Turbulent drag reduction over air- and liquid- impregnated surfaces

Modifying the texture and wetting behavior of a surface can have important consequences for drag reduction. For example, superhydrophobic surfaces, where pockets of air are trapped inside micro- or nano-scale features on a non-wetting solid surface, have received much attention over the past decade as they have been shown to reduce wall shear stress in laminar channel flows^1,2 and rheometer flows^3,4 by introducing a partial slip at the air/water interface.⁵ Drag reduction using superhydrophobic surfaces has also been demonstrated in turbulent flows, numerically^6,7 and experimentally,^8–11 although the degree of success has been variable.¹² In such treatments, sustaining drag reduction hinges on the retention of air in the surface features. The air pockets fail when using complex liquids such as crude oil,¹³ when they are under high pressure,^14,15 and under high shear rates due to dissolution of vapor into the working liquid.¹⁶ Unless the impregnated vapor is replenished, for example, by electrolytic methods,¹⁷ the drag reducing properties will be lost, or in turbulent flow can lead to a drag increase due to roughness effects.

An alternative to maintaining these stable air pockets is to infuse a second liquid in the surface features. These liquid/liquid systems, which demonstrate omniphobic properties and robustness to pressure, will be stable as long as the two liquids are immiscible, the impregnating liquid preferentially wets the substrate compared to the working liquid, and interfacial tension is stronger than destabilizing body forces.^13,18,19 Recent rheometer measurements over a liquid infused surface indicate a drag reduction of up to 16% in laminar flow when the impregnating fluid is two orders of magnitude less viscous than the working fluid, with the drag benefit weakening for more viscous lubricants.²⁰ 

Here, using the same base micro-texture, we test superhydrophobic and liquid-infused surfaces in a turbulent flow. We perform experiments in a Taylor-Couette configuration, that is, flow in the annulus between two concentric cylinders, the outer of which is rotating, and demonstrate the effect of impregnating fluid viscosity on the skin friction. We show, for the first time, drag reduction in turbulent flow over a liquid-infused surface.

The Taylor-Couette flow was generated using a commercial rheometer (Brabender Rheotron), shown schematically in Figure 1(a). A stainless steel cylindrical cup (R₀ = 28.00 mm), containing test fluid of density ρ_w and dynamic viscosity μ_w, rotates in a brass bushing at angular velocity ω. The torque T, induced by the annular flow on the stationary inner cylinder (R_i = 26.02 mm), is transmitted by a bearing to a 25 mm lever arm which pushes against a 25 gram load cell (LCM systems UF1). The load cell was calibrated over the entire range by hanging precision weights from the force transmission arm, and the output signal was conditioned using a low-pass Butterworth filter and amplifier (Krohn-Hite Corporation). The accuracy of the load cell, as specified by the manufacturer, was < ±0.3% of the full-scale range, which corresponds to a torque accuracy of ±4%-10% for the laminar regime and ±1%-2% for the turbulent regime, with better accuracy at higher Reynolds number. The radial gap between the outer and inner cylinders, d = R_o − R_i = 1.982 ± 0.013 mm and the height of the inner cylinder H = 80.0 ± 0.1 mm. An acrylic ring, spaced 500 μm above the inner cylinder, was attached to the outer cylinder in order to prevent the free surface of the test liquid from rising during rotation. The fluid-filled gap underneath the inner cylinder measures approximately 15 mm in height, and we will discuss its significance with respect to end effects in Section III. The working fluid, deionised water, was maintained at constant temperature to within ±0.1 °C for each experiment using a cooling jacket. Over all tests, the temperature varied between 19 and 23 °C.

The surface of the aluminum inner cylinder was textured using a lathe and a sharp cutting tool to create threads with a pitch b ≈ 106 μm. With respect to the flow, the surface features were aligned in the streamwise direction. A confocal microscope (Olympus LEXT OLS4000) was used to measure the surface topography (see Figure 1(b)). Figure 1(c) shows three surface profiles near x/H = 0.25, 0.5, and 0.75, where x is the axial distance from the top of the cylinder. The features have a nominal height, found from averaging several profiles, of h = 75 ± 3 μm, and have an average cross-sectional area of 3470 ± 150 μm², which will be used to estimate the expected oil weight.

