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PHYSICS OF FLUIDS 21, 085103 ?2009?

Drag reduction in turbulent ?ows over superhydrophobic surfaces
Robert J. Daniello, Nicholas E. Waterhouse, and Jonathan P. Rothstein
Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, 160 Governors Drive, Amherst, Massachusetts 01003, USA

?Received 11 February 2009; accepted 13 July 2009; published online 26 August 2009? In this paper, we demonstrate that periodic, micropatterned superhydrophobic surfaces, previously noted for their ability to provide laminar ?ow drag reduction, are capable of reducing drag in the turbulent ?ow regime. Superhydrophobic surfaces contain micro- or nanoscale hydrophobic features which can support a shear-free air-water interface between peaks in the surface topology. Particle image velocimetry and pressure drop measurements were used to observe signi?cant slip velocities, shear stress, and pressure drop reductions corresponding to drag reductions approaching 50%. At a given Reynolds number, drag reduction is found to increase with increasing feature size and spacing, as in laminar ?ows. No observable drag reduction was noted in the laminar regime, consistent with previous experimental results for the channel geometry considered. The onset of drag reduction occurs at a critical Reynolds number where the viscous sublayer thickness approaches the scale of the superhydrophobic microfeatures and performance is seen to increase with further reduction in viscous sublayer height. These results indicate superhydrophobic surfaces may provide a signi?cant drag reducing mechanism for marine vessels. ? 2009 American Institute of Physics. ?DOI: 10.1063/1.3207885?
I. INTRODUCTION

The development of techniques which produce signi?cant drag reduction in turbulent ?ows can have a profound effect on a variety of existing technologies. The bene?ts of drag reduction range from a reduction in the pressure drop in pipe ?ows to an increase in fuel ef?ciency and speed of marine vessels. Drag reduction in turbulent ?ows can be achieved through a number of different mechanisms including the addition of polymers to the ?uid,1 the addition of bubbles2 or air layers,3,4 compliant walls,5 and riblets.6 We will demonstrate that superhydrophobic surfaces can be used as a new passive technique for reducing drag over a wide range of Reynolds numbers from laminar7,8 to turbulent ?ows. Superhydrophobic surfaces were originally inspired by the unique water repellent properties of the lotus leaf.9 They are rough, with micro or nanometer-sized surface features. In the Cassie state, illustrated in Fig. 1, the chemical hydrophobicity of the material prevents the water from moving into the space between the peaks of the rough surface, resulting in the air-water interface which is essentially shear-free. The resulting surface possesses a composite interface where momentum transfer with the wall occurs only at liquid-solid and not the liquid-vapor interfaces. Recent synthetic superhydrophobic surfaces have been developed which are perfectly hydrophobic, obtaining contact angles that can approach ? = 180° with no measurable contact hysteresis.9,10 It should be noted that the extreme contact angles available with superhydrophobic surfaces result from their superhydrophobic topography rather than chemical hydrophobicity; contact angles on smooth surfaces of the same chemistry are much lower. Philip11,12 and Lauga and Stone13 provide analytical
1070-6631/2009/21?8?/085103/9/$25.00

solutions for laminar Poiseuille ?ows over alternating slip and no-slip boundary conditions, such as those existing above a submerged superhydrophobic surface. These results provide an analytical solution predicting and quantifying drag reduction resulting from slip/no-slip walls, in laminar ?ows. Ou and Rothstein7,14 demonstrated that superhydrophobic surfaces produce drag reduction and an apparent slip, corresponding to slip lengths of b = 25 ?m, at the wall in laminar ?ows as a direct result of the shear-free air-water interface between surface microfeatures. Here the slip length is de?ned using Navier’s slip model where the slip velocity, u0, is proportional to the shear rate experienced by the ?uid at the wall u0 = b

???u . ???y

?1?

These results have been extended to a variety of superhydrophobic surface designs and ?ow geometries.8,15 A thorough overview of the no-slip boundary condition is given by Lauga et al.16 Ybert et al.17 examined scaling relationships for slip over superhydrophobic surfaces. For a superhydrophobic surface in the Cassie state, they showed slip length to increase sharply with decreasing solid fraction and increasing effective contact angle.17 However, Voronov et al.18,19 demonstrated that for hydrophobic surfaces, there is not necessarily a positive correlation between increased contact angle and slip length. Fundamentally, the effective reduction in solid-liquid boundary as a superhydrophobic drag reduction mechanism should be independent of whether the ?ow is laminar or turbulent. In turbulent ?ows, a thin viscous-dominated sublayer exists very near to the wall. It extends to a height, measured in terms of wall units, viscous lengths, of y + = y / ???w / ? = 5.20 Where y is the height above the wall, ?
? 2009 American Institute of Physics

21, 085103-1

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085103-2
a)

Daniello, Waterhouse, and Rothstein

Phys. Fluids 21, 085103 ?2009?

