2 Experimental set-up and techniques

The experiments have been performed in a small, high-speed water tunnel described in Gopalan & Katz (Reference Gopalan and Katz2000) and Liu & Katz (Reference Liu and Katz2006). The flow is driven by two 15 HP (maximum) centrifugal pumps located 5 m below the test section, and passes through a settling tank, an electromagnetic flow meter, a settling chamber containing honeycombs and screen as well as a 9 : 1 contraction before entering the test section. Components relevant to the present study, as they are installed in the 406 mm long, 61 mm high and 50  mm wide transparent test section, are sketched in figure 1(a). The mean tunnel speeds (flow rate divided by the tunnel cross-section), $U_{m}$ , are between 2 and 20[[()[]!--\jatstusp--[]()]] m[[()[]!--\jatstusp--[]()]] s^-1. At the entrance to the test section, the bottom window contains a series of machined spanwise tripping grooves, which are located 165 mm (9–22 boundary layer heights) upstream of the SHSs. The purpose of these grooves is to force early boundary layer transition to turbulence, as shown in prior studies (Liu & Katz Reference Liu and Katz2013). The 152 mm long and 50 mm wide SHS is flush mounted on the same wall. The coordinate system is also shown, with $x$ , $y$ and $z$ denoting the streamwise, wall-normal and spanwise directions, respectively, and with $x=0$ coinciding with the leading edge of the SHS, all consistent with the coordinates used in the Introduction.

The SHSs have been created on both porous stainless steel bases and non-porous aluminium bases, denoted as $\text{SHS}_{Por}$ and $\text{SHS}_{Al}$ , respectively. The $\text{SHS}_{Por}$ is manufactured by spray coating a mixture of a poly methyl methacrylate (PMMA) binder and fluorinated polyhedral oligomeric silsesquioxane (F-POSS) using procedures described in Srinivasan et al. (Reference Srinivasan, Chhatre, Mabry, Cohen and McKinley2011). The reason for using a porous substrate as a base is to provide a means for continuously replenishing the micro air pockets that are gouged away and entrained by the flow. A sample scanning electron micrograph (SEM) of the porous bases prior to spraying is shown in figure 1(b). They have a permeability of $0.27{-}0.98~{\rm\mu}\text{m}^{2}$ ( $1/700$ – $1/200$ acfm $\cdot$ cp $\cdot$ inch/psid/ft $^{2}$ ) and porosity of 17–26 %, as specified by the manufacturers. The size of particles forming the bases as seen in the SEM images (since the manufacturer does not provide them) varies from 20 to $40~{\rm\mu}\text{m}$ . The porous bases are tightly glued to six $6.4\times 6.4~\text{mm}^{2}$ support legs and 5.2 mm wide edges, leaving ${\sim}70\,\%$ of their underside exposed to an air chamber. This chamber is connected through valves to a compressor and a vacuum pump, allowing us to set it at desired pressures $P_{1}$ . The pressure in the test section, $P_{2}$ , is also controlled by connecting the same compressor and vacuum pump to an air–water interface located in a chamber well above the test section (Gopalan & Katz Reference Gopalan and Katz2000). The pressure difference across the porous wall, ${\rm\Delta}P=P_{2}-P_{1}$ can be varied and is monitored by a pressure transducer.

The $\text{SHS}_{Al}$ is manufactured by polishing an aluminium base manually, and then etching it in hydrochloric acid, boiling it in water and coating it with (heptadecafluoro-1,1,2,2-tetrahydrodecyl) trichlorosilane using vapour deposition. Further details are provided in Yang et al. (Reference Yang, Zhang, Xu, Men, Zhu and Zhou2011). Two different subtypes of $\text{SHS}_{Al}$ are involved in this study. For the first, surface polishing is performed in random directions and the resulting surface is denoted $\text{SHS}_{Alr}$ . For the second, denoted $\text{SHS}_{Alx}$ , the chemical treatment and silanization procedure are kept the same, but the polishing is intentionally aligned in the $x$ direction forming preferentially streamwise grooves that have spatially non-uniform spacing and depth. The average spacing, as determined by a laser interferometry, is $110~{\rm\mu}\text{m}$ . A sample SEM image of $\text{SHS}_{Alx}$ is shown in figure 1(c). For these samples, ${\rm\Delta}P$ is used to denote the pressure difference between $P_{2}$ and atmosphere pressure when we examine the effects of compressing the air layer on the skin friction. To characterize the SHSs before the water tunnel tests, the static contact angles and sliding angles of water droplets on the surfaces are measured. The contact angles are calculated by recording images of static ${\sim}4~\text{mm}$ ( ${\sim}270~{\rm\mu}\text{l}$ ) water droplets situated on the SHSs, e.g. figure 1(d). The sliding angles are determined by slowly tilting the SHSs until the droplets start moving and rolling off of the surfaces.

Figure 2.

Illustrations of (a) the dual view in-line digital holographic microscopy set-up for high-resolution velocity measurements; and (b) the holographic set-up for imaging the air layer or ‘plastron’ present on the SHSs.

