rsta.royalsocietypublishing.org Review Cite this article: Golovin KB, Gose JW, Perlin M, Ceccio SL, Tuteja A. 2016 Bioinspired surfaces for turbulent drag reduction. Phil. Trans. R. Soc. A 374: 20160189. http://dx.doi.org/10.1098/rsta.2016.0189 Accepted: 27 April 2016 One contribution of 12 to a theme issue ‘Bioinspired hierarchically structured surfaces for green science’. Subject Areas: materials science Keywords: drag reduction, turbulence, superhydrophobic, biomimetics, hierarchical, slip Author for correspondence: Anish Tuteja e-mail: [email protected]† Present address: Biointerfaces Institute, University of Michigan, 2800 Plymouth Road, A186 NCRC, Building 10, Ann Arbor, MI 48109, USA. Bioinspired surfaces for turbulent drag reduction Kevin B. Golovin 1 , James W. Gose 2 , Marc Perlin 2 , Steven L. Ceccio 2 and Anish Tuteja 1,3,4,† 1 Department of Materials Science and Engineering, 2 Department of Naval Architecture and Marine Engineering, 3 Department of Chemical Engineering, and 4 Department of Macromolecular Science and Engineering, University of Michigan, Ann Arbor, MI 48109, USA 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’. 1. Introduction Water effortlessly rolls off the leaf of the lotus plant [1]. Penguins seamlessly dive to ocean depths, trapping 2016 The Author(s) Published by the Royal Society. All rights reserved. on April 20, 2018 http://rsta.royalsocietypublishing.org/ Downloaded from
Transcript
rsta.royalsocietypublishing.org
ReviewCite this article: Golovin KB, Gose JW, PerlinM, Ceccio SL, Tuteja A. 2016 Bioinspiredsurfaces for turbulent drag reduction. Phil.Trans. R. Soc. A 374: 20160189.http://dx.doi.org/10.1098/rsta.2016.0189
Accepted: 27 April 2016
One contribution of 12 to a theme issue‘Bioinspired hierarchically structured surfacesfor green science’.
†Present address: Biointerfaces Institute,University of Michigan, 2800 Plymouth Road,A186 NCRC, Building 10, Ann Arbor, MI 48109,USA.
Bioinspired surfaces forturbulent drag reductionKevin B. Golovin1, James W. Gose2, Marc Perlin2,
Steven L. Ceccio2 and Anish Tuteja1,3,4,†
1Department of Materials Science and Engineering,2Department of Naval Architecture and Marine Engineering,3Department of Chemical Engineering, and 4Department ofMacromolecular Science and Engineering, University of Michigan,Ann Arbor, MI 48109, USA
In this review, we discuss how superhydrophobicsurfaces (SHSs) can provide friction drag reduction inturbulent flow. Whereas biomimetic SHSs are knownto reduce drag in laminar flow, turbulence addsmany new challenges. We first provide an overviewon designing SHSs, and how these surfaces cancause slip in the laminar regime. We then discussrecent studies evaluating drag on SHSs in turbulentflow, both computationally and experimentally. Theeffects of streamwise and spanwise slip for canonical,structured surfaces are well characterized by directnumerical simulations, and several experimentalstudies have validated these results. However, thecomplex and hierarchical textures of scalable SHSsthat can be applied over large areas generateadditional complications. Many studies on suchsurfaces have measured no drag reduction, or evena drag increase in turbulent flow. We discuss howsurface wettability, roughness effects and some newlyfound scaling laws can help explain these variedresults. Overall, we discuss how, to effectivelyreduce drag in turbulent flow, an SHS shouldhave: preferentially streamwise-aligned features toenhance favourable slip, a capillary resistance of theorder of megapascals, and a roughness no largerthan 0.5, when non-dimensionalized by the viscouslength scale.
This article is part of the themed issue ‘Bioinspiredhierarchically structured surfaces for green science’.
1. IntroductionWater effortlessly rolls off the leaf of the lotus plant[1]. Penguins seamlessly dive to ocean depths, trapping
2016 The Author(s) Published by the Royal Society. All rights reserved.
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numerous air bubbles between their feathers [2]. The diving bell spider lives life completelysubmerged, breathing in a pocket of air it has dragged beneath the surface of the water [3]. Naturehas masterfully controlled the air–water interface for millions of years. Recently, the mechanismsbehind these amazing natural occurrences of exceptional water repellency have begun to bebetter understood.
The above-mentioned examples are facilitated by trapping pockets of air within the pores ofthe outer surface of the organisms, be it hairy insect limbs, feathers or waxy leaves. One use ofthis entrapped air is to reduce the frictional drag when moving through the water, as penguinsdo when they dive [2]. The purpose of this review is to understand how entrapped air can reducethe frictional drag on an object within turbulent flow. Although much work has been done tounderstand the effects of these water-repellent surfaces in laminar flows [4], turbulence addsmany new challenges. Eddy formation, vorticity, pressure fluctuations, significant wall shear andan 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 turbulentconditions [7,8]. Thus, extending our understanding of these surfaces to non-laminar flows isessential. In the first part of this review, we discuss how surfaces can be designed to robustlytrap pockets of air under water. In the second part, the mechanism by which such water-repellentsurfaces reduce drag in laminar flows is explained. We then discuss the difficulties that arise inthe presence of turbulence, and highlight the current research being conducted to understandthis phenomenon, both computationally and experimentally. We complete the review by notingthe still-unsolved challenges that must be addressed before superhydrophobic surfaces (SHSs)should be considered for turbulent drag reduction applications.
