In this section, we collect the data from the experimental efforts in the literature to investigate the SHPo drag reduction in turbulent flows. Readers will find that the results are quite scattered under the wide range of the SHPo surface morphologies, turbulent flow types, flow systems, and measurement techniques, not revealing clear and common trends, although the scatter has subsided in recent years. Considering the close connection between the state of plastron during the flow tests and the SHPo drag reduction, more often than not we will discuss or speculate the state of plastron when reviewing the reported results of SHPo drag reduction.
Plastron and flow facilities
While the existence of plastron is an essential premise for SHPo drag reduction, in reality the plastron is easily lost under common water flows. It is a fundamental challenge that a large (long) air–water interface, e.g., s in Fig. 1b and c, which is needed for a large drag reduction, would, unfortunately, imperil the trapped air in the first place. The plastron is vulnerable also to many environmental variables, making the replication of laboratory results in field conditions especially difficult. Multiple factors cause or accelerate the plastron depletion and make the rough hydrophobic surface transition from the initial dewetted or Cassie–Baxter state, which may reduce the friction drag, to the eventual fully wetted or Wenzel state, which will likely increase the friction drag. In nature, where the water below the free surface is undersaturated with air, the plastron becomes more vulnerable as the hydrostatic pressure increases with the immersion depth, following the Henry’s law (Lei et al. 2010; Samaha et al. 2012a; Lv et al. 2014; Xu et al. 2014). In addition, if the water flows with an appreciable speed, the shear stress (Wang et al. 2015; Waxler et al. 2015a) and pressure fluctuation (Piao and Park 2015; Seo et al. 2015) may deplete the plastron dynamically, making high Reynolds number flows exceedingly more challenging for the plastron to survive, as the scarcity of successful SHPo drag reduction at high Reynolds number flows in the literature indicates. Furthermore, even when the plastron is intact, the surfactants in the flowing water (e.g., diffused out from the synthesized surface or naturally existing in the environmental water) may immobilize the water–air interface and deteriorate or negate the drag reduction (Kim & Hidrovo 2012; Peaudecerf et al. 2017; Song et al. 2018; Landel et al. 2020; Li et al. 2020a, b). The stability of plastron will be further discussed in Sect. 4.1. Unlike the numerical simulation, experimental studies have to deal with the above factors that compromise the plastron and hinder drag reduction.
For flow experiments with SHPo surfaces, the importance of plastron poses unique differences between different flow systems as they determine flow geometries and water conditions. For water tunnel, for example, while the pressure and air concentration of the water in the test section are usually not important for conventional surfaces (except having to suppress the cloud of bubbles in water if excessive), such water conditions are critically important for SHPo surfaces as they affect the drag-reducing performance directly. If the water pressure is below the surrounding air pressure, the SHPo surface will enjoy an overgrown plastron and produce an overly optimistic drag reductions, as explained in Xu et al. (2021). If used in a realistic condition, such as in lake or ocean water which is at or above the atmospheric pressure, the same SHPo surface will produce a smaller drag reduction or even a drag increase. On the other hand, if the water pressure happens to be high in the test section, water tunnel experiments may produce overly pessimistic drag reduction. This importance of water condition compels us to categorize the flow systems somewhat differently for SHPo drag reduction studies from the convention. Flow systems may be a closed system, as many water tunnels are, or an open system, such as water flume. To represent the water most commonly considered for SHPo drag reduction, such as ocean or lake, Xu et al. (2020b) used the term ‘open water.’ Open water refers to the water that has been exposed to the ambient long enough in time so that the dissolved air is in equilibrium with the ambient air. Good examples would be lake and ocean water in natural environment and towing tank among laboratory facilities. Note the open water differs from the water flowing in open channel or flume, which is open to but likely not in full equilibrium with the surrounding air. To help the readers determine (or speculate if unclear) how favorable certain experiments could have been for plastron—uniquely important for SHPo drag reduction experiments but so far not adequately addressed or observed in most studies, the types of facility setup that creates the flows (e.g., water tunnel and towing tank) are noted in Table 1 as well as the types of flow over the SHPo surface (e.g., boundary layer flow and channel flow).