Water d Air Air w Air

h

Superhydrophobic Surface

b)

FIG. 1. ?Color online? Schematic of air trapped between hydrophobic microfeatures of a superhydrophobic surface. The air-water interface produces shear-free regions resulting in a reduction in wetted area and regions that can experience signi?cant slip in ?ows. ?b? Micrograph of a superhydrophobic microridge geometry containing 60 ?m wide ridges spaced 60 ?m apart. Features are approximately 25 ?m deep.

is the kinematic viscosity, ?w is the wall shear stress, and ? is the ?uid density. In the viscous sublayer, the mean velocity increases linearly with position, u+ = y +. Changes in momentum transfer to the viscous sublayer can have a dramatic in?uence on the entire turbulent ?ow and can result in drag reduction. This effect is demonstrated in the direct numerical simulation ?DNS? studies of Min and Kim21 who imposed a ?xed, arbitrary, but not unreasonable, longitudinal slip length boundary condition in a turbulent channel ?ow. Similar work was performed by Fukagata et al.22 who related drag reduction and slip length. More recently, Martell et al.23 used DNS to study the turbulent ?ows over periodic slip/no-slip boundary conditions to simulate microposts and microridges geometries that approximate the superhydrophobic surfaces presented here. Their simulations predict a drag reduction that increases with both the microfeature spacing and the surface coverage of the shear-free air-water interface as well as with the Reynolds number.24 In addition to the presence of the shear-free interface, drag reduction mechanisms such as surface compliance and turbulent structure attenuation may also exist for micropatterned superhydrophobic surfaces. Few experimental studies have considered superhydrophobic drag reduction into the turbulent regime.25–28 In a recent experimental study, Gogte et al.25 observed drag reduction in turbulent ?ow over a hydrofoil coated with a randomly structured superhydrophobic surface produced from hydrophobically modi?ed sandpaper. Drag reductions of up to 18%, based on combined skin friction and form drag, were reported for the hydrofoil. Overall drag reduction on the hydrofoil decreased with increasing Reynolds number. However, one should note that the total drag was reported and the individual contribution of friction and form drag was not deconvoluted. The form drag of the body should increase signi?cantly with Reynolds number and could obscure the performance trend of the superhydrophobic surface which affects only skin friction drag. It is not necessarily inconsis-

tent for skin friction drag reduction to be stable or increasing with Reynolds number as predicted by the DNS simulations of Martell.24 Balasubramanian et al.28 achieved similar results for ?ow over an ellipsoidal model with a disordered superhydrophobic surface similar to that employed by Gogte et al.,25 but having smaller microfeatures. Henoch et al.29 demonstrated preliminary success in a conference proceeding noting drag reduction over 1.25 ?m spaced “nanograss” posts in the turbulent regime. Similar in physical mechanism to superhydrophobic drag reduction, air layer drag reduction, results from continuous air injection suf?cient to produce an uninterrupted vapor layer existing between the solid surface and the water. Such air layers are an active technique for producing drag reduction; they do not require chemical hydrophobicity of the surface and exist only as long as the required air injection rate is maintained. Elbing et al.3 demonstrated air layers are capable of producing nearly complete elimination of skin friction drag. The authors demonstrated the existence of three distinct regions; bubble drag reduction at low air injection rates where performance is linear with air injection rate and drag reductions up to 20% can be achieved, a transitional region at moderate injection rates, and a full air layer at large air injection rates. Once the full air layer is achieved, Elbing reported little performance increase with additional air?ow. It should be noted that drag reduction falls off with distance from the injection point until a complete air layer is achieved. Reed30 utilized millimeter sized ridges to capture and stabilize injected air and form a continuous air layer between the ridges. The author noted hydrophobic walls, with ridge features much too large ?millimeter? to produce a superhydrophobic effect, exhibited an enhanced ability to form and maintain stable air layers. Additionally, Fukuda et al.4 demonstrated an increase in drag reduction obtained when a discontinuous layer of injected bubbles are attracted by walls treated with hydrophobic paint. Geometrically, riblets appear similar to the superhydrophobic surfaces under present consideration; however, their scale and function are completely different. Riblets are ridges aligned in the ?ow direction which reduce drag in turbulent ?ows by disrupting the transverse motion of the ?uid at the surface, thereby moving near-wall turbulent structures farther from the wall.6 Unlike superhydrophobic surfaces, the grooves between riblet features are wetted by the ?uid, and function equally well for both liquids and gasses. Unfortunately, riblet geometries only perform well within a limited range of Reynolds numbers and can have derogatory effects outside of their designed range. To function, riblets must maintain a spacing, w+ = w / ???w / ?, between 10 ? w+ ? 30 wall units.31 As will be demonstrated in Secs. II–IV, the superhydrophobic microfeatures used in the present experiments are at least an order of magnitude too small to produce a riblet effect. It will be shown that the observed drag reduction is due to the presence of a shear-free air-water interface supported between microfeatures.

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085103-3

Drag reduction in turbulent ?ows over superhydrophobic surfaces

Phys. Fluids 21, 085103 ?2009?
Transparent PMMA top plate

W
PIV imaging



FIG. 2. ?Color online? Water and ethanol droplets resting on a superhydrophobic surface. The water drops stand off the surface in the Cassie state while ethanol fully wets the surface in the Wenzel state. Microridges run front to back and the air-water interfaces they support are visible under the water drops.

window

PDMS surface Bottom plate

II. EXPERIMENTAL PROCEDURE

FIG. 3. ?Color online? Cross section of ?ow cell used for PIV with a PDMS superhydrophobic surface on the bottom and a smooth acrylic surface on top. The bottom surface was interchangeable and was replaced with a number of different superhydrophobic PDMS surfaces.