2. Designing superhydrophobic surfacesA drop of water makes a contact angle, θ , when placed on a smooth, chemically homogeneoussurface, which can be determined by a force or energy balance, as first formalized by Young [9],
cos θ = γSV − γSL
γLV. (2.1)
Here, γSV is the solid surface energy, γSL is the liquid–solid interfacial tension and γLV is theliquid’s surface tension. Water adopts θ values approaching 0◦ (hydrophilic) on clean glass, andvalues near 120◦ (hydrophobic) on a perfluorinated monolayer [10]. Whereas θ represents a singlethermodynamic equilibrium, most surfaces exhibit a range of contact angles. The maximum angleis achieved as liquid initially advances on a dry substrate, and is called the advancing contactangle, 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,�θ ≡ θadv − θrec.
Surface roughness and chemical inhomogeneity are the two most common factors thatinfluence contact angle hysteresis. Wenzel [12] first formalized the effect of roughness on theobserved apparent contact angle of a droplet on a textured surface, in an equation that now bearshis name,
cos θ∗ = r cos θ . (2.2)
Here, θ∗ is the observed apparent contact angle, and r is the Wenzel roughness, defined as theratio of the actual surface area to its projected area. Hence, r always exceeds unity and enhancesthe intrinsic phobic/philic nature of the substrate.
Wenzel assumed that the rough surface was completely wetted by the liquid. Studyingcontact angles of liquids atop metal gratings, Cassie & Baxter [13] proposed another possibleconfiguration, in which part of the drop was supported by air
cos θ∗ = f1 cos θ + f2 cos π , (2.3)
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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) American Chemical Society.) Animals such as penguins [2] (e) and ducks (f ) also display water repellency owing to thetexture of their feathers (g–i) [17] (with permission). The diving bell spider lives underwater inside a plastron ( j), a bubble of airit drags beneath the water using spider silk (k,l) [3]. A water strider (m) similarly controls the air–water interface with the finehairs on its legs (n). ((a,b,m,n) From (Science 7 December 2007: vol. 318, Issue 5856, pp. 1618–1622; doi:10.1126/science.1148326).Reprinted with permission from AAAS.) (Online version in colour.)
where f1 is the total surface area of liquid in contact with a solid (with contact angle θ ) and f2 isthe surface area in contact with air (with a contact angle of π ). Assuming the wetted solid is roughand the liquid meniscus is flat, it is convenient to write [14]
cos θ∗ = rφφs cos θ + (1 − φs) cos π , (2.4)
where φs is the fraction of solid in contact with the liquid and rφ is the Wenzel roughness ofthe wetted solid. One can see that θ∗ becomes large as φs approaches zero, or when rφ � 1.0and θ > 90◦.
When θ∗ > 150◦ and �θ is low (an exact cut-off for contact angle hysteresis has never beenstandardized, but is probably no larger than �θ = 10◦), the surface is denoted as an SHS. Naturehas provided numerous SHSs to be studied, including: leaves from broccoli, lotus, lacinato kale,savoy cabbage, Indian hemp and taro plants [15] (figure 1a–d). Superhydrophobicity is alsopresent in animal features such as the feathers of ducks, geese and penguins [2,17] (figure 1e–i), aswell as the hairy legs of diving bell spiders [3] and water striders [18] (figure 1j–n). Two commonfeatures are found on all such surfaces. First, they possess hierarchical texture, with features in themicrometre and submicrometre ranges. Second, they all possess at least moderately low surfaceenergy, usually achieved by a waxy coating [1].
By mimicking these natural textures, researchers have engineered many artificial SHSs. Severalexamples of such hierarchical surfaces that have also been tested for turbulent drag reductionare 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] formednanoscale roughness on the sidewalls of the canonical geometry of posts (figure 2b–d). Similarly,Jung & Bhushan [25] self-assembled the wax responsible for the water repellency of the lotusleaf onto the tops of silicon posts (figure 2e,f ). SHSs can also be fabricated without any sortof initial, regular structure. A commercially available superhydrophobic coating, NeverWet�,combines silica nanoparticles in a silicone matrix [21] (figure 2g). Randomly textured SHSs canalso be fabricated using spray-coating [22,24,26,27] (figure 2h), thermal deposition [23] (figure 2i)
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Figure 2. Engineered superhydrophobic surfaces tested in turbulence. Hierarchical ridges (a) [19] or posts (b–d) have beenfabricated using microlithography [20]. ((a) Reprinted with permission from [19]. With permission of ASME. (b–d) Reprintedwith permission from [20]. Copyright (2009) American Chemical Society.) Nanotexturing can also be achieved using vapourdeposition techniques (e,f ). (Adapted from [20]. Copyright IOP Publishing. Reproducedwith permission. All rights reserved.)The texture of the commercial superhydrophobic spray NeverWet (g). (Reprinted from [21]. With the permission of AIPPublishing.) Other randomly structured superhydrophobic surfaces can be fabricated using spray-coating (h) [22], thermaldeposition (i) [23] or chemical etching procedures (j). ((h) Reprinted from [24]. Copyright (2011), with permission from Elsevier.(i) Reprinted from [23]. AIP Publishing.)