Table 1 Experimental studies on the SHPo drag reduction in turbulent flows presented with key parameters useful for comparisons Comprehensive collection of experimental data
Table 1 summarizes the experimental studies of SHPo drag reduction in turbulent flows available in the literature. For the past two decades, experiments were extensively performed mostly for canonical wall-bounded turbulent flows, whether external flows (i.e., boundary-layer flows) or internal flows (e.g., channel flows and Taylor–Couette flows), with the Reynolds number based on its characteristic scale (L) ranging \({\mathrm{Re}}_{L}={10}^{3}-{10}^{7}\). Since the definition (or size) of a characteristic scale depends on the flow geometry, it is not proper to compare the drag reduction results by different studies simply based on \({\mathrm{Re}}_{L}\). To overcome the problem, the frictional Reynolds number (\({\mathrm{Re}}_{\tau }\)) based on the viscous length scale is shown as well (some are estimated using certain related information found in the corresponding reference), ranging \({\mathrm{Re}}_{\tau }\) up to 7000. For the SHPo surfaces, majority of the studies used randomly distributed roughness morphology made by spray coating or chemical etching for convenience and scalability, and a relatively small number of studies adopted well-defined and ordered morphology such as grooves (trenches, ridges) and posts that require precision machining such as photolithography. An exemplary picture of the SHPo surfaces employed is shown for each reference, if reported, in Table 1 to let the readers identify the type of SHPo surfaces quickly for each study.
Organized roughness morphology
As for the SHPo surfaces with organized morphology, which helps provide the physical insights into the relationship between the surface morphology and the flow behavior, mostly microgrooves and some microposts as depicted in Fig. 1b were fabricated and tested. For example, Henoch et al. (2006) measured the drag variation of a SHPo surface with an array of tall posts spaced in nanoscale (400 nm in diameter and 1.25 μm in periodicity, \({w}^{+}\approx 0.03\)) in boundary-layer flows inside a water tunnel. While drag reductions as large as 50% were achieved in laminar flows, the drag reduction decreased to 10% when the flow became turbulent. The particle image velocimetry (PIV) measurement by Woolford et al. (2009) showed how the direction of an effective slip over the grooves affects the turbulent channel flow at \({\mathrm{Re}}_{H}=4.8\times {10}^{3}-{10}^{4}\), where H is the channel height. The SHPo surface with longitudinal grooves (i.e., grooves aligned to the streamwise direction) with GF = 0.8 produced less turbulence (by about 11%) than the smooth surface, but the same surface with transverse grooves (i.e., the grooves aligned transverse to the streamwise direction) increased the turbulence by 6.5%. Daniello et al. (2009) measured the pressure drop in turbulent channel flows with a SHPo surface having longitudinal grooves at ReH = 2000–9500 by using a millimeter-scale channel height. While keeping the gas fraction as 50% (GF = 0.5), they varied the space width (w) of the groove. Although there was no discernible drag reduction in a laminar regime, a significant drag reduction occurred in turbulent regime at ReH > 3000. Furthermore, the rate of drag reduction tended to increase as ReH increased. They suggested that for an effective drag reduction in turbulent flows, the space width of the groove should be comparable to the thickness of a typical viscous sublayer in TBL flows, that is \({w}^{+}=w/{l}_{v}\approx 5.0\) in wall unit (see Fig. 1b for the definition of roughness parameters). As addressed earlier, the viscous length scale (\({l}_{v}= \nu /{u}_{\tau }\)) generally decreases as the Reynolds number increases. Thus, the increase in the Reynolds number can allow the given groove width to work more effectively for drag reduction, as far as the plastron can be well retained despite the increased shear and turbulence. Later, Park et al. (2014) used longitudinally grooved SHPo surfaces monolithically fabricated on floating elements and flexure beams, following the silicon microlithographic process developed by Sun et al. (2015), to directly measure the skin-friction drag in TBL flows in a water tunnel, and achieved a drag reduction as high as 75% (or ~ 65% after compensating for the effect of small sample size, following Park (2015)) with GF = 0.97 at \({\mathrm{Re}}_{\tau }=250\) (or \({\mathrm{Re}}_{x}= {10}^{5}-{10}^{6}\), where x denotes the local position in the streamwise direction), which corresponds to \({w}^{+}\approx 0.9\). The amount of drag reduction increased exponentially with gas fraction at GF > 0.9, following the same trend found in laminar flows (Lee et al. 2016). On the other hand, Van Buren and Smits (2017) tested a SHPo surface with longitudinal grooves (\({w}^{+}\approx 10-80\)) in a Taylor–Couette (TC) flow and measured the variation of frictional torque corresponding to \({\mathrm{Re}}_{L}=6000-11000\), where L is the gap between two concentric cylinders. Note, however, the longitudinal grooves in concentric flows are infinitely long, artificially overcoming the shear drainage effect that is inevitable in practical conditions. Over the range of the Reynolds number, they obtained the maximum drag reduction rate of about 45% with \({w}^{+}\approx 35\). Within the range of Reynolds number, the drag reduction ratio generally increased with increase in Reynolds number. However, the trend was not simple and changed depending on the groove width (w). Very recently, Xu et al. (2021) have shown that a drag reduction up to about 30% can be achieved using similar longitudinally grooved SHPo surfaces in a high-speed towing tank experiment with GF = 0.9 and \({\mathrm{Re}}_{\tau }= 2000-5800\), which corresponds to \({w}^{+}=4.0-16.0\). They visualized the partial loss of plastron occurring at high Reynolds numbers of \({\mathrm{Re}}_{\tau }>4000\), which deteriorated the drag reduction performance, as will be discussed in more detail in Sects. 2.6 and 2.7.