The present work presents particle image velocimetry ?PIV? and pressure drop measurements of a turbulent channel ?ow over several superhydrophobic walls. The superhydrophobic surfaces were engineered with regular arrays of microridges aligned in the ?ow direction in order to systematically investigate the effect of topological changes on the velocity pro?les, slip length and drag reduction in turbulent channel ?ows. Superhydrophobic polydimethylsiloxane ?PDMS? test surfaces were cast from silicon wafer molds produced by a lithographic process. A 25 ?m layer of SU 8 photoresist ?Microchem? was spun onto bare or oxide coated silicon wafers. The substrate was then exposed through a negative mask of the desired pattern and developed to produce a mold. A micrograph of a typical wafer mold, in this case for 60 ?m microridges spaced 60 ?m apart, is shown in Fig. 1?b?. Once completed, the wafers were used to cast patches of micropatterned PDMS approximately 150 mm long which were then seamlessly joined to produce a 1 m long superhydrophobic surface. All measurements are conducted on the downstream section of the patch, minimally thirty channel half heights, ?, downstream of the nearest patch joint. Smooth test surfaces were prepared by curing PDMS on a smooth ?at cast polymethyl methacrylate ?PMMA? plate. The PDMS was treated with a highly ?uorinated silane ?Gelest, Tullytown, PA? to make it more hydrophobic, resulting in an advancing contact angle of approximately ? = 125°. Untreated PDMS having an advancing contact angle of approximately ? = 110° on a smooth surface was also used with identical results. No measurable slip lengths were observed for ?ows over smooth PDMS surfaces. It should be noted that for materials not demonstrating slip over smooth surfaces, contact angle is important to superhydrophobicity only inasmuch as it increases the maximum pressure sustainable by the three phase interface.7 Contact angle does not affect the shear-free area or the interface de?ection for a ?xed sustainable pressure, and thus should not affect the turbulent drag reduction obtained. A section of microridge superhydrophobic surface is seen in Fig. 2 with two droplets of water, sitting on top of the microfeatures, demonstrating the Cassie state, and ethanol, which wets the surface, demonstrating the Wenzel state. PIV is conducted in a rectangular channel ?ow geometry shown in Fig. 3?a?., fabricated from optically clear PMMA with a single interchangeable PDMS test surface at the bottom wall. The channel was W = 38.1 mm wide and full channel height was 2? = 7.9 mm. Reverse osmosis puri?ed water

was used as the working ?uid. Water purity does not seem to affect drag reduction results the same water was used for several weeks with no change in performance. For PIV, the water was seeded with 0.005 wt % of 11 ?m diameter hollow silvered glass spheres ?Sphericel, Potters Industries, Carlstadt, NJ?. Flow was provided under gravity from a head tank and collected for reuse. A centrifugal pump returns ?uid to maintain head level, provisions exist to run the apparatus directly from the pump although, to reduce vibrations, the pump is turned off during measurements. Static pressures within the ?ow cell were held below 5 kPa to ensure the Cassie state was maintained. Ridges were designed to prevent air from escaping at the ends to allow operation near or possibly slightly above the limit predicted by Young’s law for captive air at atmospheric pressure. The ?ow rate was measured by one of two turbine ?ow meters ?low ?ow rates FTB-603, Omega; high ?ow rates FTB-902, Omega? placed in series with the test section. It was adjusted by a throttling valve located far upstream. Reynolds number was calculated from ?ow rate and veri?ed by numerical integration of velocity pro?les when PIV pro?les of the entire channel height were accessible. PIV was conducted in the x-y plane at mid channel approximately 200–225 half heights from the inlet, far enough downstream to ensure a fully developed turbulent ?ow over the superhydrophobic surfaces. Illumination is provided by a 500 ?m wide light sheet. Images were recorded with a high-speed video camera ?Phantom 4.2? at frame rates up to 8500 frames/s and correlated with a commercial code ?DaVis, LaVision Gmbh?. Under the maximum magni?cation of our experiments, the velocities could be accurately resolved within 50 ?m from the wall. At reduced magni?cations, PIV images cover the entire channel to simultaneously observe smooth top and superhydrophobic bottom walls. Images were recorded under ambient lighting to establish wall location; for full channel measurements the true wall location is known to within 10 ?m accuracy. Up to 10 000 frames of steady state ?ow were captured, correlated and averaged to generate each velocity pro?le. Scale was established by imaging targets and veri?ed with the known height of the channel. Presently, we consider two superhydrophobic microridge geometries and the smooth PMMA top wall, which have been tested over a range of mean Reynolds numbers 2000 ? Re= 2?U / ? ? 9500. Here U is the mean ?uid velocity measured from the ?ow. Transitional effects are considered

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085103-4
Pressure tap

Daniello, Waterhouse, and Rothstein

Phys. Fluids 21, 085103 ?2009?

a)
2d W

1.50

FIG. 4. ?Color online? Cross section of ?ow cell used for pressure drop measurements. Superhydrophobic surfaces were ?tted to both the top and the bottom surfaces of the channel.

Velocity, u [m/s]

Height spacer

PDMS Superhydrophobic surfaces

1.25

1.00

0.75
1.00

0.50
0.75 0.50

to persist up to Re= 3000 for this ?ow.20 Two geometries with 50% shear-free air-water interface coverage were considered. The ?rst contains microridges d = 30 ?m wide and spaced w = 30 ?m apart ?30-30? and the second contained microridges d = 60 ?m wide and spaced w = 60 ?m apart ?60-60?. As noted, feature sizes considered range from w+ ? 2 wall units for the 30-30 ridges and remain less than w+ ? 3.5 wall units for the 60-60 ridges. These ridge spacings are an order of magnitude too small to produce a riblet effect over the present range of Reynolds numbers. Additional quanti?cation of superhydrophobic drag reduction was obtained through direct pressure drop measurements in the channel. Here, the test section was replaced with a channel having superhydrophobic surfaces on both top and bottom walls, Fig. 4. The channel height was set by the precisely machined aluminum side spacer seen in the ?gure, and the ?ow cell assembly was conducted with a calibrated wrench to maintain precise uniformity of the channel between tests, ?xing the channel aspect ratio. The channel was W = 38.1 mm wide and 2? = 5.5 mm high. Additionally, multiple data collection sessions were performed for each surface, with reassembly of the apparatus between each session. Measurements were conducted from single taps, as illustrated, over a 70 mm span more than 130? from the channel inlet. Pressure was read directly from a pair of water column manometers reading static pressures at the front and back of the test section. Water column heights were photographically recorded, the differences in column height being used to calculate the pressure drop across the test section. The manometer resolution was ?1 Pa, which resulted in pressure drop measurement uncertainty that ranged from 5% for the slowest ?ows to 0.5% for the highest Reynolds numbers tested. Flow rate was measured with a turbine ?ow meter as in the PIV experiments. Flow control and Reynolds number capabilities are identical to those used for PIV. To ensure steady state, data points were taken no more than once per minute and the ?ow rate was adjusted only incrementally between measurements. Data were collected on increasing and decreasing ?ow rate sweeps to ensure that no hysteresis was observed.
III. RESULTS AND DISCUSSION

