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Philos Trans A Math Phys Eng Sci. 2016 August 6; 374(2073): 20160189.
PMCID: PMC4928507

Bioinspired surfaces for turbulent drag reduction

Abstract

In this review, we discuss how superhydrophobic surfaces (SHSs) can provide friction drag reduction in turbulent flow. Whereas biomimetic SHSs are known to reduce drag in laminar flow, turbulence adds many new challenges. We first provide an overview on designing SHSs, and how these surfaces can cause slip in the laminar regime. We then discuss recent studies evaluating drag on SHSs in turbulent flow, both computationally and experimentally. The effects of streamwise and spanwise slip for canonical, structured surfaces are well characterized by direct numerical simulations, and several experimental studies have validated these results. However, the complex and hierarchical textures of scalable SHSs that can be applied over large areas generate additional complications. Many studies on such surfaces have measured no drag reduction, or even a drag increase in turbulent flow. We discuss how surface wettability, roughness effects and some newly found scaling laws can help explain these varied results. Overall, we discuss how, to effectively reduce drag in turbulent flow, an SHS should have: preferentially streamwise-aligned features to enhance favourable slip, a capillary resistance of the order of megapascals, and a roughness no larger than 0.5, when non-dimensionalized by the viscous length scale.

This article is part of the themed issue ‘Bioinspired hierarchically structured surfaces for green science’.

Keywords: drag reduction, turbulence, superhydrophobic, biomimetics, hierarchical, slip

1. Introduction

Water effortlessly rolls off the leaf of the lotus plant [1]. Penguins seamlessly dive to ocean depths, trapping numerous air bubbles between their feathers [2]. The diving bell spider lives life completely submerged, breathing in a pocket of air it has dragged beneath the surface of the water [3]. Nature has masterfully controlled the air–water interface for millions of years. Recently, the mechanisms behind these amazing natural occurrences of exceptional water repellency have begun to be better understood.

The above-mentioned examples are facilitated by trapping pockets of air within the pores of the outer surface of the organisms, be it hairy insect limbs, feathers or waxy leaves. One use of this entrapped air is to reduce the frictional drag when moving through the water, as penguins do when they dive [2]. The purpose of this review is to understand how entrapped air can reduce the frictional drag on an object within turbulent flow. Although much work has been done to understand the effects of these water-repellent surfaces in laminar flows [4], turbulence adds many new challenges. Eddy formation, vorticity, pressure fluctuations, significant wall shear and an unsteady three-phase interface all complicate the problem [5,6]. However, approximately 60% of the fuel used by displacement ships today is expended overcoming frictional drag in turbulent conditions [7,8]. Thus, extending our understanding of these surfaces to non-laminar flows is essential. In the first part of this review, we discuss how surfaces can be designed to robustly trap pockets of air under water. In the second part, the mechanism by which such water-repellent surfaces reduce drag in laminar flows is explained. We then discuss the difficulties that arise in the presence of turbulence, and highlight the current research being conducted to understand this phenomenon, both computationally and experimentally. We complete the review by noting the still-unsolved challenges that must be addressed before superhydrophobic surfaces (SHSs) should be considered for turbulent drag reduction applications.

2. Designing superhydrophobic surfaces

A drop of water makes a contact angle, θ, when placed on a smooth, chemically homogeneous surface, which can be determined by a force or energy balance, as first formalized by Young [9],

equation image
2.1

Here, γSV is the solid surface energy, γSL is the liquid–solid interfacial tension and γLV is the liquid's surface tension. Water adopts θ values approaching 0° (hydrophilic) on clean glass, and values near 120° (hydrophobic) on a perfluorinated monolayer [10]. Whereas θ represents a single thermodynamic equilibrium, most surfaces exhibit a range of contact angles. The maximum angle is achieved as liquid initially advances on a dry substrate, and is called the advancing contact angle, θadv [11]. As liquid recedes from the wetted solid, it adopts its minimum, or receding, contact angle, θrec. The difference between these two angles is termed contact angle hysteresis, Δθ[equivalent]θadvθrec.

Surface roughness and chemical inhomogeneity are the two most common factors that influence contact angle hysteresis. Wenzel [12] first formalized the effect of roughness on the observed apparent contact angle of a droplet on a textured surface, in an equation that now bears his name,

equation image
2.2

Here, θ* is the observed apparent contact angle, and r is the Wenzel roughness, defined as the ratio of the actual surface area to its projected area. Hence, r always exceeds unity and enhances the intrinsic phobic/philic nature of the substrate.

Wenzel assumed that the rough surface was completely wetted by the liquid. Studying contact angles of liquids atop metal gratings, Cassie & Baxter [13] proposed another possible configuration, in which part of the drop was supported by air

equation image
2.3

where f1 is the total surface area of liquid in contact with a solid (with contact angle θ) and f2 is the surface area in contact with air (with a contact angle of π). Assuming the wetted solid is rough and the liquid meniscus is flat, it is convenient to write [14]

equation image
2.4

where ϕs is the fraction of solid in contact with the liquid and rϕ is the Wenzel roughness of the wetted solid. One can see that θ* becomes large as ϕs approaches zero, or when rϕ[dbl greater-than sign]1.0 and θ>90°.

When θ*>150° and Δθ is low (an exact cut-off for contact angle hysteresis has never been standardized, but is probably no larger than Δθ=10°), the surface is denoted as an SHS. Nature has provided numerous SHSs to be studied, including: leaves from broccoli, lotus, lacinato kale, savoy cabbage, Indian hemp and taro plants [15] (figure 1ad). Superhydrophobicity is also present in animal features such as the feathers of ducks, geese and penguins [2,17] (figure 1ei), as well as the hairy legs of diving bell spiders [3] and water striders [18] (figure 1jn). Two common features are found on all such surfaces. First, they possess hierarchical texture, with features in the micrometre and submicrometre ranges. Second, they all possess at least moderately low surface energy, usually achieved by a waxy coating [1].

Figure 1.
Superhydrophobicity in Nature. Water is repelled by the lotus leaf (a) [16] owing to its micro- and nanotexture (b). Other plants exhibit similar properties, such as broccoli (c) or taro (d). ((c,d) Reprinted with permission from [15]. Copyright (2008) ...

By mimicking these natural textures, researchers have engineered many artificial SHSs. Several examples of such hierarchical surfaces that have also been tested for turbulent drag reduction are shown in figure 2. Perhaps the simplest hierarchical texture was fabricated by Prince et al. [19], who combined two different length scales of ridges (figure 2a). Lee & Kim [20] formed nanoscale roughness on the sidewalls of the canonical geometry of posts (figure 2bd). Similarly, Jung & Bhushan [25] self-assembled the wax responsible for the water repellency of the lotus leaf onto the tops of silicon posts (figure 2e,f). SHSs can also be fabricated without any sort of initial, regular structure. A commercially available superhydrophobic coating, NeverWetAn external file that holds a picture, illustration, etc.
Object name is rsta20160189-i1.jpg, combines silica nanoparticles in a silicone matrix [21] (figure 2g). Randomly textured SHSs can also be fabricated using spray-coating [22,24,26,27] (figure 2h), thermal deposition [23] (figure 2i) or chemical etching processes [28] (figure 2j). A summary of previously fabricated SHSs that have been evaluated in turbulent flow is presented in table 1. Most of these studies will be discussed in detail in the following sections. However, before diving into turbulent waters, we first review how SHSs have been shown to effectively reduce friction drag in laminar flow.

