In situ monitor of superhydrophobic surface degradation to predict its drag reduction in turbulent flow

Figure 1 demonstrates the working principle for the in situ air layer monitoring system over the employed superhydrophobic surfaces in a large-scale channel flow facility, which has been previously studied by Agrawal et al.²⁶ and Escudier et al.²⁷ A schematic representation of the channel system is shown in Fig. 1(a) (see more details in Section S1 of the supplementary material). A fully developed two-dimensional turbulent channel flow experiment can be achieved in this rig with dimensions of length(l) × width(w) × half-height(h) =  $7.2 × 0.298 × 0.0125 m 3$⁠, providing an aspect ratio (⁠ $A R = w / 2 h$⁠) of 11.92.²⁸ A test section (red rectangular highlighted) was installed $6.0 m$ downstream (⁠ $z / h = 480$⁠) from the inlet. A superhydrophobic coated PVC (polyvinyl chloride, Direct Plastics Limited, UK) substrate with dimensions 100 wide and 200 mm long (i.e., $8 h × 16 h$⁠) was flush mounted at the center of the bottom wall of the test section. The employed SHO materials (SPNC, superhydrophobic polymer-nanoparticle composite) have been recently developed and reported, as discussed in previous studies.^29,30 The detailed fabrication procedure of the current surfaces can be found in Section S2 of the supplementary material. The surface properties were characterized by a dynamic shape analyzer (Kruss, Model DSA 100E) and an optical profiler (WYKO NT1100), and the results are displayed in Table I. The highly water-repellent nature and surface roughness of the current SPNC surface have been confirmed, and these advantages are promoted by a dual-layer structure at the microscale level.³¹ The side and top walls of the test section are made of borosilicate glass to allow optical access for illuminating light and laser beams. Two pressure transducers were adopted to measure the local gauge pressure (P1, Validyne DP15) 20 mm upstream of the coated area and the pressure-drop (P2, Druck LPX-9381) across a distance of 240 mm over the test surface, respectively. Instantaneous streamwise velocity measurement was conducted by a laser Doppler velocimetry (LDV) system (Dantec dynamics, see more details in Section S3 of the supplementary material). In the current study, LDV measurements were carried out at a fixed wall-normal (fixed y/h) location to monitor the time-dependent velocity/turbulent intensity variations. The velocity and RMS (root mean square) velocity profile across the channel half-height for a no-slip boundary condition (benchmarked against direct numerical simulation results³²) are included in the supplementary material (Section S4) to provide a validation of the LDV measurement system. The real-time air layer observation setup built at the test section of the channel [highlighted red in Fig. 1(a)] is shown in Fig. 1(b). The white light source from the illuminator (Thorlabs OSL2) was reflected by the submerged SHO surface and the formed air layer. The reflection was captured as videos/images by a camera (iPhone XR, Apple, Inc.) above the channel as illustrated in Fig. 1(b). The received images were processed (cut to a constant pixel size from the center—800 × 800 pixels) by MATLAB software, and the average pixel intensity (I) was calculated using in which N is the total number of pixels (N = 640 000 for each image) and I_i is the intensity of the ith pixel. The average pixel intensity for each image was normalized by the average intensity of the image at the end of each test, which refers to the state that an air layer that has been lost entirely (I_final), in which I_n is the normalized pixel intensity. Four representative images at various time points are displayed in Figs. 1(c)–1(f) with the normalized intensity shown below each image. The brightness (reflecting intensity) of the images is observed to increase over time. This phenomenon can be attributed to the dissipation of the air layer (plastron), which reduces the effect of interference (refraction). As a result, the white silica particles, which makeup a component of the superhydrophobic coatings, appear with higher intensity in the captured images. Therefore, the observed increase in the average pixel intensity indicates the progressive loss of the air plastron and the diminishing superhydrophobicity with time. At the end of this test, the image was almost purely white, and the normalized pixel intensity reached 100% as no air-plastron survived. Example videos showing the dynamic process of air layer loss are included in the supplementary material (Movies 1 and 2).