0.25

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Normalized Channel Height, y/δ
b) 1.25

1.00

Velocity, u [m/s]

0.75
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

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Normalized Channel Height, y/δ
FIG. 5. ?a? Velocity pro?les over a microridge surface w = 60 ?m d = 60 ?m showing the development of signi?cant slip velocities with increasing Reynolds number from 2700 ??? to 8200 ???. ?Inset? Velocity pro?les near the wall demonstrating prominent slip velocities. Reynolds numbers are 2700 ???, 3900 ???, 4840 ???, 5150 ???, 6960 ???, and 8200 ???. For clarity, the modi?ed Spalding ?ts ??? from Eq. ?3? are only overlaid on the pro?les corresponding to Re= 2700 and Re= 8200. ?b? Velocity pro?les over the w = 30 ?m d = 30 ?m microridge surface demonstrate slip velocity behavior consistent with that observed on the 60-60 surface, but reduced in magnitude. Reynolds numbers range from 4970 ??? to 7930 ???. Larger feature spacing performs better for a given Reynolds number. Reynolds numbers are 4970 ???, 5400 ???, 6800 ???, 7160 ???, and 7930 ??? The modi?ed Spalding ?ts ??? are overlaid on the pro?le corresponding to Re= 7930.

just past transition are, to the limit of our measurements, equivalent to smooth pro?les at identical Reynolds numbers This is not unexpected for the data points in the laminar or transitional regime.7,14 For pressure driven ?ow between two in?nite parallel plates separated by a distance 2? the volume ?ow rate per unit depth is given by q= 2?3 dp ? ? dx

? ??

1 b . + 3 b + 2?

?

?2?

A typical set of velocity pro?les, resulting from PIV near the superhydrophobic wall for the 60-60 ridge surface is shown in Fig. 5?a? for a range of Reynolds number between 2700 ? Re ? 8200. The effect of the superhydrophobic wall is not observed for the low Reynolds number experiments. At the low Reynolds numbers, the turbulent velocity pro?les

For a given pressure gradient, dp / dx, and ?uid viscosity, ?, the volume ?ow rate can be signi?cantly enhanced only if the slip length is comparable to the channel height. Previous laminar regime studies over similar superhydrophobic microfeatures measured slip lengths of b = 25 ?m independent of Reynolds number.7 In our channel geometry, such laminar ?ow slip lengths would produce a drag reduction of around

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085103-5

Drag reduction in turbulent ?ows over superhydrophobic surfaces
3

Phys. Fluids 21, 085103 ?2009?

1%. Additionally, for small slip lengths, the expected slip velocity can be approximated by uslip = 4Ub / ? which should also be on the order of only a couple of percent of the average free stream velocity, U, and below the resolution of our PIV measurements. As the Reynolds number is increased and the ?ow becomes fully turbulent, however, a substantial slip velocity, and slip lengths greater than b ? 25 ?m, are observed along the superhydrophobic wall. The presence of an air-water interface is visually apparent on the superhydrophobic surface giving it a silvery appearance. This result, due to the differing indices of refraction and slight curvature of the interface, was observed throughout the range of testing giving us con?dence that the interface was maintained for all of the experiments reported in this paper. As the inset of Fig. 5?a? clearly shows, the magnitude of the slip velocity was found to increase with increasing Reynolds number. Similar, although less pronounced, trends were observed for the 30–30 ridge case as seen in Fig. 5?b?. Signi?cant deviation from no-slip behavior is noted past a Reynolds number of approximately Re= 4000 for both the 30-30 and 60-60 ridged cases. Above these Reynolds numbers, a nearly linear increase in the slip velocity with increasing Reynolds number was observed for each of the superhydrophobic surfaces used. A maximum slip velocity of nearly 40% the mean channel velocity, uslip / U = 0.4 was observed for the 60-60 ridged case at the highest Reynolds numbers tested. In order to determine both the shear stress and slip velocity at the smooth and superhydrophobic walls, the PIV velocity ?elds were ?t to a modi?ed Spalding equation for turbulent velocity pro?le above a ?at plate,32
+ y + = ?u+ ? uslip ? + e?2.05?e?0.41?u
+?u+ ? slip

Pressure Drop Per Unit Length, ?p/l [kPa/ m]

2

1

0

0

2000

4000

6000

8000

10000

Reynolds Number, Re

FIG. 6. Pressure drop measurements for ?ow through a rectangular channel with a smooth walls ??? and with two walls containing superhydrophobic microridges with w = 60 ?m and d = 60 ?m ???. The Colebrook line ?–—? is shown for a smooth channel.

+ ? 1 ? 0.41?u+ ? uslip ?

1 1 + + ?0.41?u+ ? uslip ??2 ? 6 ?0.41?u+ ? uslip ??3? . ?2

?3?