Table 1.
Previous studies on drag reduction in turbulent flow with corresponding geometry of surfaces, apparent contact angle and contact angle hysteresis reported, length of tested surfaces, range of friction Reynolds numbers evaluated (Reτ=δ ...
Figure 2.
Engineered superhydrophobic surfaces tested in turbulence. Hierarchical ridges (a) [19] or posts (bd) have been fabricated using microlithography [20]. ((a) Reprinted with permission from [19]. With permission of ASME. (bd) Reprinted ...

3. Superhydrophobic friction drag reduction in laminar flow

When liquid flows over a solid surface, the usual boundary condition assumed is that the velocity of the liquid must match the velocity of the solid [41]. This is typically referred to as the ‘no-slip’ condition. However, SHSs possess a fraction of air (1−ϕs) at the solid–liquid interface, which can yield a non-zero slip velocity. In fact, Navier [42] first proposed a slip velocity, us, for solid surfaces in 1823, where he suggested that the shear rate at the wall was proportional to us. Although somewhat different for a solid surface with a heterogeneous three-phase interface, one can extend this same idea. For a two-dimensional flow, this is depicted in figure 3a,b and described by

equation image
3.1

where u, v and w are the velocity components in the streamwise (x), wall-normal (y) and spanwise (z) directions, and λi is the slip length along direction i. Thus, through the incorporation of entrapped air, SHSs can produce an effective slip at the gas–liquid interface. Because less energy is lost to frictional dissipation, a non-zero us indicates that a reduction in the shear force along the surface is possible. If the average velocity along the intermittent solid–liquid and gas–liquid interfaces increases, such that it matches the bulk velocity of the flow, this would indicate 100% friction drag reduction. According to this analysis, this would be achieved by a perfect air layer (ϕs=0) [7].

Figure 3.
Laminar skin friction drag reduction using SHSs. A schematic of water atop ridges (a) and the corresponding slip that can result from the presence of air pockets (b). Water existing in a composite interface (c) atop ridges [39]. In contrast, ethanol, ...

A large streamwise slip length, λx, suggests a large velocity at the wall [4]. Ou and Rothstein conducted one of the seminal studies that first measured drag reduction using canonical SHSs [43,44]. By fabricating precisely defined microstructures, they showed the first experimental evidence of a higher velocity at the wall owing to the incorporation of air (figure 3c,d). Measuring pressure drop in a microchannel, they also found that the drag reduction increased with the fraction of air, (1−ϕs), and observed a maximum λx of the order of the size of their microfeatures (figure 3e,f). Others have observed a similar trend, both computationally and experimentally (see the review by Rothstein [4]).

For example, Lee & Kim [20] used etched silicon posts to study the slip length of SHSs with hierarchical texture elements (figure 2bd). Typically, a large fraction of air will cause a surface to be easily wetted by water; however, a hierarchical texture has been shown to delay this transition [45]. The meniscus can only advance when the local contact angle exceeds θadv. Roughened sidewalls add pressure stability by pinning the meniscus at the top of the posts. The authors produced micrometre-scale posts with nanoscale roughness on the walls in order to maintain both a very low ϕs and a greater pressure stability. Without nanostructured sidewalls, water wet the posts that were spaced more than 200 μm apart. Nanostructuring the sidewalls increased the allowable spacing to 450 μm. This technique allowed the authors to create idealized structures with 99.7% air in contact with the liquid. When evaluated in a rheometer, the measured slip lengths approached 140 μm. A slip length of λx=400 μm was observed for ridges with ϕs=0.02.

Computationally, friction drag reduction in laminar flows has been modelled by regions of slip and no-slip. Lauga & Stone [46] first applied these boundary conditions to pressure-driven Stokes flow. They defined an effective slip length to represent the overall slippage along a surface with slip and no-slip regions. Interestingly, they noted that defects within the perfect slip region cause much more friction when aligned streamwise rather than spanwise. We will return to this later to see if it holds true in turbulent flows. Not surprisingly, they found that the effective slip length increased with an increasing percentage of slip regions, i.e. more drag reduction with more entrapped air. Thus, experimentally, computationally and theoretically, drag reduction using SHSs in laminar flows is well understood. In summation, the following trends from laminar friction drag reduction are evident:

  • (1) A reduction in frictional drag arises from streamwise slip at the air–water interface.
  • (2) A higher air fraction allows for a greater shear-free area, and, consequently, higher drag reduction.
  • (3) The streamwise slip length characterizes the potential of an SHS to reduce drag, and is limited by the texture spacing present on the SHS.

4. The complications of turbulent flow

Turbulence adds many new challenges for SHSs that are absent in the laminar regime [41]. Secondary structures can form in turbulent flows, such as streaks, eddies and vortices [6,47] (figure 4ad). These structures can interact with the texture elements of SHSs, potentially mitigating any drag-reducing effects. Large pressure fluctuations can induce a wetting transition (figure 4e), and the increased wall shear stress can physically damage the potentially fragile microstructure of the SHS. Moreover, turbulence disallows solving the Navier–Stokes equations analytically, and many features of the underlying flow must be discussed statistically. Additionally, high-speed, turbulent flows may decrease the local pressure significantly, and thus may provide for direct gas removal through suction. Overall, SHSs capable of reducing drag in laminar flow will not necessarily reduce drag in turbulent flow.

Figure 4.
Turbulent flow. Unlike the laminar regime, turbulence includes complex structures like vortices (a) [6]. (From [6], with permission from Cambridge University Press.) There also exists a viscous sublayer very near the wall in a turbulent flow (b). In the ...

In turbulent flow, a thin laminar sublayer exists very close to the solid surface (figure 4b) [41]. The height (or thickness) over which this viscous sublayer extends is of the order of five wall units (y+) from the solid [50], where the wall-normal distance y is non-dimensionalized by the viscous length scale δν=ν/uτ, or y+=y/δν. Here ν is the kinematic viscosity of the fluid and uτ is the shear velocity at the wall. As a consequence of the laminar sublayer, it was initially thought that the mechanisms for turbulent and laminar drag reduction might be similar [4]. Fukagata et al. [51] also theoretically showed that changes in this laminar sublayer owing to the presence of an SHS could affect the entire turbulent boundary layer δ. Hence, the interaction between the viscous sublayer and the texture of an SHS is postulated to be critical in determining the potential drag reduction on a given SHS.

5. Computational insights into turbulent drag reduction

The simplest geometry studied in turbulent flows has been steamwise ridges. Min & Kim [48] performed insightful direct numerical simulations (DNS) by modelling this geometry as regions of slip (air pockets) and no-slip (solid surfaces) regions. These manifest as differing boundary conditions when solving the Navier–Stokes equations. By evaluating cases with purely steamwise slip, purely spanwise slip and slip in both directions, the effects of the SHS on turbulent drag reduction were uncovered. They found that slip in the streamwise direction works much in the same way as it does for laminar flows. A non-zero streamwise slip velocity directly reduces the skin friction at the wall (figure 4c).

