An experimental study on the transient runback characteristics of wind-driven film/rivulet flows

Aircraft icing is a well-known weather hazard affecting flight safety and performance. As an airplane flies under cold weather conditions, supercooled water droplets in the clouds would impinge, freeze, and form ice shapes on airframe surfaces.^1,2 The ice accretion process over airframe surfaces can be either a dry rime or wet glaze icing process, highly depending on the weather conditions, including the cloud liquid water content (LWC), atmosphere temperature, and size of the supercooled water droplets in the cloud.^3,4 Under a typical glaze icing condition with relatively warm ambient temperatures (e.g., just below the freezing point of water), higher LWC levels in the cloud, and larger supercooled water droplets, while only a portion of the impacted supercooled water droplets would be frozen into solid ice instantly upon impacting onto airframe surfaces, the rest of the impacted water would stay in the liquid and run back freely over the airframe surfaces, as driven by boundary layer airflows.^3,5,6 The runback behavior of unfrozen water film/rivulet flows on airframe surfaces can directly and indirectly influence the resultant shape of ice accretion.⁷ While the transport process of the unfrozen water would have a direct impact on the formation of the final ice shape due to its effect on the redistribution of the impinged water mass over airframe surfaces,⁸ it would also be able to influence the ice accretion process through its impacts on the surface roughness, thereby, affecting the local convective heat transfer and the ice accretion rate over the airframe surfaces.^9–12 Hansman and Turnock¹³ conducted an experimental study to examine the importance of surface water transport on the glaze ice accretion process, and confirmed that the wind-driven runback process of unfrozen water would alter the final glaze ice shape significantly. Waldman and Hu¹⁴ visualized the dynamic glaze ice accretion process on an airfoil surface with a high-speed imaging system, and revealed that the stagnated rivulet fronts/heads would act as barriers to block the surface water transport, thereby, affecting the ice accretion process over the airfoil/wing surface.

While the dynamics of thin film flows under the influence of external fields—such as gravitational,¹⁵ electric,^16,17 and temperature^18,19 fields—have been investigated comprehensively in recent years, the present study aims to provide fundamental insights into the dynamics of wind-driven film/rivulet flows and its effects on the surface water transport process pertinent to aircraft icing phenomena. It should be noted that several fundamental studies have already been conducted to elucidate the underlying physics associated with the dynamics of water film/rivulet flows over solid surfaces. For example, Hartley and Murgatroyd²⁰ conducted a theoretical study to predict the re-wetting process of surface shear-driven thin film/rivulet flows over solid surfaces. They established two film breakup criteria to predict the minimum film thickness required for re-wetting a dry patch that occurs when the excess pressure force built inside a film flow surpasses the restraining surface tension force at the stagnation point: one is based on the force balance (FB) analysis near the stagnation point of a dry patch; the other is based on the minimization of total energy (MTE) of a stable rivulet. Murgatroyd²¹ refined the FB model by adding the unbalanced shear stress between the solid–liquid and liquid–gas interfaces. A characteristic length was introduced to represent the length of the unbalanced shear stress range. The unbalanced shear stress length was determined by fitting the experimental data empirically. Penn et al.²² determined the unbalanced shear stress length via CFD simulation. El-Genk and Sable²³ refined the MTE model by considering the velocity distribution within film/rivulet flows. The velocity profile was estimated numerically by using a finite element method (FEM) for a more accurate prediction of the kinetic energy of the film/rivulet flow. The refined MTE model was then applied to predict the minimum film thickness of a falling film flow on a vertical surface. The method suggested by El-Genk and Sable²³ was further extended to determine the breakoff of the stagnated film flow subject to interfacial shear.²⁴ 

Both FB and MTE criteria were also adopted to study aircraft icing phenomena. Thompson and Marrochello²⁵ used a FB criterion to predict the rivulet formation of wind-driven water runback flows over a NACA4412 airfoil surface. Al-Khalil et al.²⁶ developed a water runback model based on the MTE criterion for aircraft icing applications. It should be noted that the stagnated rivulet head or dry patch was usually treated as a smooth flat surface for FB analysis in those previous studies. However, in reality, for wind-driven water film/rivulet flows pertinent to aircraft icing phenomena, smooth rivulet fronts would exist only when the incoming wind speed is very low (i.e., not high enough to induce surface waves at the air/water interfaces).²⁷ In general, the surface waves generated at the air/water interfaces would induce the formation of hump-shaped rivulet fronts/heads, consequently, leading to additional aerodynamic drag forces, which were not considered in the original FB analysis. The additional aerodynamic drag forces could be very substantial, even becoming dominant, in comparison to the restraining forces due to the surface tension. Thompson and Marrochello²⁵ and McAlister et al.²⁸ modified the original FB model by taking the additional aerodynamic drag forces into account in predicting the rivulet breakoff for wind-driven film/rivulet flows; however, a very simplified model was used in estimating the aerodynamic drag forces acting on the rivulet fronts/heads. Due to lack of accurate measurement techniques to quantify the transient behavior of wind-driven film/rivulet flows, many important runback characteristics of wind-driven water film/rivulet flows are still unclear.

In the present study, we report a comprehensive experimental campaign to characterize the transient behavior of water film/rivulet flows driven by boundary layer airflows over a flat plate for a better understanding of the surface water transport process pertinent to aircraft icing phenomena. The experimental study was conducted by using an open-circuit wind tunnel to generate laminar boundary layer airflows with different freestream wind speeds to drive thin water film flows over a flat test plate with different water flow rates. A novel digital image projection (DIP) system^29,30 was applied to provide non-intrusive, temporally and spatially resolved measurements of the thickness distributions of the wind-driven water film/rivulet flows over the flat plate. The measured film thicknesses distributions of the wind-driven water film/rivulet flows were further processed to determine the instantaneous runback velocity of the rivulet fronts/heads under different test conditions. A comprehensive force balance (FB) analysis was also performed to examine the variations of the relevant forces (i.e., the excess pressure forces built inside the film/rivulet flows, aerodynamic drag forces acting on the rivulet heads, and the restraining forces due to surface tension along the liquid contact lines) and to evaluate their importance in determining the runback characteristics of the wind-driven film/rivulet flows.

