared with the convex ones. Additionally, as the global Reynolds number is increased, we see the absolute values of $n_0$ slightly increasing in the early portion of the Flat region. As expected, as ${\mathcal {R}}$ is increased and the available depth of the riblets increases, the effective origin of the velocity profiles is also found to be larger, especially for the concave $\kappa _2 = {\mathcal {R}}$ cases. Among the ${\mathcal {R}} = 0.5$ family, for all Reynolds numbers and on both sides, the $n_0$ of the [0.5, 0.0] and [0.5, 0.0] samples have similar magnitudes in the LET and early portion of the Flat region and toward the second half, the magnitude of the $n_0$ of the [0.5, 0.5] sample becomes larger than the [0.5, 0.0] sample. Owing to the limited resolution of the three-dimensional printer (as listed in table 1), while the height of all the samples are smaller than their design height, the height of the [0.5, 0.5] sample is slightly smaller than the other two samples and, thus, for this sample, the magnitude of $n_0$ can reach to a maximum of between $53\,\%\!-\!70\,\%/39\,\%\!-\!60\,\%$ of the measured $A$ in the middle of the suction/pressure side of the riblets. For the [0.5, 0.0] and [0.5, -0.5]samples, the depth of $n_0$ reaches between $50\,\%\!-\!64\,\%/38\,\%\!-\!46\,\%$ and $38\,\%\!-\!53\,\%/24\,\%\!-\!41\,\%$ of their respective measured $A$ in the middle of the suction/pressure sides. As the depth of the samples (i.e. ${\mathcal {R}}$ ) increases, while we see the absolute value of $n_0$ slightly increasing, the ratio of $\vert n_0 \vert /A$ keeps decreasing. For the ${\mathcal {R}} = 1.5$ family, on the suction/pressure sides the $\vert n_0 \vert /A$ can reach as high as $43\,\%/33\,\%$ as seen for the [1.5, 1.5] sample. Similarly, for the ${\mathcal {R}} = 1.0$ family, on the suction/pressure sides, the maximum $\vert n_0 \vert /A$ in the middle of the plate is found to be around $50\,\%/42\,\%$ , respectively. As a result, for sharper riblets, we expect that a larger volume of the fluid inside the grooves to be ina near stagnant condition compared with the shallower grooves where the effective origin can penetrate more than $50\,\%$ of the height of the riblet.

The trend in the effective origin, $n_0$ , of the spanwise-averaged velocity profiles in the streamwise direction can be explained further with the idea of flow retardation inside the grooves (key drag reduction mechanism in the laminar regimes) that leads to the creation of a layer of stagnant (or slow-moving) fluid (figure 12), which has been shown in previous numerical simulations (Chu & Karniadakis Reference Chu and Karniadakis1993; Choi et al. Reference Choi, Moin and Kim1993; Raayai-Ardakani & McKinley Reference Raayai-Ardakani and McKinley2017). Here, as the BL develops along the plate and inside the riblets, the magnitude of the effective origin of the $\langle u \rangle$ (or $\langle u_{s} \rangle$ ) profiles slowly increases until reaching a maximum. From this point, first a layer of slow-moving fluid starts to develop inside the grooves and grows along the length of the plate. This layer due to its near-zero velocity does not communicate with the rest of the flow and acts as a blockage, pushing the moving fluid to the outside, effectively squeezing the flow between the higher $n_0$ and the inviscid outer flow and at times leading to a faster rate of increase in $m$ . This moves the location of $n_0$ outward resulting in a decrease in the magnitude of the $n_0$ . This continues until close to the vicinity of the trailing edge where the velocity in the $y$ direction, $v$ , outside the grooves changes direction toward the sample (instead of away from the sample; see figure 6 f) and effectively pushes the flow inside the grooves and increases the magnitude of $n_0$ . In this region, the flow is more attached to the surface with larger $m$ values (as seen in § 3.4.1), and also larger values of $\langle C_f \rangle (x)$ than the earlier parts of the Flat region where the magnitude of $n_0$ reaches a local maximum (as discussed in § 3.3).

Figure 12. Schematic rendering of the evolution of the effective origin of the velocity profiles (top left) along the grooves of riblet samples for a hypothetical riblet (top right) and the respective local velocity profiles at five points (and their mirror images, as shown on the top-right riblet profile with dots with the same colours) in the spanwise direction of the riblet and their respective $\langle u \rangle$ at streamwise locations (a--f) along the sample.

In some cases, the quiescent flow layer can lead to a slight recirculation in the flow (Raayai-Ardakani & McKinley Reference Raayai-Ardakani and McKinley2017), which can result in the viscous diffusion terms in $\unicode {x2461}$ of (3.4) to become positive, leading to ${\mathcal {G}}\gt 0$ in the latter part of the sample as seen in § 3.4.1. Comparing figures 11, SI.8, 10 and SI.7, one can see that, for cases with ${\mathcal {G}}\gt 0$ areas in the Flat region, the ${\mathcal {G}}\gt 0$ region is very close to where the magnitude of the $n_0$ reaches a local minimum. In this region, with a very slow-moving fluid inside the grooves, the contribution from term $\unicode {x2462}$ is likely very small and ${\mathcal {G}}$ becoming even slightly positive can serve as an indication of the potential for existence of flow reversal in these cases.

