3.3.1. Increased slope of the mean velocity profile

Figure 8(a) shows a hot-wire velocity profile above a smooth wall ($\cdot \!\cdot \!\cdot \!\cdot \!\cdot \!\cdot$), for which the slope in the logarithmic region is well known to be described by the von Kármán constant $\kappa \approx 0.40$. The riblet-flow profiles in figure 8(a) are only shown in the spanwise uniform region above the roughness sublayer (figure 5) and we assume a virtual origin at the mid-point of the groove ($\ell _t=k/2$) for these profiles. The highlighted logarithmic region in figure 8(a) shows a clear trend of increasing slope with increasing riblet size for this choice of the virtual origin. The outer layers of the flow above the present large riblets and smooth wall are therefore not similar. Figure 8(a) additionally shows a velocity profile above sandpaper roughness from Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016b), with the same definition of the virtual origin, as an example of rough-wall flows that commonly have the same $\kappa \approx 0.40$ as smooth-wall flows.

In figure 8(b) we measure the slope of the hot-wire mean profile by fitting a line to the logarithmic region. A reduction in $\kappa$ with increasing $s^+$ is clearly visible for any virtual origin between the groove bottom and the riblet crest (error bars). Even with the origin at the riblet crest (top of the error bars), $\kappa \approx 0.34$ for the largest riblets. To illustrate the effect of the origin on the apparent local value of $\kappa = (z^+ \mathrm {d}U^+/\mathrm {d}z^+)^{-1}$, consider a fixed measurement location in the logarithmic layer. A higher origin reduces the distance $z^+$ between the measurement location and the virtual origin, increasing $\kappa$. In a similar manner, the positional adjustment applied to the hot-wire locations $z$ in § 3.1, which accounts for the wall and probe-support movement under aerodynamic load, also slightly increased $\kappa$ relative to the raw data. The friction velocity used for the riblet profiles is measured with a drag balance (§ 2.1) and could include an error. If we were to reduce $u_\tau$ to match the DNS profiles in figure 5, the apparent local value of $\kappa$ would reduce further.

In hot-wire boundary-layer studies $u_\tau$ and the virtual origin are generally found by forcing $\kappa =0.4$ (or a close value like $\kappa =0.384 \pm 0.005$ as determined by Nagib & Chauhan Reference Nagib and Chauhan2008). However, with $u_\tau$ fixed by drag-balance measurements and $z$-positional errors corrected by comparing to spectra and mean profiles from DNS (§ 3.1), the position of the virtual origin is the only variable affecting $\kappa$. Forcing $\kappa =0.4$ requires the origin to be about $0.7k$ above the riblet crest (see grey dots in the inset of figure 8b), which seems unrealistic. Other studies of roughness (e.g. Raupach et al. Reference Raupach, Antonia and Rajagopalan1991; Grimmond & Oke Reference Grimmond and Oke1999) and small $s^+$ riblets (e.g. Luchini et al. Reference Luchini, Manzo and Pozzi1991; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a) use various definitions, but all place the virtual origin below the roughness crest, as summarised by Chung et al. (Reference Chung, Hutchins, Schultz and Flack2021). Therefore, reasonable adjustments to the virtual origin or to $u_\tau$ cannot force $\kappa \approx 0.4$, which provides confidence that $\kappa$ is indeed a function of $s^+$ for the present large riblets, as shown in figure 8(b).

3.3.2. Outer-layer similarity

Breugem, Boersma & Uittenbogaard (Reference Breugem, Boersma and Uittenbogaard2006) and Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010) observe reduced $\kappa$ above the permeable walls of porous surfaces at low friction Reynolds numbers ($\delta ^+ \lesssim 678$ and $\delta ^+ \lesssim 432$, respectively) and suggest the changes to the mean profile may be due to the wall-normal permeability of their surfaces. Small riblets can be interpreted as behaving similar to a porous surface, with an effective porosity that increases with riblet size (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011b; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a). Reasoning for reduced $\kappa$ above permeable surfaces may therefore also be applicable to riblets. However, Manes et al. (Reference Manes, Poggi and Ridolfi2011) show that insufficient separation between inner and outer length scales explains the trend of reducing $\kappa$ in the studies by Breugem et al. (Reference Breugem, Boersma and Uittenbogaard2006) and Suga et al. (Reference Suga, Matsumura, Ashitaka, Tominaga and Kaneda2010), in which the roughness sublayer based on Reynolds stresses induced by the wall porosity likely extends into the logarithmic layer.