Conversely, spanwise slip was found to cause an increase in overall drag. Min and Kim observed a drag increase caused by enhanced near-wall vortices, in the presence of pure spanwise slip (figure 4d). Martell et al. [52] reached a similar conclusion in their computations (figure 5a,b). They noted that the shift of the near-wall vortices has advantages and disadvantages. Fortunately, the structures are not modified, merely shifted, and, therefore, turbulence theory still applies when analysing the SHSs in channel flow. Unfortunately, the shift of the structures is downward, causing drag to increase (figure 5b,c). Min and Kim and Martell et al. also studied conditions with both spanwise and streamwise slip, which would best approximate an actual experimental case. Overall, both groups found that the streamwise reduction in skin friction can outweigh the drag increase caused by spanwise slip. Thus, even with spanwise slip, it is still possible for SHSs to reduce drag in turbulent flow.

Figure 5.
Turbulent drag reduction: computational ridges. For ridges computed using regions of slip and no slip, the slip velocity increases with increasing regions of slip (a) [52]. The reduction in friction drag also manifests as a reduction in the computed Reynolds ...

Busse & Sandham [54] reported similar results, but graphed their data according to a neutral curve in (λx, λz) space (figure 5dh). Along this curve, there was no net change in drag, as the deleterious effects of spanwise slip were cancelled exactly by the benefits of streamwise slip. In their analysis for An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i5.jpg greater than 5 (i.e. to the right of the neutral curve in figure 5d), drag reduction was always observed. They found that spanwise fluctuations in the turbulent statistics were limited by the peak value of the corresponding profile. Thus, the drag increase caused by spanwise slip was found to plateau. In contrast, the streamwise slip always reduces the intensity of turbulent fluctuations, and thus frictional drag steadily decreased with increasing An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i6.jpg. In fact, they predicted that drag reduction would always be observed when An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i7.jpg, regardless of the magnitude of An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i8.jpg. Jelly et al. [53] reported similar findings with their DNS results (figure 5c). In their work, the authors showed that the Reynolds stresses differed over the regions of slip and no slip. In fact, they found that over 70% of the skin friction on the no-slip regions (solid surface) was a direct result of the Reynolds stresses caused by the presence of the slip regions (air pockets). Overall, computational studies conclude that turbulent drag reduction is possible when streamwise slip outweighs spanwise slip, and hence experimentalists must fabricate SHSs with geometries that favour streamwise slip.

6. Experimental findings for ridges and posts

Experimentally, the simple geometry of ridges has allowed experimentalists to validate the above computational insights. Daniello et al. [39] measured the skin friction along streamwise ridges of varying geometry in a turbulent microchannel (figure 6a). For low-Reynolds number (Re) flow, no drag reduction was observed, which agrees with theoretical predictions. As Re was increased, a large reduction in the skin friction was measured. This reduction roughly doubled when both the upper and lower surfaces of the microchannel were superhydrophobic, streamwise ridges. The authors also noted that the onset of drag reduction corresponded to a texture size that approached the viscous sublayer thickness. Eventually, their drag reduction saturated at a value of approximately 50%. However, their ridges had a ϕs=0.5, and thus no more drag reduction was expected. Their results highlight an important point: SHSs with ϕs=0.5 will exhibit a moderately low θ* and Δθ[dbl greater-than sign]0°. Using equation (2.4) gives θ*=139°. Thus, drag reduction of the order of 10–20% may be possible even with surfaces that would not be considered superhydrophobic. Indeed, Park et al. [29] have shown that turbulent drag reduction of the order of 5% was possible when ϕs=0.7, and, alternately, as high as a 75% reduction in friction drag was achieved when ϕs=0.03 (figure 6b,c).

Figure 6.
Turbulent drag reduction: experimental ridges and posts. When evaluated in a turbulent microchannel flow, streamwise ridges produced large reductions in drag that increased with increased Re (a) [39]. (Reprinted from [39]. With permission of AIP publishing.) ...

Woolford et al. [31] conducted an important study measuring the skin friction on four different configurations of ridges: longitudinal and non-wetted, longitudinal and wetted, transverse and non-wetted, and transverse and wetted. The measured friction experienced in turbulent flow, and a baseline comparison, are shown in figure 6d. For both the wetted cases, as well as the transverse superhydrophobic case, an increase in drag was observed. This agrees with the predictions of Min & Kim [48] and Busse & Sandham [54], i.e. that spanwise slip increases the frictional resistance. Woolford et al. used ridges with ϕs=0.2, but it would be worthwhile to have additional experiments with a larger λx or λz to help confirm the computational findings discussed above. For example, would transverse superhydrophobic ridges produce drag reduction if the ϕs was designed to be extremely small?

Henoch et al. [32] evaluated the turbulent drag reduction of a surface fabricated from ‘nanograss’, i.e. an SHS made of posts with submicrometre diameter and a height of approximately 7 μm (figure 6e). Fields of wafers were stitched together to form a large sample that was placed in a water channel at the Naval Undersea Warfare Center, Newport, RI. Using a deflection method, the hydrodynamic forces were recorded for the nanograss and a smooth PVC baseplate. Essentially, no drag on the nanograss plate was observed for speeds up to 0.6 m s−1 (figure 6f). Moreover, greater than 50% drag reduction was observed for 150≤Reτ≤600, where Reτ is the friction Reynolds number, δ/δν. With ϕs=0.06, such high drag reduction is more than feasible. However, unlike the saturation in drag reduction observed by Daniello et al. [39] and Woolford et al. [31], these authors found a decrease in drag savings as the Reynolds number was increased. At the highest speeds tested, approximately 1.3 m s−1, a moderate 15% savings in drag was observed. Combined with the drag increase observed with the spanwise ridges evaluated by Woolford et al., these studies were the first indication that complex microstructures, especially those without predominantly streamwise orientation, may not produce as large a reduction in friction as observed with streamwise ridges.

7. Drag reduction in turbulent flows using random textures: NeverWet

Turbulent drag reduction using precisely fabricated nanostructures, such as ridges and posts, has been replicated across several laboratories. Although these surfaces provide insights into the mechanisms by which SHSs can reduce drag in turbulent flows, they are inherently not scalable. Multiple research groups have independently evaluated only one scalable surface, the commercial coating NeverWet, for turbulent drag reduction [21,22,37,40]. NeverWet is a sprayable blend of hydrophobic silica nanoparticles embedded within a silicone matrix (figure 2g). Aljallis et al. [40] were the first to evaluate the efficacy of this coating, using a tow tank facility. By spraying large (1.2 m × 0.6 m) aluminium plates, which were towed in a model basin, they monitored the frictional drag over a Reynolds number range of 500 000–6 000 000, based on length (ReL). They tested two variations of this coating, one slightly rougher than the other. The resultant coefficients of drag are shown in figure 7a. Once the flow became fully turbulent (ReL>1×106), no drag reduction was observed for the smoother variant of the coating, and a significant drag increase was observed for the rougher variant. The authors claim that surface roughness, and possibly wetting, explained their lack of observed drag reduction. This would be consistent with the findings of Woolford et al. [31] for wetted ridges.

Figure 7.
Turbulentdrag reduction: NeverWet, a case study. The commercially available coating NeverWet has been independently evaluated by four different research groups. When tested in a large tow tank facility (a), two variants on the coating showed either drag ...