The location of the origin of the normal coordinate is a complicating matter for the experimental efforts in the analysis of the flow over riblet surfaces. Wallace & Balint (Reference Wallace and Balint1988) chose the geometric average of the height of the peak and trough of the riblets (midway) as the origin for their analysis. Later, as discussed in § 1, the protrusion height model was introduced (Luchini et al. Reference Luchini, Manzo and Pozzi1991; Luchini Reference Luchini1995; Bechert et al. Reference Bechert, Bruse, Hage, Van der Hoeven and Hoppe1997; Grüneberger & Hage Reference Grüneberger and Hage2011) as the origin of the velocity profiles below the level of the grooves. Mainly calculated using a Stokes flow analysis inside the grooves, the difference between the location of the protrusion heights seen by the streamwise and spanwise motions have been used to find correlation for the drag reduction values in turbulent flows (Bechert et al. Reference Bechert, Bruse, Hage, Van der Hoeven and Hoppe1997; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011; Wong et al. Reference Wong, Camobreco, García-Mayoral, Hutchins and Chung2024). Here, with the finite length of the sample and the variations expected in the streamwise direction, as well as the laminar nature of the flow, we expect the distribution of $n_0$ to also vary along the length and, thus, instead of using a Stokes flow approach or a linear estimation, we use the fitting process to extract the effective origin.

There are similarities between the idea of the effective origin and the other definitions, and overall, locally the velocity profiles of shear-reducing riblets experience a lower $n_0$ (higher magnitude) than shear-increasing ones. For example, in the case of [1.5, 1.5] at ${ Re}_{L} = 24\,200$ and on the suction sides, the available cross-sectional area of the grooves makes it possible for the origin of the velocity profile at ${ Re}_x = 10,000$ to reach $43\, \%$ of the texture height ( $56\, \%$ of the half-spacing) and, thus, allow the $\langle u \rangle$ to take a more detached form and $m$ to get as low as $-0.057$ , ${\mathcal {G}}$ to get to visibly lower values compared with the smooth reference (see figure SI.7(Ci)) and ultimately close to $30\, \%$ local shear reduction (figure SI.4(Cc)). In turn on the pressure side of [1.0, 0.0] at ${ Re}_{L} = 12\,200$ , $n_0$ is able to reach to as low as $28\, \%$ of the measured height ( $21\, \%$ of the half-spacing) at ${ Re}_x = 6,000$ , where the velocity profile is more attached compared with the smooth reference with $m = 0.003$ , and we see a similar $\mathcal{G}$ compared with the smooth reference (figure SI.7(Cc)), recording about a $10\, \%$ shear increase (figure SI.4(Cc)). Ultimately, as demonstrated, for a limited-size body, the effect of the $n_0$ and $m$ on the frictional shear/drag changes are more intertwined where the available cross-sectional area inside the riblets, the Reynolds number, and the pressure distribution all guide the development of the BL, and additional work needs to be done to translate these two parameters into predictive tools.

3.5. Pressure drag

Somewhere between $23\,\%\!-\!36\,\%$ of the total drag experienced by the samples is attributed to the $D_{p}$ . The pressure drag is due to the finite size of the sample and the resulting pressure distribution around the entire body. Independent of the total skin-friction drag, the pressure drag also experiences alterations as a result of the presence of the riblets. This drag component is found cumulatively with the friction drag using a control volume analysis where the sum of the friction and pressure drag can be found as a total reaction force applied to the sample, $D_{{ CV}}$ , also known as the profile drag (Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023). (To avoid confusion between the profile and pressure drag, here we use a generic $D_{{ CV}}$ to represent the force calculated via the control volume method.) Therefore, $D_{p} = D_{{ CV}} - D_{f}$ . In summary, we use the Reynolds-averaged integral momentum formulation (Ferreira et al. Reference Ferreira, Gymnopoulos, Prinos, Alves and Ricardo2021; Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023; Suchandra & Raayai-Ardakani Reference Suchandra and Raayai-Ardakani2024)

(3.11) \begin{align} \mathbf{D}_{{ CV}} \, = \, - \, \rho \int \overline {(\mathbf{u} + \mathbf{u}^{\prime}) [(\mathbf{u} + \mathbf{u}^{\prime}) \cdot \mathbf{n}_S]} dS \, - \, \int p \, \mathbf{n}_S \, dS ,\end{align}

where $\mathbf{n}_S$ is the normal to the control surface and $S$ is the area of the control surface. This method follows a similar logic as that used in previous studies using wake surveys for riblets on airfoils (Caram & Ahmed Reference Caram and Ahmed1989, Reference Caram and Ahmed1991, Reference Caram and Ahmed1992; Coustols Reference Coustols1989; Subashchandar et al. Reference Subashchandar, Rajeev and Sundaram1995; Sundaram et al. Reference Sundaram, Viswanath and Rudrakumar1996; Subaschandar et al. Reference Subaschandar, Kumar and Sundaram1999; Chamorro et al. Reference Chamorro, Arndt and Sotiropoulos2013). In addition, we have previously validated this technique for the drag of a long circular cylinder (aspect ratio of 20) in the same water tunnel and with the same PIV set-up (Suchandra & Raayai-Ardakani Reference Suchandra and Raayai-Ardakani2024).