For the present riblets, the roughness sublayer based on the extent of wall-parallel inhomogeneity of the mean flow is limited to $z \lesssim z_t+k$ (figure 5). This extent of the direct influence of riblets is small in outer units for all our experimental cases (table 1), suggesting that outer-layer similarity should exist above $z \approx z_t+k$ contrary to what we observe in figure 8. One way to rule out insufficient scale separation for the present riblets would be to repeat the experiments at matched $s^+$, but at an even higher $\delta /k$ (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021), which is not possible with the current set-up. Nevertheless, unlike the surfaces that have been demonstrated to recover outer-layer similarity at larger scale separation (Krogstad & Efros Reference Krogstad and Efros2012; Flack & Schultz Reference Flack and Schultz2014; Nugroho et al. Reference Nugroho, Monty, Utama, Ganapathisubramani and Hutchins2021), the present riblets lack an obvious larger scale that could limit outer-layer similarity by farther extending the roughness sublayer of the mean flow. In this sense, the present riblets are exceptional. In fact, the scale separation for the experimental data $\delta _{99}^\prime /k \approx 47$ is roughly constant (table 1), but $\kappa$ reduces nevertheless as $\delta _{99}^{\prime +}$ and $k^+$ increase (figure 8b).

If the damped large-scale streamwise energy is interpreted as a roughness sublayer of turbulence, the affected region $z \lesssim z_t+4k$ (figure 4) agrees with estimates of the roughness-sublayer height by, for example, Jiménez (Reference Jiménez2004) and Lee & Sung (Reference Lee and Sung2007). In § 3.4 we propose a connection between the damped large-scale energy and the changes to $\mathrm {d}U^+/\mathrm {d}z^+$ and $\kappa$ that have been noted in figure 8 to persist beyond heights at which mean-flow homogeneity has recovered. However, since $\delta _{99}^\prime /k \approx 47$ is roughly constant for all experimental riblet cases (table 1), we cannot discern if the wall-normal extent of large-scale energy reduction scales with the riblet size (e.g. these large scales are damped at heights $(z-z_b) \lesssim 5k$, as in figures 3, 4) or if it scales with the boundary-layer thickness (e.g. these large scales are damped near the wall at $z\lesssim (0.1\unicode{x2013}0.2) \delta ^\prime _{99}$, as in figure 4a). Consequently, it is not clear if increasing the scale separation $\delta _{99}^\prime /k$ would restore outer-layer similarity in an increasingly large part of the logarithmic layer.

3.3.3. Roughness function

The mean velocity shifts $U^+_{smooth}-U^+$ for hot-wire profiles, shown in figure 8(c) and calculated with smooth-wall data for almost matching $\delta _{99}^{\prime +}$, are visibly decreasing in the logarithmic region for the larger riblet cases. Although the range of uncertainty in ${\rm \Delta} U^+$ measured at different heights is only about $1$, the clearly increased logarithmic slope for the large riblets challenges our fundamental understanding and expectation of outer-layer similarity. Consequently, the velocity deficit relative to smooth-wall flow ${\rm \Delta} U^+$ may not be an accurate measure of the drag change for these large riblets and, hence, cannot be easily extrapolated to full scale.

If we nevertheless measure a value for ${\rm \Delta} U^+$ at an otherwise appropriate height of $z=0.1\delta _{99}^\prime$, the drag curve in figure 8(d) closely matches that from experimental channel flow data by Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020) for $s^+ \lesssim 80$ (inferred from the pressure drop, as profile measurements are not available). At larger $s^+ \approx 100$, our riblets with tip angle $\alpha =30^\circ$ have higher drag than the data for $\alpha =53.5^\circ$ of Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020). If we force $\kappa =0.4$ by adjusting the virtual origin, the drag curve (grey dots in figure 8d) deviates significantly from the reference data at all $s^+$. The large deviation from the drag curve of similar riblets supports the assumption that a virtual origin far above the riblet crest is unphysical.