The optical set-up for performing high-resolution velocity measurements is illustrated in figure 2(a). To fully resolve the flow in the inner part of the boundary layer, we have opted to use in-line digital holographic microscopy (DHM). Since we cannot record holograms through the SHSs, we have developed a new approach to record DHM data in ‘deep’ samples, as described in detail by Ling & Katz (Reference Ling and Katz2014) In-line holographic reconstruction creates twin three-dimensional (3-D) images located symmetrically on both sides of the hologram plane. Thus, placing the hologram plane in the middle of the sample volume (i.e. to sustain submicron resolution), particles located on one side of the hologram will be reconstructed on both sides, ‘mixing’ them. To overcome this problem, we record a pair of holograms whose planes are separated by a short distance from each other. The real particle fields of both images coincide, whereas the virtual/twin ones are separated by twice the distance between planes. The software and procedures are discussed in detail and demonstrated in Ling & Katz (Reference Ling and Katz2014). The light source is an Nd–YAG laser (532 nm). Since very little energy is required for in-line DHM, only light reflected from an uncoated flat glass surface is used. The beam is spatially filtered, expanded and collimated to 5 mm diameter before illuminating the sample volume. The flow is seeded locally with 2  ${\rm\mu}\text{m}$ silver-coated glass particles. To minimize the effect of injection on the flow, the particles are injected at a low speed of less than $0.08U_{m}$ from 25 evenly distributed 100  ${\rm\mu}\text{m}$ holes located ${\geqslant}$ 100 mm (1000 injector diameters) upstream of the sample volume. The light scattered by these particles interferes with the remainder of the collimated beam to form the in-line hologram. An $8\times$  infinity-corrected, long working distance microscope objective magnifies the images, while focusing on the selected hologram planes. A cube beam splitter directs the images to two interline transfer digital cameras (Imperx ICL-B6640, $4400\times 6600$ pixels, $5.5~{\rm\mu}\text{m}~\text{pixel}^{-1}$ ) generating a pair of holograms. Their focal planes are located approximately 2 mm away from the centre of the test section, separated by 50  ${\rm\mu}\text{m}$ and at $x=70~\text{mm}$ , as shown in figure 2(a). They have calibrated spatial resolutions of 0.677 and $0.685~{\rm\mu}\text{m}~\text{pixel}^{-1}$ . The total sample area cross-section is $4.4\times 2.4~\text{mm}^{2}$ ( $x\times y$ ) and the total depth of the interrogated volume extends to 3.2  mm in the $z$ direction for most cases. However, as discussed below, this domain is divided into multiple sample volumes, which are interrogated separately.

High-speed digital holography has been used to monitor the air layer interface and to measure the entrainment rate of air from the $\text{SHS}_{Por}$ , as sketched in figure 2(b). Here, the light source is a Q-switched Nd:YLF laser (523 nm) whose output is expanded and collimated to a 10 mm diameter beam. The holograms are recorded by a high-speed camera (PCO.Dimax HD) at $20\,000~\text{frames}~\text{s}^{-1}$ and at a resolution of $11~{\rm\mu}\text{m}~\text{pixel}^{-1}$ . The sample volume is centred at $x=70$  mm, and has a streamwise length of 9.5 mm, a wall-normal height of 2.5 mm and a depth of 50 mm, the latter covering the entire depth of the water tunnel. Three sample high-speed movies of original holograms of the air layer on the $\text{SHS}_{Por}$ for $U_{m}$ varying between 2 and $6~\text{m}~\text{s}^{-1}$ , and ${\rm\Delta}P$ varying between $-4.0$ and 12 kPa, are in the supplementary data available at http://dx.doi.org/10.1017/jfm.2016.450. They provide direct confirmation that an air layer or plastron (Shirtcliffe, McHale & Newton Reference Shirtcliffe, McHale and Newton2011) is indeed attached to the $\text{SHS}_{Por}$ , and this interface fluctuates at increasing surface speeds with increasing $U_{m}$ . To measure the size distribution of bubbles and their cumulative volume, the holograms are reconstructed every 100  ${\rm\mu}\text{m}$ . The 3-D intensity field is subsequently collapsed into a single plane, where each pixel has the minimum intensity (darkest) over the entire depth. The collapsed image is thresholded and segmented to identify each bubble and obtain its size. The flux of bubbles is calculated by dividing the cumulative volume of all the bubbles in the sample volume, averaged over eight realizations, by the time that is required for them to pass through the sample volume. This time is estimated by dividing the length of the sample area by the height-dependent mean streamwise velocity of the fluid. This flux is used for estimating the rate of bubbles entrained from the upstream 70 mm of the porous surface (3500 mm²). This estimate is smaller than the actual entrainment rate since some of the bubbles might be lifted to elevations located above the sample volume. However, by tracking the vertical bubble flux at different elevations and determining that it is negligible at the top of the sample volume, and by insuring that the buoyancy alone is insufficient to lift the bubbles away from the field of view, the overwhelming majority ( ${>}$ 99 %) of the bubbles entrained from the wall remain inside the sample volume. It should also be noted that without the air layer, there are no free stream bubbles with resolvable sizes near the bottom wall. The entrainment rate of the bubbles is then divided by the surface area and $U_{m}$ to obtain the dimensionless, spatially and time-averaged air velocity $U_{air}^{\ast }$ .

As expected, $U_{air}^{\ast }$ increases with increasing $U_{m}$ and decreasing ${\rm\Delta}P$ , as shown in figure 3(a). For $U_{m}=2.0~\text{m}~\text{s}^{-1}$ ( $Re_{{\it\tau}}=1408$ , as shown later), $U_{air}^{\ast }$ is essentially zero, i.e. the wall shear stresses fall below the threshold required for entraining the air. It increases to $1.5\times 10^{-10}$ at $U_{m}=6.0~\text{m}~\text{s}^{-1}$ and ${\rm\Delta}P=-4.0$  kPa. Furthermore, using a separate axis, figure 3(a) also shows the ratio of the gas to liquid flow rates in the boundary layer, $Q_{g}/Q_{w}$ . As shown, the maximum value of this ratio is approximately $7\times 10^{-7}$ . According to Ceccio (Reference Ceccio2010) and Ferrante & Elghobashi (Reference Ferrante and Elghobashi2004), to achieve drag reduction by injecting air bubbles, this ratio has to be of the order of $10^{-3}$ or higher. Therefore, the present bubble concentration is at least three orders of magnitude lower than that required for affecting the drag force. Sample ensemble-averaged size distributions of entrained bubbles are shown in figure 3(b). The results for $U_{m}=2.0~\text{m}~\text{s}^{-1}$ is not included since it is zero, but the rest demonstrate that the number of ‘large’ bubbles ( ${>}50~{\rm\mu}\text{m}$ ) increases with entrainment rate. However, the high-speed movies confirm that both the air layer and the steady entrainment of bubbles are maintained on the $\text{SHS}_{Por}$ for more than four hours for the entire current range of $U_{m}$ and ${\rm\Delta}P$ , presumably because of the continuous replenishment of the plastron by air under the porous surface. The persistence of the entrainment for $0<{\rm\Delta}P<20$  kPa indicates that the capillary forces are sufficient for overcoming the air layer suppression by the higher pressure in the test section. To remove the air layer in some experiments, the entire space under the porous base has to be filled with water. We have not tried pressures exceeding ${\rm\Delta}P=20$  kPa, fearing that it might damage the substrate.

