2. Turbulent drag definition and its implications

The definition of skin friction, Reynolds number and their relationship with the streamwise pressure gradient is straightforward for plane turbulent channel flows, but less so for parallel flows with complex cross-section, like the 2-D structures addressed in the present work. It is therefore discussed in the following.

Let the three velocity components be denoted $(u_1,u_2,u_3)=(u,v,w)$ along the streamwise, wall-normal and spanwise $(x_1,x_2,x_3)=(x,y,z)$ direction, respectively. For fully developed plane turbulent channel flow, the wall-shear stress $\tau _w$ is a function of the time-averaged streamwise pressure gradient

(2.1)\begin{equation} \tau_w={-}\frac{\partial \bar{p}}{\partial x_1}\,\delta, \end{equation}

where $\delta$ corresponds to the half-channel height (see below) and $\overline {(\,{\cdot }\,)}$ denotes time averaging. This relation is used in the present experimental campaign (see § 3) to obtain the skin-friction coefficient

(2.2)\begin{equation} C_f=\frac{2 \tau_w}{\rho U_b^2}, \end{equation}

where $U_b$ is the bulk velocity (volume flow rate per unit width $\dot {V}'$, normalized by channel height $2 \delta$) also used for the formulation of the bulk Reynolds number

(2.3)\begin{equation} Re_b =\frac{2\delta U_b}{\nu}=\frac{\dot{V}'}{\nu}. \end{equation}

Additionally, based on $\tau _w$, the friction velocity $u_\tau =\sqrt {{\tau _w}/{\rho }}$ and the respective friction Reynolds number $Re_\tau ={u_\tau \delta }/{\nu }$ are obtained.

Equation (2.1) is derived for turbulent channels with plane walls, for which $\tau _w$ then represents the temporally and spatially averaged wall-shear stress, and $\delta$ is the univocally defined half-channel height. When the same equation is applied to non-planar surfaces, $\tau _w$ assumes the different meaning of an effective wall-shear stress, which balances the measured pressure gradient as if it were caused by a virtual flat wall placed at distance $\delta$ from the channel centreline. The definition of $\delta$ is therefore not trivial when the walls are not flat.

We choose to place the wall-normal origin of the channel at the streamwise protrusion height $h_{\parallel }$ below the structure's tip for the present study (see figure 2). This ensures that the surface structures do not induce any drag variation under laminar flow conditions, as discussed in the following. In § 5.2 and the Appendix we present the evaluated drag change for alternative channel height definitions.

Figure 2. Sketch of investigated surface structures, where $h_{\parallel }$ and $h_\perp$ represent streamwise and spanwise protrusion heights (Luchini et al. Reference Luchini, Manzo and Pozzi1991), and $h_{{avg}}$ is the averaged (meltdown) height. The half-channel height $\delta$ is defined as the distance between the channel centreline and $h_\parallel$ below the structure tip.

In order to determine $h_{\parallel }$ and $h_\perp$, a Stokes flow problem given by

(2.4)\begin{equation} -\frac{1}{\rho}\,\frac{\partial p}{\partial x_i} +\nu\,\frac{\partial^2 u_i}{\partial x_j\,\partial x_j}=0 \end{equation}

in conjunction with the conservation of mass is solved numerically. Shear flow aligned with the surface structure is considered to obtain $h_{\parallel }$, whereas (2.4) is solved for shear flow perpendicular to the structure orientation to obtain $h_{\perp }$. The protrusion heights, $h_{\parallel }$ and $h_{\perp }$ correspond to the distance below the structure tip at which the spatially averaged Stokes solution extrapolates to zero. In this configuration, (2.4) simplifies to the Laplace equation $\nabla ^2 u=0$ for the parallel flow configuration, and analytical solutions can be found in the case of 2-D structures via conformal mapping (Bechert & Bartenwerfer Reference Bechert and Bartenwerfer1989). The values stated in table 2 below for $h_{\parallel }$ and $h_{\perp }$ are computed numerically via finite elements utilizing the software FreeFem++ (Hecht Reference Hecht2012). The procedure was validated against the analytical solution and literature references (Luchini et al. Reference Luchini, Manzo and Pozzi1991; Bechert et al. Reference Bechert, Bruse, Hage, Hoeven and Hoppe1997). For the present surface structures, $h_\perp$ is located closer to the structure tip compared to $h_\parallel$, as visualized in figure 2. The figure also includes the meltdown height $h_{avg}$, which is located closer to the bottom of the surface structure.

For the investigated 2-D surface structures, the laminar channel flow solution and the Stokes flow solution for $h_{\parallel }$ coincide. Therefore, the channel height definition based on $h_{\parallel }$, as indicated in figure 2, provides a set-up in which the well-known $C_f\unicode{x2013}Re_b$ relation for laminar channel flow ($C_f=12/Re_b$) also applies for a channel with riblets. Thus, by choosing the current definition of $\delta$, we focus on the surface structure influence on turbulent drag, since the laminar drags of smooth and structured channel are identical by definition. This is in agreement with the drag-reduction prediction for the viscous regime as outlined briefly in § 1, which suggests zero drag change for vanishing viscous-scaled riblet size (Luchini et al. Reference Luchini, Manzo and Pozzi1991).

Although the drag force of a specific surface for a given flow rate is unambiguous, as it can be e.g. measured with a force sensor, its dimensionless representation does depend on the length scales used for normalization. As can be seen from (2.1) and (3.2), the choice of $\delta$ enters the definitions of $\tau _w$ and $C_f$ and, through $\tau _w$, also affects all friction-related quantities such as $l_g^+$. Note that $Re_b$ is independent of the choice of $\delta$ since it is determined directly by the volume flow rate.

3. Test facility

The channel flow is generated by a blower-type wind tunnel (Mehta & Bradshaw Reference Mehta and Bradshaw1979). The schematic set-up of the facility is shown in figure 3. The flow is driven by a radial fan with a large operational range $40\unicode{x2013}6000\,{\mathrm {m^3}}\,{\mathrm {h}}^{-1}$, and progresses through a supply pipe into a large settling chamber. The air is blown towards the back wall of the settling chamber, which is opposite the nozzle directing into the actual test section, in order to ensure a homogeneous flow distribution. The air flows through five grids embedded in wooden frames and a honeycomb flow straightener on its way through the settling chamber towards the test section. The arrangement of the radial fan outside the settling chamber minimizes the blockage and decouples the vibrations generated by the radial fan. A nozzle of contraction ratio 6 : 1 connects the settling chamber with the rectangular duct test section. There are 21 pairs of pressure taps (diameter 0.3 mm) spaced at intervals of 200 mm located along the side walls of the test section in order to measure the streamwise pressure gradient. Changes in ambient conditions are accounted for by tracking the inlet and outlet temperatures via PT100 thermocouples with maximum error below 0.1 K, and the ambient pressure $p_{{amb}}$ and humidity $a_q$ using Adafruit BMP 388 and BME 280 sensors, respectively. The BMP sensor has absolute accuracy 0.5 hPa, while the BME sensor resolves $a_q$ with error ${\pm }3\,\%$.