Random roughness morphology
Although the well-ordered morphology has the advantage to show the correlation between the roughness morphology and the flow behaviors more clearly, currently it is not possible in practice to produce such organized SHPo surfaces over a relatively large surface area. Thus, many researches prepared a SHPo surface with random shape and distribution of roughness elements and tested them in various flow conditions. Zhao et al. (2007) used an anodic oxidation method to make a large SHPo plate of aluminum and measured the turbulent friction force acting on it in a water tunnel. However, they did not observe any significant drag reduction in the turbulent regime. Aljallis et al. (2013) applied a sprayable hydrophobic silica nanoparticles coating (now commercially available as NeverWet®) on an aluminum plate (1.2 m × 0.6 m) and tested in a high-speed towing tank. They measured the friction drag over \({\mathrm{Re}}_{L}={10}^{5}-{10}^{7}\) (L: plate length) and obtained the drag reduction as much as about 30% on the SHPo plate with a high gas fraction (i.e., thick plastron). As the ReL increased, however, the amount of drag reduction decreased and a slight drag increase was measured at higher ReL. This was explained by the wetting (depletion of trapped air) of the surface that effectively increased the surface roughness (consult Fig. 1c). By measuring the velocity field over the same type of SHPo surface, Zhang et al. (2015b) measured drag reduction of 10–24% in a TBL flow at \({\mathrm{Re}}_{\tau }=300-500\), while Hokmabad & Ghaemi (2016) measured a slight (3–5%) increase in the near-wall mean velocity profile (i.e., slip) on the SHPo surface compared to a smooth surface in a channel flow (ReH = 9600). Bidkar et al. (2014) measured drag reduction on a SHPo surface of random-textured roughness coated with fluorosilane (FAS, according to the authors) in a TBL flow at relatively high Reynolds numbers (\({\mathrm{Re}}_{\tau }=2000-4000\)). They obtained 20–30% of drag reduction and remarked that the non-dimensional surface roughness \({k}^{+}={R}_{\mathrm{avg}}/{l}_{v}\), where \({R}_{\mathrm{avg}}=(\int {h}_{a}dx)/S\) (S: length of the sample) is the arithmetic average roughness height (Farshad and Pesacreta 2003; Bidkar et al. 2014, see Fig. 1c), should be much (at least by one order of magnitude) smaller than the viscous sublayer thickness in wall unit for a successful drag reduction. Srinivasan et al. (2015) tested a similar SHPo surface in TC flow up to \({\mathrm{Re}}_{\Omega }=r{i\Omega g}/\nu = 8\times {10}^{4}\) (equivalent to \({\mathrm{Re}}_{L}\)), where \(\Omega\) is the rotational speed, ri is the radius of inner cylinder, and g is the gap between inner and outer cylinders of the TC setup. The drag reduction increased as the Reynolds number increased (about 22% at the highest \({\mathrm{Re}}_{\Omega }\) tested). Importantly, a greater drag reduction was obtained when the SHPo surface was not fully submerged so that the plastron was connected to the surrounding air and well maintained. Ling et al. (2016) performed a digital holographic microscopy to measure the drag in TBL flows (\({\mathrm{Re}}_{\tau }=700-4500\)) over several SHPo surfaces with varying random-texture characteristics. About 35% of drag reduction was measured along with a clear slip velocity when the root mean square of roughness height in wall unit was smaller than unity, i.e., \({k}_{\mathrm{rms}}^{+}\) \(=\sqrt{\int {h}^{2}dx}/S\) < 1.0. However, about 10% of drag increase was obtained when the roughness increased to \({k}_{\mathrm{rms}}^{+}>1.0\). Even in the case of smaller roughness (\({k}_{\mathrm{rms}}^{+}<1.0)\), the turbulence in the inner part of the boundary layer was found enhanced compared with the smooth surface. Gose et al. (2018) applied four different morphologies of random textures to the turbulent channel flow at \({\mathrm{Re}}_{H}=1\times {10}^{4}-3\times {10}^{4}\) and obtained a wide variation in drag—from 90% drag reduction to 90% drag increase, depending on the roughness characteristics. To consider the intermediate wetting transition, i.e., between the full Cassie–Baxter and Wenzel states, occurring at a high-Re flow, they suggested a scaling relation between the drag reduction and the product of roughness height (\({k}^{+}\)) and contact-angle hysteresis (i.e., the difference between the advancing and receding contact angles of water on the surface) at high pressure; both the roughness height and contact-angle hysteresis should be minimized to achieve an appreciable turbulent drag reduction. Relatedly, note in this review we have introduced the degree of wetting or air–water interface penetration in defining the roughness parameters of SHPo surface, as illustrated in Fig. 1b and c, to account for the intermediate wetting.