We further show a fully rough asymptote for reference in figure 8(d), even though riblets do not experience pressure drag, which is generally considered a prerequisite for fully rough behaviour. Riblets appear to approach fully rough behaviour (Jiménez Reference Jiménez2004), but, as discussed by Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020), only for a finite range of large $s^+$ before the drag curve departs from the fully rough asymptote. The same trend can be discerned in the present data, although at higher $s^+$ for the sharper geometry ($\alpha =30^\circ$ instead of $\alpha =53.5^\circ$). Furthermore in § 3.5 we see that the friction-scaled velocity at the crest, which would be constant in the fully rough regime, instead continues to increase with increasing $s^+$. Direct numerical simulation values of ${\rm \Delta} U^+$ in figure 8(d) agree well with the experiments, within the expected uncertainties.

3.4.1. Spectral shortcut mechanism

The energy carried by dispersive Reynolds shear stresses increases with increasing $s^+$ (figure 6b), which correlates with both the energy reduction around $\lambda _x \approx 5 \delta ^\prime _{99}$ (figures 3, 4) and the reduction of $\kappa$ (figure 8b). However, the dispersive fluctuations of secondary flows cannot be measured by a single probe and, therefore, cannot be discerned from the spectra in figure 3. We nevertheless hypothesise that the reduction of large-scale streamwise energy by riblets compared with smooth-wall flow in experiments (discussed in § 3.1) is the result of a spectral shortcut as described by Finnigan (Reference Finnigan2000, figure 14) for wake production in plant canopies: Following Goldstein & Tuan (Reference Goldstein and Tuan1998), the interaction of large flow structures ($\lambda _y \gtrsim s$) with riblets induces smaller flow structures of spanwise size $\lambda _y=s$, which appear as secondary motions in the time average and enhance cross-flow fluctuations in the roughness sublayer. Therefore, even without streamwise pressure drag that could damp the large-scale motions, riblets transfer energy directly from large-scale streamwise fluctuations to $s$-scaled Reynolds stresses of secondary flows. This accumulation of energy in the dispersive component at $\lambda _y=s$ is demonstrated, for example, by the cross-plane streamfunction contours in figure 7, that scale with the riblet spacing for all groove sizes. The interplay between turbulent and secondary motions is further supported by the study of Modesti et al. (Reference Modesti, Endrikat, Hutchins and Chung2021), who decompose the drag change ${\rm \Delta} U^+$ and show that riblet surfaces with a large drag penalty due to secondary flows have a relatively low drag penalty due to purely turbulent velocity fluctuations and vice versa.

Zampiron, Cameron & Nikora (Reference Zampiron, Cameron and Nikora2021) describe a similar mechanism for the generation of secondary flows by triangular streamwise ridges. They decompose the stress balance equation for particle image velocimetry data and conclude that secondary flows extract energy from cross-flow velocity fluctuations, which in turn receive energy from turbulent streamwise fluctuations. Furthermore, the near-wall large-scale motions in the sandpaper study (isotropic roughness) of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016b) are much less damped than over our riblets (figure 4), suggesting that anisotropic roughness elements (with some defined direction, like riblets) strengthen secondary motions. A clearer picture will have to await high-Reynolds-number measurements of secondary motions in the wall-parallel or cross-stream plane above riblets.

Figure 9.

Premultiplied energy difference to smooth-wall flow $k_x^+ k_y^+ ( E^+ - E^+_{{smooth}} )$ above large riblets with $s^+=100,150$. Streamwise velocity fluctuations in (a,c,e) and Reynolds shear stress in (b,d,f). Two-dimensional spectra in (a–d) are at $z-z_t=0.1s$. Contour levels range from blue (negative) to red (positive) between the limits $k_x^+ k_y^+ (E^+_{uu} - E^+_{uu,{smooth}} ) = \pm 0.22$ in steps of $0.04$ (a,c) and $k_x^+ k_y^+ (-E^+_{uw} + E^+_{uw,{smooth}} ) = \pm 0.09$ in steps of $0.02$ (b,d). The three integration regions for (e,f) are sub-riblet scales at $\lambda _y \lesssim s$ and $\lambda _x<\infty$, super-riblet (larger-than-riblet) scales at $\lambda _y \gtrsim 2s$ and the dispersive component at $\lambda _x=\infty$ and $\lambda _y=s$ including smaller scale harmonics. The threshold $2s$ is chosen for the super-riblet scales to only select the largest flow structures in the domain and avoid including energy at $\lambda _y\approx s$ that spills into adjacent modes.