Figure 3. Schematic of the experimental facility with respect to wind tunnel components and measurement instrumentation.

On the suction side of the radial fan, an inlet pipe of diameter $D$ is installed. A zigzag tripping is located at the pipe entrance $32D$ upstream of an orifice flow meter of diameter $d$ and diameter ratio $\beta ={d}/{D}$, across which the pressure drop ${\rm \Delta} p_o$ is measured and related to the orifice flow rate

(3.1)\begin{equation} \dot{V_o}=\frac{C}{\sqrt{1-\beta^4}}\,\epsilon\, \frac{\rm \pi}{4}\,d^2 \sqrt{\frac{2\,{\rm \Delta} p_o}{\rho_{in}}}. \end{equation}

The coefficients $C$ and $\epsilon$ are derived from empirical expressions (International Organization for Standardization 2003b), where $C$ depends on the diameter ratio $\beta$ and the pipe Reynolds number $Re_D$ (Sattary & Reader-Harris Reference Sattary and Reader-Harris1996). As a result, the orifice flow rate $\dot {V_o}$ is computed with the iterative procedure outlined in norm EN (International Organization for Standardization 2003a), since $Re_D$ itself is dependent on $\dot {V_o}$. The mass flow rate $\dot {m}$ through the pipe (and thus also the channel test section) is obtained based on multiplication of the volume flow rate with the air density $\rho _{{in}}$ deduced from the ideal gas law including humidity effects and a temperature measurement $T_{{in}}$ at the pipe inlet.

In order to keep the dissipative losses generated by the orifice flow meter within reasonable limits and to avoid compressibility effects, a maximum differential pressure ${\rm \Delta} p_{o{max}} = 625\,{\rm Pa}$ across the orifice is considered, thereby limiting the maximum achievable flow rate for a given $d$. For the sake of covering a range of the bulk Reynolds number $4.5 \times 10^3 < Re_b <8.5 \times 10^4$ (this corresponds to $150 \lesssim Re_\tau \lesssim 2000$ for the smooth wall case) in the test section, two different orifice flow meters are installed with inlet pipe diameters $D = 100$ mm and $D = 200$ mm, respectively. Each custom-manufactured annular orifice measuring chamber can be equipped with orifice plates of varying inner diameter $d$. The configurations are specified in table 1. The orifice's pressure drop is measured with one of two Setra 239D (125 Pa and 625 Pa full-scale) unidirectional differential pressure transducers with accuracy 0.07 % of the full scale, switching automatically depending on ${\rm \Delta} p_o$.

Table 1. Specifications of the different orifice flow meter configurations. Introduced markers are used in figures 4–10.

The resulting (effective) wall-shear stress in the considered portion of the channel test section is determined by evaluating the streamwise pressure gradient at a given flow rate. Based on the measured pressure gradient $\varPi =-({{\rm \Delta} \bar {p}}/{{\rm \Delta} x})$ and the measured mass flow rate $\dot {m} =\dot {V_o}\rho _{in}$, the skin-friction coefficient is computed via

(3.2)\begin{equation} C_f=\frac{8\varPi \delta^3 W^2 \rho}{\dot{m}^2}, \end{equation}

where $\rho$ denotes the density at the test section's outlet, and $W$ is the test section width. For the pressure measurement, an MKS Baratron 698A unidirectional differential pressure transducer with 1333 Pa maximum range and accuracy 0.13 % of the reading is employed. In order to measure the pressure drop at all streamwise pressure taps, a Scanivalve multiplexer is used that switches mechanically between the individual pressure taps such that all pressure differences are measured with the same high-accuracy pressure transducer.

The test section consists of a flat rectangular duct of aspect ratio ${\rm AR}=12$, with $W= 0300$ mm and $\delta =12.6$ mm in the smooth reference case. It extends in the streamwise direction $L=3950$ mm, corresponding to $L=313.5\delta$. Two plates for the top and bottom walls, as well as two monolithic rods for the side walls, form the test section. The duct walls are made of polished aluminium of average roughness $Ra= 0.4$ mm, and the top and bottom walls consist of three segments (950 mm, 1500 mm, 1500 mm) that can be exchanged individually.

The bulk Reynolds number in the test section is constant along its length and computed via

(3.3)\begin{equation} Re_b=\frac{\dot{m}}{W\rho \nu}. \end{equation}

Two trip strips are installed at the inlet of the test section along the whole channel width. Each strip protrudes 1.6 mm ($0.13\delta$) from the wall and extends for 0.3 mm in the streamwise direction, resulting in 13 % cross-section blockage. The trip provides fully-developed turbulent flow conditions for $Re_b>4500$ starting $160\delta$ downstream of the tripping location (Güttler Reference Güttler2015).

The measurement uncertainty is quantified via Gaussian error propagation at 95 % confidence level, as outlined in the Appendix of Gatti et al. (Reference Gatti, Güttler, Frohnapfel and Tropea2015) for the utilized facility. The accumulated uncertainty of $C_f$ is below ${\pm }2.7\,\%$, where the biggest uncertainty contribution stems from the orifice flow rate measurement with ${\pm }1.7\,\%$. Assuming the error in the flow rate measurement to be systematic, and considering only the error arising from the measurement instrumentation (i.e. random error sources), the measurement uncertainty for drag change measurements ${\rm \Delta} C_f/C_{f0}$ (where $C_{f0}$ denotes the smooth reference value at matched $Re_b$) reduces to $0.4\,\%$ (Güttler Reference Güttler2015).

4. Experimental matrix

For the present investigation, 8 sets of 2-D trapezoidal-grooved geometries of varying height $h$ and spacing $s$ are considered (see figure 2 for geometrical definitions), which are installed on both channel walls symmetrically in the most downstream $119 \delta$ of the test section. All cases are listed in table 2. The naming convention is introduced following the distinction between riblets and ridges outlined in the Introduction: an ID starting with dr refers to potentially drag-reducing riblets with $l_g/\delta <0.17$, and purely drag-increasing ridges that exceed $l_g/\delta >0.17$ are labelled di.