Three configurations of this textured surface were tested, as illustrated in Figure 1(a), called “hydrophilic,” “superhydrophobic,” and “liquid-infused,” respectively. For the control case (hydrophilic), the surface was made wettable so that the test liquid completely impales the texture, referred to as the Wenzel state.²¹ The aluminum surface was first cleaned in an ultrasonic bath of acetone (99.9%, Sigma-Aldrich) and then in an air-filled plasma cleaner (Harrick Plasma), each for approximately twenty minutes. This surface preparation rendered the surface strongly hydrophilic, as illustrated in Figure 2(a).

For the superhydrophobic case, following Kim et al.,²² the same cylinder was then placed in an ethanol-based solution containing Masurf FS100 (Mason Chemical Co.), a phosphate ester with a mixed length of fluorinated alkyl chains, resulting in strong repellency to water drops due to the entrapment of air, referred to as the Cassie-Baxter state²³ (see Figure 2(b)).

For the liquid-infused cases, the same surface used in superhydrophobic tests was first infused with perfluorinated oils that have a high chemical affinity for the fluorinated aluminum.²² Specifically, a thick coat of oil was pipetted, for the more viscous oil (μ_w/μ₀ = 1/30), or dip-coated, for less viscous oil (μ_w/μ₀ = 1/1.5), onto the surface and allowed to drain vertically by gravity. Over a sufficiently long time, the excess oil overlying the features is expected to drain, leaving only pockets of oil trapped inside the textures.²⁴ After testing these two surfaces, a final surface was tested where the infused oil was heptane, which has a lower viscosity than water (μ_w/μ₀ = 2.7). To make the surface wetting with respect to alkanes, the fluorination treatment was removed in the plasma cleaner and the surface was functionalized with n-Octadecyltrichlorosilane (OTS). The physical and chemical properties of all test surfaces are summarized in Table I.

Capillary pressure should sustain the trapped oil since the characteristic Bond number, Bo = Δρgb²/γ, which represents the ratio of body to capillary forces, is O(10⁻²) and O(10⁻³) when the cylinder is immersed in air and water, respectively. The interfacial tension γ ≈ 15–20 × 10⁻³ N/m for the oils with respect to air, and approximately 51–56 × 10⁻³ N/m for the oils with respect to water, as measured using the pendant drop method. The dynamic viscosity of all test liquids was independently measured at the test temperature using a parallel plate rheometer (Anton-Paar GmbH. Physica MCR 301).

To confirm that oil was retained in the cavities during the experiments, and that the drainage of the overlying oil was complete, the cylinder was infused with the most viscous oil and the total weight was monitored using a scale with 10 mg accuracy (Ohaus, Explorer Pro) until it converged. Comparing the converged weight to the dry cylinder weight, the mass of the oil was found to be approximately 700 mg, which is within 10% of the expected mass as inferred by the estimated cross-sectional area of the grooves. The difference is probably due to details regarding the oil interface shape, and because a thin overlying layer of O(1) μm is expected to remain even after the bulk drainage has ceased.

It should be noted that the less viscous fluorinated oil is somewhat volatile in air, and so for those experiments the cylinder was dip-coated in a bath of oil and then immediately submerged in the test fluid, allowing the excess to drain to the gap underneath the cylinder, since the oil is denser than water. The mass of excess oil is very small compared to the volume of water in the bottom gap, and so does not affect the results.

Due to the especially volatile nature of heptane, considerable care was taken to ensure that the surface features were fully impregnated with liquid for each experiment. Figure 3 shows time lapse images of the threaded cylinder in air as the heptane evaporates from the features over the course of ∼40 s. Since it takes approximately 4 s for the cylinder to be dip-coated with heptane and then fully submerged in the test water, we estimate that the features are approximately 90% full of oil at the start of each experiment. In between each measurement, the heptane was rinsed off using ethanol, or the fluorinated oil was removed using an appropriate solvent (Krytox Remover, Miller-Stephenson Chemical Co.). The cylinder was then allowed to dry in a warm oven. Before application of a new oil, the weight of the cylinder was measured to confirm that there was no residue from the previous measurement.