The experimental study was performed in an open-circuit, low-speed wind tunnel available at the Aerospace Engineering Department of Iowa State University. The wind tunnel has a test section with a dimension of 200 mm × 150 m× 500 m (width × height × length) and four optically transparent walls. Figure 1 shows the experimental setup used to quantify the dynamic runback process of wind-driven water film/rivulet flows over a flat test plate. As shown schematically in Fig. 1, a flat test plate, which is 250 mm in length and 150 mm in width, is made of a hard plastic material (i.e., VeroWhitePlus, RGD835) and manufactured by using a rapid prototyping machine (i.e., Stratasys 3D printer). The upper surface of the test plate was polished with fine sandpaper (i.e., up to 2000 grit) and special plastic polishes to achieve a very smooth, glossy finish. During the experiments, the test plate was flush mounted to the bottom wall of the wind tunnel test section. Thirteen water outlet holes were uniformly distributed near the entrance of the test plate. A digital gear pump (Cole-Parmer 75211–30) was used to drive water from a reservoir to go through the water injection holes to form a water film over the test plate. While the diameter of the water injection holes is 2.0 mm, the distance between the neighboring holes is 5.0 mm. As a result, the nominal width of the water film over the test plate is $D=60 mm$⁠. The flow rate of the water film flow was controlled by adjusting the settings of the digital gear pump and monitored by using a digital flowmeter (Omega FLR 1010 T-D). After exhausted from the water injection holes, the water film would run back over the test plate as driven by the boundary layer airflow.

It was found that surface water runback processes over ice accreting airframe surfaces would be conducted in different modes, depending on the speed of the incoming airflow and the thickness of the water film over the surface. As described in Liu et al.,²⁷ the evolution of the instability at the air/water interface of a wind-driven film flow could be categorized into three regimes: (1) no obvious surface wave generation at the air/water interface when the incoming wind speed is relatively low (i.e., $U∞<10 m/s$⁠); (2) formation of well-organized, two-dimensional surface waves at medium wind speeds (i.e., $10<U∞<15 m/s$⁠); and (3) formation of complex three-dimensional surface waves when the incoming wind speed is sufficiently high (i.e., $U∞>15 m/s$⁠). In order to examine the effects of surface waves generated at the air/water interfaces on the characteristics of wind-driven film/rivulet flows in different flow regimes, the freestream speed of the incoming airflows over the test plate was changed from U_∞ = 5 to U_∞ = 20 m/s for different test cases. Following up the work of Liu et al.,²⁷ the flow rates of the water flow injected onto the test plate were set to be $Q$ =100 and 200 ml/min during the experiments.

Before injecting water mass over the test plate, a high-resolution digital particle image velocimetry (PIV) system was used to characterize the boundary layer airflow over the test plate. Figure 2 shows some typical PIV measurement results (i.e., time-averaged velocity profiles) at the center of the test plate (i.e., at 100 mm downstream of the water injection holes). Before the injection of water film over the test plate, the velocity profiles of the boundary layer airflows at the center of the test section were found to be fit well by Blasius solution.³¹ After injecting water over the test plate, a thin water film/rivulet flow would form over the test plate. It should be noted that, while wavy surface waves were found to be generated at the air/water interface for the test cases with relatively high incoming wind speeds (i.e., U_∞ > 15 m/s), since the thickness of the wind-driven film/rivulet flow over the test plate is very thin (i.e., being less than 1.0 mm), the characteristics of the boundary layer airflows over the test plate were found to be almost unchanged in comparison to those shown in Fig. 2. Therefore, the water film/rivulet flows over the test plate are assumed to be driven by laminar boundary layer winds for the test cases of the present study.

A digital image projection (DIP) system was used in the present study to achieve time-resolved measurements of the thickness distributions of the water film over the test plate in order to quantify the transient runback behavior of the wind-driven water film/rivulet flows. As described in Zhang et al.,³⁰ the DIP technique is based on the principle of structured light triangulation in a fashion similar to a stereo vision technique, but replacing one of the cameras for stereoscopic imaging with a digital projector. A Dell DLP projector (M109S) was used to in the present study to project a grid pattern onto the test plate. The projected digital images have 7 ×7 pixel grids, corresponding to about $1.0×1.0 mm$ in the physical domain over the test plate. A digital camera was used to acquire images of the projected grids over the test surface. The frame rate of the digital camera was set to 30.0 Hz with 2.0 ms exposure time. The digital camera and the projector were synchronized by using a digital delay generator. A DIP measurement window was selected to locate right downstream of the water injection holes with approximately $100×60 mm$ in size. The DIP reference images were acquired without having water film/rivulet on the test plate. After switching on the gear pump to inject water onto the test plate, 600 frames (i.e., with a time duration of 20 s) of the projected grid images (i.e., the distorted images with the wind-driven water film/rivulet flow on the test plate) were acquired for each test case. A cross-correlation algorithm³² was used to determine the displacement maps of the projected grids in the distorted images against their original positions in the reference images. With the displacement-to-thickness conversion ratio map determined via a calibration procedure similar to that described in Zhang et al.,³⁰ the film thickness distribution of the water film/rivulet flow over the test plate was determined quantitatively based on the displacement maps obtained from cross-correlation-based DIP image processing. With the same DIP system as the one used in the present study, Zhang et al.³⁰ measured the 3D shape of a spherical cap with known dimensions for the DIP measurement validation. The uncertainty level of the DIP measurements was found to be about 0.02 mm, which is about 0.5% of the nominal height of $H=4.00 mm$⁠. Further information about the technical basis of the DIP technique and its applications to quantify wind-driven surface water flows pertinent to aircraft icing phenomena can be found in our recently published papers.^12,30,33

It should be noted that, in order to enhance light reflection over the free surface of the water film/rivulet flow for improved quality of the acquired DIP images (i.e., for a better DIP measurement accuracy), a small amount of flat white latex dye (i.e., ∼1.0% in volume fraction) was premixed with the water in the reservoir. After adding the latex dye into the water, while the viscosity of the working liquid was found to be almost unchanged (i.e., ±2% in variation), its surface tension was found to be reduced from σ ≈ 0.072 to σ ≈ 0.042 N/m. The advancing contact angle of liquid droplets over a tilted test plate was measured by using a shadowgraph-based imaging technique, which was found to be $θa≈ 80°$⁠.