Table 2. Dimensions of the investigated geometries as introduced in figure 2. Here, $P$ denotes the perimeter, $l_g$ is the square root of the groove area as introduced by García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011a), $h_{\parallel }$, $h_{\perp }$ are the streamwise and spanwise protrusion heights measured from the tip as defined by Luchini et al. (Reference Luchini, Manzo and Pozzi1991), and $\eta _c$ represents a constant hydraulic channel height relation $\delta _{hyd}/\delta =\text {const.}$ (see (5.6) for the definition of $\delta _{hyd}$) obtained a posteriori from figure 10.

The number part of the ID represents the ratio of the wavelength $s$ and the structure width $s_r$, i.e. dr_1a, …, dr_1c are sawtooth riblets, while di_13 are widely spaced ridges. Both $h$ and $s$ are varied systematically over one order of magnitude. For the interested reader, note that the mean flow data of di_13 up to $Re_b=5 \times 10^4$ is discussed in von Deyn et al. (Reference von Deyn, Schmidt, Örlü, Stroh, Kriegseis, Böhm and Frohnapfel2021). Moreover, set dr_2 corresponds to the one discussed in Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020).

In doing this, the evolution of the drag change with respect to a smooth surface is studied starting from very small drag-reducing riblets submerged in the viscous sublayer up to widely spaced drag-increasing ridges that protrude into the logarithmic flow region and are known to induce large-scale secondary motions (Medjnoun et al. Reference Medjnoun, Vanderwel and Ganapathisubramani2020). Note that di_2 was designed purposely as an enlarged trapezoidal riblet geometry. To quantify the increase in wetted surface area, the perimeter $P$ to spanwise wavelength ratio $P/s$ is included in table 2. For each geometry, the Stokes flow solution (see § 2) is computed to obtain the protrusion height values $h_{\parallel }$ and $h_{\perp }$.

The changes in skin-friction drag ${\rm \Delta} C_f$ are obtained by comparing two consecutive experiments: first, a smooth wall measurement used as a common reference for all structured cases was conducted, followed by skin-friction measurements of the structured plates. The smooth data are fitted with a polynomial function of fifth order for each orifice configuration stated in table 1, enabling a comparison at constant flow rate between smooth and structured cases. All measurements are carried out in the most downstream third 1500 mm (or $119\delta$) portion of the test section, allowing $194 \delta$ for flow development. The pressure taps in the second segment are used as a reference to confirm reproducibility between different measurements. The investigated structures were milled in 1500 mm-long aluminium plates with a high precision CNC milling-machine, with the exception of sets dr_1a and dr_1b, which consist of a riblet foil manufactured by 3M glued onto aluminium plates. The net half-channel height $\tilde {\delta }_{avg}$, defined as the distance between the channel centreline and the average structure height $h_{avg}$ (see figure 2), is adjusted to match the smooth reference value $\delta =12.6$ mm. In doing so, the net fluid volume in the channel is kept identical between different cases. This set-up was chosen initially because we considered $\tilde {\delta }_{avg}$ an appropriate channel height for the comparison among different types of riblets and ridges. As will be discussed in § 5.2, a more physically sound choice for interpreting the measurements is $\delta$ based on $h_{\parallel }$. Thanks to the use of dimensionless numbers, the choice of channel height in the experimental set-up can be converted easily to other choices in the data evaluation.

The geometrical parameters stated in table 2 were verified via optical (Sensofar S neox) and tactile measurements (perthometer Mahr MarSurf PCV). Next to the geometrical parameters $s$, $h$, $s_r$ and $\alpha$ as defined in figure 2, the square root of the groove area $l_g$ and the perimeter $P$ are included in table 2.

5. Measurement results

In order to assess the skin-friction drag and identify respective drag regimes, the pressure-drop measurement results obtained for all surface structures specified in table 2 and the smooth reference are analysed in the following.

5.1. Skin-friction measurements

The skin-friction coefficient $C_f$ (see (2.2)) as a function of $Re_b$ is presented in figure 4. The smooth results are shown as black markers with respect to the well-known correlation proposed by Dean (Reference Dean1978). The different markers denote the varying orifice flow meter sizes employed to measure the volumetric flow rate (see § 3). Overall, the reference results show very good agreement with the Dean correlation with detected deviations below $2.7\,\%$, i.e. within the measurement uncertainty margin. Also note that the data collected by Dean (Reference Dean1978) to derive his famous correlation are subject to significant scatter.

Figure 4. Skin-friction coefficient $C_f$ as a function of the bulk Reynolds number $Re_b$. Different markers indicate different orifice diameters as introduced in table 1. Riblets (dr, drag-reducing) are depicted in red, and ridges (di, purely drag increasing) are shown in blue. The reference measurements (black markers) are shown in comparison to the correlation proposed by Dean (Reference Dean1978). The horizontal and vertical error bars represent the measurement uncertainty for exemplary data points.

The colour code is introduced following the distinction between riblets and ridges: riblet data are depicted with red symbols in different shades, while ridge data are shown in shades of blue. Lighter colours indicate physically smaller riblets, or more widely spaced ridges. Sets dr_1a, dr_1b, dr_1c and dr_2 are at least partially located below the smooth reference in the Nikuradse-type diagram of figure 4, indicating that these surfaces reduce skin-friction drag in this particular Reynolds number range in the present facility. The drag-reducing effect scales in viscous units (Bechert et al. Reference Bechert, Bruse, Hage, Hoeven and Hoppe1997), thus the drag-reducing regime of the different structures is shifted to lower $Re_b$ with increasing ID/colour intensity (a,b,c and 2) due to their increasing physical size. All drag-increasing structures (in blue) reveal a monotonically decreasing $C_f$ with increasing $Re_b$, indicating that none of the investigated geometries reaches an apparent fully rough (i.e. $C_f$ independent of $Re_b$) flow state in the investigated Reynolds number range.

The physically larger riblet sets dr_1c ($l_g/\delta =0.0533$) and dr_2 ($l_g/\delta =0.0302$) are one order of magnitude larger than the smallest riblet set dr_1a ($l_g/\delta =0.0049$). They transition from a drag-reducing to a drag-increasing behaviour within the present Reynolds number range, exhibiting a region of constant $C_f$ for a limited range of Reynolds numbers before eventually showing $C_f(Re_b)$ behaviour similar to that of the drag-increasing ridges. A trace of nearly constant $C_f$ is also present for di_2 ($l_g/\delta =0.2401$) at very low Reynolds numbers, which hints at a similarity between the two surface structures di_2 and dr_2.