or chemical etching processes [28] (figure 2j). A summary of previously fabricated SHSs that havebeen evaluated in turbulent flow is presented in table 1. Most of these studies will be discussedin detail in the following sections. However, before diving into turbulent waters, we first reviewhow SHSs have been shown to effectively reduce friction drag in laminar flow.
3. Superhydrophobic friction drag reduction in laminar flowWhen liquid flows over a solid surface, the usual boundary condition assumed is that the velocityof 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 canyield a non-zero slip velocity. In fact, Navier [42] first proposed a slip velocity, us, for solid surfacesin 1823, where he suggested that the shear rate at the wall was proportional to us. Althoughsomewhat different for a solid surface with a heterogeneous three-phase interface, one can extendthis same idea. For a two-dimensional flow, this is depicted in figure 3a,b and described by
us = λx
∣∣∣∣
dudy
∣∣∣∣, vs = 0 and ws = λz
∣∣∣∣
dwdy
∣∣∣∣, (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 ofentrapped air, SHSs can produce an effective slip at the gas–liquid interface. Because less energyis lost to frictional dissipation, a non-zero us indicates that a reduction in the shear force alongthe surface is possible. If the average velocity along the intermittent solid–liquid and gas–liquidinterfaces increases, such that it matches the bulk velocity of the flow, this would indicate 100%
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Table 1. Previous studies on drag reduction in turbulent flowwith corresponding geometry of surfaces, apparent contact angleand contact angle hysteresis reported, length of tested surfaces, range of friction Reynolds numbers evaluated (Reτ = δ/δν )and observed drag reduction (positive) or drag increase (negative). Presentation of studies using Reτ allows for a moremeaningful comparison across differing experimental set-ups, and also facilitates comparisons with computational work.
friction drag reduction. According to this analysis, this would be achieved by a perfect air layer(φs = 0) [7].
A large streamwise slip length, λx, suggests a large velocity at the wall [4]. Ou and Rothsteinconducted one of the seminal studies that first measured drag reduction using canonical SHSs[43,44]. By fabricating precisely defined microstructures, they showed the first experimentalevidence of a higher velocity at the wall owing to the incorporation of air (figure 3c,d). Measuringpressure drop in a microchannel, they also found that the drag reduction increased with thefraction 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 (seethe review by Rothstein [4]).
For example, Lee & Kim [20] used etched silicon posts to study the slip length of SHSs withhierarchical texture elements (figure 2b–d). Typically, a large fraction of air will cause a surface tobe 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. Roughenedsidewalls add pressure stability by pinning the meniscus at the top of the posts. The authorsproduced micrometre-scale posts with nanoscale roughness on the walls in order to maintainboth a very low φs and a greater pressure stability. Without nanostructured sidewalls, water wetthe posts that were spaced more than 200 µm apart. Nanostructuring the sidewalls increased theallowable spacing to 450 µm. This technique allowed the authors to create idealized structureswith 99.7% air in contact with the liquid. When evaluated in a rheometer, the measured sliplengths 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 ofslip and no-slip. Lauga & Stone [46] first applied these boundary conditions to pressure-drivenStokes flow. They defined an effective slip length to represent the overall slippage along a surface
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Figure 3. Laminar skin friction drag reduction using SHSs. A schematic of water atop ridges (a) and the corresponding slip thatcan result from the presence of air pockets (b). Water existing in a composite interface (c) atop ridges [39]. In contrast, ethanol,a low surface tension liquid, fully wets such a texture. (Reprinted from [39]. With permission of AIP publishing.) The velocityprofile measured for the ridges shown in (c), with three different fractions of air (1 − φs) (d). The velocity shown at y/H = 0is the slip velocity. The effect of channel depth (e) and air fraction (f ) on the observed drag reduction, for ridges evaluated inlaminar flow. Note that the highest drag reduction observed corresponds to the maximum air fraction (1 − φs), and also thelargest slip velocity. ((d,e) Reprinted from [43]. With the permission of AIP Publishing. (f ) Reprinted from [44]. With permissionof AIP Publishing.) (Online version in colour.)