Roughness morphology and plastron
Although the existence of plastron is an essential premise for SHPo drag reduction in the first place, most of the early studies did not pay proper attention to the condition of plastron during the experiments, contributing to the widely scattered data in the literature even for laminar flows (Lee et al. 2016). In fact, it is not possible to fully understand or analyze experimental data of SHPo drag reduction without well-documented states of plastron throughout the reported experiments. Despite the lack of full information on the plastron state for most experimental studies of turbulent SHPo drag reduction, Table 1 still reveals a general trend that SHPo surfaces with longitudinal grooves resulted in a drag reduction most consistently, while those with random roughness produced less consistent results including drag increase. For example, considering most random roughness surfaces have roughness scale in the range below 10 μm or \({\varvec{O}}(1) {\mu} {\text{m}}\), which would lead to a relatively small slip length of \(\lambda ={\varvec{O}}(1) {\mu} {\text{m}}\) or smaller, the surprisingly large drag reductions measured on them in many reports are likely by the plastron overgrown during the flow experiments. The air–water interfaces on the overgrown plastron would be much larger than the roughness scale, i.e., \({s}_{a,0}\gg {P}_{a}\) in Fig. 1c, resulting in a much larger slip length than predictable from the roughness scale, i.e., \(\lambda \gg {P}_{a}\). This deviation was unmistakable even for the simpler experiments in laminar flows; see Fig. 6b in Lee et al. (2016). As a good example of the overgrown plastron in turbulent flows, Li et al. (2020a, b) used a confocal microscope image to reveal the air–water interface between water–asperity contact points (corresponding to \({s}_{a,0}\) in Fig. 1c), which would determine the slip length (\(\lambda \sim {s}_{a,0})\), is much larger than the geometric roughness value (\({P}_{a}\sim {k}_{a}\)) of the given surface, i.e., \({(\lambda \sim s}_{a,0}\gg {P}_{a}\)), in their water tunnel experiments. Conversely, on the same surfaces the plastron may be thinned down (corresponding to \({s}_{a,1}\) and \({s}_{a,2}\) in Fig. 1c) under different flow conditions, such as the open water of natural environment where the water is mostly undersaturated. In such unfavorable but more realistic conditions, the effective slip length would become smaller than the expected from the roughness scale, i.e., \({(\lambda \sim s}_{a,2}\ll {P}_{a}\)), in a fashion opposite to how the overgrown plastron increased the effective slip length. In addition and importantly, the asperities of roughness that impale into the water would increase the drag, negating or even overtaking the plastron-based drag reduction, especially in turbulent regime (Aljallis et al. 2013; Xu et al. 2020b, 2021). In contrast, on well-defined morphologies the plastron would not be overgrown or thinned down much, as illustrated in Fig. 1b, whether the flow condition is favorable or unfavorable to the plastron, respectively, as far as the plastron exists. Note not only the slip interface width would remain constant throughout the intermediate states, i.e., \({s}_{1}\) and \({s}_{2}\) in Fig. 1b, but also the intermediate states are typically only transient anyway. Furthermore, the asperities that impale into the water would not create a form drag if the morphology is of longitudinal grooves. Much of the varying conditions of the flow that would change the thickness of plastron (i.e., slip interface width and morphology impalement height) on random morphologies would only change the curvature of water–air interfaces (not shown in Fig. 1 for simplicity) on organized morphologies. Since the curvature of pinned interfaces does not affect the slip length much (Lee et al. 2016), the organized morphologies provide a consistent slip length, explaining why longitudinal grooves resulted in more consistent drag reduction data than random roughness surfaces.