The measurement results of the differently spaced ridge configurations reveal that narrower-spaced ridges (darker blue colour) of the same element height produce more skin-friction drag compared to wider-spaced ridges, i.e. skin-friction drag increases with increasing perimeter $P$ to wavelength $s$ ratio $P/s$. However, we observe that the increase in $C_f$ is not simply proportional to the ratio $P/s$. A further discussion of this geometrical parameter is presented in § 5.3.4.

Locally, e.g. for set dr_1c at $Re_b\approx 3\times 10^4$, small (${<}2\,\%$) differences of $C_f$ between the different orifice flow meter configurations can be observed. As specified in table 1, the different orifice configurations are necessary to cover the full range of $Re_b$. The observed scatter in the data points is related to systematic uncertainty arising from the flow rate measurement, and is within the expected measurement uncertainty discussed in § 3. Also note that the flow rate uncertainty is a function of the orifice pressure drop ${\rm \Delta} p_o$, increasing nonlinearly for small ${\rm \Delta} p_o$. Therefore, the overlap of $C_f$ measurements obtained with different orifices is particularly challenging, since one of them must work towards its lower $Re_b$ limit of applicability, as specified in table 1, i.e. at low ${\rm \Delta} p_o$ and larger uncertainty. However, the offset in $C_f$ induced by the flow rate measurement is independent of the investigated individual surface structure. Therefore, the offset error cancels when evaluating the relative drag change ${(C_f-C_{f0})}/{C_{f0}}={{\rm \Delta} C_f}/{C_{f0}}$ or ${\rm \Delta} U^+$ discussed in the following subsections. In consequence, the resulting uncertainty for these relative quantities is significantly smaller.

5.2. Implications of the channel height definition

As discussed in § 2, the arbitrary choice of the effective half-channel height directly affects $C_f$ and many other flow quantities. In the present work, the wall-normal origin has been placed a streamwise protrusion height $h_{\parallel }$ below the surface crest, which is the position where the far-wall Stokes flow oriented in the mean turbulent flow direction predicts zero average velocity. For non-planar surfaces, the resulting half-channel height $\delta$ differs from the average half-channel height $\tilde {\delta }_{{avg}}$, which is defined as the distance between the average structure height $h_{{avg}}$ (see figure 2) and the channel centreline. This definition is commonly used for rough surfaces due to its relatively easy accessibility (Chan et al. Reference Chan, Macdonald, Chung, Hutchins and Ooi2015).

In order to show that the present choice of the effective channel height is particularly suited to assess the effect of 2-D structures on turbulent drag, we compare how the relative drag change varies when it is computed assuming half-channel heights $\delta$ and $\tilde {\delta }_{{avg}}$, respectively. For the sake of clarity, quantities computed with $\tilde {\delta }_{{avg}}$ are denoted with $\widetilde {(\,{\cdot }\,)}$.

The corresponding results for ${\rm \Delta} C_f/C_{f0}$ are presented in figure 5 for the riblet data sets, where figure 5(a) shows ${\rm \Delta} \tilde {C}_f / \tilde {C}_{f0}$. Utilizing $\tilde {\delta }_{{avg}}$ as half-channel height, i.e. placing the wall-normal origin at the average (meltdown) height, corresponds to comparing $\tilde {C}_f$ of the structured surface against $\tilde {C}_{f0}$ of a reference channel with the same cross sectional area and at the same $Re_b$. The physically small ($h<0.3$ mm, i.e. $h/\delta <0.0235$) riblet sets dr_1a, dr_1b and dr_2 show the expected negative ${\rm \Delta} \tilde {C}_f/\tilde {C}_{f0}$, in agreement with previous wall-shear stress balance results of Bechert et al. (Reference Bechert, Bruse, Hage, Hoeven and Hoppe1997). Compared at the same viscous-scaled riblet size, one would expect the geometrically similar but physically larger data set dr_1c ($h/\delta =0.0703$) to behave similarly to dr_1a and dr_1b. However, this is clearly not the case, as shown in figure 5(a), where ${\rm \Delta} \tilde {C}_f/\tilde {C}_{f0}$ is generally positive for dr_1c. Moreover, the zero crossing ${\rm \Delta} \tilde {C}_f/\tilde {C}_{f0}=0$ for data sets dr_1b and dr_2 does not occur at $l_g^+\approx 17$, as commonly reported for riblets (García-Mayoral et al. Reference García-Mayoral, Gómez-De-Segura and Fairhall2019; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021). Therefore, even though ${\rm \Delta} \tilde {C}_f/\tilde {C}_{f0}$ is an easily accessible quantity and thus might seem favourable at first glance, considering the drag change at matched cross-sectional area and flow rate does not result in collapsing drag curves for different data sets, and yields evidence contrasting the present understanding of riblets from the literature.

Figure 5. Relative drag change ${\rm \Delta} C_f /C_{f0}$ versus the viscous-scaled square root of the groove cross-sectional area $l_g^+$, with zoomed view on ${\rm \Delta} C_f/C_{f0}<0$. (a) Wall-normal origin placed at the averaged structure (meltdown) height $h_{avg}$, where $\widetilde {(\cdot )}$ is used to denote the change of the wall-normal origin to $h_{avg}$ above the structure valley. (b) Wall-normal origin placed at $h_\parallel$ below the structure crest. The vertical error bars represent the measurement uncertainty for exemplary data points. The horizontal error bars are negligible in this representation.