with slip and no-slip regions. Interestingly, they noted that defects within the perfect slip regioncause much more friction when aligned streamwise rather than spanwise. We will return to thislater to see if it holds true in turbulent flows. Not surprisingly, they found that the effectiveslip length increased with an increasing percentage of slip regions, i.e. more drag reductionwith more entrapped air. Thus, experimentally, computationally and theoretically, drag reductionusing SHSs in laminar flows is well understood. In summation, the following trends from laminarfriction 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 flowTurbulence 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 4a–d). These structures can interact with the texture elements of SHSs, potentiallymitigating any drag-reducing effects. Large pressure fluctuations can induce a wetting transition(figure 4e), and the increased wall shear stress can physically damage the potentiallyfragile microstructure of the SHS. Moreover, turbulence disallows solving the Navier–Stokesequations analytically, and many features of the underlying flow must be discussed statistically.Additionally, high-speed, turbulent flows may decrease the local pressure significantly, and thusmay provide for direct gas removal through suction. Overall, SHSs capable of reducing drag inlaminar flow will not necessarily reduce drag in turbulent flow.
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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 thewall in a turbulent flow (b).In the streamwise direction, slip in turbulence works much the same way as in laminar flow (c) [48]. A non-zeroλx results in aslip velocity at thewall. However, in the spanwise direction, vortices are brought closer to thewall, increasing the effective drag(d). (Reprinted from [48] with permission from AIP.) Moreover, large pressure fluctuations will exist in turbulence (e), such asthose computed over posts [49]. From blue to red, the fluctuations increase from−10 to+10 wall units. For the case shown,the meniscus was assumed to be flat. (Reprinted from [49], with permission from Cambridge University Press.)
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 wallunits (y+) from the solid [50], where the wall-normal distance y is non-dimensionalized by theviscous 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 thoughtthat 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 anSHS could affect the entire turbulent boundary layer δ. Hence, the interaction between the viscoussublayer and the texture of an SHS is postulated to be critical in determining the potential dragreduction on a given SHS.
5. Computational insights into turbulent drag reductionThe simplest geometry studied in turbulent flows has been steamwise ridges. Min & Kim [48]performed insightful direct numerical simulations (DNS) by modelling this geometry as regionsof slip (air pockets) and no-slip (solid surfaces) regions. These manifest as differing boundaryconditions when solving the Navier–Stokes equations. By evaluating cases with purely steamwiseslip, purely spanwise slip and slip in both directions, the effects of the SHS on turbulent dragreduction were uncovered. They found that slip in the streamwise direction works much in thesame way as it does for laminar flows. A non-zero streamwise slip velocity directly reduces theskin friction at the wall (figure 4c).
Conversely, spanwise slip was found to cause an increase in overall drag. Min and Kimobserved a drag increase caused by enhanced near-wall vortices, in the presence of pure spanwiseslip (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
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Figure 5. Turbulent drag reduction: computational ridges. For ridges computed using regions of slip andno slip, the slip velocityincreases with increasing regions of slip (a) [52]. The reduction in friction drag also manifests as a reduction in the computedReynolds stresses (b). (The symbols differentiate the curves and do not represent data points.) (Reprinted from [52] withpermission of AIP Publishing.) The Reynolds stresses are not decreased uniformly (c), and the regions of slip (air pockets) causeincreased stresses above the no-slip (solid surface) regions, especially for the 〈R+13 〉 and 〈R+23 〉 components of the total stresstensor [53]. (Reprinted from [53] with permission of AIP Publishing.) For surfaces exhibiting both streamwise and spanwise slip,curves of equivalent drag can be formed (d) [54]. Here negative numbers denote drag reduction (green region), and, therefore,after λ+
x ≈ 5, drag reduction is always predicted, regardless of the spanwise slip. This can be visualized by the streamwisevelocity fluctuations a distance y+ ≈ 10 from the wall, for cases with no slip (e), purely spanwise slip (f ), purely streamwiseslip (g) and steamwise and spanwise slip (h). ((d–h) Reprinted from [54] with permission of AIP Publishing.)
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 withboth 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 dragincrease caused by spanwise slip. Thus, even with spanwise slip, it is still possible for SHSs toreduce drag in turbulent flow.
Busse & Sandham [54] reported similar results, but graphed their data according to a neutralcurve in (λx, λz) space (figure 5d–h). Along this curve, there was no net change in drag, as thedeleterious effects of spanwise slip were cancelled exactly by the benefits of streamwise slip. Intheir analysis for λ+
x = λx/δν 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 turbulentstatistics were limited by the peak value of the corresponding profile. Thus, the drag increasecaused by spanwise slip was found to plateau. In contrast, the streamwise slip always reducesthe intensity of turbulent fluctuations, and thus frictional drag steadily decreased with increasingλ+
x . In fact, they predicted that drag reduction would always be observed when λ+x ≥ 5, regardless
of the magnitude of λ+z . 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 slipand no slip. In fact, they found that over 70% of the skin friction on the no-slip regions (solidsurface) was a direct result of the Reynolds stresses caused by the presence of the slip regions (airpockets). Overall, computational studies conclude that turbulent drag reduction is possible whenstreamwise slip outweighs spanwise slip, and hence experimentalists must fabricate SHSs withgeometries that favour streamwise slip.