After being carefully calibrated and validated, the DIP system was utilized to characterize the wind-driven water film/rivulet flows over the flat test plate under different test conditions. Figure 3 shows some typical DIP measurement results to reveal the transient behaviors of the wind-driven film/rivulet flows in terms of the measured film thickness distributions over the test plate. During the experiments, the freestream speed of the incoming airflow was changed from $U∞=5$ to $U∞=20 m/s$⁠. While very similar water runback characteristics were also observed for the test cases with a higher water flow rate $Q=200 ml/min$⁠, only the measurement results for the test case with $Q=100 ml/min$ are shown here for conciseness.

As revealed clearly in Fig. 3(a), after turning on the pump to inject water flowing over the test plate at a time t = 0, a rather uniform water film flow was found to be generated over the test plate for the test case with relatively low airflow velocity (i.e., the freestream wind speed $U∞=5 m/s$⁠). While the free surface of the wind-driven film flow was found to be rather flat with the averaged film thickness near the film front being about 2.2 mm (i.e., H_ave. ≈ 2.2 mm), no obvious surface waves were observed at the water/air interface. The wind-driven film flow was found to have a rather straight front contact line, which was moving slowly (i.e., having an averaged runback velocity of $V¯$_runback$.$≈ 4.5 mm/s, which was estimated based on the time needed for the film/rivulet front to flow over the measurement window) as driven by the boundary layer airflow with $U∞=5 m/s$⁠.

As the freestream velocity of the airflow was increased to $U∞=10 m/s$⁠, while the averaged film thickness of the wind-driven water film flow near the film front was found to be reduced to about 1.5 mm (i.e., H_ave ≈ 1.5 mm), well-organized surface waves were found to be generated at the air/water interface for this test case, as shown clearly in Fig. 3(b). The surface waves were found to be sinusoidal-shaped and have rather uniform shape along the spanwise direction, i.e., having very similar characteristics as the two-dimensional (i.e., 2D) surface waves observed in the experimental study of Liu et al..²⁷ It was also found that, while the wind-driven water film flow runs back with a much higher averaged moving speed (i.e., the averaged moving velocity of the front contact line was increased to $V¯$_runback.≈ 30 mm/s, which is about six times higher than that with $U∞=5 m/s$⁠), the front contact line was found to move forward in a stumbling motion. Instead of advancing smoothly and continuously, the front contact line of the wind-driven film flow was found to stop moving from time to time, coupling well with the propagation of the surface waves generated at the air/water interface. More specifically, the film front/head was found to progress very slowly or to not move at all (i.e., in stagnation) when the film front/head has a relatively uniform film thickness distribution. As the crests of the surface waves approach the film front contact line to cause the formation of a high-rise film front/head, the film front contact line was found to move forward very rapidly (i.e., in acceleration) to a further downstream location. Then, the moving speed of the film front/head was found to slow down again (i.e., in deceleration) until next crests of the surface waves approached the film front/head to make the film front/head run back in a stumbling motion with obvious “acceleration-and-deceleration” cycles. The characteristic period of the acceleration-and-deceleration cycles was found to be 0.227 s, for the test case of $U∞=10 m/s$⁠, which correlated well with the dominant frequency of the 2D surface waves (i.e., f ≈ 4.4 Hz) generated at the air/water interface of the wind-driven film flow.

As shown clearly in Fig. 3(c), when the incoming airflow velocity was further increased to $U∞=15 m/s$⁠, the wind-driven water film flow over the test plate was found to become much thinner with its film thickness being only about 0.8 mm (i.e., $H≈ 0.8 mm$⁠). Instead of forming well-organized 2D waves, the surface waves generated at the air/water interface were found to be evolved into much more complex three-dimensional (i.e., 3D) wave structures. Due to rapid growth of the 3D surface waves, the front contact line of the wind-driven film flow was found to be deformed significantly along the spanwise direction, and two rivulet flows, which have much faster moving speeds at the rivulet heads, were found to be generated at the two sides of the wind-driven film flow, as shown clearly in Fig. 3(c). The fast transition of surface waves from 2D waves to more complex 3D waves at relatively high wind speeds (i.e., $U∞$ ≥15 m/s) was also reported in the previous studies of Craik³⁴ and Liu et al.²⁷ Due to the generation of 3D surface waves, the runback motion of the wind-driven water film/rivulet flow was found to become much more unsteady. While the averaged runback speed of the film front/head was found to increase significantly, reaching up to $V¯$_runback$.$≈ 100 mm/s, the “acceleration-and-deceleration” runback feature was found to become more noticeable for this test case. While the film/rivulet heads were found to move forward in a more obvious stumbling motion, the characteristic period of the acceleration-and-deceleration cycles was found to decrease to 0.135 s for this test case, which is much shorter than for the test case with a lower speed of $U∞=10 m/s$⁠.