Figure 3.

(a) Normalized entrainment rate of air ( $U_{air}^{\ast }$ and $Q_{g}/Q_{w}$ ) from the $\text{SHS}_{Por}$ upstream of the sample area for several pressure differences across the porous wall; (b) the measured size distribution of the entrained bubbles.

The optical set-up of figure 2(a) with the same magnification and sample area is used to evaluate and quantify the topographies of each $\text{SHS}_{Por}$ and $\text{SHS}_{Alr}$ . The local roughness height $h(x)$ is estimated directly from the projection of the roughness peaks on the holograms. The holograms are recorded while the facility is running, filled with water, at the same velocities as those used during the actual measurements. By selecting a threshold of intensity to include roughness elements which are nearly in focus, the projected ridgeline is tracked, as illustrated in the inserts in figure 4(a). For the sample holograms of $\text{SHS}_{Por}$ and $\text{SHS}_{Alr}$ presented in the inserts, $U_{m}=2~\text{m}~\text{s}^{-1}$ . Figure 4(a) also shows the two corresponding histograms of $h(x)$ . This procedure is aimed at characterizing the roughness height under conditions that best represent the surface shape during the velocity and turbulence measurements. Thus, the $h(x)$ accounts for the presence of the air layer. Since the histograms of $h(x)$ appear to be nearly Gaussian, we opt to characterize the roughness height using the root mean square value of $h(x)$ , and denote it as $k_{rms}=(\int h^{2}\,\text{d}x)^{0.5}/L$ , where $L$ is the sample length. For the present measurements, the magnitude of $k_{rms}$ of the $\text{SHS}_{Por}$ ranges from 4.8 to 20.4  ${\rm\mu}\text{m}$ and that of the $\text{SHS}_{Alr}$ is 10.9  ${\rm\mu}\text{m}$ , with an uncertainty of approximately 1  ${\rm\mu}\text{m}$ . Figure 4(b) shows the cumulative distributions of $h(x)$ for five values of $k_{rms}$ . The holograms are not suitable for characterizing the roughness height of $\text{SHS}_{Alx}$ since the roughness elements are not projected outward. Therefore, for $\text{SHS}_{Alx}$ , we utilize a laser interferometer with a resolution of approximately 1  ${\rm\mu}\text{m}$ in both $x$ and $z$ directions to measure the height distribution. The values of $k_{rms}$ of the $\text{SHS}_{Alx}$ in the flow measurement area ( $x=70$  mm) is 8.9  ${\rm\mu}\text{m}$ . To obtain an estimate of the uncertainty in $k_{rms}$ , the laser interferometer has also been applied to obtain independent measurements of roughness for one of the $\text{SHS}_{Por}$ and the $\text{SHS}_{Alr}$ . The results give $k_{rms}$ values that are lower by 10–20 % than those obtained from the holograms. Hence, this difference is used as a rough estimate of our uncertainty in roughness measurements.

Figure 4.

(a) Sample probability density functions of the roughness height $h(x)$ for one of the $\text{SHS}_{Por}$ ( $k_{rms}=7.8~{\rm\mu}\text{m}$ , case no. 8) and the $\text{SHS}_{Alr}$ (case no. 4). The insets show corresponding sample holograms. Dotted lines: $h(x)$ , solid lines: mean roughness height ( $y=0$ ) and dashed lines: $y=2k_{rms}$ . (b) Cumulative distribution of $h(x)$ for several $k_{rms}$ .

The present surface geometries and composition prevent us from providing a reliable scale for the roughness spacing, especially if one wants to compare surfaces. Spraying inherently involves a multiscale surface pattern. Thus, evaluating the surface only based on the observed peaks is questionable, since much smaller crevices might affect the trapping of air as well. We estimate the mean spacing between roughness elements by counting the total number of maxima that exceed two local standard deviations of the local mean height. Assuming they are distributed randomly within the hologram depth of focus ${\sim}200~{\rm\mu}\text{m}$ (without reconstruction), the characteristic roughness wavelength ${\it\alpha}$ for the $\text{SHS}_{Por}$ and $\text{SHS}_{Alr}$ falls in $90~{\rm\mu}\text{m}<{\it\alpha}<100~{\rm\mu}\text{m}$ . This value does not differ significantly among different samples, even though the magnitude of $k_{rms}$ does change. The magnitude and distribution of ${\it\alpha}$ should also be accounted for while characterizing the efficacy of the SHS or roughness effects. However, due to the higher confidence in its magnitude, we opt to characterize the surfaces based on $k_{rms}$ . The randomly distributed roughness also prevents us from quantifying the gas fraction for the different surfaces and flow conditions.