As an alternative, figure 5(b) shows ${\rm \Delta} C_f / C_{f0}$. Note that $\delta$ is generally smaller than $\tilde {\delta }_{avg}$ for structured surfaces. Thus utilizing $\delta$ as half-channel height corresponds to comparing $C_f$ of the structured surface against $C_{f0}$ of a reference channel with a slightly larger cross-sectional area and at the same $Re_b$. The data presented in figure 5(b) can be related directly to the $\tilde {\delta }_{{avg}}$-based scaling via the expressions:

(5.1a,b)\begin{equation} \frac{{\rm \Delta} C_f}{C_{f0}}=\left(\frac{{\rm \Delta} \tilde{C}_f}{C_{f0}}+1\right) \frac{\delta^3}{\tilde{\delta}_{avg} ^3}-1,\quad l_g^+= \tilde{l}_g^+\sqrt{\frac{\delta}{\tilde{\delta}_{avg}}}. \end{equation}

In comparison to figure 5(a), the differences are largest for the physically large riblets. This is due to the fact that in the case of the physically smallest set, dr_1a ($h/\delta =0.0072$), $\delta$ is only 0.2 % smaller compared to $\tilde {\delta }_{{avg}}$, while for dr_1c ($h/\delta =0.0703$), the two heights differ by 2.2 %. In figure 5(b), the expected ${\rm \Delta} C_f/C_{f0}$ behaviour is recovered. All riblet data sets yield negative ${\rm \Delta} C_f/C_{f0}$ for small $l_g^+$, and a better data collapse around the zero crossing at $l_g^+\approx 17$ is obtained. This indicates that the $h_{\parallel }$-based definition of $\delta$ is better suited to compare data sets of strongly varying physical size at the additional expense of evaluating the Stokes flow solution.

In the drag-reducing regime, sets dr_1a, dr_1b and dr_2 exhibit the expected ${\rm \Delta} C_f/C_{f0}$ trends, while set dr_1c yields 38 % less maximum ${\rm \Delta} C_f/C_{f0}$ compared to the geometrically similar sets dr_1a and dr_1b. This is a result of the challenging manufacturing process: instead of the desired sharp tips, set dr_1c has rounded tips with 70 mm curvature radius, i.e. $R/s=0.07$. In this respect, data set dr_1c agrees well with the findings of Walsh (Reference Walsh1990), who reported 40 % decrease of ${\rm \Delta} C_f/C_{f0}$ for tip radius to spanwise spacing ratio $R/s=0.08$. Note that only the maximum ${\rm \Delta} C_f/C_{f0}$ is affected by the tip rounding, while the optimum ${\rm \Delta} C_f/C_{f0}$ still occurs at $l_g^+\approx 11$, as discussed previously by García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011a). It is noteworthy that sets dr_1a and dr_1b are expected to collapse due to their geometric similarity. The visible deviations and slightly smaller ${\rm \Delta} C_f/C_{f0}$ of dr_1b can be an effect of local imperfections observable visually on the 3M riblet foils.The influence of further channel height definitions on the present data evaluation is shown in the Appendix.

5.3. Drag regimes

As introduced with the schematic in figure 1, the data evaluation in terms of roughness function ${\rm \Delta} U^+$ allows us to distinguish different drag regimes. Figure 6 shows ${\rm \Delta} U^+$ against $l_g^+$ for all investigated cases. Here, ${\rm \Delta} U^+$ is retrieved from $C_f$ and $C_{f0}$ measured at the same $Re_b$ via the relationship

(5.2)\begin{equation} {\rm \Delta} U^+=\frac{1}{\kappa} \ln{\sqrt{\frac{C_f}{C_{f0}}}}+ \sqrt{\frac{2}{C_{f0}}}-\sqrt{\frac{2}{C_{f}}}, \end{equation}

which is obtained from the Prandtl–von Kármán friction relation as described in Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020), i.e. by approximating the change ${\rm \Delta} U_c^+$ of centreline velocity, unknown for the present experiment, with the change ${\rm \Delta} U_b^+$ of bulk velocity. This methodology is also employed in other studies dealing with pressure-drop measurements of flows over irregular roughness (Barros, Schultz & Flack Reference Barros, Schultz and Flack2018; Flack, Schultz & Barros Reference Flack, Schultz and Barros2020). The viscous-scaled mean velocity profile retrieved from DNS data (see von Deyn et al. Reference von Deyn, Schmidt, Örlü, Stroh, Kriegseis, Böhm and Frohnapfel2021), which is available for the case di_13, hints at the applicability of the described framework also for large ridges.

Figure 6. Roughness function ${\rm \Delta} U^+$ versus $l_g^+$. The black solid line represents the fully rough behaviour (see (5.4)) with $\kappa =0.39$ and $B=-7.3$. The red solid line represents the viscous friction prediction with ${\rm \Delta} U^+=({(h_\parallel -h_\perp )}/{l_g}) l_g^+= -0.14 l_g^+$ computed for set dr_1a. Additionally, the Colebrook roughness function is included for reference as a black dashed line (Colebrook et al. Reference Colebrook, Blench, Chatley, Essex, Finnieome, Lacey, Williamson and Macdonald1939). The vertical error bars represent the measurement uncertainty for exemplary data points. The horizontal error bars are negligible in this representation.

5.3.1. Drag-reducing regime

The viscous prediction introduced by Luchini et al. (Reference Luchini, Manzo and Pozzi1991) for the physically smallest riblet set dr_1a is also included in figure 6 and can be expressed as

(5.3)\begin{equation} {\rm \Delta} U^+=\frac{h_\parallel{-}h_\perp}{l_g}\,l_g^+= -0.14 l_g^+ \end{equation}

for the viscous regime (García-Mayoral et al. Reference García-Mayoral, Gómez-De-Segura and Fairhall2019). The measurement results show excellent agreement with this viscous prediction for $1< l_g^+<7$, thus confirming its applicability as also demonstrated previously by Grüneberger & Hage (Reference Grüneberger and Hage2011). As noted before, the smaller drag reduction achieved with the geometrically similar data set dr_1b is probably related to local imperfections of this riblet foil. For increasing $l_g^+$, ${\rm \Delta} U^+$ deviates from the predicted behaviour and the well-known breakdown of the viscous regime occurs, leading to an increase of ${\rm \Delta} U^+$ past the point of maximum drag reduction. In agreement with literature data, the drag-reducing regime extends up to $l_g^+\approx 17$, where ${\rm \Delta} U^+=0$ is found.

5.3.2. Fully rough drag regime

For even larger $l_g^+$, the fully rough drag regime follows. In this regime, the riblet surfaces, which cannot induce any pressure drag due to their streamwise invariant surface, reveal an apparent fully rough behaviour (typically attributed to the dominance of pressure drag on roughness elements). The data sets dr_1b, dr_1c and dr_2 show remarkably good agreement. In addition, data set di_2 – which is geometrically similar to dr_2 – also reaches into this region and partially overlaps with dr_2. We note that this collapse between riblets and ridges is obtained only for a data evaluation based on the half-channel height $\delta$. The alternative channel height definitions discussed in § 5.2 and in the Appendix do not yield this collapse. For reference, the Colebrook roughness function (Colebrook et al. Reference Colebrook, Blench, Chatley, Essex, Finnieome, Lacey, Williamson and Macdonald1939) is also included in figure 6, which does not capture the observed trends as it is also found for irregular roughness (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021).