Here, we assume that we are far enough from the riblets that the three-dimensional nature of the velocity profile has subsided. We place the boundaries of the control volumes on a (i) plane prior to the leading edge at the earliest possible available location at around $x/L \approx -0.35$ , (ii,iii) two parallel planes far from the BL on either side of the sample at $y/L = \pm 0.12$ , and (iv) a plane after the trailing edge at $x_{{ TE}}\gt L$ (similar to the procedure followed by Fu & Raayai-Ardakani Reference Fu and Raayai-Ardakani2023). We fix the first three boundaries and use 40 control volumes with 40 different $x_{{ TE}}$ to calculate the $D_{{ CV}}$ for all the control volumes and present the mean of the values and their $95\, \%$ confidence intervals in figure 4. Here, we assume that at the earliest available point before the leading edge ( $x/L \approx -0.35$ ) and far from the BL ( $y/L = \pm 0.12$ ) the pressure is $p_{\infty } = 0$ and perform the directional integration following the procedure discussed in § 2.4.3 to find the pressure distribution needed in (3.11).

Figure 13. Distribution of pressure past the trailing edge of the plates at $x/L = 1.05$ for families of (a,b,c) ${\mathcal {R}} = 0.5$ , (d,e,f) ${\mathcal {R}} = 1.0$ and (g,h,i) ${\mathcal {R}} = 1.5$ for three global Reynolds numbers. The pressure distribution of the smooth reference is shown with a solid black line.

In the presence of the riblets, samples only experience a marginal difference in the momentum distribution crossing the control volume boundaries (i) and (iv) at the leading and trailing edges, while the pressure difference between the two planes, $p_{\textit{TE}} - p_{\infty }$ (as shown in figure 13), experiences a clear difference between the riblet-covered samples and the smooth reference. For the samples experiencing reductions in the pressure drag, this difference comes out in the form of pressure recoveries past the trailing edge (see figure 13) that ultimately enhances the overall drag reduction of the samples.

Riblets affect the near-wall BL that results in differences in the $\langle u \rangle$ of the riblet samples compared with the smooth reference. On the one hand, we have non-zero $\unicode{x1D6E5} \langle \tau _{n=0} \rangle$ that cumulatively affects the skin-friction portion of the total drag. On the other hand, the changes in the velocity profiles affect the location of the edge of the BL and hence, the BL thickness. Here, we calculate the BL thickness in terms of $\delta _{99}$ where $u(x,\delta _{99},z) = \langle u \rangle (x,\delta _{99})= 0.99 U(x)$ (and, equivalently, in the leading edge $u_{s}(x,\delta _{99},z) = \langle u_{s} \rangle (x,\delta _{99},z) = 0.99 U(x)$ ) and plot the results for the suction and pressure sides of all the cases in figures 14 and SI.9. The $\delta _{99}$ values are with respect to the $n=0$ or the location of the peak of the riblets, which is the same as the location of the boundary of the smooth sample and are normalised by $x/\sqrt {{ Re}_x}$ . Asseen in figures 14 and SI.9, in the presence of the riblets and with the available space inside the grooves for the velocity profiles to develop, the thickness of the BL on both sides are smaller than the case of the smooth reference. Especially with larger $m$ on the pressure sides, the BL thickness on the pressure side is even smaller than the suction side. Thus, overall, the edge of the BL as seen by the inviscid flow is different and also slightly thinner for the riblet samples compared with the smooth reference, which can lead to lower pressure drops at the trailing edge of the samples as well (similar behaviour has been reported in the numerical simulations of Mele et al. Reference Mele, Tognaccini, Catalano and de Rosa2020).

Figure 14. Distribution of the BL thickness, $\delta _{99}$ , normalised by local $x/\sqrt {{ Re}_x}$ for all the riblet samples of ${\mathcal {R}} = 1.0$ on the suction and pressure side for all tested Reynolds numbers. The $\delta _{99}$ of the smooth reference on the suction and pressure side are shown with solid and dashed black lines, respectively. Locations of $x_{{ LET}}$ and $x_{{ Flat}}$ are marked by grey dotted and dash-dotted vertical lines.

Lastly, this reduction in the overall thickness of the fictitious boundary of the sample and the BL seen by the outer inviscid flow can be a substantial help to samples that cannot capture a reduction in the frictional drag compared with the smooth surface. Example of that is the [1.0, 0.0] sample thatfrom a frictional point is not able to reduce the drag force, however, as seen in figure 9(b,e,h), with a large extent of the plate experiencing $m$ values larger than the smooth reference, the $\delta _{99}$ of this sample experiences enough reduction on both sides, especially in the early part of the Flat region (see figure 14 b,e,h) to experience noticeable levels of pressure recovery (figure 13 d,e,f) at the trailing edge and ultimately becoming drag-reducing (figure 4).