The Spalding equation is an empirical ?t to experimental turbulent velocity pro?le data that covers the entire wall region through the log layer.33 This allows the ?t to be applied farther into the channel, to determine the wall shear stress more accurately using a greater number of data points than would be available within the viscous sublayer. Wall shear stress enters the equation in the de?nition of the velocity, u+, and position y +, in wall units. To account for slip, each instance of the velocity in wall units, u+ = u?? / ?w, in the Spal+ . The ?t ding equation was replaced by the difference u+ ? uslip was performed by a numerical routine given an initial value for slip velocity extrapolated from a coarse linear ?t of nearwall data points. An initial wall shear stress was determined by minimizing the error in the ?t. Subsequent iterations were performed on wall slip velocity and wall shear stress to minimize the standard error of the ?t over the interval 0 ? y + ? 50. The resulting ?ts were accurate to better than 4% at a 95% con?dence interval. The results were not appreciably different if the ?t is taken to y + = 100. The size of the PIV correlation window was chosen to be 0.2 mm. For the frame rates used, the resulting particle displacements within the correlation window were typically much less than 25% of the window in the viscous sublayer and less than 33% of the window everywhere for Reynolds

numbers less than Re ? 4500. Large particle displacements were observed far from the wall at the highest Reynolds numbers, however, no noticeable effects were observed on the resulting pro?les. As seen in Fig. 5, the resulting ?ts of Eq. ?3? to the velocity pro?les are excellent with and without slip, which instills con?dence in the values of shear stress calculated from the velocity gradient extrapolated to the wall, ?w = ??????u / ???y ??y=0. The maximum slip velocity and observed wall shear stress reductions correspond to slip lengths of b ? 70 ?m for the 30-30 microridges and b ? 120 ?m for the 60-60 microridges. Larger slip velocities and slip lengths were measured for turbulent ?ow past superhydrophobic surfaces with larger microfeature spacings even as the percentage of shear-free interface was kept constant at w / ?w + d? = 0.5, as has been observed in the laminar ?ow measurements over superhydrophobic surfaces.14 This observation is consistent previous laminar ?ow studies7,14 and with the predictions of DNS in turbulent ?ows.24 Additionally, Ybert et al.17 showed through a scaling argument that in laminar ?ows one expects the slip length to scale linearly with the microfeature spacing, b ? ?w + d?. In Fig. 6, direct measurements of the pressure drop per unit length of channel, dp / l, are shown for a smooth PDMS surface and the superhydrophobic surface containing 60 ?m ridges spaced 60 ?m apart in an identical channel. The result predicted by the Colebrook equation34 for a perfectly smooth channel of the same dimension is plotted for reference. The pressure drop per unit length is directly related to the channel geometry and the wall shear stress, dp / l = ?w?1 + 2? / W? / ?, so it provides a second method for measuring drag reduction. Signi?cant drag reduction is initially noted by a leveling off of the in the pressure drop during the transition from laminar to turbulent ?ow between Reynolds numbers of 2000 ? Re ? 3000. These data indicate a delay in the transition to fully developed turbulent ?ow. Additionally, for Reynolds numbers greater than Re ? 3000 the pressure drop over the surperhydrophobic surface grows at roughly half of the rate of pressure drop over the smooth

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085103-6
6

Daniello, Waterhouse, and Rothstein

Phys. Fluids 21, 085103 ?2009?

0.015

5

Wall Shear Stress, τw [kPa]

4

Friction Coefficient, Cf
4000 6000 8000 10000

0.010

3

2

0.005

1

0 2000

0.000 0 2000 4000 6000 8000 10000

Reynolds Number, Re

Reynolds Number, Re

FIG. 7. Wall shear stress measured from PIV as a function of Reynolds number for a channel with a single superhydrophobic surface. Results are presented for both the smooth top wall ??? and the superhydrophobic bottom wall containing w = 30 ?m wide ridges spaced d = 30 ?m apart ?b?. Drag reduction is seen only on the superhydrophobic wall, the smooth wall being in good agreement with the Colebrook prediction for a smooth channel ?—?.

surface. The Colebrook line, accurately ?ts the turbulent ?ow data from the smooth surface, and the predicted laminar ?ow result passes through the microridge data in the laminar region below Re ? 2200. This result is consistent with those predicted by Eq. ?2? and observed by PIV. As noted before there is no measurable drag reduction or slip velocity for the present channel geometry in the laminar regime. Further insight comes from the full channel PIV where smooth and superhydrophobic surfaces may be simultaneously observed at the same mean channel Reynolds numbers. Wall shear stress, calculated from the modi?ed Spalding ?ts, is shown in Fig. 7 for the smooth and superhydrophobic surfaces. Again the Colebrook line for a channel of the same dimensions is shown for comparison. Shear stress reduction on the superhydrophobic wall follows the same trends observed from pressure measurements in Fig. 6. Little signi?cant drag reduction is observed Re ? 3000 with a marked reduction in rate of shear stress increase for Re ? 5000. The smooth wall behaves as expected for an entirely smooth channel, as indicated by the good agreement with the Colebrook line. In Fig. 8, the wall shear stresses, ?w, calculated from the Spalding ?t to the velocity pro?les and from pressure measurements of smooth, 30-30 and 60-60 channels are nondimensionalized to form a coef?cient of friction, C f = 2?wall / ?U2, and plotted as a function of Reynolds number. For comparison, the Colebrook prediction of friction coef?cient for the present perfectly smooth channel is superimposed over the data in Fig. 8. Friction coef?cient was selected to account for small variations in channel height existing between the pressure drop and PIV experiments. As previously indicated, the friction coef?cients of the smooth wall, calculated from PIV, and that of the smooth channel, determined from pressure drop, are in good agreement with each other as well as with the Colebrook prediction. At low Reynolds numbers, in the absence of any quanti?able slip at