We note that the dissolution of the air-plastron occurs even when the surfaces are immersed in quiescent water, especially at very high pressure.^8,19 Therefore, prior to conducting turbulent tests, a “no-flow” hydrostatic test over the employed SHO surface was performed to obtain more understanding of the interaction between the air-plastron and the local absolute pressure. Figure 2 shows the variations in the normalized reflecting pixel intensity as a function of time at various gauge pressures (P_gauge) over the surface. In the first 60 min, I_n was increasing with time for all the pressures studied and the higher the pressure the faster the increment. This is primarily due to the thinning of the air layer (i.e., air mass transfer from the plastron to water),¹³ so the reflection pixel intensity became higher. From 60 to 180 min, the evolution of I_n tends to reach a constant state (thermodynamic equilibrium at the air–water interface⁹) for P_gauge = 3.65, 4.53, and 5.50 kPa. However, for the other three higher gauge pressures, I_n increased continuously with time, and the entire air layer could be lost if the test was long enough (anticipated from the non-zero slope of the curve at the final measurement time). Given this observation, P_gague = 5.50 kPa was treated as a critical pressure²⁵ beyond which all the air-plastrons would gradually breakup and the wetting mode transforms from Cassie–Baxter to Wenzel. Thus, most of the following turbulent flow experiments were conducted at a P_gauge < 5.50 kPa so as to keep the local gauge pressure over the test surface below this critical pressure. In this way, the drag reduction decay resulting from air layer loss can be monitored independently. Moreover, the critical pressure (P_gague = 5.50 kPa) is closely related to the surface topology. According to the Laplace equation [Eq. (3)], P_gague was used to determine the radius of trapped air bubbles (R), where $Δ P$ represents the pressure difference at the air–water interface. In this case, $Δ P$ is assumed to be equal to P_gauge, as we consider the air pressure in the plastron to be constant and equal to atmospheric pressure. $γ = 72.4$ mN/m is the surface tension of water in air. (Note, we measured the actual surface tension of the tap water used in the current study via a surface tensiometer.) The estimated length is calculated as 26 μm, which is consistent with the average roughness result from surface characterization in Table I.

To confirm the feasibility of the current approach in turbulent flow, pressure-drop measurements were conducted to show the correlation between drag reduction (DR) and I_n. The Fanning friction factor (⁠ $f = τ w / 0.5 ρ U b 2$⁠) was employed to characterize the drag reduction and calculated by the mean wall shear stress (⁠ $τ w = Δ Pwh / ( l ( w + 2 h ) )$⁠), in which ρ is the water density, U_b is the bulk velocity, and $Δ P$ is the mean pressure-drop across the tested surface. A comparison between the calculated friction factor against Re_h (⁠ $= ρ U b h / μ$⁠, in which μ is the viscosity of water) over reference (smooth) and SHO surfaces in which Pope's correlation is also included is shown in Fig. 3(a).³³ To avoid any potential degradation of the SHO surface, the duration of this pressure-drop measurement is only 1 min in each case. The tests were performed on a single SHO sample and repeated ∼3 times. In the Re_h range from 1500 to 3000, a 5%–10% drag reduction is achieved via the current surfaces as shown in Fig. 3(a). It is noted that at Re_h = 4270, there was no drag reduction measured as the friction factor collapsed to the smooth data line. Due to a high local absolute pressure at this Re_h, the air layer over the surface can barely survive to create a slip boundary, and therefore, no drag-reducing effect was achieved. Focusing on a low Re_h and P_gague, a long-term pressure-drop measurement was carried out simultaneously with air layer monitoring (in situ). The drag ratio (⁠ $f SHO / f smooth$⁠, the friction factor ratio between superhydrophobic and smooth surfaces) and normalized intensity (I_n) were plotted as a function of the flow time in Fig. 3(b). $f SHO / f smooth$ stayed constant in the first 40 min and increased with flow time afterward, indicating a time-dependent degradation of the SHO surfaces. Meanwhile, a consistent trend of the normalized intensity showed that the reflecting intensity was increasing because of the loss of air-plastron (transform from the Cassie–Baxter to Wenzel state). The loss rate of air-plastron was significantly higher than the hydrostatic case at the same P_gague if comparing Re_h = 3165 in Fig. 3(d) and P_gague = 5.50 kPa in Fig. 2, for example. Since a turbulent flow was created, the mass transfer from air to water was forced by the viscous and turbulent shear (i.e., the mass transfer pattern changed from dissolution to turbulent forced convection). Although both techniques capture the surface degradation (mostly reversible, see details in Section S6 of the supplementary material), the drag ratio provides quantitative results, and the pixel intensity was more straightforward with a very simple setup. Drag reduction (DR $= 1 − f SHO / f smooth$⁠) and air-plastron loss (⁠ $I n , loss$⁠) over SHO surfaces against flow time at various Re_h and the corresponding P_gague were presented in Figs. 3(c) and 3(d), respectively. $I n , loss$ is a parameter detected from the pixel intensity results and directly represents the level of air-plastron loss, as described as in which I_initial is the initial average intensity measured at the start of the test. As discussed, DR decays along with time and a higher gauge pressure would lead to a faster degradation. It is noteworthy that some drag enhancements can be observed at the end of the first 3 tests when the air layer is almost gone. This is attributed to the surface roughness of the SHO sample since there are still peaks and valleys with the maximum size of ∼300 μm though the average roughness is as low as 30 μm, as shown in Table I. Consistency between Figs. 3(c) and 3(d) (DR and $I n , loss$⁠) confirms that this approach is valid to predict drag reduction of the SHO surfaces qualitatively. Although previous studies have made significant efforts to discuss the influence of hydrostatic pressure and shear stress on superhydrophobic surface degradation, these two parameters have rarely been considered together.^12,13,20 In Figs. 3(e) and 3(f), the drag reduction and air-plastron loss (⁠ $I n , loss$⁠) at different P_gague but equivalent wall shear stresses (τ_w) are compared. Clearly, the one with high P_gague demonstrated a faster superhydrophobic longevity deterioration. Thus, it is clear that the normal and shear force generated from the turbulent flow can independently accelerate the transformation from the Cassie–Baxter to Wenzel state for SHO surfaces.