The DIP measurement results given in Fig. 3(d) reveal clearly that, as the airflow velocity was further increased to $U∞=20 m/s$⁠, the front contact line of the water film was found to be teared off and broken into multiple segments right after water was injected onto the test plate. As a result, multiple rivulet flows were found to be generated downstream of water injection holes. Corresponding to the much higher airflow velocity for this test case, while the wind-driven film/rivulet flows were found to run back much faster with the averaged runback speed (⁠$V¯$_runback ≈ 200 mm/s), the thickness of the film/rivulet flows on the test plate was found to be further decreased to become less than ∼0.5 mm(i.e., $H<0.5 mm$⁠). In contrast, the widths of the rivulets were found to grow gradually as they moved downstream. Some of the rivulets were even found to merge with the neighboring rivulets as they grew wider enough to wet the dry patches between the rivulets. The merging of neighboring rivulets was found to cause obvious meandering of the rivulet heads along their moving paths. While the wind-driven rivulet flows were also observed to run back in stumbling motions, since the runback motions of the wind-driven film/rivulet flows were found to become much more chaotic, no obvious dominant frequencies can be easily identified to represent the characteristic period of the acceleration-and-deceleration cycles for the wind-driven rivulet flows.

In order to characterize the stumbling motion of wind-driven water film/rivulet flows as they run back over the test plate more clearly and quantitatively, the DIP measurement results were further processed to extract the moving trajectories of the film fronts or rivulet heads of the film/rivulet flows, thereby, determining the instantaneous runback speed of the wind-driven film/rivulet flows under different test conditions. As shown schematically in Fig. 4, the locations of the front contact lines of the wind-driven water film/rivulet flows were determined based on the instantaneous DIP measurement results with a threshold film thickness value of 0.05 mm (i.e., the red line highlighted in Fig. 4 with the measured film thickness of H = 0.05 mm). Then, the utmost downstream location of the front contact line at each frame of the instantaneous DIP measurements was identified, e.g., the points of A and A′ indicated in Figs. 4(a) and 4(b). Finally, the displacement of the utmost positions of the front contact line over the time interval between two successive frames was used to determine the instantaneous runback velocity of the wind-driven film/rivulet flow (i.e., V=Δl/Δt).

Figure 5 gives the results extracted from the DIP measurements to quantify the transient runback behavior of the wind-driven film/rivulet flow for the test case with a freestream wind speed $U∞=5 m/s$ and $Q=100 ml/min$⁠. While Fig. 5(a) shows a typical instantaneous DIP measurement result to reveal the front line of the wind-driven film flow over the test plate, the extracted film thickness profile along the centerline of the wind-driven film flow [i.e., in the plane highlighted in Fig. 5(a)] at different time instants is shown in Fig. 5(b). It can be seen quantitatively that, when the incoming airflow velocity was relatively low (i.e., $U∞=5 m/s$⁠), the free surface of the wind-driven film/rivulet flow was found to be quite flat without forming any obvious surface waves at the air/water interface. Based on the time evolution of the utmost location of the film front given in Fig. 5(c), the wind-driven film/rivulet flow was found to take about 20 s to move through the DIP measurement window (i.e., $80 mm$ along the flow direction) for this test case. As shown quantitatively from the estimated runback velocity of the film front given in Fig. 5(d), while the wind-driven film flow over the test plate was found to move very slowly (i.e., having an averaged runback speed $V¯$_runback$$ ≈ 4.0 mm/s), the instantaneous runback velocity of the film front shown in Fig. 5(d) was found to be always greater than zero (i.e., V_runback > 0), indicating that the wind-driven film flow would progress continuously (i.e., the runback film front was never in stagnation) for the test case with $U∞=5 m/s$⁠. Based on the measured runback speed data given in Fig. 5(d), the acceleration rate of the moving front contact line of the wind-driven film flow can also be estimated, which was found to be on the order of 10⁻⁴ m/s², indicating that the wind-driven film flow under the test condition of $U∞=5 m/s$ could be considered a “quasi-steady” film flow. As described in Hartley and Murgatroyd²⁰ in developing the minimum film thickness criteria for shear-driven thin film flows, the shear force exerted by the incoming airflow at the air/water interface should be able to surpass the restraining surface tension force under the test conditions of $U∞=5 m/s$ and $Q=100 ml/min$ in order to drive the film front moving forward.

Figure 6 gives the DIP measurement results for the test case with the wind speed being increased to $U∞=10 m/s$⁠, while the water flow rate was kept at the same level of $Q=100 ml/min$⁠. It can be seen clearly that, due to the formation of well-organized surface waves at the air/water interface, the film thickness of the wind-driven film flow was found to vary significantly along the flow direction, as revealed clearly in Fig. 6(b). Corresponding to the higher incoming wind speed for this case, the wind-driven film/rivulet flow was found to move much faster in comparison to the test case with a lower incoming wind speed of $U∞=5 m/s.$ More specifically, the averaged runback velocity of the film/rivulet flow driven by the boundary layer airflow with $U∞=10 m/s$ was found to increase to $V¯$_ave$.$≈ 14 mm/s, which is about 3.5 times higher than that of the case with $U∞=5 m/s.$ As revealed clearly by moving trajectories of the front contact line in the middle plane of the wind-driven film/rivulet flow given in Fig. 6(c), the film front was found to stop moving from time to time, resulting in the formation of multiple flat plateaus in the film front moving trajectories for this test case. Since the film front was in stagnation from time to time, the instantaneous runback velocity of the film front was found to become zero frequently, as shown clearly in Fig. 6(d). As a result, the wind-driven water film flow was found to move forward in a stumbling runback motion with obvious “acceleration-and-deceleration” cycles, as described above. In summary, the runback characteristics of the wind-driven film flow for this test case was found to become very different from those with a lower wind speed of $U∞=5 m/s$⁠. Corresponding to the formation of surface waves at the air/water interface, the wind-driven film flow was found to run back over the test plate in an obvious stumbling runback motion with the film front moving forward in acceleration-and-deceleration cycles.