The good collapse of all experimental data indicates that the combination of $l_g^+$ and a half-channel height definition based on $h_\parallel$ are particularly well-suited for the description of the fully rough drag regime that extends up to $l_g^+\approx 40$. Note that choosing $s^+$ or $h^+$ leads to worse agreement between data sets compared to $l_g^+$. The corresponding relation between ${\rm \Delta} U^+$ and $l_g^+$ corresponds to the one of a classical fully rough surface (Perry et al. Reference Perry, Schofield and Joubert1969) with $l_g^+$ as the roughness length scale

(5.4)\begin{equation} {\rm \Delta} U^+(l_g^+)=\frac{1}{\kappa} \ln{l_g^+}+B. \end{equation}

Employing a von Kármán constant $\kappa =0.39$ yields an additive constant $B=-7.3$. A comparison with the corresponding relation for the Nikuradse sand grain roughness ${\rm \Delta} U^+(k_s^+)=({1}/{\kappa }) \ln {k_s^+}-3.5$ (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021) results in an equivalent sand grain roughness

(5.5)\begin{equation} k_s^+= l_g^+{/}4.4 \end{equation}

for the investigated surface structures. The equivalent sand grain roughness of the identified fully rough regime of riblets and ridges appears thus to be predictable based on the geometrical properties of the surface, in particular $l_g$ and $h_\parallel$. This is a remarkable result since the identification of such links between surface topography and the hydraulic roughness scale $k_s$ (and thus the related drag) is one of the key issues in roughness research due to large uncertainties in the existing predictive correlations (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021). In analogy to (5.3), (5.5) or (5.4) can thus be employed directly to predict the drag of 2-D streamwise-aligned trapezoidal-grooved surface structures in a certain region of $l_g^+$, here $17\lessapprox l_g^+\lessapprox 40$.

5.3.3. Non-monotonic regime

Beyond $l_g^+\approx 40$, ${\rm \Delta} U^+$ exhibits a weaker increase with $l_g^+$ than in the preceding fully rough drag regime. The present data sets confirm the deviation from the logarithmic increase of ${\rm \Delta} U^+$ around $l_g^+\approx 40$ reported previously by Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020) (based on data set dr_2) for additional riblet (data set dr_1c) and ridge (data set di_2) structures.

Then ${\rm \Delta} U^+$ increases mildly for $200< l_g^+<1000$. Before this mild increase is reached, data sets di_2 and di_4 reveal a particular behaviour, most striking for di_2, that is reported for the first time. Following the deviation from the fully rough regime, ${\rm \Delta} U^+$ reaches a local maximum at $l_g^+\approx 60$ before it decreases towards a local minimum at $l_g^+\approx 100\unicode{x2013}200$. Set di_4 also exhibits this non-monotonic behaviour in ${\rm \Delta} U^+$, with a local minimum around $l_g^+\approx 85$.

Based on these observations, an additional regime can be introduced where ${\rm \Delta} U^+$ decreases unexpectedly and which we term the non-monotonic regime. This regime can be identified in only two of the present data sets, and it remains to be investigated in future studies whether this non-monotonic behaviour of ${\rm \Delta} U^+$ featuring local maxima and minima is a universal feature of 2-D longitudinally aligned surface structures.

5.3.4. Hydraulic channel height regime

The observed mild increase of ${\rm \Delta} U^+$ for $200< l_g^+<1000$ for the ridge cases corresponds to a Reynolds number range in which the data sets di_2, …, di_13 exhibit $C_f$ curves that seem retrievable by shifting the $C_{f0}$ versus $Re_b$ curve upwards (see figure 4). In other words, the $C_f$ curves of the structured surfaces could be derived by multiplying the smooth reference by a constant factor. Such a procedure is comparable to the common definition of the hydraulic diameter $D_{{hyd}}=4 A/P$, where $A$ corresponds to the duct cross-section and $P$ to the wetted perimeter (Schiller Reference Schiller1923). Pirozzoli (Reference Pirozzoli2018) showed that this classic definition is not applicable for ducts with high aspect ratios, and presented an alternative formulation for the hydraulic diameter. However, this formulation assumes a constant distribution of wall-shear stress across the span, which is a fair assumption for ducts whose cross-section varies mildly and regularly, but clearly does not hold for the present trapezoidal-grooved surfaces.

In order to assess whether the present data can be described by a concept similar to a hydraulic diameter approach, we evaluate a hydraulic half-channel height $\delta _{hyd}$ a posteriori for each surface structure and $Re_b$. If $\delta _{hyd}$ assumes a constant value independently of $Re_b$, then the friction behaviour in this regime can be described solely by the ratio $\delta _{hyd}/\delta$ and the smooth wall $C_{f0}(Re_b)$ correlation, in analogy with the well-established hydraulic diameter concept for low-aspect-ratio ducts.

For each $(C_f,Re_b)$ pair measured for a considered 2-D surface, we compute $\delta _{hyd}$ by answering the question: ‘what would be the channel height of a smooth channel driven at the same flow rate that yielded the same skin-friction coefficient?’ Therefore, $\delta _{hyd}$ is expressed based on the ratio $C_{f0}/C_f$ via (3.2), such that

(5.6)\begin{equation} \delta_{hyd} = \delta \left(\frac{C_{f0}}{C_f}\right)^{{1}/{3}}. \end{equation}

In analogy to the definition of the smooth wall reference channel height $\delta$ (based on an identical $C_f(Re_b)$ correlation for smooth and structured walls under laminar flow conditions), $\delta _{hyd}$ represents the channel height of a smooth wall reference channel with the same friction coefficient under turbulent flow conditions. The ratio $\eta =\delta _{hyd}/\delta$ can thus be interpreted as a measure for the effective change in hydraulic channel height due to turbulence.

Figure 7 shows $\eta$ as a function of $l_g^+$. Riblets that yield ${\rm \Delta} C_f/C_{f0}<0$ (see figure 5) result in values $\eta >1$, while the opposite occurs for $l_g^+>17$ (where ${\rm \Delta} C_f/C_{f0}>0$), where turbulence is modified such that $\delta _{hyd}$ is smaller than $\delta$ ($\eta <1$).