The location of $y=0$ is selected as the mean roughness height, consistent with previous rough walls studies (Brzek et al. Reference Brzek, Cal, Johansson and Castillo2008; Chan et al. Reference Chan, Macdonald, Chung, Hutchins and Ooi2015). It is indicated by solid lines in the insets of figure 4(a). To minimize the potential effects of form drag, the wall friction and slip velocity are calculated at the top of the roughness, which is selected as the elevation where the cumulative distribution of $h(x)$ reaches 95 %, namely $y=2k_{rms}$ . The location of this elevation is indicated by the dashed lines in the insets of figure 4(a) and in figure 4(b). While one could also define the top of the roughness as the point where the cumulative distribution of $h(x)$ reaches e.g. 90 % or 99 %, the impact of this choice on the uncertainties in slip velocity and wall friction is discussed and accounted for later. The present choice for $y=0$ facilitates displaying the velocity and stress profiles at the top of the roughness, namely at $y=2k_{rms}$ . Altering this reference height within the roughness domain has not led to collapse of data, but has caused loss of data near the point where the wall stress is evaluated. It has negligible effect on the mean profile in the log region and does not change the overall trends.

To calculate the velocity field from the high-resolution holograms obtained from the set-up described in figure 2(a), the following data analysis procedures are employed. First, to reconstruct only the real particle images, we use a phase based reconstruction (Denis et al. Reference Denis, Fournier, Fournel and Ducottet2005; Ling & Katz Reference Ling and Katz2014), which involves a spatial convolution of both the intensity and phase (complex amplitude) distribution in (one of) the hologram plane with a Rayleigh–Sommerfeld kernel (Katz & Sheng Reference Katz and Sheng2010). The phase distribution is estimated iteratively by propagating the wave field back and forth between the two hologram planes using diffraction theory (Denis et al. Reference Denis, Fournier, Fournel and Ducottet2005). This phase based reconstruction is performed every 13  ${\rm\mu}\text{m}$ in depth to generate a series of closely spaced planes containing real images only. Then, following Sheng, Malkiel & Katz (Reference Sheng, Malkiel and Katz2008) and Talapatra & Katz (Reference Talapatra and Katz2013), the 3-D fields are segmented to generate the spatial distribution of particles, followed by particle tracking to match particle traces in the hologram pairs. Between 6000 and 10 000 particles pairs are typically matched in each of the $4.4\times 2.4\times 3.2~\text{mm}^{3}\;(x\times y\times z)$ sample volumes. Matching involves seven criteria, including similarity of particle size, shape and intensity, as well as smoothness of the velocity field and agreement with guess 2-D vectors generated using standard PIV cross-correlations of images created by compressing the entire volume into a plane. Symbols $u$ , $v$ and $w$ are used to denote the instantaneous velocity components in the $x$ , $y$ and $z$ directions respectively. The magnitudes of $u$ and $v$ are calculated from the in-plane centroids of the particles and $w$ is calculated based on locations of minimum intensity within the elongated traces of particle in the spanwise direction. The accuracy of $w$ is lower compared to $u$ and $v$ but could be improved by locating the centre of the particles using edge detection (Talapatra & Katz Reference Talapatra and Katz2013) or correlations among the elongated traces (Ling & Katz Reference Ling and Katz2014). However, as the focus of this paper is on the distributions of $u$ and $v$ , additional effort is not invested in improving the accuracy of $w$ . The sample volume is divided into multiple small windows with a size of $10{\it\delta}_{{\it\upsilon}}\times 1{\it\delta}_{{\it\upsilon}}\times 10{\it\delta}_{{\it\upsilon}}\;(x\times y\times z)$ for $U_{m}=2~\text{m}~\text{s}^{-1}$ and $20{\it\delta}_{{\it\upsilon}}\times 1{\it\delta}_{{\it\upsilon}}\times 20{\it\delta}_{{\it\upsilon}}\;(x\times y\times z)$ for $U_{m}=6~\text{m}~\text{s}^{-1}$ . Only the windows containing particles are included in the statistical analysis for each volume. More than 1000 instantaneous velocity fields are processed and ensemble-averaged locally for each window to obtain the mean (dented as $U$ and $V$ ) and the corresponding Reynolds normal and shear stress components, $\langle u^{\prime }u^{\prime }\rangle$ , $\langle v^{\prime }v^{\prime }\rangle$ and $\langle u^{\prime }v^{\prime }\rangle$ . Results are then spatially averaged in the $x$ and $z$ directions to obtain data that are not dependent on the local roughness patterns. Spatially averaged values are denoted with an over bar, e.g. $\overline{U}$ , $\overline{\langle u^{\prime }v^{\prime }\rangle }$ , and etc.