Figure 7 shows the DIP measurements under the test condition with a wind speed of $U∞=15 m/s$ and water flow rate of $Q=200 ml/min$⁠. Due to a stronger aerodynamic shear force exerted by the airflow with higher speed, the front contact of the film flow was found to break down rapidly as water was injected on the test plate, causing the formation of multiple rivulets over the test plate, as shown clearly in Fig. 7(a). As revealed clearly by the measured film thickness profile extracted along the center of rivulet #2, while the wind-driven film/rivulet flow was found to become much thinner (i.e., the averaged film thickness becoming less than 0.7 mm), irregular surface waves (i.e., 3D surface waves) were found to be generated at the air/water interface to induce much more complicated film thickness distributions for the wind-driven film/rivulet flow. For the three rivulet flows formed on the test plate, even though they were driven by the same incoming airflow of $U∞=15 m/s$⁠, the rivulet head with narrower rivulet width (e.g., rivulet #3) was found to move much faster than those with greater rivulet width (i.e., rivulet #1 and #2). As shown clearly in Fig. 7(d), corresponding to the formation of complicated 3D surface waves at the air/water interface, the moving speeds of the rivulet heads were found to fluctuate significantly with the instantaneous runback speeds, even becoming zero sometimes for all the three rivulet flows. As described above, the fluctuating frequency revealed in the time evolution profiles of the instantaneous rivulet heads' runback speed (i.e., the frequency of the acceleration-and-deceleration runback cycles) was also found to increase monotonically as the wind speed increases. It should also be noted that, while the DIP measurement results for the test case with the airflow speed being further increased to $U∞=20 m/s$ also reveal very similar runback characteristics for the wind-driven film/rivulet flow, the measurement data are not presented here for conciseness. In summary, the DIP measurement results reveal clearly that, as the incoming wind speed increases, while more complex 3D surface waves were found to be generated at the air/water interface, the wind-driven film flow became highly unsteady with the front contact line breakoff rapidly to form multiple isolated rivulet flows. As driven by the airflow over the test plate, the rivulet flows were found to be meandering and merging with the neighboring rivulets with the rivulet heads moving forward in stumbling motion with obvious “acceleration-and-deceleration” cycles.

Figure 8 gives the instantaneous DIP measurement results near a moving rivulet head at three selected instants, which can be used to clearly reveal the coupling of the surface waves generated at the air/water interface with a stumbling runback motion of the wind-driven flow/rivulet flows. While Fig. 8(a) gives the measured runback speed of the rivulet head under the test conditions of $U∞=10 m/s$ and $Q=200 ml/min$⁠, Figs. 8(b)–8(d) show the “zoom-in” view of the film thickness distributions near the rivulet head at three selected moments (i.e., right before and after the rivulet head was in stagnation). It is well known that, for a wind-driven film/rivulet flow, the summation of the aerodynamic drag force acting on the rivulet head (i.e., mainly form drag) and the aerodynamic shear force (i.e., skin friction force) at the air/water interface should be greater than the restraining force due to surface tension in order to drive the film/rivulet flow moving forward. As shown clearly in Fig. 8(b), at time t = t₁ (i.e., right before the rivulet head was in stagnation), while a crest of the surface wave was moving toward the rivulet front, the rivulet flow was found to have a low-rise rivulet head with its film thickness being smaller than that at the rivulet body. Since the aerodynamic drag force acting on the low-rise rivulet head would be relatively small, the formation of such a low-rise rivulet head is not favorable to generate enough aerodynamic drag force to overcome the restraining surface tension force. As a result, the rivulet head was found to be in a decelerated runback motion at the instant t = t₁ (i.e., the moving speed of the rivulet head becoming smaller and smaller), as shown clearly in Fig. 8(a).

The DIP measurement result given in Fig. 8(c) reveals clearly that, while the rivulet head was in stagnation (i.e., the runback speed being zero), the crest of the surface waves arrived at the rivulet front at the instant t = t₂. As a result, the film thickness at the rivulet head was found to be increased substantially, causing the formation of a high-rise hump at the rivulet head with its film thickness becoming much greater than that at the rivulet body. Corresponding to the formation of a high-rise hump-shaped rivulet head, the aerodynamic drag force acting on the rivulet head would become much greater, in comparison to the case with a low-rise rivulet head shown in Fig. 8(b). The rapidly increased aerodynamic drag force acting on the rivulet head would be very favorable to surpass the restraining surface tension force to push the rivulet head moving forward. Therefore, the runback of the wind-driven rivulet flow was found to be accelerated after time t = t₂, as shown clearly by the runback velocity data given in Fig. 8(a).

As the rivulet front reached a further downstream location at the instant $t=t3$⁠, since the rivulet head was found to become relatively flat again as shown clearly in Fig. 8(d), the corresponding aerodynamic drag force acting on the rivulet head would become much smaller again, resulting in the deceleration of the runback motion for the wind-driven film/rivulet flow. This would lead to a new deceleration-and-acceleration cycle for the rivulet head of the wind-driven rivulet/film flow to run back over the test plate. In summary, the DIP measurement results given in Fig. 8 reveal clearly that, by changing the local film thickness distributions near the rivulet head dynamically, the surface waves generated at the water/air interface would induce significant variations of the aerodynamic drag force acting on the rivulet head. As a result, the wind-driven film/rivulet flow was found to run back in a stumbling motion with the rivulet head moving forward in obvious deceleration-and-acceleration cycles.

The DIP measurement results given above reveal clearly that, as the surface waves were found to be generated at the air/water interface at relatively high incoming wind speeds (e.g., $U∞≥10 m/s)$⁠, the wind-driven water film/rivulet flows would run back in a stumbling motion with the film/rivulet heads moving forward in deceleration-and-acceleration cycles. By leveraging the quantitative DIP measurement results, a comprehensive force balance (FB) analysis is performed in the present study in order to gain further insight into the underlying physics pertinent to the stumbling runback motion of wind-driven water film/rivulet flows.