3.4.2. Spectral shortcut at DNS-resolved scales

The present hot-wire experiments resolve long enough flow structures to reveal a reduction of energy at scales $\lambda _x^+ \approx$10 000–50 000 (figure 3e,f) that exceed typical numerical domains. However, some of the flow structures that interact with sufficiently large riblets to create secondary flows belong to the near-wall cycle ($\lambda _x^+ \approx 1000$ and $\lambda _y \gtrsim s$) and are resolved by both hot-wire measurements and DNS. Consequently, the energy reduction relative to smooth-wall flow observed in experiments (figure 4) is also evident at the largest scales resolved in the present DNS ($\lambda _x^+\lesssim 2000$, table 2).

In figure 9 we subtract spectra from the DNS over a smooth wall from those over large riblets at matched heights above the virtual origin ($\ell _t=k/2$). Blue contours in figure 9(a,c) indicate a reduction of streamwise velocity fluctuations by riblets at super-riblet scales (i.e. larger-than-riblet scales) $\lambda _y \gtrsim 2s$, and red contours indicate an increase of energy at sub-riblet scales $\lambda _y\lesssim s$. Figure 9(e) shows profiles of the integrated energy difference for the super-riblet scales, the sub-riblet scales and for the dispersive component. The solid lines for super-riblet scales in figure 9(e) are below zero, indicating large-scale energy reduction in DNS that is qualitatively the same as that seen in the hot-wire profiles (dotted smooth-wall profiles are above the solid riblet profiles in figure 4). In both data sets, energy of super-riblet scales is reduced at heights of at least $z-z_t \lesssim 2s$ or, equivalently, $z-z_b \lesssim 2.5s=5k$. This extent is much greater than that of the dispersive fluctuations, which are also shown in figure 9(e).

3.4.3. Spectral shortcut affects the mean velocity

In DNS the energy reduction is likewise visible in Reynolds shear stresses (figure 9b,d,f). Large riblets reorganise some of the super-riblet-scale Reynolds shear stress to the dispersive component discussed in § 3.2 and shown again in figure 9(f), which only extends to heights $z \lesssim z_t+0.5s$. Figure 9(f) therefore suggests that the spectral shortcut created by large riblets reorganises the turbulence by reducing super-riblet-scale Reynolds shear stresses at $z \lesssim z_t+2s$ and increasing dispersive shear stresses closer to the wall at $z \lesssim z_t+0.5s$. Although not measured in experiments, we can assume based on DNS that the largest $s^+$ riblets, shown in figure 4(a), reorganise Reynolds shear stress from super-riblet scales at $z/\delta _{99}^\prime =0.1$ in the logarithmic layer to dispersive stresses of $s$-scaled secondary flows below the logarithmic layer. The net reduction of Reynolds shear stress in the logarithmic layer is relatively small, but nevertheless accompanied by a clear increase of local $\mathrm {d}U^+/\mathrm {d}z^+$ or, correspondingly, a reduction of $\kappa$ as observed in figure 8. Such a connection between damped large-scale energy and reduced $\kappa$ is likewise demonstrated in numerical experiments by Jiménez (Reference Jiménez2022), in which the longest streaks in channel flow are artificially damped.

The even larger flow structures around $\lambda _x \approx 5\delta _{99}^\prime$ in figure 3 are also damped by the spectral shortcut that acts on super-riblet scales, but they might not affect $\mathrm {d}U^+/\mathrm {d}z^+$, because such large flow structures do not contribute significantly to Reynolds shear stress at heights in the logarithmic layer $z \ll \lambda _x$ (Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017; Kwon & Jiménez Reference Kwon and Jiménez2021) and are therefore ordinarily considered inactive at those heights (Townsend Reference Townsend1976).

3.5.1. Turbulence inside very large grooves

The flow in the groove of small drag-reducing riblets is dominated by viscosity, but the wall at the bottom of very large grooves creates a secondary peak of sustained $u$ fluctuations inside the groove, which we capture in spectra of both DNS and hot wire (figure 3e). The secondary peak at $\lambda _x^+ \approx 1000$ and $(z-z_b)/k \approx 0.18$ in the DNS, i.e. about $13$ wall units above the groove bottom, suggests the formation of a nascent near-wall cycle inside the groove with $s^+ \gtrsim 100$, which is also partially visible for $s^+ = 63$ in figure 3(c). The emergence of a nascent near-wall cycle inside the groove of single-scale trapezoidal riblets with $s^+ \gtrsim 100$ agrees with the spanwise extent required to sustain near-wall turbulence in numerical experiments of narrow but doubly periodic smooth-wall channels (Jiménez & Moin Reference Jiménez and Moin1991), even though the riblet grooves with no-slip side walls further constrain the flow. The half-height riblet in figure 7(j) displaces turbulence upwards in the groove compared with the single-scale riblets with the same spacing in figure 7(d) and presumably delays the formation of secondary near-wall turbulence to $s^+ \gtrsim 200$.