For $\text{SHS}_{Por}$ , the measurement domain starts from $y=0$ , and the first data point included in this paper is located at $y={\it\delta}_{{\it\upsilon}}$ . This point is located below the tip of the roughness elements. For $\text{SHS}_{Alr}$ , the measurement domain starts from $y=k_{rms}$ and the first data point is located at $y=k_{rms}+{\it\delta}_{{\it\upsilon}}\approx 2k_{rms}$ . For $\text{SHS}_{Alx}$ , we cannot examine the space between the roughness elements. Therefore, the location of $y=0$ can only be estimated based on observed particles attached to the surface. Hence, the first data point is located at $y\approx 2k_{rms}+1{\it\delta}_{{\it\upsilon}}$ . The values of the mean spatially averaged viscous stress, ${\it\tau}^{{\it\mu}}$ , are calculated using ${\it\tau}^{{\it\mu}}={\it\mu}\partial \overline{U}/\partial y$ , where ${\it\mu}=1\times 10^{-3}~\text{kg}~\text{m}^{-1}~\text{s}^{-1}$ . Since the mean velocity profiles for the smooth walls, the $\text{SHS}_{Por}$ with $k_{rms}\leqslant {\it\delta}_{{\it\upsilon}}$ and the $\text{SHS}_{Alr}$ are nearly linear at $y\leqslant 5{\it\delta}_{{\it\upsilon}}$ , the values for $y<3{\it\delta}_{{\it\upsilon}}$ are calculated by linearly fitting the mean velocity profiles based on the $y\leqslant 5{\it\delta}_{{\it\upsilon}}$ data. However, for the SHSs with $k_{rms}>{\it\delta}_{{\it\upsilon}}$ , the mean profiles are not linear at $y\leqslant 5{\it\delta}_{{\it\upsilon}}$ . Thus, ${\it\tau}^{{\it\mu}}$ at $y={\it\delta}_{{\it\upsilon}}$ (first point) is not available, second-order finite differencing is used for $y=2{\it\delta}_{{\it\upsilon}}$ , and 5 points are used for higher elevations. The total stress ${\it\tau}_{t}={\it\tau}^{{\it\mu}}+{\it\tau}^{R}$ is determined by adding ${\it\tau}^{{\it\mu}}$ and the spatially averaged Reynolds shear stress, ${\it\tau}^{R}=-{\it\rho}\overline{\langle u^{\prime }v^{\prime }\rangle }$ , where ${\it\rho}=1\times 10^{3}~\text{kg}~\text{m}^{-3}$ . The wall viscous stress ${\it\tau}_{w}^{{\it\mu}}$ , wall Reynolds shear stress ${\it\tau}_{w}^{R}$ and total wall friction ${\it\tau}_{w}$ are determined from the corresponding stresses at $y=$ 0 for smooth walls, $y=2k_{rms}$ for $\text{SHS}_{Por}$ and $\text{SHS}_{Alr}$ and at $y=2k_{rms}+1{\it\delta}_{{\it\upsilon}}$ for $\text{SHS}_{Alx}$ . To estimate the slip velocity, we calculate the mean and spatially averaged velocity at $y=2k_{rms}$ , and denote it by $\overline{U}_{s}$ .

The uncertainty in velocity measurements is based on prior studies where many of the present tools for applying digital holography to perform near-wall velocity measurements were introduced (Sheng et al. Reference Sheng, Malkiel and Katz2008; Talapatra & Katz Reference Talapatra and Katz2013). In both studies, the uncertainty is evaluated by testing how well the measurements satisfy the continuity equation. Using their results, and considering the resolution of the current set-up ( $0.68~{\rm\mu}\text{m}~\text{pixel}^{-1}$ ), the uncertainty in $x{-}y$ motions is approximately 0.5 pixel, corresponding to an uncertainty in the instantaneously interpolated $u$ and $v$ of $0.01U_{m}$ . The corresponding uncertainty in the velocity gradient used for calculating the viscous stress is an order of magnitude higher, but decreases back to approximately 1 % when ensemble averaged. However, a bigger contributor to wall stress uncertainty involves the selection of elevation for evaluating the wall stress for the ‘rough’ SHSs. It is estimated by calculating the differences between the stresses at $y=2k_{rms}$ and those at $y=2.5k_{rms}$ and $1.5k_{rms}$ , namely on both sides of the roughness peak, and selecting the larger of the two differences as an uncertainty.

Table 1.

A summary of experimental conditions and measured parameters for all tests included in this paper. CA denotes the contact angle, SA the sliding angle and DR the drag reduction. Values of CA in parenthesis are contact angles measured after the experiment and exposing the wall to air. The same symbols apply for all the figures.

High-resolution velocity measurements have been performed for 20 cases, and results for 13 of them are included in this paper. As listed in table 1, they include $\text{SHS}_{Por}$ , $\text{SHS}_{Al}$ and smooth walls, with $U_{m}$ varying from 2 to $6~\text{m}~\text{s}^{-1}$ , pressure differences in the range $-0.3\leqslant {\rm\Delta}P\leqslant 20$  kPa, root mean square roughness height varying between $4.8\leqslant k_{rms}\leqslant 20.4~{\rm\mu}\text{m}$ and two streamwise locations. For control measurements above smooth walls, the SHSs are exchanged with smooth polyvinyl chloride bases and the measurement locations are kept identical as those of the SHSs. Table 1 also provides results of contact and sliding angle measurements performed before each experiment, which are denoted as CA and SA, respectively. The contact angle after the experiment and exposing the wall to air is also provided for some of the cases (in parenthesis). Each flow measurement is started after running the facility continuously at a particular condition for at least one hour. Each data acquisition lasts approximately two hours. We have also noticed that after running for more than four hours at $U_{m}=6.0~\text{m}~\text{s}^{-1}$ , the roughness height of the $\text{SHS}_{Por}$ decreases significantly, presumably due to mechanical degradation and entrainment of the sprayed polymer. For example, for case no. 12, the initial $k_{rms}$ value is $20.4~{\rm\mu}\text{m}$ , but after running for two hours at $U_{m}=6~\text{m}~\text{s}^{-1}$ prior to data acquisition, $k_{rms}$ decreases to $14.5~{\rm\mu}\text{m}$ . During the two hour data acquisition period, the values of $k_{rms}$ decrease from 14.5 to $13.7~{\rm\mu}\text{m}$ . Values measured at the end of the experiments are used in subsequent discussions. This phenomenon does not occur at $U_{m}=2.0~\text{m}~\text{s}^{-1}$ for both the $\text{SHS}_{Por}$ and/or the $\text{SHS}_{Al}$ .