FIG. 8. Coef?cient of friction for various surfaces calculated from both PIV and pressure measurements. Smooth surfaces ??? and superhydrophobic surfaces containing w = 30 ?m wide microridge spaced d = 30 ?m apart ?b? are shown for PIV measurements of a channel with a single superhydrophobic wall. Pressure drop measurements from channels with two smooth walls ??? and two superhydrophobic walls containing w = 30 ?m and d = 30 ?m microridges ??? and w = 60 ?m d = 60 ?m microridges ??? are also shown. The predictions of the friction coef?cient for a smooth channel are also shown ?—? in both the laminar and turbulent regimes. Transition occurs around Re= 2100.

the superhydrophobic wall, the coef?cient of friction for all cases tracks with that of the smooth-walled channel. At larger Reynolds numbers, where slip velocities are observed, the coef?cients of friction of the superhydrophobic surfaces were found to lie well below those of the smooth channels. The drag reduction was found to increase with increasing Reynolds number, becoming more signi?cant for Re ? 5000 as observed in the pressure measurements. The PIV measurements of the channel with a 30-30 superhydrophobic microridge surface on one wall and a smooth no-slip surface on the opposing wall show a somewhat smaller drag reduction than that which is noted by pressure drop along with two superhydrophobic walls. This result is likely due to differences in the ?ow cell geometry, speci?cally, the presence of the smooth wall in the PIV measurements, which was necessary to have transparency for ?ow visualization. The smooth wall has a higher wall shear stress than the superhydrophobic surface resulting in an asymmetric velocity pro?le and an increase in the turbulence intensity near the smooth wall. These observations were also made by Martell et al.24,23 for a DNS of channel ?ow with a single superhydrophobic wall. Observed drag reductions and slip velocities are in good agreement with predictions for a DNS at Re? = 180, corresponding to an experimental Re= 5300 in the PIV data. DNS slightly over predicts slip velocity, and slightly under predicts drag reduction at 11% and reports enhanced performance with increasing microfeature size, as observed in the experiments. It should also be noted that DNS of Martell et al.24,23 does not include interface de?ection or compliance effects. Drag reduction calculated from PIV data are in excellent agreement with the slip length boundary condition DNS of Min and Kim21 and predictions of Fukagata et al.22 for streamwise slip. Both groups reported approximately

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085103-7
a)

Drag reduction in turbulent ?ows over superhydrophobic surfaces
40
"

Phys. Fluids 21, 085103 ?2009?

Drag Reduction [%]

30

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20

!
0 2000 4000 6000 8000 10000

!

10

%

0

Reynolds Number, Re

!

"

#

$

%

'()*+,-. 0123(45 "#

b) 75

Drag Reduction [%]

50

25

FIG. 10. The microridge spacing in wall units, w+, as a function of Reynolds number. The data are taken from PIV measurements from a channel with a single superhydrophobic surface of w = 30 ?m and d = 30 ?m microridges ?b? and from pressure measurements for ?ow through a channel with two superhydrophobic walls containing w = 30 ?m and d = 30 ?m microridges ??? and w = 60 ?m and d = 60 ?m microridges ???. A spacing of w+ = 5 corresponds to the thickness of the viscous sublayer. Only points in the turbulent regime are shown.

0 0 2000 4000 6000 8000

Reynolds Number, Re
FIG. 9. Drag reduction as a function of Reynolds number for a channel with ?a? a single superhydrophobic wall w = 30 ?m d = 30 ?m ?b? and ?b? two superhydrophobic walls containing w = 30 ?m and d = 30 ?m microridges ??? and w = 60 ?m and d = 60 ?m microridges ???.

21% drag reduction21,22 at the same dimensionless slip length and friction Reynolds number observed in the present experiments at Re= 5300. Given the challenges of directly matching DNS and experiments, these results are quite encouraging. The turbulent drag reduction, DR = ??no-slip ? ?SH? / ?no-slip, was computed as the percent difference in shear stress at the superhydrophobic and no-slip wall and is presented in Fig. 9 as a function of Reynolds number. Drag reduction is presented rather than slip length because the slip length is dif?cult to quantify from the pressure drop measurements in turbulent ?ows. The slip length calculated from PIV data is insigni?cant in the laminar region and obtains a maximum value greater than b = 70 ?m for 30-30 and greater than b = 120 ?m for 60-60 ridges. In the present experiments, a maximum drag reduction in approximately 50% was observed for both microridge geometries once a suitably high Reynolds number was achieved. Drag reduction is initiated at a critical Reynolds number in the turbulent regime. For the microridges under present consideration, the critical Reynolds number was determined to be Recrit ? 2500. This Reynolds number is at or just past the transition to turbulent ?ow. This observation, along with the noted lack of drag reduction in the laminar regime, suggests that the underlying physical cause of the observed turbulent drag reduction must relate to the unique structure of wall-bounded turbulent ?ow. The physical origins of the critical Reynolds number for the onset of drag reduction can be understood by analyzing the relevant length scales in the ?ow. If the drag reduction and the slip length were dependent on the microridge geom-