It should be noted that Hartley and Murgatroyd²⁰ developed a FB criterion to predict the dry-patch rewetting process for thin film flows driven by surface shear. The breakoff of the stagnated rivulet head to re-start the runback process for a wind-driven rivulet/film flow investigated in the present study is very similar to the dry-patch rewetting process studied by Hartley and Murgatroyd.²⁰ Therefore, a similar methodology was adopted in the present study for the FB analysis of wind-driven film/rivulet flows.

As revealed clearly in Fig. 8(c), while the rivulet head of a wind-driven film/rivulet flow was in stagnation, due to the arrival of crest of surface waves at the rivulet front, a high-rise hump would be formed at the rivulet head with its film thickness becoming noticeably greater than that at the rivulet body. Therefore, at the moment when the rivulet head was in stagnation, the wind-driven film/rivulet flow was simplified to having a high-rise hump-shaped rivulet head and relatively flat rivulet body, as shown schematically in Fig. 9.

It should also be noted that, for the dry-patch rewetting process studied by Hartley and Murgatroyd,²⁰ only two relevant forces were considered in the FB analysis: the excess pressure force built inside the film head and the restraining force due to surface tension. Hartley and Murgatroyd²⁰ suggested that the dry patch would be re-wetted when the excess pressure force surpasses the restraining surface tension force. More recently, McAlister et al.²⁸ conducted an FB analysis to model the breakoff of wind-driven rivulet/droplet flows in microgravity and terrestrial-gravity conditions. While an empirical model was suggested to estimate the aerodynamic drag forces acting on rivulets/droplets, the excess pressure force built inside the rivulet heads was not considered in the FB analysis of McAlister et al.²⁸ In the present study, a comprehensive FB analysis is performed to examine all the relevant forces in order to elucidate the underlying physics pertinent to the breakoff of the stagnated rivulet head to re-start the runback motion for the wind-driven rivulet/film flow.

For the wind-driven film/rivulet flows investigated in the present study, three different forces were believed to play very important roles in determining the runback characteristics of the film/rivulet head as revealed from the DIP measurement results given schematically in Fig. 9, which includes the excess pressure force built inside the rivulet head of a wind-driven film/rivulet flow, F_P; aerodynamic drag force (i.e., form drag) acting on the rivulet head exerted by the incoming airflow, F_D; and the restraining force due to surface tension along the front contact lines, F_σ. Since the rivulet head was assumed to be in stagnation, the front contact line of the wind-driven film/rivulet flow would not be able to move forward until the summation of the aerodynamic drag force acting on the rivulet head and the excess pressure force built inside the rivulet head becomes greater than the restraining surface tension force, which can be expressed as follows

As described in the FB analysis of Hartley and Murgatroyd²⁰ and McAlister et al.,²⁸ the restraining force due to surface tension at the rivulet head can be estimated as

where L is the contact line length at the rivulet head; $θr$ and $θa$ denote the receding and advancing contact angles of the film/rivulet contact line, respectively; and $σ$ is surface tension coefficient.

As revealed clearly by the DIP measurement results given above, the characteristic lengths of the rivulet heads for the wind-driven film/rivulet flows on the test plate of the present study (i.e., rivulet width and length) are usually much larger than the film thickness. Therefore, as shown schematically in Fig. 9, the receding contact angle of the contact line at the back of the rivulet head would approach zero, making $cosθr ≈ 1.0$⁠. Meanwhile, while the hump-shaped rivulet head is three-dimensional in shape, since the rivulet head usually has a very large width-to-height ratio (i.e., $W/H>5$ as revealed from the DIP measurement results given in Fig. 8(c), where $W$ and $H$ are the width and height of the rivulet head, respectively), the estimation of the restraining force due to surface tension can be simplified from a 3D problem to a 2D problem. Therefore, the restraining forces due to surface tension at the rivulet head can be approximated as

As described above, $σ=0.04 N/m$ for the working liquid used in the present study (i.e., water pre-mixed with 1.0% latex dye). The advancing contact angle at the front contact line of the wind-driven film/rivulet flows was measured to be $θa≈ 80°$⁠. Therefore, the restraining surface tension force at the rivulet head would be almost independent of the incoming wind speed and water flow rate of the wind-driven film/rivulet flows over the test plate.

Similar to the description of Hartley and Murgatroyd,²⁰ as the rivulet head was in stagnation (i.e., the runback speed becoming zero), the liquid fluid in a stream tube inside a surface shear-driven film flow with an incoming flow velocity $uw$ would reduce gradually to zero at the rivulet head, resulting in an excess pressure built inside the film/rivulet head, which can be expressed as

Therefore, the total excess pressure force built inside the rivulet head can be estimated by integrating the excess pressure across the entire film flow field, and can be express as

where $ρw$ is the liquid density, which is 1.0 g/ml for the water used in the present study.

As described above, since the wind-driven film/rivulet flows investigated in the present study usually have a very large width-to-height ratio (i.e., $W/H>5$⁠), they can be approximated as two-dimensional flows. As described in Al-Khalil et al.,²⁶ the velocity distribution within a rivulet section can be determined by solving the Laplace equation as follows:

Equation (6b) is the no-slip condition at the water–solid interface, Eq. (6c) assumes the velocity profile is symmetric along the spanwise direction in relation to the rivulet centerline. Equation (6d) indicates the shear stress, $τa$⁠, at the air/water interface is continuous, where $r$ is the radial direction (i.e., normal to the local water–air interface). The shear stress at the air/water interface is assumed to be a constant over the rivulet surface and can be estimated based on the Blasius solution of $τa=0.664/Rex$ for different test conditions. $μw$ in Eq. (6d) is the viscosity of water. Al-Khalil et al.²⁶ solved the above equations numerically, and found that, as the width-to-height ratio of the film flow became greater than five (i.e., $W/H>5$⁠), the velocity distribution inside the rivulet flow would be simplified by using a linear velocity profile of

Therefore, the excess pressure force built inside the rivulet head could be expressed as

As shown clearly in Eq. (8), the excess pressure head built inside the rivulet head of the wind-driven film/rivulet flow would increase quadratically with the increasing aerodynamic shear stress acting on the air/water interface as expected. It should also be noted that the excess pressure head built inside the rivulet head would increase cubically with the film thickness, indicating that a thicker film flow would be more favorable in generating enough excess pressure force at the film/rivulet head to surpass the restraining surface tension force in order to enable the film/rivulet front to move forward.