Zhang et al. [37] studied the drag on a NeverWet-coated substrate in their large-scale flow facility using particle image velocimetry (PIV). They observed increasing drag reduction with increasing Reynolds number, from 10% at Reτ=329 to 24% at Reτ=467. This was evidenced by the increased velocity measured far from the wall (figure 7b). They also showed that the turbulent structures were significantly reduced in the presence of the non-wetted NeverWet surface (figure 7c,d). It is important to recall that this drag reduction mechanism is absent in the laminar case, but was predicted by the computational works cited above. However, the drag reduction observed here and by Aljallis et al. [40] only occurred at low speeds. Computationally, however, the drag reduction has been predicted to increase with Reynolds number.

Very recently, Hokmabad et al. [21] used PIV to study the flow over a NeverWet-coated substrate in channel flow. In disagreement with the two previous studies, the authors found no increase in velocity near the wall of the SHS, i.e. no drag reduction. Interestingly, the authors forced the NeverWet to wet by using surfactants, and also found no change in the velocity profile of the wetted surface. This is similar to the case of SH-2 (smoother) from Aljallis et al. [40], but dissimilar to SH-1, the rougher variant. When fully wetted, any hydrodynamically rough SHS should behave like a rough wall, and increase the overall drag [23]. Surprisingly, Hokmabad et al. found that the non-wetted NeverWet surface did show a decrease in Reynolds shear stress for 20≤y+≤80 (figure 7e). This would indicate drag reduction, and would mechanistically agree with the computational work of Min & Kim [48], Martell et al. [52] and Busse & Sandham [54].

We also recently evaluated the NeverWet coating in a fully developed turbulent channel flow (figure 7f 57). Turbulent drag reduction has not been reported above a height-based Reynolds number of 10 000, and we also found no discernible difference in skin friction between a NeverWet-coated sample and a smooth base plate. In fact, at the higher Reynolds numbers tested, a slight increase in friction was observed, indicating a transitionally rough surface. Our lack of observed drag reduction matches that of SH-2 measured by Aljallis et al. [40], whose measurements of the NeverWet coating also matched the friction coefficient of an uncoated baseline sample at higher Reynolds numbers.

The resulting drag reduction of the four independent studies on NeverWet are presented as a function of Reτ in figure 7g. At low-friction Reynolds number, NeverWet has produced a 10–30% reduction in drag. However, the drag reduction always disappeared at higher friction Reynolds numbers, although the cut-off in Reτ would appear to be dependent on the flow facility. The most likely reasons for this discrepancy are the differing pressures in each test set-up, how quickly the pressure exceeds the capillary resistance of the NeverWet coating, as well as the viscous length scale of the flow.

8. Changes in superhydrophobic surfaces owing to pressure and roughness

Much like the biological SHSs shown in figure 1, many engineered SHSs, including the NeverWet coating, exhibit a random texture (figure 2hj). This topography is markedly different from the simple geometries of ridges and posts. The obvious question is: how does random texture affect the possibility of SHS drag reduction in turbulent flow? The first important difference, from both laminar drag reduction and the use of ridges or posts, is wetting (figure 8a). Water flowing over ridges exists in only two idealized states: fully wetted or fully non-wetted. Admittedly, there always exists a fraction of the solid surface ϕs that is wetted by the liquid. However, ϕs is effectively independent of the pressure [59] until the capillary resistance is overcome, and wetting occurs, or equivalently ϕs→1.0. For laminar drag reduction, this is not a problem as the pressure experienced during flow can be easily calculated, and the SHS designed accordingly [4]. For example, Lee & Kim [20] designed surfaces with ϕs=0.03 (figure 3bd), and were able to measure λx≈140 μm. Such large slip lengths could easily reduce drag in turbulent flow, except the surface would wet almost immediately owing to the large pressure fluctuations in the presence of turbulence. Seo et al. [49] have also shown that large stagnation pressures, of the order of 100 times the mean pressure of the flow, develop near the leading edge of the texture features on an SHS (figure 4e).

Figure 8.
The effect of pressure on ϕs. For surfaces with random texture, the meniscus is positioned at some equilibrium height that balances the local pressure with the capillary resistance of the surface. This can leave a significant portion of the surface ...

As the pressure on an SHS is increased, the meniscus will penetrate further into the texture of the surface [58]. This penetration will cause an increase in ϕs once the meniscus reaches any previously non-wetted asperities. In this case, the area of slip, and, therefore, the possibility of drag reduction, is a function of pressure. Even for laminar flows, Ybert et al. [60] have shown that the slip length decreases with increasing pressure as a result of meniscus curvature. Srinivasan et al. [22] previously used confocal microscopy to image the meniscus of an SHS directly by dyeing the water and solid surface separately (figures 2h and and88b). For this image, ϕs≈0.12. An increase in ϕs dictates a change in the apparent contact angle according to the Cassie–Baxter relation, equation (2.4). The effect of pressure on θ* has been shown for an evaporating droplet atop micropillars in figure 8c [58]. As the droplet evaporated, the Laplace pressure within the droplet increased following 2γLV/R, where R is the radius of the droplet. Thus, even before the wetting transition was observed around t=16 min, θ* had decreased by almost 20°. Thus, for SHSs to be effective in turbulent flow, ϕs must remain low even at significant pressures. For example, at a depth of 1 m, the hydrostatic pressure is of the order of 10 kPa, causing the turbulent stagnation pressure fluctuations on the texture elements of an SHS to be of the order of 1 MPa [49].

If ϕs can vary with pressure, how will turbulent drag reduction be affected by SHSs that are either more or less resistant to changes in ϕs? Bidkar et al. [23] evaluated the skin friction on many different SHSs (figure 2i) that both increased (figure 9a) and decreased (figure 9b) the overall drag coefficient. To explain these results, the authors compared the observed drag reduction with the root-mean-squared roughness of the surfaces, k, non-dimensionalized by δν (k+). The resulting drag measurements for five of their surfaces are shown in figure 9c. For comparison, they also plotted the viscous sublayer, and 1/10th the viscous sublayer. Only for surfaces below one-tenth of the viscous sublayer was drag reduction observed. Above this threshold, the roughness elements of the SHS interacted with the boundary layer, causing viscous dissipation and form drag (figure 8a). They concluded that surfaces must exhibit k+<0.5 for turbulent drag reduction to be observed. However, recalling the findings summarized in figure 8, the surface must also exhibit a sufficiently large pressure resistance, or else wetting or an increase in ϕs will occur. This may, in turn, reduce or negate the effect of streamwise slip (λx=0). Contact angles measured following the flow experiments were only minimally reduced, indicating that wetting alone could not explain the absence of observed drag reduction, or the increase in observed drag, for some of the tested samples.

Figure 9.
The effect of roughness on turbulent drag reduction. Not all SHSs are capable of reducing friction in turbulent flow. For example, Bidkar et al. [23] measured the frictional drag on many different SHSs, both ones that increased drag (a) or decreased drag ...