For each test case, the outer cup was spun up to speed and then maintained at that speed for 10 s before measuring the torque on the inner cylinder, averaged over 20 s. At least five trials were conducted for each different test surface. The results for the control case (hydrophilic) are shown in Figure 4. The skin friction coefficient C_f is given by

where the shear stress $τ=T/ ( 2 π R i 2 H )$⁠, and the Reynolds number Re = ρ_wUd/μ_w. Also plotted is the theoretical curve for a laminar Taylor-Couette flow given by

where the term in parenthesis represents a modification to the parallel plate solution due to curvature effects. Taylor-Couette data for the case of a rotating outer cylinder/stationary inner cylinder configuration are rather limited, and so we compare with the original results by Taylor.²⁵ The influence of curvature on the turbulence profile, characterized by the curvature ratio d/R₀, can be significant and so the closest value from Taylor’s data (d/R₀ = 0.0776), was selected, which is within 10% of our curvature ratio of 0.0706.

The Reynolds number range tested, approximately 1700–9000, was limited by the maximum rotational speed of the motor and drive system (about 1500 rpm). As shown in Figure 4, the C_f versus Re curve exhibits a change in shape around Re = 4000, where the flow transitions from a laminar to a turbulent state, characterized by an abrupt increase in torque level and fluctuation. The measured friction coefficient is higher than expected due to torque contributions from the bottom of the cylinder. This additional torque contribution was measured by carefully filling only the gap underneath the inner cylinder with the test fluid. The measured torque was corrected by subtracting this tare value, and the corrected data match the expected results well, where the small deviation from Taylor’s data might be attributed to the crudeness of this correction. Unlike most Taylor-Couette devices, which use a counter-bore underneath the inner cylinder to entrap a low friction air cushion, the inner cylinder in this setup has a smooth, flat bottom. Therefore, the wetting condition and torque contribution from the bottom surface should be the same for all test surfaces, and it is sufficient to use a simple tare correction.

The viscous length scale, $δ v = ν w / τ / ρ w$⁠, is approximately 19 μm at Re = 5900, decreasing to 14 μm at Re = 9000. That is, 5.6 ≤ b⁺ ≤ 7.6 and 3.9 ≤ h⁺ ≤ 5.4, where b⁺ = b/δ_v and h⁺ = h/δ_v, which raises the possibility that there may be a contribution to the drag reduction purely by a riblet effect. Walsh²⁶ reports drag reduction for V-shaped grooves when h⁺ < 25 and b⁺ < 30, with the maximum drag reduction of 8% occurring for h⁺ = 10 and b⁺ = 15. Based on his experiments, though, the expected riblet induced drag reduction for our roughness should only be 1%–2%. Furthermore, this effect will only occur for the untreated hydrophilic surface, and therefore any drag reduction observed for the superhydrophobic and liquid-impregnated surfaces will be in addition to this contribution.

Figure 5(a) shows the raw torque measurements for all surfaces and Figure 5(b) shows the drag reduction for the air- and liquid-infused cases as a function of Reynolds number in the turbulent regime. The drag reduction, DR, is defined in terms of the torques as

where T₀ and T are the measured control and treated cylinder torques, respectively, and T_0,a represents the annular torque of the control surface, in which the bottom contribution has been subtracted. Since the bottom torque should be the same for all test surfaces, this gives a better estimate of the drag reduction over the treated portion of the cylinder. The vertical error bars represent one standard deviation of all torque measurements, which were repeated at least five times for each different test surface. The horizontal error bars for the Reynolds number result mainly from scatter in the test fluid temperature. In between each experiment, the cylinder was removed, fresh oil was applied for the liquid-infused cases, and the cylinder was reinstalled in the apparatus.