An empirical model was suggested by McAlister et al.²⁸ for analyzing the aerodynamic forces acting on the rivulet heads of wind-driven rivulet/droplet flows under microgravity and terrestrial-gravity conditions. While very similar hump-shaped rivulet heads were also observed by McAlister et al.,²⁸ the rivulet heads were simplified as semi-ellipsoids with semi-axis height, H (i.e., the maximum water depth); semi-minor axis length, b (i.e., the rivulet width); and semi-major axis, a (i.e., the rivulet length) of the elliptical area on the test plate. McAlister et al.²⁸ suggested that the aerodynamic drag force acting on a rivulet head can be estimated as

where $CD$ is the drag coefficient of the hump-shaped rivulet head; $Afrontal$ is the projected frontal aera of the rivulet head exposed to the airflow. After performing a comprehensive numerical study to characterize the airflows over semi-ellipsoids with different aspect ratios, McAlister et al.²⁸ suggested that the $CD$ value would be about 0.6 (i.e., $CD ≈ 0.6$⁠) for hump-shaped rivulet heads.

In the present study, the empirical model suggested by McAlister et al.²⁸ was adopted to estimate the aerodynamic drag forces acting on the hump-shaped rivulet heads of the wind-driven film/rivulet flows. It should be noted that, for the wind-driven rivulets/droplets studied by McAlister et al.,²⁸ the rivulet flows had very big rivulet heads with much narrower rivulet bodies or the rivulet heads were found to be completely separated from their bodies to become isolated droplets. Therefore, McAlister et al.²⁸ suggested that projected frontal aera of the rivulet head exposed to the airflow could be estimated by the projected area of the semi-ellipsoid in the airflow direction, i.e., $Afrontal$ ≈ $π2bH$⁠. However, due to a much larger wetted aera for wind-driven film/rivulet flows over the test plate pertinent to aircraft icing phenomena, the wind-driven film/rivulet flows investigated in this study were found to have relatively small rivulet heads and more significant rivulet bodies, as revealed clearly by the DIP measurement results given in Fig. 8. Therefore, the actual frontal area of the hump-shaped rivulet head exposed to the incoming airflow would be the projected area of the rivulet head subtracting that of the rivulet body (i.e., $Afrontal=Ahead−Abody$⁠), which is indicated by the shaded area given in Fig. 9(b). Furthermore, since the rivulet head usually has a very large width-to-height ratio (i.e., $W/H>5$ as revealed by the DIP measurement results), the projected frontal area of the rivulet head exposed to the incoming airflow can be further simplified as

where W is the rivulet width. $Hrh¯$ and $Hrb¯$ are the average film thicknesses at the rivulet head and rivulet body, which can be determined based on the DIP measurement results near the rivulet head, similar to those shown in Fig. 8.

As expressed in Eq. (1), a stagnated rivulet head would re-start the runback process if the summation of the aerodynamic drag force acting on the film/rivulet head and the excess pressure force built inside the rivulet head becomes greater than the restraining force due to the surface tension. By substituting Eqs. (3), (8), and (9) intoEq. (1), the FB criteria for the breakoff of a stagnated rivulet head to re-start the runback process can be expressed as

It should be noted that, for the test cases with relatively low incoming airflow speed (e.g., the test case with $U∞=5 m/s$⁠), since the film front of the wind-driven film/rivulet flow was found to be quite flat with almost the same film thickness at the film head and the body of the film flow, the term $(Hrh¯−Hrb¯$⁠) in Eq. (11) would become zero. Therefore, the aerodynamic drag force acting on the film font/head would become zero. As a result, the excess pressure force built inside the film front of the wind-driven film flow would need to become greater than the restraining surface tension force in order to push the front contact line of the wind-driven film/rivulet flow to move forward. Therefore, based on the FB criteria given in Eq. (11), the minimum film thickness of the wind-driven film flow should satisfy the following condition:

which is exactly the expression suggested by Hartley and Murgatroyd²⁰ in predicting the minimum film thickness for a surface shear-driven film flow to re-wetting the dry patches.

More specifically, for the test case under the test conditions of $U∞=5 m/s$ and $Q=100 ml/min$ (i.e., the test case shown in Fig. 5), while the advancing angle at the front contact line of the wind-driven film flow was found to be about 80° (i.e., $θa$ ≈ $80°),$ the surface tension coefficient of the working fluid is $σ=0.04 N/m.$ The aerodynamic shear stress at the air/water interface can be estimated based on the Blasius solution of $τa=0.664/Rex$⁠. $Rex$ was estimated to be about 200 000 (i.e., $Rex ≈ 200 000$⁠) at the center of the test plate based on the relationship $δ=5.0x/Rex$⁠, where the boundary layer thickness $δ$ was determined based on the PIV measurement results given in Fig. 2. Therefore, the minimum film thickness predicted by using the formula given in Eq. (12) was found to be about 1.84 mm (i.e., H_critic ≈ 1.84 mm). As revealed clearly by the DIP measurement results given in Fig. 5(b), the measured film thickness of the wind-driven film flow over the test plate was found to be greater than 2.1 mm (i.e., H > 2.1 mm). It confirms that the wind-driven film flow over the test plate is indeed thicker than the required minimum film thickness predicted by the FB analysis criteria given in Eq. (12).