3.5.2. Phenomenological model for the crest velocity

In the following, we propose a phenomenological power law for the mean streamwise velocity at the height of the riblet tips $\bar {u}_t^+ \propto A_p/A_w (s^+)^{1/4}$, where the total wall plan area is $A_p$ and the wetted area $A_w$. Figure 10(b) shows that the power law is consistent with values from our DNS for riblets with $s^+ \gtrsim 60$ and it therefore appears to capture the relevant flow physics in large riblet grooves.

Figure 10.

(a) Sketch of turbulent mixing in the groove of large riblets. (b) Streamwise velocity at the height of the riblet tips shown for the single-scale trapezoidal riblets, ${A_w/A_p=1.77}$ and for the two-scale riblets ${A_w/A_p=2.15}$ (the additional riblet increases the wetted area by a factor of $1.22$). Solid lines show the laminar crest velocity for the two riblet shapes (as described by Modesti et al. Reference Modesti, Endrikat, Hutchins and Chung2021). Dashed lines are the power laws as labelled.

Similar, but distinctly different scaling laws have been developed for roughness by Brutsaert (Reference Brutsaert1975) and for superhydrophobic surfaces by Seo & Mani (Reference Seo and Mani2016), among others. Seo & Mani (Reference Seo and Mani2016) developed a $1/3$-power law for the slip velocity of superhydrophobic surfaces with no-slip posts. They assume that a Prandtl–Blasius laminar-like boundary layer forms on each of the no-slip sections, which are spaced apart in the streamwise direction. This assumption is, of course, not valid for streamwise uniform riblets. The surface-renewal arguments of Brutsaert (Reference Brutsaert1975) are developed for roughness, where temporarily stagnant eddies form in between the roughness elements. Here, diffusion limits the transport of a scalar from the stagnant eddies to the surface. Streamwise uniform riblets do not create such regions of stagnant flow, but viscous diffusion still limits the transport of momentum from the convecting eddies to the surface, an idea we use below.

For large riblet grooves, we envision an incipient turbulent Richardson–Kolmogorov cascade that mixes the momentum vigorously, except in the viscous sublayer enveloping the wetted wall. The well-mixed flow is assumed to have a velocity $\langle u \rangle = \bar {u}(z) + \tilde {u}(y,z) \approx \bar {u}_t$ that is uniform everywhere in the groove except in the wetted viscous boundary layer. In this unobstructed spectral region, the Kolmogorov cascade passes energy from size $s$ to the dissipation length $\eta \equiv (\nu ^3/\epsilon )^{1/4}$, where $\epsilon = K u_\tau ^3/s$ is the dissipation rate of $s$-sized structures, with a constant $K$ of $O(1)$. We assume $C\eta$ to scale the thickness of the viscous layer inside the groove with another constant $C$ of $O(1)$ (sketched in figure 10a). With the velocity varying linearly inside the viscous layer, we can express the surface-perpendicular gradient by $\bar {u}_t / (C\eta )$. The drag is thus

(3.1)\begin{equation} \frac{D}{\rho} = A_w \nu \langle \frac{\partial u}{\partial n} \rangle_w \approx A_w \nu \frac{\bar{u}_t}{C \eta} = A_w \nu \frac{\bar{u}_t}{C} \left(\frac{\nu^3}{Ku_\tau^3/s} \right)^{{-}1/4}, \end{equation}

where $\langle \,\boldsymbol {\cdot}\, \rangle _w = A_w^{-1} \int _w (\,\cdot\,)\,\mathrm {d}S$ averages over the riblet wall and $n$ is locally normal to the surface. The wall-shear stress $\tau _w/\rho = D/(\rho A_p) \equiv u_\tau ^2$ and, therefore, to within an ${O}(1)$ constant,