Very recently, Seo & Mani [61] have used DNS to show that, for moderate values of ϕs, λx and λz can be represented by a single, effective slip length. Only for very small values of ϕs, such as the ridges of Park et al. [29], will λx and λz differ drastically. This matches well with the model of Fukagata et al. [51]. However, we note that streamwise ridges, by design, will have differing streamwise and spanwise slip lengths, and are perhaps a special case. For the randomly textured surfaces evaluated in figures 79, without any preferential streamwise or spanwise orientation, a single, effective slip length makes intuitive sense. Overall, scalable, random SHSs can still produce meaningful turbulent drag reduction, but their efficacy will be lessened owing to spanwise slip and roughness effects. This has been modelled by Fukagata et al., computed by several groups, and experimentally confirmed by several researchers.

Several other research groups have evaluated randomly structured SHSs in turbulent flow with varying degrees of success (table 1). Gogte et al. [30] coated Joukovsky (also spelled as Joukowski) hydrofoils with SHSs fabricated from modified sandpaper. Two SHSs, with k=8 μm and k=15 μm, provided 18% drag reduction at low speeds, and as little as 3% at higher speeds. For all cases, higher drag reduction was observed with the smoother SHS (An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i9.jpg and 0.1, for the two surfaces, respectively). Lu et al. [34] prepared SHSs from a blend of fluorosilanes, carbon nanotubes and a fluorosilicone matrix. When tested in a turbulent microchannel, they observed as high as 53% drag reduction, although they attributed the drag reduction to a delay in the onset of turbulence. Zhao et al. [38] coated a large, flat plate with this same coating, and measured the friction in a recirculating water tunnel. In the turbulent regime, essentially no drag reduction was observed for all measured flow speeds. This matches well with the work on NeverWet performed by Aljallis et al., where more than 30% friction drag reduction was observed in transitional turbulence, and no drag reduction (or even drag increase) was observed in the turbulent regime (figure 7b). Peguero & Breuer [36] measured the drag reduction of ridges, posts and a random SHS fabricated from hydrophobic particles dusted on a sandblasted aluminium substrate. No discernible drag reduction or increase could be deduced from the authors’ measurements. Jung & Bhushan [25] measured turbulent drag reduction in a microchannel using hierarchically structured posts (figure 2e,f). Drag reduction as high as 30% was achieved.

Although experimentalists have measured drag reduction using SHSs with varying degrees of success, a more detailed understanding has started to emerge. Overall, experimental results of SHSs in turbulent flow yield the following conclusions.

  • (1) SHS drag reduction in turbulent flow is possible, especially when the texture elements are aligned in the streamwise direction.
  • (2) Wetting owing to pressure fluctuations may remove the regions of slip, and thus mitigate any drag-reducing properties of the SHS, and may even cause drag increase if the surface's roughness interacts with the viscous sublayer.
  • (3) Even when the air pockets remain on the SHS, spanwise slip and roughness effects can cause an increase in the observed drag.
  • (4) An effective SHS for reducing drag in turbulent flow should have streamwise-aligned structures, with a capillary resistance of the order of MPa and a k+≤0.5.

9. The effect of Reynolds number on turbulent drag reduction

We conclude this review by discussing one final, somewhat open-ended, topic. Assuming one has fabricated an SHS capable of turbulent drag reduction, how does the drag reduction change with Reynolds number? There have been many different answers to this question, both experimentally and computationally. Min & Kim [48] have shown computationally that the non-dimensionalized slip length, An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i10.jpg, increased with increasing skin friction. The drag reduction should be enhanced as the Reynolds number is increased. This trend was also found by Busse & Sandham [54]. Experimentally, some researchers have observed increasing drag reduction with increasing Re, especially in rheometers. Barbier et al. [62] fabricated a series of hierarchical SHSs from anodized aluminium. They observed as much as 20% drag reduction in their turbulent rheometer set-up when their largest feature size was spaced 100 μm apart. They note that the observed drag reduction was always higher at higher Reynolds number, which they attributed to a delay in the onset of turbulence. Srinivasan et al. [22] coated rheometer rotors with an SHS fabricated by spray-coating (figure 2h). At Re<20 000, they observed no drag reduction in their Taylor–Couette flow. However, for Re≥30 000, significant drag reduction was observed. They also derived a skin friction law that incorporated wall slip, which yielded the relation that An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i11.jpg scales with Re1/2. Good agreement was shown between their experiments and this relation.

In microchannels, Jung & Bhushan [25], Daniello et al. [39] and Lu et al. [34] observed increasing drag reduction with increasing Reynolds number (figure 6a). However, it has been proposed that, to properly characterize turbulent drag reduction over SHSs, the flow must be allowed to adjust to the new boundary condition imposed by the regions of slip and no slip [63,64]. The larger the k, the further downstream before the flow has adjusted fully. For example, around 30δ is required when δ/k<25 [63]. For microchannels, this distance can be longer than the entire length of the channel. Thus, measurements in microchannels could inaccurately portray the full effects of turbulence over SHSs.

Accordingly, in large-scale, turbulent boundary layer flow facilities, the opposite trend has been observed experimentally. Both Aljallis et al. [40] and Hokmabad et al. [21] found essentially no skin friction drag reduction for the NeverWet coating at high Re. In both these studies, a small amount of drag reduction was observed, but only at the lowest Reynolds numbers tested. Similar trends have been reported for other SHSs, such as some of the surfaces tested by Bidkar et al. [23] (figure 9). However, recall from figure 8 that increasing the pressure can raise ϕs. For channel flow, the pressure drop is proportional to the Reynolds number. Thus, it is uncertain whether increasing the Reynolds number mitigated the observed drag reduction in some studies owing to pressure, increased wall shear stress or if the effect was due to a specific Reynolds number dependence. Most likely, all three will affect the efficacy of an SHS to reduce drag in turbulent flow. To the best of our knowledge, there has not been a fully developed turbulent channel or turbulent boundary layer flow that produced increased drag reduction with increased Re. In fact, all studies have shown that the drag reduction decreased with Re, so much so that drag increased beyond baseline resistance at some critical Reynolds number. Conversely, the rheometer and microchannel studies have shown no drag reduction until the onset of turbulence, and then increased drag reduction as Re increased.

The discrepancy alluded to above also highlights the fact that decreasing δν alters both An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i12.jpg and k+. However, the effect of altering An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i13.jpg with Re is not straightforward. For example, Kim & Min [48] have hypothesized, and Fukagata et al. [51] have shown, that the effects of streamwise and spanwise slip act independently. Therefore, an increase in Re could cause more spanwise slip without increasing the streamwise slip, which would decrease the observed drag reduction. Moreover, the meniscus height will change as a function of pressure (figure 8c), effectively changing the k that the flow sees. Thus, even if the effects of increasing Re on An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i14.jpg and An external file that holds a picture, illustration, etc.
Object name is rsta20160189-i15.jpg are well understood, there is still much work to be done uncovering the effects of Re on k. Seo et al. [49] have started investigating these effects by allowing the meniscus to fluctuate during their turbulent DNS calculations (figure 4e). They found that the large stagnation pressures that were generated near the leading edge of their posts were lessened when the meniscus was free to move. However, because the meniscus was pinned at the top of the posts, and posts possess a negligible k+, it is unlikely that the deleterious effects of increasing Re on k+ were captured in their calculations. Very recently, Crowdy [65] extended his previous work [66] by deriving an analytical formula for the slip length of superhydrophobic ridges with a curved air–water interface. For ridges with ϕs=0.1, when the angle of the meniscus curved from flat (0°) to −30° (into the texture), the slip length decreased by approximately 14%. Recall that, on randomly textured SHSs, the meniscus can be highly curved from the texture alone (figure 8a). It is very encouraging, then, that significant amounts of drag reduction have been observed on SHSs with random texture, even though the meniscus may be highly curved on these random surfaces (figure 9).