etry and channel dimensions alone, as is the case in laminar ?ows, then we would expect to ?nd the drag reduction and slip length to be independent of Reynolds number. In turbulent ?ows, however, there is a third length scale of importance, the thickness of the viscous sublayer which extends out to y + = 5. Although the viscous sublayer thickness remains ?xed in wall units, in dimensional form the thickness of the viscous sublayer decreases with increasing Reynolds number as y vsl = 5??? / ?w. Close to the wall, where viscous stresses dominate, the analytical solutions of Philip11,12 show that the in?uence of the shear-free air-water interface extends to a distance roughly equal to the microridge spacing, w, into the ?ow. Thus for the superhydrophobic surface to impact the turbulent ?ow, the microridge spacing must approach the thickness of the viscous sublayer, w ? y vsl, or in other words w+ = y + ? 5. As seen in Fig. 10, the microfeature spacing in wall units is at least w+ ? 0.75 for all the 30-30 surfaces tested and w+ ? 2.4 for the 60-60 surfaces. The w+ values are calculated from shear stress measured at the superhydrophobic surface. This means that the microfeature spacing is minimally 15%–50% of viscous sublayer thickness almost immediately after the turbulent transition. Hence for 30-30 and 60-60 ridges, drag reduction is noticed almost as soon as a turbulent ?ow develops. In laminar ?ows, signi?cant drag reduction is noted at feature to height ratios comparable to those seen with the present feature size and viscous sublayer thickness.14 A similar scaling has been observed for turbulent ?ow over wetted, rough surfaces, where the effects of roughness are not observed until the size of the roughness exceeds the thickness of the viscous sublayer.32 As the Reynolds number increases and the thickness of the viscous sublayer is further reduced, the presence of the superhydrophobic surface will more strongly in?uence the velocity pro?le within the viscous sublayer and reduce the momentum transferred from the ?uid to the wall and the vorticity of the ?uid at the edge of the viscous sublayer. Turbulence intensity is thereby reduced, increasing the drag reduction. One therefore expects that saturation of the turbulent drag reduction is likely in the

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085103-8

Daniello, Waterhouse, and Rothstein
1 2

Phys. Fluids 21, 085103 ?2009? P. S. Virk, “Drag reduction fundamentals,” AIChE J. 21, 625 ?1975?. W. C. Sanders, E. S. Winkel, D. R. Dowling, M. Perlin, and S. L. Ceccio, “Bubble friction drag reduction in a high-Reynolds-number ?at-plate turbulent boundary layer,” J. Fluid Mech. 552, 353 ?2006?. 3 B. R. Elbing, E. S. Winkel, K. A. Lay, S. L. Ceccio, D. R. Dowling, and M. Perlin, “Bubble-induced skin-friction drag reduction and the abrupt transition to air-layer drag reduction,” J. Fluid Mech. 612, 201 ?2008?. 4 K. Fukuda, J. Tokunaga, T. Nobunaga, T. Nakatani, T. Iwasaki, and Y. Kunitake, “Frictional drag reduction with air lubricant over a super-waterrepellent surface,” J. Mar. Sci. Technol. 5, 123 ?2000?. 5 S. Hahn, J. Je, and H. Choi, “Direct numerical simulation of turbulent channel ?ow with permeable walls,” J. Fluid Mech. 450, 259 ?2002?. 6 D. W. Bechert, M. Bruse, W. Hage, J. G. T. van der Hoeven, and G. Hoppe, “Experiments in drag-reducing surfaces and their optimization with an adjustable geometry,” J. Fluid Mech. 338, 59 ?1997?. 7 J. Ou, J. B. Perot, and J. P. Rothstein, “Laminar drag reduction in microchannels using ultrahydrophobic surfaces,” Phys. Fluids 16, 4635 ?2004?. 8 P. Joseph, C. Cottin-Bizonne, J.-M. Benoit, C. Ybert, C. Journet, P. Tabeling, and L. Bocquet, “Slippage of water past superhydrophobic carbon nanotube forests in microchannels,” Phys. Rev. Lett. 97, 156104 ?2006?. 9 D. Quere and M. Reyssat, “Non-adhesive lotus and other hydrophobic materials,” Philos. Trans. Roy. Soc. A 366, 1539 ?2008?. 10 L. Gao and T. McCarthy, “A commercially available perfectly hydrophobic material,” Langmuir 23, 9125 ?2007?. 11 J. R. Philip, “Integral properties of ?ows satisfying mixed no-slip and no-shear conditions,” Z. Angew. Math. Phys. 23, 960 ?1972?. 12 J. R. Philip, “Flows satisfying mixed no-slip and no-shear conditions,” Z. Angew. Math. Phys. 23, 353 ?1972?. 13 E. Lauga and H. A. Stone, “Effective slip in pressure-driven Stokes ?ow,” J. Fluid Mech. 489, 55 ?2003?. 14 J. Ou and J. P. Rothstein, “Direct velocity measurements of the ?ow past drag-reducing ultrahydrophobic surfaces,” Phys. Fluids 17, 103606 ?2005?. 15 R. Truesdell, A. Mammoli, P. Vorobieff, P. van Swol, and C. J. Brinker, “Drag reduction on a patterned superhydrophobic surface,” Phys. Rev. Lett. 97, 044504 ?2006?. 16 E. Lauga, M. P. Brenner, and H. A. Stone, in Handbook of Experimental Fluid Dynamics, edited by J. Foss, C. Tropea, and A. L. Yarin ?Springer, New York, 2007?. 17 C. Ybert, C. Barentin, C. Cottin-Bizonne, P. Joseph, and L. Bocquet, “Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries,” Phys. Fluids 19, 123601 ?2007?. 18 R. S. Voronov, D. V. Papavassiliou, and L. L. Lee, “Review of ?uid slip over superhydrophobic surfaces and its dependence on the contact angle,” Ind. Eng. Chem. Res. 47, 2455 ?2008?. 19 R. S. Voronov, D. V. Papavassiliou, and L. L. Lee, “Boundary slip and wetting properties of interfaces: Correlation of the contact angle with the slip length,” J. Chem. Phys. 124, 204701 ?2006?. 20 S. B. Pope, Turbulent Flows ?Cambridge University Press, Cambridge, UK, 2003?. 21 T. Min and J. Kim, “Effects of hydrophobic surface on skin-friction drag,” Phys. Fluids 16, L55 ?2004?. 22 K. Fukagata, N. Kasagi, and P. Koumoutsakos, “A theoretical prediction of friction drag reduction in turbulent ?ow by superhydrophobic surfaces,” Phys. Fluids 18, 051703 ?2006?. 23 M. B. Martell, J. B. Perot, and J. P. Rothstein, “Direct numerical simulation of turbulent ?ow over ultrahydrophobic surfaces,” J. Fluid Mech. 620, 31 ?2009?. 24 M. Martell, in Department of Mechanical and Industrial Engineering ?University of Massachusetts, Amherst, 2008?, p. 127. 25 S. Gogte, P. Vorobieff, R. Truesdell, A. Mammoli, F. van Swol, P. Shah, and C. J. Brinker, “Effective slip on textured superhydrophobic surfaces,” Phys. Fluids 17, 051701 ?2005?. 26 C. Henoch, T. N. Krupenkin, P. Kolodner, J. A. Taylor, M. S. Hodes, A. M. Lyons, C. Peguero, and K. Breuer, “Turbulent drag reduction using superhydrophobic surfaces,” Collection of Technical Papers: Third AIAA Flow Control Conference, 2006, Vol. 2, p. 840. 27 K. Watanabe, Y. Udagawa, and H. Udagawa, “Drag reduction of Newtonian ?uid in a circular pipe with highly water-repellent wall,” J. Fluid Mech. 381, 225 ?1999?.