Instantaneous streamwise velocity (u) measurements have been carried out at fixed wall-normal locations (fixed y/h) using the LDV system to obtain information on the local velocity field. Two representative locations were selected to perform the velocity measurement as a function of the flow time. At each location, velocity samples were collected for 220 min until the surface was unable to produce any drag reduction (air layer dissolved/achieved Wenzel state). The mean velocity (⁠ $u ¯$⁠) and RMS velocity fluctuations (⁠ $u ′$⁠) were calculated using the software (BSA flow software) of the LDV system. The mean velocity normalized by wall units (⁠ $u + = u ¯ / u τ$⁠) against flow time at two different wall locations (⁠ $y final +$ = 3.4 and $y final +$ = 122) is shown in Fig. 4(a), in which $u τ = τ w / ρ$ is the friction velocity estimated by pressure-drop and $y final +$ represents the normalized (by wall units) wall locations at the end of the test. (The physical spatial location has not changed; however, the frictional velocity changed due to the loss of the air layer.) u⁺ at $y final +$ = 3.4 (within the viscous sublayer where the turbulent Reynolds stresses do not significantly contribute to the flow³³) exhibited an increase along with flow time, indicating the wall shear stress was increasing as $τ w = μ ( d u / d y )$⁠. The air-water contact area decreases due to the dissolution of the air-plastron so the overall shear stress increases. However, a reverse trend of u⁺ was observed at $y final +$ = 122 for which the normalized velocity decreased with time to keep mass flow balanced. Thus, it could be expected that the entire velocity profile along the channel height shows a downward shift after the degradation of the surfaces. It is worth pointing out that the value $u ′ / u ¯$ at both locations was constant the whole time, as shown in Fig. 4(b). This demonstrates that the turbulent intensity was not affected by the SHO surfaces/slip boundary. Figure 4(c) presents a comparison of the wall shear stress calculated from the near wall velocity (⁠ $u ¯$ at $y final +$ = 3.4) and the pressure-drop across the SHO surface. Consistent with Fig. 4(a), the shear stress increased with flow time since the air-plastron was being lost. Moreover, these two calculations were almost equal with a very slight shift downwards to the right (τ_w was slightly higher estimated from $Δ P$⁠). This could be attributed to the pressure-drop being measured at a cross-sectional area in which only 13.7% was actually coated as SHO so that the shear stress was over-estimated. Finally, in Fig. 4(d), the slip length (b) estimated by the near wall velocity [using Eq. (5)] was plotted vs flow time, Note that the maximum slip length result here (62.5 μm) is less than half of the results we measured in the previous study using the same SHO surface (which was above 140 μm) in a laminar flow via a rheometric device.³⁴ This is mainly because of the significantly higher absolute pressure that limits the initial plastron thickness of the current surface. The decay of slip length illustrated the degradation of SHO surfaces. Similar results have also been reported using different SHO surfaces in a rheometer-based laminar flow.³⁵ However, the normal stress (pressure) was much lower so it took ∼8 h for the sample to lose all superhydrophobicity. As an additional indicator, the normalized pixel intensity (I_n) in Fig. 4(d) also demonstrates the SHO surface degradation though the sensitivity is low at the beginning of the test.