10. Conclusion and outlook

In this review, we have tried to summarize the vibrant field of turbulent drag reduction using biomimetic SHSs. Much like the unpredictability of turbulence itself, many of the drag reduction studies have produced unexpected results, such as an increase in drag where a reduction was expected. Wetting, roughness effects and spanwise slip seem to be the leading culprits that have hampered some surfaces from providing turbulent friction reduction. For the field to move forward, three important questions need to be answered.

First and foremost, what is the optimal microstructure for turbulent drag reduction, and how can this texture be created on a large scale, and modelled computationally? Micrometre-scale ridges aligned in the streamwise direction appear to reduce drag unambiguously. However, as all such surfaces are either fabricated in a cleanroom or require moulds and are fabricated similarly, it is unlikely that this type of texture will be viable for a realistic large-scale application. Moreover, if the recent experimental studies are any indication, materials scientists prefer making surfaces with novel, random textures that are capable of reducing drag in turbulent flow, rather than exploring how to scale up the proven technology of ridges. Although computationally challenging, numerical modellers can investigate the effects of randomness on drag reduction in turbulent flow. Results in laminar flow with streamwise and spanwise randomness seem promising [67], but the largest form drag contributions will arise owing to large variations in the wall-normal direction [68]. Yet, there have been no computational studies investigating this type of geometry such as for ridges or posts of differing heights.

Second, how can one understand the contrasting results of drag reduction in small-scale and large-scale flow facilities, especially with regard to the observed dependence of drag reduction on Reynolds number? Arguably, the biggest impact of drag-reducing SHSs would be their successful application on external flows of marine vehicles, or internal flows of pipelines, for example. The discrepancies between the results of microchannels, rheometers, towing tanks and larger flow channels need to be either remedied or understood further. To date, only streamwise ridges have consistently shown drag reduction in large- and small-scale test facilities (table 1).

Finally, what is the expected longevity of an SHS that does indeed reduce frictional drag in turbulent flow? Once the surfaces work consistently, and many of the excellent studies cited here exhibit surfaces that do, how long will the drag reduction persist? For example, diffusion of N2 into water is known to limit the longevity of the air pockets entrained by the diving bell spider [69,70]. To this point, how should SHSs be designed so as to not increase drag once the air pockets have been removed? Based on the results of Bidkar et al. [23], we have proposed here that a roughness of k+≤0.5 is required for turbulent drag reduction. This threshold may need to be reduced if one considers the fully wetted case that may result over time.

Alternatively, much work has been done on shark-skin inspired surfaces, known as riblets [19,62,7175]. These bioinspired surfaces modify the near-wall vorticity during turbulent flow, reducing skin friction in the fully wetted Wenzel state. As such, they could prove an attractive, underlying texture for SHSs to display once the air pockets have been removed. Prince et al. [19] and Barbier et al. [62] have made the first attempts at combining these two technologies (figure 2a), but with limited success. As the optimal length scales for the two technologies are orders of magnitude apart, it remains to be seen whether the two mechanisms can be combined synergistically or drag reduction by riblets can only succeed drag reduction by SHSs once the air pockets have been removed.

Contrastingly, what strategies exist for restoring or replenishing the air depleted from the SHS, and, consequently, the slip region? Several excellent studies have already shown that electrochemical gas generation [76,77], Leidenfrost heating [78,79] or even thermodynamically stable water vapour [80,81] can restore the air pockets of a fully wetted SHS. Incorporating some of these failsafe mechanisms may help SHSs transition from the laboratory to actual marine environments. Overall, if we wish to take full advantage of surfaces that entrap air pockets, as penguins, spiders and lotus leaves have done for millennia, much work still needs to be done.

Acknowledgements

The authors thank Hangjian Ling and Dr Joseph Katz at Johns Hopkins University for their insightful discussions. The authors are also indebted to Dr Ali Mani, at Stanford University, and Dr Stephano Leonardi, at the University of Texas at Dallas, for their excellent suggestions and comments.

Authors' contributions

K.B.G., J.W.G., M.P., S.L.C. and A.T. wrote the manuscript. K.B.G. fabricated the NeverWet-coated substrate, which was evaluated by J.W.G.

Competing interests

The authors have no competing interests.

Funding

We thank Dr Ki-Han Kim and the Office of Naval Research for financial support under grant no. N00014-12-1-0874.