Figure 7. Hydraulic half-channel height ratio $\eta = \delta _{hyd}/\delta$ obtained from (5.6) versus $l_g^+$. Grey lines indicate values for constant $\eta$, referred to as $\eta _c$. The corresponding values are included in table 2. The vertical error bars represent the measurement uncertainty for exemplary data points. The horizontal error bars are negligible in this representation.

For high values of $l_g^+$, the ratio $\eta =\delta _{hyd}/\delta$ tends to a surface-specific constant value, which we denote as $\eta _c$. The corresponding values are included in table 2. The observed trend indicates that the friction behaviour of drag-increasing ridges can be described by a hydraulic reference channel height in the high-Reynolds-number regime (or high $l_g^+$ regime), which we term the hydraulic channel height regime. Widely spaced ridges such as di_13 saturate to $\eta _c$ at $l_g^+ \approx 300$, whereas the narrow-spaced ridges undergo a transient region and approach $\eta _c$ at higher $l_g^+$: di_2, …, di_8 at $l_g^+ \approx 500$. Given the discrepancy for the onset of the hydraulic channel height regime in terms of $l_g^+$, $l_g^+$ does not seem to be the suitable scaling to determine its onset. The existence of such a scaling remains to be investigated in future studies. We note that the observed hydraulic channel height regime is not in agreement with a constant friction increase ${\rm \Delta} C_f/C_0$ beyond the fully rough drag regime postulated by Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020).

In order to obtain a complete predictive framework for the drag behaviour in the hydraulic channel height regime, an a priori definition of $\eta _c$ would be required. The classical hydraulic diameter approach is built on the idea that geometrical parameters such as the perimeter increase $P/s$ determine the drag behaviour. Such parameters are considered in the following, and figure 8 shows $\eta _c$ as a function of different geometrical surface parameters, namely the relative increase of wetted surface area $P/s$, the square root of the groove area normalized by the tip distance $l_g/s$, and the reciprocal of the structure spacing normalized by the half-channel height $\delta /s$. The geometrical similarity of the surfaces for which we were able to achieve the hydraulic channel height regime does not allow general predictive statements, albeit that clear trends can be recognized.

Figure 8. Constant hydraulic channel height ratio $\eta _c$ obtained in the hydraulic channel height regime as a function of different geometrical surface properties. Larger markers indicate increasing $s/\delta$, with marker colours representing the individual data sets as introduced in figure 10. (a) Perimeter increase $P/s$; (b) $l_g/s$; (c) reciprocal spanwise wavelength $\delta /s$.

Figure 8(a) reveals a nearly linearly decreasing $\eta _c$, and thus higher drag, with increasing $P/s$. For the limiting case of wide-spaced small ridges, for which the increase of wetted perimeter becomes negligibly small, $P/s$ tends to 1. One would thus expect to retrieve $\eta _c=1$ at $P/s=1$, which is fulfilled approximately with the observed trend in figure 8(a).

Since $l_g$ is a length scale commonly applied to capture and unify different riblet shapes, $l_g/s$ can be understood as a measure of groove shape versus ridge spacing. The available four data points from the present study appear to follow a nonlinear decrease $\eta (l_g/s)$ in figure 8(b). The limit $\eta _c=1$ is reached in this representation for increasing $s$ at fixed $h$, decreasing $h$ at fixed $s$, or a combination of both. Again, this limiting data case is retrieved approximately by the observed data trend.

The constant hydraulic channel height ratio $\eta _c$ as a function of the reciprocal spanwise wavelength $s/\delta$ is plotted in figure 8(c) such that the limiting case of a smooth wall (${\eta _c=1}$) is located at $\delta /s=0$. In this representation, a linear trend is also observed, which agrees reasonably well with the limiting value for a smooth wall.

Overall, all investigated surface structure features seem equally suitable to obtain a predictive correlation for $\eta _c$. Additional ridge configurations of varying cross-sectional shape need to be investigated in future studies in order to tackle a possible identification of a unified empirical or physics-informed predictive correlation for the hydraulic channel height ratio induced by ridges. This remaining challenge of an a priori definition for the hydraulic channel height ratio $\eta$ can be compared to ‘the bottleneck in our ability to make full-scale predictions of drag’ (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021) in roughness research by means of the equivalent sand grain roughness (which is also a hydraulic quantity) based on geometrical surface properties.

5.3.5. Hydraulic channel height difference

To further assess the change of the hydraulic channel height induced by different surface structures in the drag decreasing and drag-increasing regimes, we introduce the hydraulic channel height difference defined as

(5.7)\begin{equation} {\rm \Delta} h_{hyd}=\delta_{hyd}-\delta, \end{equation}

which is visualized in figure 9. Here, ${\rm \Delta} h_{hyd}$ assumes positive values if $\eta >1$ and is otherwise negative, thus offering an alternative way to visualize changes in $C_f$.

Figure 9. Sketch to illustrate the definition of the hydraulic channel height difference ${\rm \Delta} h_{hyd} = \delta _{hyd}-\delta$. Note that ${\rm \Delta} h_{hyd}$ can assume positive and negative values depending on the drag regime.

Figure 10 shows ${\rm \Delta} h_{hyd}$ normalized by the structure height $h$ as a function of $Re_b$ for all investigated surface structures. In this normalization, all physically large ridge structures show small negative values, indicating that the reduction of the effective channel height for turbulent flow conditions is larger than $h_{\parallel }$, i.e. larger than the one that would occur in a laminar flow, but still much smaller than the maximum height of the structures. In contrast, the physically smaller riblet surface structures result in values ${\rm \Delta} h_{hyd}/h$ that in modulo can be significantly larger than 1. This indicates an interesting property of riblets: the origin for $\delta _{hyd}$ can be located below the riblet valleys in the drag-reducing regime, and above the riblet tips in the drag-increasing regime.

Figure 10. Hydraulic channel height difference ${\rm \Delta} h_{hyd}$ defined in analogy to the protrusion height (see sketch in figure 9 for definitions) normalized with the structure height $h$ as a function of $Re_b$. Same markers as in figure 7. The horizontal and vertical error bars represent the measurement uncertainty for exemplary data points.

In figure 10, constant values of ${\rm \Delta} h_{hyd}/h$ indicate the presence of the hydraulic channel height regime discussed above. In this plot versus $Re_b$, it appears as if this regime is also reached for the physically largest riblet case, dr_1c. However, the resulting visualization in figure 6 suggests that this might still be a transient behaviour. Therefore, this surface structure is not considered to have reached the hydraulic channel height regime within the $Re_b$ range of our experimental facility. In fact, we expect a decrease of ${\rm \Delta} h_{hyd}/h$ for dr_1c beyond the fully rough regime, resulting in values more similar to the ones observed for ridges.