Also listed in table 1 are the ${\it\tau}_{w}^{{\it\mu}}$ , ${\it\tau}_{w}^{R}$ , ${\it\tau}_{w}$ , $\overline{U}_{s}$ and their associated uncertainties. The values of $u_{{\it\tau}}=({\it\tau}_{w}/{\it\rho})^{1/2}$ and ${\it\delta}_{{\it\upsilon}}$ based on ${\it\tau}_{w}$ are provided in table 1. These values are used for the inner scaling. In the rest of the paper, a superscript $^{+}$ is used for quantities that are normalized by $u_{{\it\tau}}$ and ${\it\delta}_{{\it\upsilon}}$ , a subscript $_{0}$ for quantities measured above the smooth wall (the baseline) and the combination of the superscript $^{+}$ and subscript $_{0}$ for quantities normalized by $u_{{\it\tau}0}$ and ${\it\delta}_{{\it\upsilon}0}$ . For all cases, in addition to ${\it\tau}_{w}$ , we also estimate the wall friction by a logarithmic fit to the mean velocity profile in the regions where values of $y\partial \overline{U}/\partial y$ are nearly unchanged, which fall in the range of $80{\it\delta}_{{\it\upsilon}}<y<180{\it\delta}_{{\it\upsilon}}$ for cases no. 1 to no. 6 and no. 13, $50{\it\delta}_{{\it\upsilon}}<y<180{\it\delta}_{{\it\upsilon}}$ for cases no. 7 to no. 9 and $50{\it\delta}_{{\it\upsilon}}<y<350{\it\delta}_{{\it\upsilon}}$ for cases no. 10 to no. 12. Results, denoted as ${\it\tau}_{w}^{Log}$ , along with the maximum value of the total stress across the entire boundary layer, denoted as ${\it\tau}_{t}^{max}$ , are also listed in table 1. There are mismatches between the log layer based estimate of wall stress and the directly measured value. Implications of these findings are discussed later in this paper.

Because the high-resolution measurements only cover the inner part of the turbulent boundary layer, 2-D PIV has also been used to obtain the entire boundary profile, including the missing wake region. It has been performed at a lower magnification ( $5.4~{\rm\mu}\text{m}~\text{pixel}^{-1}$ ), using a larger sample area $(36\times 24~\text{mm}^{2},x\times y)$ . The centres of the sample areas ( $x=70~\text{mm}$ ) coincide with those of high-resolution measurements. These 2-D PIV measurements have also been performed after running at a particular experimental condition continuously for one hour. Data acquisitions typically last for 30 min, during which more than 250 pairs of images are captured. Standard PIV cross-correlations using in-house software (Roth & Katz Reference Roth and Katz2001) with a window size of $64{\it\delta}_{{\it\upsilon}}\times 16{\it\delta}_{{\it\upsilon}}$ ( $x\times y$ ) for $U_{m}=2~\text{m}~\text{s}^{-1},128{\it\delta}_{{\it\upsilon}}\times 32{\it\delta}_{{\it\upsilon}}$ ( $x\times y$ ) for $U_{m}=6~\text{m}~\text{s}^{-1}$ and 50 % overlap are used to calculate the velocity, resulting in a characteristic grid spacing of $32{\it\delta}_{{\it\upsilon}}\times 8{\it\delta}_{{\it\upsilon}}$ $(x\times y)$ for $U_{m}=2~\text{m}~\text{s}^{-1}$ and $64{\it\delta}_{{\it\upsilon}}\times 16{\it\delta}_{{\it\upsilon}}$ ( $x\times y$ ) for $U_{m}=6~\text{m}~\text{s}^{-1}$ . The boundary layer thicknesses, ${\it\delta}_{99}$ , as listed in table 1, are based on the elevation where 99 % of the maximum velocity $(U_{0})$ is reached. Note that $U_{0}$ is slightly higher (typically by 8 %–12 %) than $U_{m}$ owing to the boundary layer-induced blockage. As is evident, unlike the total wall stress, the effects of the SHSs on ${\it\delta}_{99}$ are small, possibly since the present fetch $(4{-}9{\it\delta}_{99})$ is limited. The magnitudes of the drag reduction are defined as $DR=({\it\tau}_{w0}-{\it\tau}_{w})/{\it\tau}_{w0}$ , where ${\it\tau}_{w0}$ is the value obtained for the smooth wall at the same $U_{m}$ and very similar ${\it\delta}_{99}$ . To calculate the velocity deficit or increase in the log region, ${\rm\Delta}U^{+}$ , the mean velocity profile is fitted with $\overline{U}^{+}=(1/{\it\kappa})\ln y^{+}+{\rm\Delta}U^{+}+B_{0}$ , where $B_{0}$ is the value obtained for the corresponding smooth wall. For subsequent discussion, values of $Re_{{\it\delta}}=U_{0}{\it\delta}_{99}/{\it\upsilon}$ , $Re_{{\it\tau}}=u_{{\it\tau}}{\it\delta}_{99}/{\it\upsilon}$ , $k_{rms}^{+}$ , ${\rm\Delta}U^{+}$ , $\overline{U}_{s}^{+}$ and DR are also included in table 1. Note that when normalized, $\overline{U}_{s}^{+}={\it\lambda}_{x}^{+}$ for the cases where ${\it\tau}_{w}={\it\tau}_{w}^{{\it\mu}}$ , and $\overline{U}_{s}^{+}<{\it\lambda}_{x}^{+}$ for the cases where ${\it\tau}_{w}>{\it\tau}_{w}^{{\it\mu}}$ or ${\it\tau}_{w}^{R}>0$ .