References

1. Bhushan B, Jung YC 2011. Natural and biomimetic artificial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction. Prog. Mater. Sci. 56, 1–108. (doi:10.1016/j.pmatsci.2010.04.003)
2. Davenport J, Hughes RN, Shorten M, Larsen PS 2011. Drag reduction by air release promotes fast ascent in jumping emperor penguins—a novel hypothesis. Mar. Ecol. Prog. Ser. 430, 171–182. (doi:10.3354/meps08868)
3. Neumann D, Kureck A 2013. Composite structure of silken threads and a proteinaceous hydrogel which form the diving bell wall of the water spider Agyroneta aquatica. SpringerPlus 2, 223 (doi:10.1186/2193-1801-2-223) [PMC free article] [PubMed]
4. Rothstein JP. 2010. Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109. (doi:10.1146/annurev-fluid-121108-145558)
5. Durbin PA, Reif BAP 2011. Statistical theory and modeling for turbulent flows, 2nd edn Chichester, UK: Wiley.
6. Liepmann D, Gharib M 1992. The role of streamwise vorticity in the near-field entrainment of round jets. J. Fluid Mech. 245, 643–668. (doi:10.1017/S0022112092000612)
7. Ceccio SL. 2010. Friction drag reduction of external flows with bubble and gas injection. Annu. Rev. Fluid Mech. 42, 183–203. (doi:10.1146/annurev-fluid-121108-145504)
8. Mäkiharju SA, Perlin M, Ceccio SL 2012. On the energy economics of air lubrication drag reduction. Int. J. Naval Architect. Ocean Eng. 4, 412–422. (doi:10.2478/ijnaoe-2013-0107)
9. Young T. 1805. An essay on the cohesion of fluids. Phil. Trans. R. Soc. 95, 65 (doi:10.1098/rstl.1805.0005)
10. Schulman F, Zisman WA 1952. The spreading of liquids on low-energy surfaces. V. Perfluorodecanoic acid monolayers. J. Colloid Sci. 7, 465–481. (doi:10.1016/0095-8522(52)90030-5).
11. de Gennes PG, Brochard-Wyart F, Quere D 2004. Capillarity and wetting phenomena: drops, bubbles, pearls, waves. Berlin, Germany: Springer.
12. Wenzel RN. 1936. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 28, 988 (doi:10.1021/ie50320a024)
13. Cassie ABD, Baxter S 1944. Wettability of porous surfaces. Trans. Faraday Soc. 40, 546–551. (doi:10.1039/tf9444000546)
14. Marmur A. 2004. The lotus effect: superhydrophobicity and metastability. Langmuir 20, 3517–3519. (doi:10.1021/la036369u) [PubMed]
15. Mockenhaupt B, Ensikat H-J, Spaeth M, Barthlott W 2008. Superhydrophobicity of biological and technical surfaces under moisture condensation: stability in relation to surface structure. Langmuir 24,13591–13597. (doi:10.1021/la802351h) [PubMed]
16. Kota AK, Kwon G, Tuteja A 2014. The design and applications of superomniphobic surfaces. NPG Asia Mater. 6, e109 (doi:10.1038/am.2014.34)
17. Srinivasan S, Chhatre SS, Guardado JO, Park K-C, Parker AR, Rubner MF, McKinley GH, Cohen RE 2014. Quantification of feather structure, wettability and resistance to liquid penetration. J. R. Soc. Interface 11, 20140287 (doi:10.1098/rsif.2014.0287) [PMC free article] [PubMed]
18. Lepore E, Giorcelli M, Saggese C, Tagliaferro A, Pugno N 2013. Mimicking water striders’ legs superhydrophobicity and buoyancy with cabbage leaves and nanotube carpets. J. Mater. Res. 28, 976–983. (doi:10.1557/jmr.2012.382)
19. Prince JF, Maynes D, Crockett J 2014. Pressure drop measurements in turbulent channel flow over superhydrophobic surfaces with riblets. In Proc. ASME 2014 12th Int. Conf. on Nanochannels, Microchannels and Minichannels, Chicago, IL, 3–7 August 2014, p. V001T008A004. New York, NY: ASME.
20. Lee C, Kim CJ 2009. Maximizing the giant liquid slip on superhydrophobic microstructures by nanostructuring their sidewalls. Langmuir 25, 12812–12818. (doi:10.1021/la901824d) [PubMed]
21. Hokmabad BV, Ghaemi S 2016. Turbulent flow over wetted and non-wetted superhydrophobic counterparts with random structure. Phys. Fluids 28, 015112 (doi:10.1063/1.4940325)
22. Srinivasan S, Kleingartner JA, Gilbert JB, Cohen RE, Milne AJB, McKinley GH 2015. Sustainable drag reduction in turbulent Taylor–Couette flows by depositing sprayable superhydrophobic surfaces. Phys. Rev. Lett. 114, 014501–014501-5 (doi:10.1103/PhysRevLett.114.014501) [PubMed]
23. Bidkar RA, Leblanc L, Kulkarni AJ, Bahadur V, Ceccio SL, Perlin M 2014. Skin friction drag reduction in the turbulent regime using random-textured hydrophobic surfaces. Phys. Fluids 26, 085108 (doi:10.1063/1.4892902)
24. Srinivasan S, Chhatre SS, Mabry JM, Cohen RE, McKinley GH 2011. Solution spraying of poly(methyl methacrylate) blends to fabricate microtextured, superoleophobic surfaces. Polymer 52, 3209–3218. (doi:10.1016/j.polymer.2011.05.008)
25. Jung YC, Bhushan B 2010. Biomimetic structures for fluid drag reduction in laminar and turbulent flows. J. Phys. Condensed Matter 22, 035104 (doi:10.1088/0953-8984/22/3/035104) [PubMed]
26. Golovin K, Lee DH, Mabry JM, Tuteja A 2013. Transparent, flexible, superomniphobic surfaces with ultra-low contact angle hysteresis. Angew. Chem. Int. Ed. 52,13007–13011. (doi:10.1002/anie.201307222) [PubMed]
27. Srinivasan S, Choi W, Park K-C, Chhatre SS, Cohen RE, McKinley GH 2013. Drag reduction for viscous laminar flow on spray-coated non-wetting surfaces. Soft Matter 9, 5691 (doi:10.1039/c3sm50445j)
28. Li L, Huang T, Lei J, He J, Qu L, Huang P, Zhou W, Li N, Pan F 2015. Robust biomimetic-structural superhydrophobic surface on aluminum alloy. ACS Appl. Mater. Interfaces 7, 1449–1457. (doi:10.1021/am505582j) [PubMed]
29. Park H, , Guangyi S, , Kim CJ.2013. Turbulent drag reduction on superhydrophobic surfaces confirmed by built-in shear sensing. In Proc. 2013 IEEE 26th Int. Conf. on Micro Electro Mechanical Systems (MEMS), Taipei, Taiwan, 20–24 January 2013, pp. 1183–1186. New York, NY: IEEE.
30. Gogte S, Vorobieff P, Truesdell R, Mammoli A, van Swol F, Shah P, Brinker CJ 2005. Effective slip on textured superhydrophobic surfaces. Phys. Fluids 17, 051701 (doi:10.1063/1.1896405)
31. Woolford B, Prince J, Maynes D, Webb BW 2009. Particle image velocimetry characterization of turbulent channel flow with rib patterned superhydrophobic walls. Phys. Fluids 21, 085106 (doi:10.1063/1.3213607)
32. Henoch C, , Krupenkin T, , Kolodner P, , Taylor J, , Hodes M, , Lyons A, , Charles P, , Breuer K.2006. Turbulent drag reduction using superhydrophobic surfaces. In Proc. 3rd AIAA Flow Control Conf., San Francisco, CA, 5–8 June 2006. Reston, VA: American Institute of Aeronautics and Astronautics.
33. Tian H, Zhang J, Wang E, Yao Z, Jiang N 2015. Experimental investigation on drag reduction in turbulent boundary layer over superhydrophobic surface by TRPIV. Theor. Appl. Mech. Lett. 5, 45–49. (doi:10.1016/j.taml.2015.01.003)
34. Lu S, Yao ZH, Hao PF, Fu CS 2013. Drag reduction in turbulent flows over superhydrophobic surfaces with micro-nano textures. Mech. Eng 35, 20–24. (doi:10.6052/1000-0879-13-098)
35. Watanabe K, Udagawa H 2001. Drag reduction of non-Newtonian fluids in a circular pipe with a highly water-repellent wall. AIChE J. 47, 256–262. (doi:10.1002/aic.690470204)
36. Peguero C, Breuer K 2009. On drag reduction in turbulent channel flow over superhydrophobic surfaces. In Advances in turbulence XII (ed. B Eckhardt), pp. 233–236. Heidelberg, Germany: Springer.