Over the Reynolds number range tested, the drag reduction remained fairly constant for the different surface treatments. The liquid-infused surface with the highest viscosity did not provide any drag benefit, but the oil with μ_w/μ₀ = 1/1.5 reduced the skin friction by approximately 5%. The heptane-infused surface (μ_w/μ₀ = 2.7 = 1/0.37) performed the best, reducing skin friction by 14%, compared to the 10% for the superhydrophobic case. It should be noted that upon removal of the superhydrophobic cylinder from the test apparatus, there were distinct dry patches, where the surface maintained a Cassie-Baxter state, and distinct wet patches, where the surface had failed into a Wenzel state, which may explain why it did not perform better than the heptane-infused surface.

The dominant mechanism of drag reduction for superhydrophobic surfaces is a finite average slip velocity $u s$ at the air/water interface, although some drag reduction can result from modifications to the near-wall turbulence dynamics for longitudinal grooves.^7,27 It is common to describe these surfaces by an apparent slip length, δ_s, defined by the Navier slip boundary condition as $δ s = u s d u s / d y − 1$⁠, which is the effective distance into the surface where the no-slip condition would be satisfied by a linear extension of the wall velocity gradient. One can estimate the slip length in turbulent Taylor-Couette flow by fitting a semi-empirical logarithmic friction law and assuming a simple offset to the velocity profile in the viscous sublayer.¹¹ Following this procedure, the superhydrophobic surface is characterized by a slip length of δ_s = 71 ± 48 μm, and the liquid-infused surfaces with viscosity ratios μ_w/μ_o = 1/1.5 and 1/0.37 are characterized by slip lengths of δ_s = 38 ± 33 μm and 138 ± 55 μm, respectively.

A possible failure mode that arises for liquid-infused surfaces is shear driven drainage of the oil.²⁸ The drainage rate for a long longitudinal groove is constant, and the dewetting speed is given by Cτh/μ₀, where C is a function of groove aspect ratio b/h and oil contact angle.²⁸ For a rectangular groove of these dimensions, the model predicts that, at the average shear stress, the heptane would take approximately 20 min to drain helically around the cylinder. We expect this drainage time to be a conservative estimate due to the higher ratio of surface area to volume for a triangular groove. For one sweep through the Reynolds number range, the surface is exposed to shear for only about 3.5 min, which implies that our conclusions are not significantly affected by lubricant drainage. From a practical standpoint, we expect that the oil in this streamwise groove configuration will eventually be depleted due to shear, and then the drag-reducing properties will be lost. Different physical and chemical techniques, however, such as appropriately spaced drainage barriers,²⁸ patterned wettability,²⁹ or surfactant-induced Marangoni stresses,³⁰ may help to retain the lubricant indefinitely.

In this study, we considered the wall friction generated in turbulent Taylor-Couette flow over a generalized fluid-infused micro-textured surface, where the impregnating fluid was either air or a liquid that is immiscible with respect to the external flow. For the liquid-impregnated surfaces, it was observed that the torque decreased as the oil viscosity decreased, and the best surface performed even better than a superhydrophobic surface, which failed partly over the course of the experiment. For the geometry studied here, that is, streamwise aligned ridges with a cross-stream wavelength of approximately 5-8 viscous units, it was found that for the liquid-infused surface the viscosity ratio between the water and the infused liquid should be O(1) for some drag reduction to occur. A drag reduction of 14% was achieved for a liquid that has a viscosity about 1/3rd that of water. Current work is focused on studying the longevity of liquid-infused surfaces, and means for promoting lubricant retention under high shear conditions.

We acknowledge Philseok Kim, Tak Sing Wong, Ying Liu, and Joanna Aizenberg for their assistance with the surface preparation, Frederik Brasz and Craig Arnold for use of the confocal microscope, and Jason Wexler, Ian Jacobi, Howard Stone, and Marcus Hultmark for helpful discussions. This work was supported by the Office of Naval Research through MURI Grant Nos. N00014-12-1-0875 and N00014-12-1-0962 (Program Manager Dr. Ki-Han Kim).