limit of very large Reynolds numbers where the microridges are much larger than the viscous sublayer. In this limit, the drag reduction should approach a limit of DR = w / ?d + w? as momentum is only transferred from the solid fraction of the superhydrophobic surface and the viscous sublayer is thin enough that the no-slip and shear-free portions of the surface can be considered independently. For the present shear-free area ratios, this limit would be 50%. This is consistent with both the asymptotic value of our PIV and pressure drop measurements. Drag reduction results shown in Fig. 9 appear consistent with this hypothesis, the 60-60 ridges already appearing to plateau. As the critical Reynolds number will decrease with increasing feature spacing, coarser superhydrophobic surfaces will begin to perform better at lower Reynolds numbers. It is therefore expected that equivalent drag reduction performance will be accessible to much ?ner microfeature spacings at higher Reynolds numbers. With ?ne superhydrophobic surfaces, little drag reduction may be evident until the viscous sublayer shrinks signi?cantly, well past transition. This result appears promising for possible commercial applications of this technology. This is because small feature spacing results in a more robust superhydrophobic surface capable of maintaining a coherent air-water interface at larger static pressures, while at the same time ships that might bene?t from such surfaces operate at Reynolds numbers signi?cantly greater than those interrogated in the present experiments.

IV. CONCLUSIONS

Signi?cant drag reduction has been measured by PIV and direct pressure measurements in turbulent ?ows over superhydrophobic microridge surfaces. No signi?cant drag reduction or slip velocities were noted in the laminar regime, consistent with theoretical predictions of laminar ?ow superhydrophobic drag reduction and previous experimental studies. This and the slip velocities observed at the wall demonstrate that the drag reduction is due to the presence of a shear-free interface. Slip velocities and drag reductions were found to increase with Reynolds number, the latter appearing to plateau at the highest Reynolds numbers tested. This drag reduction is found to increase more quickly with increasing feature spacing for equal shear-free area ratio. Our experiments suggest that viscous sublayer thickness is the correct height scaling for these surfaces and there exists a critical Reynolds number reached as the viscous sublayer thickness approaches microfeature size, when the onset of drag reduction will occur. Additional experiments and numerical simulations are currently underway to investigate this hypothesis.
ACKNOWLEDGMENTS

The authors wish to acknowledge the Of?ce of Naval Research for the support provided for this research under Grant No. N00014-06-1-0497. The authors would also like to thank M. Martell and B. Perot for helpful discussions and suggestions.

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085103-9
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Drag reduction in turbulent ?ows over superhydrophobic surfaces

Phys. Fluids 21, 085103 ?2009?

A. K. Balasubramanian, A. C. Miller, and O. K. Rediniotis, “Microstructured hydrophobic skin for hydrodynamic drag reduction,” AIAA J. 42, 411 ?2004?. 29 C. Henoch, T. N. Krupenkin, P. Kolodner, J. A. Taylor, M. S. Hodes, A. M. Lyons, C. Peguero, and K. Breuer, Turbulent Drag Reduction Using Superhydrophobic Surfaces ?AIAA, Reston, VA, 2006?. 30 J. C. Reed, “Using grooved surfaces to improve the ef?ciency of air injection drag reduction methods in hydrodynamic ?ows,” J. Ship Res. 38,

133 ?1994?. D. B. Goldstein and T.-C. Tuan, “Secondary ?ow induced by riblets,” J. Fluid Mech. 363, 115 ?1998?. 32 F. M. White, Viscous Fluid Flow ?McGraw-Hill, Boston, MA, 2006?. 33 D. B. Spalding, “A single formula for the “Law of the Wall”,” Trans. ASME, J. Appl. Mech. 28, 455 ?1961?. 34 L. F. Moody, “Friction factors for pipe ?ow,” ASME-Transactions 66, 671 ?1944?.
31

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