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PVC base platenanograss tilesfit to base-plate drag50% of base-plate drag
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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]. Withpermission of AIP publishing.) Using a precisely fabricated shear-sensing apparatus (b), the effect of the air fraction (1 − φs)has also been measured (c) [29]. It is encouraging to see substantial (10–20%) reductions in drag with low fractions of air,especially from a longevity standpoint. (Reprinted from [55] with permission.) Only non-wetted streamwise ridges producesuch reductions in friction. Transverse ridges, dry or wetted, as well as wetted streamwise ridges all produce drag increase (d).The baseline curve shown is for 0.507Re−0.3 and the data are taken fromWoolford et al. [31]. Micro-fabricated two-dimensionalstructures (e) have also been shown to produce drag reduction in turbulent flow (f ) [32]. (Reprinted with permission.) Note thediffering Re dependencies observed: the measured drag reduction increased (a), remained constant (d) or decreased (e), withincreasing Reynolds number.
6. Experimental findings for ridges and postsExperimentally, the simple geometry of ridges has allowed experimentalists to validate the abovecomputational insights. Daniello et al. [39] measured the skin friction along streamwise ridges ofvarying 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 boththe 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 thatapproached the viscous sublayer thickness. Eventually, their drag reduction saturated at a valueof approximately 50%. However, their ridges had a φs = 0.5, and thus no more drag reduction wasexpected. Their results highlight an important point: SHSs with φs = 0.5 will exhibit a moderatelylow θ∗ and �θ � 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 whenφs = 0.7, and, alternately, as high as a 75% reduction in friction drag was achieved when φs = 0.03(figure 6b,c).
Woolford et al. [31] conducted an important study measuring the skin friction on four differentconfigurations of ridges: longitudinal and non-wetted, longitudinal and wetted, transverse andnon-wetted, and transverse and wetted. The measured friction experienced in turbulent flow, anda baseline comparison, are shown in figure 6d. For both the wetted cases, as well as the transversesuperhydrophobic case, an increase in drag was observed. This agrees with the predictions
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of Min & Kim [48] and Busse & Sandham [54], i.e. that spanwise slip increases the frictionalresistance. Woolford et al. used ridges with φs = 0.2, but it would be worthwhile to have additionalexperiments 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 wasdesigned 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 ofapproximately 7 µm (figure 6e). Fields of wafers were stitched together to form a large samplethat was placed in a water channel at the Naval Undersea Warfare Center, Newport, RI. Using adeflection method, the hydrodynamic forces were recorded for the nanograss and a smooth PVCbaseplate. 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, whereReτ is the friction Reynolds number, δ/δν . With φs = 0.06, such high drag reduction is more thanfeasible. However, unlike the saturation in drag reduction observed by Daniello et al. [39] andWoolford et al. [31], these authors found a decrease in drag savings as the Reynolds number wasincreased. At the highest speeds tested, approximately 1.3 m s−1, a moderate 15% savings in dragwas observed. Combined with the drag increase observed with the spanwise ridges evaluatedby Woolford et al., these studies were the first indication that complex microstructures, especiallythose without predominantly streamwise orientation, may not produce as large a reduction infriction as observed with streamwise ridges.
7. Drag reduction in turbulent flows using random textures: NeverWetTurbulent drag reduction using precisely fabricated nanostructures, such as ridges and posts,has been replicated across several laboratories. Although these surfaces provide insights into themechanisms 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 commercialcoating NeverWet, for turbulent drag reduction [21,22,37,40]. NeverWet is a sprayable blend ofhydrophobic 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 sprayinglarge (1.2 m × 0.6 m) aluminium plates, which were towed in a model basin, they monitored thefrictional drag over a Reynolds number range of 500 000–6 000 000, based on length (ReL). Theytested two variations of this coating, one slightly rougher than the other. The resultant coefficientsof drag are shown in figure 7a. Once the flow became fully turbulent (ReL > 1 × 106), no dragreduction was observed for the smoother variant of the coating, and a significant drag increasewas observed for the rougher variant. The authors claim that surface roughness, and possiblywetting, explained their lack of observed drag reduction. This would be consistent with thefindings of Woolford et al. [31] for wetted ridges.
Zhang et al. [37] studied the drag on a NeverWet-coated substrate in their large-scale flowfacility using particle image velocimetry (PIV). They observed increasing drag reduction withincreasing Reynolds number, from 10% at Reτ = 329 to 24% at Reτ = 467. This was evidencedby the increased velocity measured far from the wall (figure 7b). They also showed that theturbulent structures were significantly reduced in the presence of the non-wetted NeverWetsurface (figure 7c,d). It is important to recall that this drag reduction mechanism is absent in thelaminar case, but was predicted by the computational works cited above. However, the dragreduction 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-coatedsubstrate in channel flow. In disagreement with the two previous studies, the authors found noincrease in velocity near the wall of the SHS, i.e. no drag reduction. Interestingly, the authorsforced the NeverWet to wet by using surfactants, and also found no change in the velocity profileof the wetted surface. This is similar to the case of SH-2 (smoother) from Aljallis et al. [40], butdissimilar to SH-1, the rougher variant. When fully wetted, any hydrodynamically rough SHS
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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 for20 ≤ y+ ≤ 80 (figure 7e). This would indicate drag reduction, and would mechanistically agreewith 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 channelflow (figure 7f [57]). Turbulent drag reduction has not been reported above a height-basedReynolds number of 10 000, and we also found no discernible difference in skin friction betweena NeverWet-coated sample and a smooth base plate. In fact, at the higher Reynolds numberstested, a slight increase in friction was observed, indicating a transitionally rough surface. Ourlack of observed drag reduction matches that of SH-2 measured by Aljallis et al. [40], whosemeasurements of the NeverWet coating also matched the friction coefficient of an uncoatedbaseline sample at higher Reynolds numbers.