6. Concluding remarks

The high-precision pressure-drop measurements carried out on eight 2-D trapezoidal- grooved surface structures provide a comprehensive description of the occurring drag regimes spanning the well-understood drag-reducing regime, where 2-D structures are commonly termed riblets, up to the drag-increasing regime of ridge-type roughness, where such structures are usually termed ridges. Varying $Re_b$, the spanwise wavelength $s$ and the structure height $h$ over one order of magnitude allow to study the skin friction $C_f$ and related drag change, expressed as the roughness function ${\rm \Delta} U^+$, with respect to the smooth reference in the range $1< l_g^+<1340$. In doing so, we investigate four 2-D surfaces with $l_g/\delta <0.17$ that we term riblets, and four surfaces exceeding $l_g/\delta \approx 0.17$ that we refer to as ridges.

Investigating the drag change of three geometrically similar riblets of varying physical sizes highlights the importance of the choice of the effective channel height. In the present study, the wall-normal origin is placed at $h_\parallel$ below the crest of the 2-D structures. The streamwise protrusion height $h_\parallel$ is obtained by solving an inexpensive Poisson equation. The resulting effective channel height $\delta$, utilized to translate the measured pressure gradient into an equivalent wall-shear stress of a virtual flat wall, is smaller than the average channel height. The present definition ensures that the relation $C_f=12/Re_b$, valid for laminar flow over flat surfaces, also holds for laminar flow above the structured ones. In this way, all surface-induced changes of $C_f$ measured in turbulent channel flows stem from turbulent effects only. With this choice of $\delta$, the drag-change curves collapse remarkably well both in the drag-reducing regime (all structures achieve maximum drag reduction at $l_g^+ \approx 11$ and cross ${\rm \Delta} C_f = 0$ at $l_g^+\approx 17$) and in the drag-increasing regime up to $l_g^+ \approx 40$.

The challenge of choosing a physically sound value of $\delta$ is shared by many recent studies (Chan-Braun, Garcí-A-Villalba & Uhlmann Reference Chan-Braun, Garcí-A-Villalba and Uhlmann2011; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021; Ibrahim et al. Reference Ibrahim, Gómez-De-Segura, Chung and García-Mayoral2021) involving surface structuring, not only of the 2-D kind addressed here. Even numerical studies, where $\tau _w$ can be determined directly without resorting to (2.1), face this ambiguity as the different problem of the virtual wall placement, typically required to represent spanwise-averaged turbulence statistics. For instance, Ibrahim et al. (Reference Ibrahim, Gómez-De-Segura, Chung and García-Mayoral2021) propose a virtual wall placement relying upon turbulent statistics obtained a posteriori. Evaluation of the effective wall-shear stress for DNS of rough wall flows (Forooghi et al. Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017) often also relies on a posteriori methods. Naturally, any a priori method has clear advantages for experimental studies in which the full flow field information is not available. More importantly, one has to be aware of the fact that different definitions of the effective channel height could lead to different interpretations of $C_f$ variations, especially when the scale separation between structure size and boundary layer thickness is not very large. Since it is possible to re-evaluate ${\rm \Delta} C_f$ for different channel heights following (5.1a,b), all data published in this field should contain a clear statement about the equivalent channel height (or virtual origin) definition. Sensitivities of $C_f$ to other equivalent channel height definitions can then be evaluated easily.

Enabled by the present choice of $\delta$, four main outcomes can be drawn from the analysis of the structure-induced drag changes of the present study. First, the viscous prediction of riblet drag reduction based on the difference of the streamwise ($h_\parallel$) and spanwise ($h_\perp$) protrusion heights (Luchini et al. Reference Luchini, Manzo and Pozzi1991) agrees very well with the measured drag-change curve of the physically smallest riblet set dr_1c for $1< l_g^+<7$. Also, for larger structures such as dr_1c, the expected drag-reducing effect known from the literature (see, for instance, Walsh Reference Walsh1990; Bechert et al. Reference Bechert, Bruse, Hage, Hoeven and Hoppe1997) is obtained.

Second, past the drag-reducing regime ($l_g^+ \geq 17$) and up to $l_g^+\approx 40$, the drag-change curves in terms of the roughness function ${\rm \Delta} U^+$ plotted against $l_g^+$ are found to collapse onto one single curve that agrees well with the fully rough behaviour or rough wall surfaces characterized by a logarithmic increase of the roughness function ${\rm \Delta} U^+$. Interestingly, data for geometrically similar riblets and ridges are found to collapse in this regime, confirming that the distinction between riblets and ridges is just mere nomenclature. For the equivalent sand grain roughness $k_s$ of the identified (apparent) fully rough regime, the empirical relation $k_s=l_g/4.4$ is found, making the drag in this regime predictable. Since riblets and ridges cannot experience any pressure drag (the dominance of which is typically associated with fully rough drag behaviour), it remains to be understood which flow phenomena induce this particular phenomenon.

Third, an unexpectedly rich drag behaviour is observed for the first time beyond the fully rough regime ($l_g^+ \geq 40$). The departure from the fully rough behaviour, whose onset was already observed by Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020) relying upon the same data set dr_2, is confirmed distinctly here via the additional riblet geometry dr_1c. The new data show that such deviation from the fully rough regime is very complex, with ${\rm \Delta} U^+$ exhibiting local maxima and minima for two sets of ridges, di_2 and di_4. Thus we term this regime, found for $50< l_g^+<200$ for the investigated geometries, the non-monotonic regime.

Finally, a drag regime beyond $l_g^+=200$ is identified in which a hydraulic channel height ratio (similar to a hydraulic diameter concept) is sufficient to describe the drag behaviour of ridges. Riblets might also eventually reach such a regime, but this is not the case for the presently investigated riblet shapes within the Reynolds number range of the experimental facility. The hydraulic channel height ratio $\eta$ is a quantity obtained a posteriori that compares the effective hydraulic channel height under turbulent flow conditions to its laminar counterpart, which is exactly the equivalent channel height $\delta$ employed for the present study. In the hydraulic channel height regime, $\eta$ assumes a constant but surface-specific value $\eta _c$. Relating this hydraulic surface property to geometrical surface properties remains an open task for which data for different ridge geometries are required.