Two facts need to be considered when comparing the presently measured wall friction to that of much larger surfaces. First, there is a relatively short distance ( $4{-}9{\it\delta}_{99}$ ) from the beginning of the SHSs to the sample area. Prior studies have reported an overshot of the wall friction after transitioning from a smooth to a rough wall and an undershot after transitioning from a rough to a smooth wall (Antonia & Luxton Reference Antonia and Luxton1971). Further downstream, the wall friction adjusts to the new boundary condition after a distance that increases with decreasing ${\it\delta}_{99}/k$ . For example, a relatively short distance of $2{-}3{\it\delta}_{99}$ is required for ${\it\delta}_{99}/k>1000$ (Saito & Pullin Reference Saito and Pullin2014), and a much longer distance of approximately $30{\it\delta}_{99}$ is needed for ${\it\delta}_{99}/k<25$ (Lee & Sung Reference Lee and Sung2007). If the roughness effects are dominant (as will be shown to occur in some cases), one would expect a rather quick transition considering ${\it\delta}_{99}/k_{rms}>100$ for current samples. There is limited information about the transition from a smooth wall to an SHS. However, a recent numerical study by Park (Reference Park2015) reports undershoots of the wall friction and adjustment distances to constant wall friction of $5{\it\delta}_{99}$ to $20{\it\delta}_{99}$ for SHSs with streamwise grooves of different wavelengths at $Re_{{\it\tau}}\sim 200$ . Most of the present measurements are performed at substantially higher Reynolds numbers and with random roughness. Moreover, due to this short distance, the outer region does not reach equilibrium condition, namely, the velocity profile in the log layer does not fully adjust to the local stress in the inner part of the boundary layer. For transition from a smooth to a rough wall, the required distance for mean velocity to be self-preserved varies from $10{\it\delta}_{99}$ to $40{\it\delta}_{99}$ (Saito & Pullin Reference Saito and Pullin2014). Overshoots of Reynolds stresses in this transition region have also been reported (Lee & Sung Reference Lee and Sung2007; Saito & Pullin Reference Saito and Pullin2014). Hence, in discussing the results, our main focus is on the inner region while highlighting discrepancies between the inner and outer regions. Furthermore, data are presented to demonstrate the streamwise evolution of the wall friction and Reynolds stresses in the outer part of the boundary layer. Second, without the plastron, the current SHSs fall in the transitionally rough to rough regimes (Schultz & Flack Reference Schultz and Flack2007) and form drag might affect the total stress at the spatially averaged top of the roughness elements. However, based on computational results found in Chan et al. (Reference Chan, Macdonald, Chung, Hutchins and Ooi2015), for a transitionally rough surface with $k^{+}=10$ , the total stress normalized by the wall friction near the roughness tip increases from approximately 0.87 to 0.97 as ${\it\delta}_{99}/k$ increases from 9 to 25. The viscous stress contributes approximately 60 % in both cases. In the current study, considering that ${\it\delta}_{99}/k_{rms}>100$ and $k_{rms}^{+}<5$ , it is reasonable to expect that the total stress near the roughness tip represents the wall friction.

Figure 5.

Mean velocity profiles for the baseline (smooth wall) cases. Symbols are consistent with those in table 1. Grey dashed lines and symbols are obtained from 2-D PIV and black symbols show DHM data.

4 Discussion and conclusions

In this study, digital holographic microscopy is used for characterizing the profiles of mean velocity, viscous and Reynolds shear stresses, as well as turbulence level in the inner part of turbulent boundary layers over several super-hydrophobic surfaces. Two types of SHSs are involved, namely $\text{SHS}_{Por}$ generated by spray coating hydrophobic material on porous bases and $\text{SHS}_{Al}$ created by using vapour deposition to coat etched solid aluminium bases. The magnitudes of $Re_{{\it\tau}}$ range from 693 to 4496 and $k_{rms}^{+}$ vary from 0.43 to 3.28. Experiments are also repeated at different streamwise locations and ambient pressures as well as with aluminium surfaces that are either polished randomly or along the streamwise direction. The wall shear stress is estimated from the sum of the viscous and Reynolds shear stress at the top of roughness elements, and the mean slip velocity is obtained from the mean profile at the same elevation.

The data show that the near-wall momentum transport involves a competition between two opposing effects, namely skin friction reduction by the SHSs, and an increase in Reynolds stresses with increasing roughness effects. Their relative significance depends on the values of $k_{rms}^{+}$ . As $k_{rms}^{+}$ increases from 0.43 to 3.28, the near-wall stresses transition from drag reduction, when the viscous stress dominates, to drag increase when the Reynolds shear stress is the primary contributor. For $k_{rms}^{+}<1$ , the SHSs cause a reduction of drag as well as an upward shift of the mean velocity profiles. In the log region, this upward shift is lower than that in the inner layer, a phenomenon observed before in both numerical simulations (e.g. Min & Kim Reference Min and Kim2004) and experimental measurements (e.g. Woolford et al. Reference Woolford, Prince, Maynes and Webb2009). These changes are accompanied with increases in $\overline{\langle u^{\prime }u^{\prime }\rangle }^{+}$ and $\overline{\langle v^{\prime }v^{\prime }\rangle }^{+}$ , the latter only slightly, in the inner part of the boundary layer, and shifts of their peaks closer to the wall. Roughness effects, motion of the air–water interface, spatial non-uniformity and even the non-equilibrium conditions might play a role in the increase of the turbulence level. When $k_{rms}^{+}=1.71$ , it appears that there is a balance between drag reduction by the SHS and an increase by the roughness. For $k_{rms}^{+}=3.28$ , the roughness becomes dominant, causing an increase in wall friction, a downward shift in the mean velocity profile and increases in $\overline{\langle u^{\prime }u^{\prime }\rangle }^{+}$ and $\overline{\langle v^{\prime }v^{\prime }\rangle }^{+}$ close to the wall. Consistent with prior experimental studies involving measurements of wall friction (Bidkar et al. Reference Bidkar, Leblanc, Kulkarni, Bahadur, Ceccio and Perlin2014), it appears that the transition between drag reduction to increase occurs when $k_{rms}$ falls in the $1\leqslant k_{rms}^{+}\leqslant 2$ range. This observation should presumably refer to the height of roughness elements extending above the air layer. This transition might be affected by roughness spacing and gas fraction, which are not measured in the present study.