The resulting drag reduction of the four independent studies on NeverWet are presented as afunction 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 Reynoldsnumbers, although the cut-off in Reτ would appear to be dependent on the flow facility. The mostlikely reasons for this discrepancy are the differing pressures in each test set-up, how quickly thepressure exceeds the capillary resistance of the NeverWet coating, as well as the viscous lengthscale of the flow.
8. Changes in superhydrophobic surfaces owing to pressure and roughnessMuch like the biological SHSs shown in figure 1, many engineered SHSs, including the NeverWetcoating, exhibit a random texture (figure 2h–j). This topography is markedly different from thesimple geometries of ridges and posts. The obvious question is: how does random texture affectthe possibility of SHS drag reduction in turbulent flow? The first important difference, from bothlaminar drag reduction and the use of ridges or posts, is wetting (figure 8a). Water flowingover 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, andwetting occurs, or equivalently φs → 1.0. For laminar drag reduction, this is not a problem asthe 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 3b–d), and were ableto 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 inthe presence of turbulence. Seo et al. [49] have also shown that large stagnation pressures, of theorder of 100 times the mean pressure of the flow, develop near the leading edge of the texturefeatures on an SHS (figure 4e).
As the pressure on an SHS is increased, the meniscus will penetrate further into the textureof the surface [58]. This penetration will cause an increase in φs once the meniscus reaches anypreviously non-wetted asperities. In this case, the area of slip, and, therefore, the possibility ofdrag reduction, is a function of pressure. Even for laminar flows, Ybert et al. [60] have shown thatthe slip length decreases with increasing pressure as a result of meniscus curvature. Srinivasanet al. [22] previously used confocal microscopy to image the meniscus of an SHS directly by dyeingthe water and solid surface separately (figures 2h and 8b). For this image, φs ≈ 0.12. An increasein 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 atopmicropillars in figure 8c [58]. As the droplet evaporated, the Laplace pressure within the dropletincreased following 2γ LV/R, where R is the radius of the droplet. Thus, even before the wettingtransition was observed around t = 16 min, θ∗ had decreased by almost 20◦. Thus, for SHSs to beeffective in turbulent flow, φs must remain low even at significant pressures. For example, at adepth of 1 m, the hydrostatic pressure is of the order of 10 kPa, causing the turbulent stagnationpressure fluctuations on the texture elements of an SHS to be of the order of 1 MPa [49].
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Figure 8. The effect of pressure onφs. For surfaceswith random texture, themeniscus is positioned at some equilibriumheightthat balances the local pressure with the capillary resistance of the surface. This can leave a significant portion of the surfacewetted (a) [23]. (Reprinted from [23].With the permission of AIP Publishing.) A significantly differentφs is observed at differingpressures, which will affect the possible drag reduction (b,c). Attempts to visualize the meniscus in situ show that, even forsurfaces with high contact angle and negligible hysteresis,φs ≈ 0.12 [22] (b). (Reprinted with permission from [22]. Copyright(2015) by the American Physical Society.) The higher φs will cause the contact angle to decrease, which was observed for anevaporating droplet atop polydimethylsiloxane (PDMS) micro-posts (c). Thus, even before a wetting transition occurs, the airfraction (1 − φs) available to cause slip on an SHS will be reduced by an increase in pressure [58].
If φs can vary with pressure, how will turbulent drag reduction be affected by SHSs that areeither more or less resistant to changes in φs? Bidkar et al. [23] evaluated the skin friction on manydifferent SHSs (figure 2i) that both increased (figure 9a) and decreased (figure 9b) the overall dragcoefficient. To explain these results, the authors compared the observed drag reduction with theroot-mean-squared roughness of the surfaces, k, non-dimensionalized by δν (k+). The resultingdrag measurements for five of their surfaces are shown in figure 9c. For comparison, they alsoplotted 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 roughnesselements of the SHS interacted with the boundary layer, causing viscous dissipation and formdrag (figure 8a). They concluded that surfaces must exhibit k+ < 0.5 for turbulent drag reductionto be observed. However, recalling the findings summarized in figure 8, the surface must alsoexhibit 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 measuredfollowing the flow experiments were only minimally reduced, indicating that wetting alone couldnot explain the absence of observed drag reduction, or the increase in observed drag, for some ofthe tested samples.