As the importance of plastron became better understood over the years, most of the recent experimental studies checked its existence during the flow tests, commonly by looking for the bright silvery sheen that appears due to the total internal reflection of light on the air–water interface. However, while it indicates the existence of plastron, the bright appearance does not inform the thickness of plastron. Even for organized morphologies, on which the plastron is more stable and the resulting slip stays more constant than on the random roughness, an appreciable deviation is expected if the air–water interfaces depin from the top of asperities and slide down in between them (Lee et al. 2016; see also Fig. 1b), suggesting the importance of keeping a proper plastron with pinned interfaces for successful drag reduction. Analytical studies and empirical evidences indicate that minute details of roughness asperities may affect the pinning of air–water interface and, thus, the status of plastron, which determines how slippery the surface is for overlying flows, e.g., slip length. Generally, sharp edges, including reentrant edges (Xu et al. 2020b), help keep the interface pinned on top of the roughness feature longer under unfavorable conditions until the onset of wetting starts to compromise the slip effect. Recently, Yu et al. (2021) developed a convenient (only with naked eyes) observation technique that can detect the depinning of interface, not just the existence of plastron, on microgrooves, making the monitoring of the plastron status throughout flow experiments more practical.
Drag reduction vs. slip length
Although highly relevant to design SHPo surfaces for applications, the correlation between the slip length and drag reduction for turbulent flows has been studied mostly by numerical simulations, lacking experimental corroboration. Despite the difficulty, however, organized roughness morphologies, such as the longitudinal grooves, could provide a useful physical insight into the correlation. For example, testing in TBL flows in a water tunnel, Park et al. (2014) measured how the drag ratio (i.e., drag on a sample surface relative to that on a smooth surface) on longitudinal grooves decreases (i.e., drag reduction increases) with increase in gas fraction (GF) and also with the pitch (P) of the groove. Although the drag reduction in turbulent flows is not solely determined by the slip length (see Sect. 3 for detailed discussion), the trend of increasing drag reduction with increase in GF and P was attributed to the slip length (\(\lambda\)) that increases with increase in GF and P, similar to the trend in laminar flows (Lee et al. 2016). Testing in a Taylor–Couette flow, Srinivasan et al. (2015) measured the drag reduction on spray-coated random roughness and suggested a scaling that the skin-friction coefficient (cf) would follow \({c}_{f} \sim 1/{({\lambda }_{s}^{+})}^{2}\), where \({\lambda }_{s}^{+}={\lambda }_{s}{u}_{\tau }/\nu\) is the streamwise slip (\({\lambda }_{s}\)) expressed in wall unit, also attributing the drag reduction to the increased slip length, especially expressed in wall unit. This scaling trend is corroborated by the numerical studies (Fukagata et al. 2006; Busse et al. 2012; Park et al. 2013; Jung et al. 2016), as plotted in Fig. 2, which also includes the results from the TBL flows performed in a water tunnel (Park et al. 2014) (after compensating for the effect of small size), underneath a motorboat (Xu et al. 2020b), and in towing tank (Xu et al. 2021). The results of the TC flows by Hu et al. (2017) and Van Buren and Smits (2017) are not plotted because we could not deduce their values of \({u}_{\tau }\) from the reported data. The data sets that involved deteriorated plastron (existing in the boat and towing tank studies) are excluded in the graph for a fair comparison with the numerical data, which assumed a perfect plastron. While the water tunnel and towing tank data show the relationship between the drag ratio and dimensionless slip length that resembles the numerical predictions, the boat data show a scatter. The scatter for the boat test is not surprising if one considers the uncontrollable flow conditions of the field tests. Here, the streamwise slip length (\({\lambda }_{s}\)) was estimated using the analytical relationship confirmed for laminar flows \(\frac{{\lambda }_{s}}{P}=-\frac{1}{\pi }ln\left[\mathrm{cos}\frac{\pi GF}{2}\right]\) (Lauga and Stone 2003; Lee et al. 2008), and the slip length in wall unit (\({\lambda }_{s}^{+}\)) was estimated using the friction velocity (\({u}_{\tau }\)) obtained from the corresponding Reynolds numbers.