37. Zhang J, Tian H, Yao Z, Hao P, Jiang N 2015. Mechanisms of drag reduction of superhydrophobic surfaces in a turbulent boundary layer flow. Exp. Fluids 56, 1–13. (doi:10.1007/s00348-015-2047-y)
38. Zhao J-P, Du X-D, Shi X-H 2007. Experimental research on friction-reduction with super-hydrophobic surfaces. J. Mar. Sci. Appl. 6, 58–61. (doi:10.1007/s11804-007-7007-3)
39. Daniello RJ, Waterhouse NE, Rothstein JP 2009. Drag reduction in turbulent flows over superhydrophobic surfaces. Phys. Fluids 21, 085103 (doi:10.1063/1.3207885)
40. Aljallis E, Sarshar MA, Datla R, Sikka V, Jones A, Choi C-H 2013. Experimental study of skin friction drag reduction on superhydrophobic flat plates in high Reynolds number boundary layer flow. Phys. Fluids 25, 025103 (doi:10.1063/1.4791602)
41. White FM. 2016. Fluid mechanics, 8th edn New York, NY: McGraw-Hill Education.
42. Navier C. 1823. Memoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Inst. France 6, 389–440.
43. Ou J, Rothstein JP 2005. Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17, 103606 (doi:10.1063/1.2109867)
44. Ou J, Perot B, Rothstein JP 2004. Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 4635 (doi:10.1063/1.1812011)
45. Bartolo D, Bouamrirene F, Verneuil É, Buguin A, Silberzan P, Moulinet S 2006. Bouncing or sticky droplets: impalement transitions on superhydrophobic micropatterned surfaces. Europhys. Lett. 74, 299 (doi:10.1209/epl/i2005-10522-3)
46. Lauga E, Stone HA 2003. Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77. (doi:10.1017/S0022112003004695)
47. Durbin PA, Reif BAP 2001. Statistical theory and modeling for turbulent flows. Chichester, UK: Wiley.
48. Min T, Kim J 2004. Effects of hydrophobic surface on skin friction drag. Phys. Fluids 16, L55–L58. (doi:10.1063/1.1755723)
49. Seo J, García-Mayoral R, Mani A 2015. Pressure fluctuations and interfacial robustness in turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 783, 448–473. (doi:10.1017/jfm.2015.573)
50. Pope SB. 2011. Simple models of turbulent flows. Phys. Fluids 23, 011301 (doi:10.1063/1.3531744)
51. Fukagata K, Kasagi N, Koumoutsakos P 2006. A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces. Phys. Fluids 18, 051703 (doi:10.1063/1.2205307)
52. Martell MB, Rothstein JP, Perot JB 2010. An analysis of superhydrophobic turbulent drag reduction mechanisms using direct numerical simulation. Phys. Fluids 22, 065102 (doi:10.1063/1.3432514)
53. Jelly TO, Jung SY, Zaki TA 2014. Turbulence and skin friction modification in channel flow with streamwise-aligned superhydrophobic surface texture. Phys. Fluids 26, 095102 (doi:10.1063/1.4894064)
54. Busse A, Sandham ND 2012. Influence of an anisotropic slip-length boundary condition on turbulent channel flow. Phys. Fluids 24, 055111 (doi:10.1063/1.4719780)
55. Park H, Sun G, Kim CJ 2014. Superhydrophobic turbulent drag reduction as a function of surface grating parameters. J. Fluid Mech. 747, 722–734. (doi:10.1017/jfm.2014.151)
56. Author A. 2015. Mechanisms of drag reduction of superhydrophobic surfaces in a turbulent boundary layer flow. Exp. Fluids. 59, 179.
57. Gose JW, Golovin K, Tuteja A, Ceccio SL, Perlin M Submitted Experimental investigation of turbulent skin-friction drag reduction along superhydrophobic materials.
58. Papadopoulos P, Mammen L, Deng X, Vollmer D, Butt H-J 2013. How superhydrophobicity breaks down. Proc. Natl Acad. Sci USA 110, 3254–3258. (doi:10.1073/pnas.1218673110) [PubMed]
59. Ishino C, Okumura K, Quéré D 2004. Wetting transitions on rough surfaces. Europhys. Lett. 68, 419–425. (doi:10.1209/epl/i2004-10206-6)
60. Ybert C, Barentin C, Cottin-Bizonne C, Joseph P, Bocquet L 2007. Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19, 123601 (doi:10.1063/1.2815730)
61. Seo J, Mani A 2016. On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces. Phys. Fluids 28, 025110 (doi:10.1063/1.4941769)
62. Barbier C, , Jenner E, , D’Urso B.2014. Large drag reduction over superhydrophobic riblets. (http://arxiv.org/abs/1406.0787. )
63. Lee S-H, Sung HJ 2007. Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall. J. Fluid Mech. 584, 125–146. (doi:10.1017/S0022112007006465)
64. Saito N, Pullin DI 2014. Large eddy simulation of smooth–rough–smooth transitions in turbulent channel flows. Int. J. Heat Mass Transf. 78, 707–720. (doi:10.1016/j.ijheatmasstransfer.2014.06.088)
65. Crowdy DG. 2016. Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7. (doi:10.1017/jfm.2016.88)
66. Crowdy D. 2010. Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22, 121703 (doi:10.1063/1.3531683)
67. Samaha MA, Vahedi Tafreshi H, Gad-el-Hak M 2011. Modeling drag reduction and meniscus stability of superhydrophobic surfaces comprised of random roughness. Phys. Fluids 23, 012001 (doi:10.1063/1.3537833)
68. White FM. 1991. Viscous fluid flow. New York, NY: McGraw-Hill.
69. Brooks AP. 2010. Fluid mechanics of bubble capture by the diving bell spider. Cambridge, MA: Massachusetts Institute of Technology.
70. Seymour RS, Matthews PG 2013. Physical gills in diving insects and spiders: theory and experiment. J. Exp. Biol. 216, 164–170. (doi:10.1242/jeb.070276) [PubMed]
71. Garcia-Mayoral R, Jimenez J 2011. Drag reduction by riblets. Phil. Trans. R. Soc. A 369, 1412–1427. (doi:10.1098/rsta.2010.0359) [PubMed]
72. Bixler GD, Bhushan B 2013. Fluid drag reduction with shark-skin riblet inspired microstructured surfaces. Adv. Funct. Mater. 23, 4507–4528. (doi:10.1002/adfm.201203683)
73. Friedmann E, Richter T 2010. Optimal microstructures drag reducing mechanism of riblets. J. Math. Fluid Mech. 13, 429–447. (doi:10.1007/s00021-010-0033-y)
74. Bixler GD, Bhushan B 2013. Shark skin inspired low-drag microstructured surfaces in closed channel flow. J. Colloid Interface Sci. 393, 384–396. (doi:10.1016/j.jcis.2012.10.061) [PubMed]
75. Bechert DW, Bruse M, Hage W, Van Der Hoeven JGT, Hoppe G 1997. Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. J. Fluid Mech. 338, 59–87. (doi:10.1017/S0022112096004673)
76. Lee C, Kim C-J 2011. Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction. Phys. Rev. Lett. 106, 014502-1–014502-4. (doi:10.1103/PhysRevLett.106.014502) [PubMed]
77. Xu M, Sun G, Kim C-J 2014. Infinite lifetime of underwater superhydrophobic states. Phys. Rev. Lett. 113, 136103-1–136103-5. (doi:10.1103/PhysRevLett.113.136103) [PubMed]
78. Vakarelski IU, Patankar NA, Marston JO, Chan DY, Thoroddsen ST 2012. Stabilization of Leidenfrost vapour layer by textured superhydrophobic surfaces. Nature 489, 274–277. (doi:10.1038/nature11418) [PubMed]
79. Vakarelski IU, Chan DY, Thoroddsen ST 2014. Leidenfrost vapour layer moderation of the drag crisis and trajectories of superhydrophobic and hydrophilic spheres falling in water. Soft Matter 10, 5662–5668. (doi:10.1039/c4sm00368c) [PubMed]
80. Jones PR. et al 2015. Sustaining dry surfaces under water. Sci. Rep. 5, 12311 (doi:10.1038/srep12311) [PMC free article] [PubMed]
81. Patankar NA.2015. Thermodynamics of sustaining gases in the roughness of submerged superhydrophobic surfaces. (http://arxiv.org/abs/1505.06233. )

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