Very recently, Seo & Mani [61] have used DNS to show that, for moderate values of φs, λx andλz can be represented by a single, effective slip length. Only for very small values of φs, such asthe ridges of Park et al. [29], will λx and λz differ drastically. This matches well with the modelof Fukagata et al. [51]. However, we note that streamwise ridges, by design, will have differingstreamwise and spanwise slip lengths, and are perhaps a special case. For the randomly texturedsurfaces evaluated in figures 7–9, without any preferential streamwise or spanwise orientation, asingle, effective slip length makes intuitive sense. Overall, scalable, random SHSs can still producemeaningful turbulent drag reduction, but their efficacy will be lessened owing to spanwise slipand 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 flowwith varying degrees of success (table 1). Gogte et al. [30] coated Joukovsky (also spelled asJoukowski) hydrofoils with SHSs fabricated from modified sandpaper. Two SHSs, with k = 8 µmand 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 (k+
max = 0.2 and 0.1, forthe two surfaces, respectively). Lu et al. [34] prepared SHSs from a blend of fluorosilanes, carbonnanotubes and a fluorosilicone matrix. When tested in a turbulent microchannel, they observed
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0.2 0.4 0.6 0.8 1.0 1.2axial location from leading edge (m)
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Figure9. The effect of roughness on turbulent drag reduction. Not all SHSs are capable of reducing friction in turbulent flow. Forexample, Bidkar et al. [23] measured the frictional drag on many different SHSs, both ones that increased drag (a) or decreaseddrag (b). Some surfaces, such as sampleno. 7 in (a), initially reduced thedrag, but then roughness effectsmitigated the reductionat higher Reynolds numbers. The authors proposed that only surfaces with a k ≤ 1/10th the viscous sublayer could reducedrag in turbulence (c). Surfaces with k+ ≥ 0.5 will interact with the flow, eliminating any skin friction reduction. Note that,although k is purely a surface property, k+ also depends on the flow. Thus, SHSs may only reduce drag for specific turbulentflow conditions. (Reprinted from [23]. With the permission of AIP Publishing.)
as high as 53% drag reduction, although they attributed the drag reduction to a delay in the onsetof turbulence. Zhao et al. [38] coated a large, flat plate with this same coating, and measured thefriction in a recirculating water tunnel. In the turbulent regime, essentially no drag reduction wasobserved for all measured flow speeds. This matches well with the work on NeverWet performedby Aljallis et al., where more than 30% friction drag reduction was observed in transitionalturbulence, 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 randomSHS fabricated from hydrophobic particles dusted on a sandblasted aluminium substrate. Nodiscernible drag reduction or increase could be deduced from the authors’ measurements.Jung & Bhushan [25] measured turbulent drag reduction in a microchannel using hierarchicallystructured posts (figure 2e,f ). Drag reduction as high as 30% was achieved.
Although experimentalists have measured drag reduction using SHSs with varying degreesof success, a more detailed understanding has started to emerge. Overall, experimental results ofSHSs in turbulent flow yield the following conclusions.
(1) SHS drag reduction in turbulent flow is possible, especially when the texture elementsare aligned in the streamwise direction.
(2) Wetting owing to pressure fluctuations may remove the regions of slip, and thus mitigateany drag-reducing properties of the SHS, and may even cause drag increase if thesurface’s roughness interacts with the viscous sublayer.
(3) Even when the air pockets remain on the SHS, spanwise slip and roughness effects cancause an increase in the observed drag.
(4) An effective SHS for reducing drag in turbulent flow should have streamwise-alignedstructures, with a capillary resistance of the order of MPa and a k+ ≤ 0.5.
9. The effect of Reynolds number on turbulent drag reductionWe conclude this review by discussing one final, somewhat open-ended, topic. Assuming one hasfabricated an SHS capable of turbulent drag reduction, how does the drag reduction change withReynolds number? There have been many different answers to this question, both experimentallyand computationally. Min & Kim [48] have shown computationally that the non-dimensionalizedslip length, λ+
x , increased with increasing skin friction. The drag reduction should be enhancedas 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,
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especially in rheometers. Barbier et al. [62] fabricated a series of hierarchical SHSs from anodizedaluminium. 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 dragreduction was always higher at higher Reynolds number, which they attributed to a delay inthe onset of turbulence. Srinivasan et al. [22] coated rheometer rotors with an SHS fabricated byspray-coating (figure 2h). At Re < 20 000, they observed no drag reduction in their Taylor–Couetteflow. However, for Re ≥ 30 000, significant drag reduction was observed. They also derived a skinfriction law that incorporated wall slip, which yielded the relation that λ+
x scales with Re1/2. Goodagreement was shown between their experiments and this relation.
In microchannels, Jung & Bhushan [25], Daniello et al. [39] and Lu et al. [34] observed increasingdrag 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 adjustto the new boundary condition imposed by the regions of slip and no slip [63,64]. The largerthe k, the further downstream before the flow has adjusted fully. For example, around 30δ isrequired when δ/k < 25 [63]. For microchannels, this distance can be longer than the entire lengthof the channel. Thus, measurements in microchannels could inaccurately portray the full effectsof turbulence over SHSs.