Drag reduction vs. Reynolds number
Figure 3 shows all the drag reduction results experimentally obtained on SHPo surfaces in turbulent flows, following Table 1. The data are presented in the form of drag ratio as function of the friction Reynolds number (\({\mathrm{Re}}_{\tau }\)). It should be noted that the effect of roughness size is not explicitly compared in the figure because the main goal is to show the effect of \({\mathrm{Re}}_{\tau }\) for a given SHPo surface. The drag ratio is widely scattered in the range between around 0.3 (i.e., drag reduction by 70%) and nearly 1.5 (i.e., drag increase by 50%) over the Reynolds numbers up to \({\mathrm{Re}}_{\tau } \simeq 7000\). The broad spectrum of the data is attributed to the plastron highly disturbed by the stronger inertia (and agitation) and shear stress in turbulent flows, as well as to the different mechanism of drag reduction, compared with laminar flows (see Sect. 3 for details). The drag ratio is also substantially dependent on the type of roughness morphology, such as organized vs. random roughness, as well as the size and shape of the morphology. Some studies confirmed the increase in the drag reduction, i.e., decrease in drag ratio, with an increase in \({\mathrm{Re}}_{\uptau }\) (Daniello et al. 2009; Srinivasan et al. 2015; Zhang et al. 2015b; Rajappan et al. 2019; Li et al. 2020a, b; Xu et al. 2020b, 2021), agreeing with the numerical and theoretical predictions (Min and Kim 2004; Fukagata et al. 2006; Park et al. 2013). All of them visually confirmed the existence of plastron during the experiments. However, other studies reported the opposite trends. For example, the drag reduction effect diminished, i.e., drag ratio increased, with an increase in \({\mathrm{Re}}_{\tau }\) and even became negative (i.e., drag ratio > 1.0) at high Reynolds numbers (Aljallis et al. 2013; Ling et al. 2016; Xu et al. 2020b, 2021), which was explained with increased wetting and hydrodynamic roughness at higher Reynolds number (\({\mathrm{Re}}_{\tau }\gtrsim 2000\)). Most of them reported observation of diminished or depleted plastron at high speeds. Note two of the above studies (Xu et al. 2020b, 2021) predictably confirmed both the positive and negative trend by testing longitudinally grooved and random roughness surfaces side by side simultaneously in given flows.
To review the effect of slip directions on the turbulent drag reduction, the drag ratio on the SHPo surfaces with longitudinal grooves (LG) and random roughness (Ra) is plotted separately in Fig. 3b and c, respectively. Additionally, the data for transverse grooves (TG) are plotted along with LG in Fig. 3b, and the data for posts (Po) are with Ra in Fig. 3). Figure 3b shows all the longitudinal groove (LG) surfaces produced a drag reduction (drag ratio < 1.0) consistently, although the degree of reduction varies significantly. Unlike those who tested one or several surfaces at one Reynolds number (Woodford et al. 2009; Park et al. 2014; Gose et al. 2020), those who tested a given surface over a range of Reynolds numbers (Daniello et al. 2009; Xu et al. 2020b, 2021) found the drag ratio would decrease as the frictional Reynolds number increases, corroborating the numerical studies (Fukagata et al. 2006; Martell et al. 2010; Park et al. 2013; Lee et al. 2015). The reversed tendency (i.e., drag increase with Reynolds number) at \({\mathrm{Re}}_{\tau }\gtrsim 4000\) in Xu et al. (2021), drawn with a lighter color, was caused by the shear-driven wetting (Wexler et al. 2015a), not by any flow mechanism. In contrast, Fig. 3c shows that the drag ratio data on the random roughness (Ra) surfaces were scattered significantly. Despite the large scatter, we can see a slight trend that the drag ratio would generally increase as the frictional Reynolds number increases, contradicting the numerical studies. The results suggest that the effect of the streamwise slip was compromised or overshadowed by the equal spanwise slip due to the non-directionality of the randomly arranged surface morphology. However, it also suggests that the hydrodynamic roughness effect becomes dominant on random textures (Ling et al. 2016), which are also more prone to wetting, and added a significant drag (Aljallis et al. 2013). While included, the surfaces of transverse grooves (TG) in Fig. 3b and posts (Po) in Fig. 3c have not produced enough data to allow us to discuss the correlation between the drag ratio and the frictional Reynolds number. Posts are considered to be closer to random roughness than longitudinal grooves in terms of slip directions although their truncated roughness height belongs to organized roughness in terms of hydrodynamic roughness.