(3.2)\begin{equation} u_\tau^2 \propto \frac{A_w}{A_p} \nu \bar{u}_t \left( \frac{u_\tau^3}{s\nu^3} \right)^{1/4} \Rightarrow \bar{u}_t^+{\propto} \frac{A_p}{A_w} (s^+)^{1/4}. \end{equation}

Figure 10(b) shows that the crest velocity of the largest of our riblets indeed follows this trend, whereas the Stokes prediction for the crest velocity holds for smaller ($s^+ \lesssim 30$) riblets, $\bar {u}_t^+ = \ell _u^+ \propto s^+$, where $\ell _u$ is the streamwise protrusion height (Bechert & Bartenwerfer Reference Bechert and Bartenwerfer1989; Luchini et al. Reference Luchini, Manzo and Pozzi1991).

The crest velocity is only marginally affected by the secondary half-height riblet inside the groove at the present riblet sizes (figure 10b). For small sizes, the flow in the groove is dominated by viscosity and the secondary riblet has no noticeable effect on turbulence in the groove. In this regime, the $1/4$-power law does not apply and the difference in $A_w$ between the two surfaces does not significantly affect $\bar {u}_t$. At large riblet sizes with turbulence in the groove, the secondary riblet affects flow structures up to a height well above the riblet crest, where dispersive stresses in figure 6 are reduced compared with the single-scale riblets with the same spacing. The crest velocity, however, only captures the reorganisation of turbulent flow inside the groove. Therefore, even at $s^+=63$ of our largest two-scale riblets, the crest velocity does not yet reflect the geometry differences. At much greater riblet sizes, we would expect the half-sized additional riblets to reduce the crest velocity in proportion to the increased $A_w$ in (3.2).

3.5.3. Approximating ${\rm \Delta} U^+$

For very small riblets ($s^+ \sim 0$), the roughness function is approximately constant from the crest upwards, as outlined by García-Mayoral et al. (Reference García-Mayoral, Gómez-de-Segura and Fairhall2019) following the slip-lengths argument by Luchini (Reference Luchini1996). Therefore, the crest (slip) velocity, along with the position of the virtual origin, describes the drag change ${\rm \Delta} U^+$ relative to a smooth wall (Ibrahim et al. Reference Ibrahim, Gómez-de-Segura, Chung and García-Mayoral2021). For larger riblets, the drag change inferred at the crest is not an accurate measure, in part because the outer layers of the flow above the larger riblets studied here are not similar to smooth-wall flow. Furthermore, Gatti et al. Reference Gatti, von Deyn, Forooghi and Frohnapfel2020 do not observe fully rough behaviour, where the skin-friction coefficient would be independent of the Reynolds number. The present data also show a departure from the solid straight line in figure 8(d).

Nevertheless, we crudely estimate ${\rm \Delta} U^+ \approx U^+_{t,{Smooth}} - \bar {u}_t^+$ based on the velocity $\bar {u}_t^+ \approx a(A_p/A_w)(s^+)^{1/4}$ at the crest of large riblets ($s^+\gtrsim 60$), with an empirical constant $a$. For the present trapezoidal riblets, $a(A_p/A_w) \approx 2.7$ (figure 10b). The corresponding smooth-wall velocity at the height of the riblet crest is $U^+_{t,{Smooth}} \approx \ln (\ell _t^+)/0.4 + 5.1$ if $\ell _t \gtrsim 12$ as for our large riblets (table 2, assuming $\ell _t=k/2$). The approximation

(3.3)\begin{equation} {\rm \Delta} U^+(s^+) \approx \ln(\ell_t^+)/0.4 + 5.1 - a(A_p/A_w)(s^+)^{1/4}, \quad \mathrm{where}\ s^+ \gtrsim 60, \end{equation}

neglects variations in the roughness function above the crest and, therefore, overestimates ${\rm \Delta} U^+$ for our DNS cases by $[1.63, 0.95, 0.54]$ at $s^+=[63, 100, 150]$ and for the hot-wire cases by $[1.87, 1.24, 0.98, 0.71]$ at $s^+=[67, 102, 132, 165]$. Being valid for very large riblets, the approximation (3.3) approaches the value measured at $z_c^+$ (DNS) or $0.1\delta ^\prime _{99}$ (hot wire) as the riblet size increases.