Accordingly, in large-scale, turbulent boundary layer flow facilities, the opposite trend hasbeen observed experimentally. Both Aljallis et al. [40] and Hokmabad et al. [21] found essentiallyno skin friction drag reduction for the NeverWet coating at high Re. In both these studies, a smallamount of drag reduction was observed, but only at the lowest Reynolds numbers tested. Similartrends 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 increasingthe 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 bestof our knowledge, there has not been a fully developed turbulent channel or turbulent boundarylayer flow that produced increased drag reduction with increased Re. In fact, all studies haveshown that the drag reduction decreased with Re, so much so that drag increased beyond baselineresistance at some critical Reynolds number. Conversely, the rheometer and microchannel studieshave shown no drag reduction until the onset of turbulence, and then increased drag reductionas Re increased.
The discrepancy alluded to above also highlights the fact that decreasing δν alters both λ+x
and k+. However, the effect of altering λ+x with Re is not straightforward. For example, Kim &
Min [48] have hypothesized, and Fukagata et al. [51] have shown, that the effects of streamwiseand spanwise slip act independently. Therefore, an increase in Re could cause more spanwiseslip 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), effectivelychanging the k that the flow sees. Thus, even if the effects of increasing Re on λ+
x and λ+z 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 theirturbulent DNS calculations (figure 4e). They found that the large stagnation pressures that weregenerated near the leading edge of their posts were lessened when the meniscus was free tomove. However, because the meniscus was pinned at the top of the posts, and posts possess anegligible k+, it is unlikely that the deleterious effects of increasing Re on k+ were captured in theircalculations. Very recently, Crowdy [65] extended his previous work [66] by deriving an analyticalformula for the slip length of superhydrophobic ridges with a curved air–water interface. Forridges with φs = 0.1, when the angle of the meniscus curved from flat (0◦) to −30◦ (into thetexture), 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).
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10. Conclusion and outlookIn this review, we have tried to summarize the vibrant field of turbulent drag reduction usingbiomimetic SHSs. Much like the unpredictability of turbulence itself, many of the drag reductionstudies have produced unexpected results, such as an increase in drag where a reduction wasexpected. Wetting, roughness effects and spanwise slip seem to be the leading culprits thathave hampered some surfaces from providing turbulent friction reduction. For the field to moveforward, three important questions need to be answered.
First and foremost, what is the optimal microstructure for turbulent drag reduction, and howcan this texture be created on a large scale, and modelled computationally? Micrometre-scaleridges 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 fabricatedsimilarly, 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 makingsurfaces with novel, random textures that are capable of reducing drag in turbulent flow, ratherthan exploring how to scale up the proven technology of ridges. Although computationallychallenging, numerical modellers can investigate the effects of randomness on drag reductionin turbulent flow. Results in laminar flow with streamwise and spanwise randomness seempromising [67], but the largest form drag contributions will arise owing to large variations inthe wall-normal direction [68]. Yet, there have been no computational studies investigating thistype 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 andlarge-scale flow facilities, especially with regard to the observed dependence of drag reduction onReynolds number? Arguably, the biggest impact of drag-reducing SHSs would be their successfulapplication on external flows of marine vehicles, or internal flows of pipelines, for example. Thediscrepancies between the results of microchannels, rheometers, towing tanks and larger flowchannels need to be either remedied or understood further. To date, only streamwise ridges haveconsistently 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 inturbulent flow? Once the surfaces work consistently, and many of the excellent studies cited hereexhibit surfaces that do, how long will the drag reduction persist? For example, diffusion of N2into 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 pocketshave been removed? Based on the results of Bidkar et al. [23], we have proposed here that aroughness of k+ ≤ 0.5 is required for turbulent drag reduction. This threshold may need to bereduced 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[19,62,71–75]. 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 areorders of magnitude apart, it remains to be seen whether the two mechanisms can be combinedsynergistically or drag reduction by riblets can only succeed drag reduction by SHSs once the airpockets have been removed.
Contrastingly, what strategies exist for restoring or replenishing the air depleted from theSHS, and, consequently, the slip region? Several excellent studies have already shown thatelectrochemical gas generation [76,77], Leidenfrost heating [78,79] or even thermodynamicallystable water vapour [80,81] can restore the air pockets of a fully wetted SHS. Incorporatingsome of these failsafe mechanisms may help SHSs transition from the laboratory to actualmarine environments. Overall, if we wish to take full advantage of surfaces that entrap airpockets, as penguins, spiders and lotus leaves have done for millennia, much work still needsto be done.
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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.Acknowledgements. The authors thank Hangjian Ling and Dr Joseph Katz at Johns Hopkins University for theirinsightful discussions. The authors are also indebted to Dr Ali Mani, at Stanford University, and Dr StephanoLeonardi, at the University of Texas at Dallas, for their excellent suggestions and comments.
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