Due to limited total length of the SHSs, $8{-}20{\it\delta}_{99}$ , all the present boundary layers do not reach equilibrium conditions. Hence, for the drag reduction cases, the directly measured local wall stress is lower than that calculated from a fit to the mean velocity profile in the log layer. The latter decreases gradually from the smooth wall value to that of the SHS as the boundary layer develops with streamwise distance. The evolving conditions also introduce wall-normal gradients in the total stress profiles, with the near-wall values being lower than the maximum values at the edge of the roughness/viscous sublayers. This maximum also decreases with streamwise distance. Collapse of the normal and shear Reynolds stress components onto those of the smooth wall in the outer part of the log layer is also consistent with the boundary being under non-equilibrium conditions. There is also a mismatch between the local wall stress and that estimated from a log layer fit to the velocity profile for cases where roughness effects dominate. However, in these cases, the log fitted values are higher than both the local values and those of the smooth wall. Furthermore, all the Reynolds stress components and the total stress in the log region are much higher than those of the smooth wall. These observations are consistent to recent results of numerical simulations, which also observe an overshoot of the velocity fluctuation in the log region shortly after a transition from a smooth to a rough wall (Lee & Sung Reference Lee and Sung2007; Saito & Pullin Reference Saito and Pullin2014). Clearly, the changing boundary conditions propagate in the wall-normal direction much faster above the rough wall than the SHS which reduces drag.

Figure 17.

Symbols (table 1) show the measured drag reduction as a function of dimensionless slip velocity in comparison to theoretical prediction (lines) based on (1.2), using the empirical values of $F({\it\lambda}_{z}^{+})$ of Busse & Sandham (Reference Busse and Sandham2012). Circles show DNS results by Park et al. (Reference Park, Park and Kim2013a ) for broadly spaced streamwise grooves.

Several other effects have also been observed. First, increasing the pressure in the facility to ${\rm\Delta}Pk_{rms}/{\it\sigma}>1$ appears to suppress the air layer, and presumably exposes the roughness elements to the liquid. Hence, the turbulence level and the shear stress in the inner part of the boundary layer increase. As one would expect, the effect of pressure on drag reduction cannot be ignored. Second, for the etched aluminium surfaces, aligning the surface groves in the streamwise direction causes higher drag reduction than a randomly polished surface. Finally, we conclude this paper by discussing the relationship between slip velocity, based on values measured at the top of the roughness and the drag reduction for all the present SHSs. Figure 17 is a plot of DR versus $\overline{U}_{s}^{+}$ , where each case is represented by the symbols listed in table 1 and used throughout this paper. It also shows the theoretical predictions by Busse & Sandham (Reference Busse and Sandham2012), based on (1.2) and the empirical function $F({\it\lambda}_{z}^{+})=16/(4+{\it\lambda}_{z}^{+})-1$ . The model results are provided for two relevant values of $Re_{{\it\tau}}$ , both for ${\it\lambda}_{z}=0$ and ${\it\lambda}_{z}={\it\lambda}_{x}$ . However, for the experimental data, ${\it\lambda}_{x}^{+}$ is replaced by $\overline{U}_{s}^{+}$ at the top of the roughness. That means that we assume that the present measurements are equivalent to a hypothetical case for which the air layer surface is aligned with the top of the roughness, and the viscous stress there is equal to the total stress. As discussed before, $\overline{U}_{s}^{+}\leqslant {\it\lambda}_{x}^{+}$ and the equal sign is only valid when ${\it\tau}_{w}^{R}=0$ . As discussed in Busse & Sandham (Reference Busse and Sandham2012), the introduction of spanwise slip and an increase in $Re_{{\it\tau}}$ reduces the extent of drag reduction. The ${\it\lambda}_{z}=0$ case predictions agree with the DNS results of Park et al. (Reference Park, Park and Kim2013a ) for an SHS consisting of long and broadly spaced streamwise grooves, which presumably involve limited spanwise slip. However, as is evident, except for the ${\rm\Delta}Pk_{rms}/{\it\sigma}=2.47$ test, all of the present cases that involve drag reduction fall close to the predicted values assuming ${\it\lambda}_{z}^{+}={\it\lambda}_{x}^{+}$ . While not surprising for the randomly distributed roughness, it raises questions for the $\text{SHS}_{Alx}$ results, where the non-uniformly distributed grooves are preferentially aligned in the streamwise direction. It is difficult to assess why $\text{SHS}_{Alx}$ gives a drag reduction and slip length that are higher than those of the randomly aligned $\text{SHS}_{Alr}$ , but still fall on the ${\it\lambda}_{z}^{+}={\it\lambda}_{x}^{+}$ curve. If the streamwise slip is preferentially higher, one would expect that the drag reduction would also fall above the ${\it\lambda}_{z}^{+}={\it\lambda}_{x}^{+}$ curve. There are several possible reasons for this trend, such as effects of the non-uniform spacing, which may allow some spanwise slips and/or roughness effects, as evidenced by the elevated Reynolds stress at the top of the grooves (figure 14). We have not performed measurements using other groove spacings or depths, so the significance of this observation remains unclear. Only three experimental cases deviate significantly from the predicted values, all of which involve an increasing role of roughness. Two of them are the $k_{rms}^{+}=1.71$ and 3.28 cases and the third is the $\text{SHS}_{Alx}$ with ${\rm\Delta}Pk_{rms}/{\it\sigma}=2.47$ , namely when the plastron is partially suppressed (the drag is still lower than the smooth wall value) by increasing the pressure in the test facility. Hence, for situations where the roughness effect is not dominant, i.e. the Reynolds stress at the top of the roughness is much lower than the viscous stress, the present measurements confirm the theoretical relationship between drag reduction and slip length for a turbulent boundary layer over an SHS. To the best of our knowledge, the present study provides the first simultaneous direct measurement of both slip velocity and drag reduction, allowing such a comparison.