The reasons for the more favorable drag reduction on LG than on Ra above can also be understood by two aspects: the effect of slip direction and the stability of plastron in flows. The slip on a SHPo surface is the most effective when it is aligned to the flow direction. Any slip not in the streamwise direction is known to mitigate the drag-reduction performance (Min and Kim 2004; Fukagata et al. 2006; Busse and Sandham 2012), and the mitigation becomes more severe as the Reynolds number (or the slip length in wall unit) increases. For grooved SHPo surfaces in a laminar flow, the slip length along the transverse direction is generally a half of that along the streamwise direction as analytically predicted (Lauga and Stone 2003) and experimentally confirmed (Lee et al. 2016). However, as alluded earlier, the effects of the flow direction on the slip length and drag are not as simple in turbulent flows. For example, Woolford et al. (2009) reported a drag reduction of about 11% on longitudinal grooves (P = 32 μm; GF = 80%) placed in streamwise direction but a drag increase of 6.5% in transverse direction. Although the favorability of streamwise direction over transverse direction is common for both laminar and turbulent flows, the experimental data for turbulent flows are still scarce to reveal any quantifiable comparison. For the plastron stability in flows, the random-textured roughness is known to be generally more vulnerable to the depletion than the well-organized roughness. Aljallis et al. (2013), Xu et al. (2020b), and Xu et al. (2021) visually confirmed the partial loss of plastron on their SHPo plate with the loss being more pronounced at higher flow speeds, which impart a larger shear stress on the plastron. This partial loss is illustrated with the subscript 1 and 2 in Fig. 1c. In addition, as the Reynolds number increases, the corresponding viscous length scale (\({l}_{v}= \nu /{u}_{\tau }\)) becomes smaller, making the dimensionless length scales of a given roughness geometry, \({P}_{a}^{+}={P}_{a}/{l}_{v}\) ~ \({k}_{a}^{+}={k}_{a}/{l}_{v}\), larger and inducing a drag increase once the average wetted roughness height in wall unit (\({h}_{a}^{+}\)) becomes large enough to become hydrodynamically rough (Bidkar et al. 2014). Note a stronger turbulent flow not only increases the wetted roughness height \({h}_{a}\) by thinning the plastron but additionally makes the increased \({h}_{a}\) hydrodynamically even rougher by decreasing the viscous length scale. This issue will be discussed more in Sect. 3.2. In comparison, for organized textures, such as grooves and posts, the wetted roughness height (\(h\)) in wall unit would be quite small, i.e., \({h}^{+}\) \(\lesssim\) O(1), and considered to be hydrodynamically smooth (Nikuradse 1933) as far as the air–water interface is pinned (much more persistently than the random textures) at the texture top edges, i.e., \({h}_{o}\) in Fig. 1b. Lastly, although the increased pressure (or its fluctuation) in a high-Re turbulent flow would promote sagging of the air–water interface and accelerate its depinning (Kim and Park 2019), the effect of decreased slip length by the curved interface (Crowdy 2016) is considered much smaller compared with the negative effect by the above roughness effect. Unlike the random roughness surfaces, where a significant wetting would lead to an overall drag increase, the longitudinal grooves still produced an overall drag reduction of an appreciable amount (~10%) even when around a half of surface was found wetted (Xu et al. 2021), likely because of less roughness height effect. Based on the studies reported until today and especially the studies that predictably confirmed both of the opposite trends in same flows (Xu et al. 2020b, 2021), we believe the drag ratio would decrease (i.e., drag reduction would increase) with increase in \({\mathrm{Re}}_{\tau }\) on a given SHPo surface as predicted by most numerical studies, as far as the plastron is properly maintained. The contradicting trend found with most of the random roughness surfaces is most likely and simply by their lack of ability to maintain the plastron rather than by any fluid dynamic mechanism, as explained in Aljallis et al. (2013).
As suggested in Sect. 2.1, the effect of flow geometry on SHPo drag reduction, including the trend of drag ratio with \({\mathrm{Re}}_{\uptau }\), tends to be greater in turbulent flows than in laminar flows. For channel flows in water tunnel, obtained for \({\mathrm{Re}}_{\tau }<1000\) as shown in Fig. 3d, the drag ratio (especially for Ra) tends to increase with increase in \({\mathrm{Re}}_{\tau }\). As the Reynolds number increases, the plastron would thin down and the wetted roughness height would increase, i.e., \({h}_{a,0}\to {h}_{a,1}\to {h}_{a,2}\) in Fig. 1c, degrading the drag-reducing capability of the surface. For TBL flows in water tunnel, obtained mostly for \({\mathrm{Re}}_{\tau }<2000\) as shown in Fig. 3e, meaningful drag reductions are obtained up to \({\mathrm{Re}}_{\tau }\simeq 5000\), showing drag ratio decreasing with increase in \({\mathrm{Re}}_{\tau }\). For Taylor–Couette (TC) flows, obtained for \({\mathrm{Re}}_{\tau }<4000\) as shown in Fig. 3f, drag ratio is shown to decrease with \({\mathrm{Re}}_{\tau }\). Compared with most other flows, TC flow can provide a uniquely favorable environment to keep the plastron, for example, by connecting the plastron to the outside air or circumventing the shear drainage effect. Finally, for TBL flows in open water shown in Fig. 3g, the data were obtained in a wider range of \({\mathrm{Re}}_{\tau }\) (= 1000–7000), and the dependency of LG and Ra on \({\mathrm{Re}}_{\tau }\) is clearly distinguished. Since the water in open water environment (e.g., towing tank and boat) is saturated or undersaturated with air, not providing the supersaturated water or any other favorable condition the laboratory flows may generate whether intentionally or unintentionally, the advantage of LG over Ra is unmistakably evident. Before the wetting transition occurs by high \({\mathrm{Re}}_{\tau }\), i.e., at \({\mathrm{Re}}_{\tau }\) < 5000, the decrease of drag ratio with increase in \({\mathrm{Re}}_{\tau }\) agrees with the prediction by numerical studies (Min and Kim 2004; Fukagata et al. 2006; Park et al. 2013), emphasizing the importance of keeping the plastron to achieve the drag reduction.
Turbulent drag reduction on liquid-infused surfaces
Slippery liquid-infused porous surface (SLIPS), or more generally liquid- or lubricant-infused surface (LIS), is a hydrophobized textured surface with its texture filled with a low-surface-energy liquid (Wong et al. 2011), in contrast to the SHPo surface whose texture is filled with a gas. The use of liquid instead of gas makes the lubricating layer significantly more robust against the loss by molecular diffusion or pressure difference across the interface—the main advantage of LIS compared with SHPo surfaces. Although the infused liquid on LIS cannot provide the very small viscosity ratio (~ 1/50 for air-to-water) the concept of SHPo drag reduction is footed on, various studies have nevertheless explored LIS for drag reduction, anticipating the liquid–liquid interface between the infused liquid (typically oil) and the bulk liquid flow would lead to a drag reduction. While not effective in laminar flows, where drag reduction is solely by the viscosity ratio, LIS was found capable of an appreciable drag reduction in turbulent flows. However, the large viscosity (typically similar to or larger than water) of the lubricants that are acceptable for LIS generally results in a smaller slip length and drag reduction compared with the air-filled SHPo surface for the same roughness geometry, as reported by Arenas et al. (2019) and Chang et al. (2019). There have been only a few successful experiments to test LIS in turbulent flows. Rosenberg et al. (2016) tested the LIS of longitudinal grooves (a spiral to be exact) in turbulent TC flows and obtained a drag reduction up to 14%. In a follow-up study, Van Buren & Smits (2017) tested the LIS with improved longitudinal grooves in turbulent TC flows and obtained a drag reduction up to 35%. The optimal width of fluid–fluid interface (groove width) that led to the maximum drag reduction was found to be \({w}^{+}\simeq 35\), which is significantly larger compared with the SHPo surface in other flows; we will discuss this issue in Sect. 3. Importantly, note all the experiments that reported an appreciable drag reduction with LIS (Rosenberg et al. 2016; Van Buren and Smits 2017) were performed in TC system, which uniquely avoids the shear-driven loss of the infused fluid, whether liquid or air, but such a loss is most likely unavoidable in most practical applications, unfortunately. While more stable against diffusion and pressure, the liquid lubricant on LIS is generally more susceptible to the shear-driven drainage (Wexler et al. 2015a; Fu et al. 2019), compared with the air plastron on SHPo surfaces. Several strategies have been proposed to enhance the oil retention for LIS (Lee et al. 2019; Chen et al. 2020), but despite the general improvement they fall far short of overcoming the high shear expected in common drag reduction applications. In employing a LIS for hydrodynamic drag reduction, it should be noted that the ratio of oil viscosity to the bulk fluid flow is an important parameter to determine the functionality, and an effective drag reduction cannot be expected in laminar flows if the infused fluid is more viscous than the bulk fluid. Although the LIS has not been widely tested in turbulent flows and more data are required, it is expected that the effective drag reduction can be obtained in the case of a turbulent flow due to the additional effects such as the modification of the flow structures similarly to the SHPo surfaces, as suggested by recent numerical studies (Fu et al. 2017; Arenas et al. 2019; Chang et al. 2019).