As revealed clearly by the DIP measurements given above, as the incoming airflow speed became high enough (e.g., $U∞≥10 m/s$⁠), well-organized 2D or more complicated 3D surface waves were found to be generated at the air/water interface of the wind-driven film/rivulet flows, The formation of surface waves would affect the film thickness distributions near the rivulet/heads significantly. As shown clearly in Fig. 8, as the crests of the surface waves arrived at the film/rivulet head of a wind-driven film/rivulet flow, high-rise humps would be formed at the rivulet heads (i.e., the value of $(Hrh¯−Hrb¯$⁠) would increase substantially), which would result in a rapid increase in the aerodynamic drag forces acting on the high-rise rivulet heads.

In order to reveal the importance of the three relevant forces to the breakoff of the stagnated rivulet head to re-start the runback process more clearly and quantitatively, the DIP measurement results obtained under different test conditions were used to estimate the three relevant forces described above. Table I summarizes the test conditions used for the FB analysis and the estimated restraining surface tension force $fσ$⁠, excess pressure force $fP$⁠, and the aerodynamic drag force $fd$ acting on the stagnated rivulet head per unit span of the wind-driven film/rivulet flow.

As revealed clearly from the estimated force data listed in Table I, corresponding to the formation of much thinner wind-driven film/rivulet flows over the test plate at higher incoming wind speeds, the excess pressure force built inside the film/rivulet head was found to be much smaller in comparison to the other two forces (i.e., being less than 20% of the restraining surface tension force) for all the investigated cases. On the other hand, the aerodynamic drag force acting on the hump-shaped rivulet head was found to play a dominant role in overcoming the restraining surface tension force for the breakoff of the stagnated film/rivulet head to re-start the runback process. More specifically, since the aerodynamic drag force acting on the high-rise film/rivulet head was found to be much greater than the restraining surface tension force, the stagnated film/rivulet heads would be pushed to move forward abruptly, resulting in a rapid increase in the runback velocity of the film/rivulet heads (i.e., the acceleration in the runback process), which is similar to those revealed from the instant $t=t2$ to the instant $t=t3$ shown in Fig. 8.

Right after the breakoff of the stagnated rivulet head to re-start the runback process, the film thickness at the film/rivulet head was found to decrease substantially as revealed clearly by the DIP measurement results given in Fig. 8(d). Therefore, a relatively flat, low-rise rivulet head would be formed as the film/rivulet head moved to a further downstream location, resulting in a much smaller aerodynamic force acting on the low-rise film/rivulet head. However, as expressed in Eq. (3), since the restraining surface tension force is only a function of the advancing contact angle of the front contact, which is almost independent of the incoming wind speed and the water flow rate, the restraining surface tension force was found to stay almost unchanged during the runback process. Since the much smaller aerodynamic force acting on the low-rise film/rivulet head could not overcome the restraining surface tension force, the runback speed of the rivulet head would decrease rapidly with the runback process for the wind-driven film/rivulet flow in deceleration until the next crest of the surface waves arriving at the rivulet front to form a new high-rise rivulet head again (i.e., the process revealed from the instant $t=t1$ to the instant $t=t2$ given in Fig. 8).

An experimental study was conducted to examine the dynamic runback characteristics of wind-driven water film/rivulet flows over a flat test plate in order to elucidate the underlying physics of the surface water transportation process pertinent to aircraft icing phenomena. The experimental study was conducted in an open-circuit wind tunnel to generate boundary layer airflows with different freestream wind speeds to drive thin water film flows over a flat test plate. During the experiments, a digital image projection (DIP) system was used to achieve non-intrusive, whole-field measurements of the film thickness distributions over the test plate to quantify the transient runback behavior of the wind-driven water film/rivulet flows under different testing conditions.

It was revealed clearly that, as the wind speed over the test plate is relatively low (i.e., $U∞<10 m/s$⁠), the wind-driven water film flow was found to be rather steady with a uniform film thickness distribution over the test plate. The film front was found to move forward slowly, but continuously, and no obvious surface waves were found to be generated at the air/water interface. However, as the wind speed became high enough (i.e., $U∞≥10 m/s$⁠), the “steady” runback motion of the wind-driven film flows was found to be interrupted by the presence of surface waves at the air/water interface. The thickness of the water film flow was found to decrease monotonically as the wind speed increases. While the averaged moving speed of the film/rivulet front was observed to increase rapidly with increase in wind speed, the wind-driven film flows were found to run back in a stumbling motion with the film/rivulet heads moving forward in obvious deceleration-stagnation-acceleration cycles. As the wind speed was further increased to higher values (i.e., $U∞≥15 m/s$⁠), while the surface waves generated at the air/water interface were found to transit rapidly from well-organized 2D structures to more complicated 3D structures, the front contact lines of the wind-driven film flows were found to be deformed, stretched, and broken down eventually, resulting in the formation of multiple rivulets to transport the water mass swiftly to further downstream locations in isolated channels. While the water runback process was found to become more and more chaotic, the rivulet heads were found to be meandering along their moving trajectories. The wind-driven rivulet flows were found to become wider as moving downstream results in the merging of neighboring rivulets at further downstream locations.

A comprehensive force balance (FB) analysis was also performed to examine the variations of the relevant forces (i.e., the excess pressure force built inside the film/rivulet head, aerodynamic drag force acting on the film/rivulet head, and the restraining surface tension forces acting along the front contact lines) in order to elucidate the underlying physics pertinent to the changes of the runback characteristics of the wind-driven film/rivulet flows under different test conditions. Simplified formulas were developed to provide quantitative estimations of the relevant forces in order to evaluate their importance in the breakoff of stagnated film/rivulet heads to re-start the runback process. It was revealed clearly that, before the formation of surface waves at the air/water interface (i.e., for the test cases with relatively low incoming wind speed), while the aerodynamic drag force acting on the flat front of the wind-driven film flow was very small, the excess pressure force built inside the film front was found to play a significant role in surpassing the restraining surface tension force in order to push the film flow to move forward. However, for the test cases with relatively high incoming wind speed (i.e., after surface waves were found to be generated at the air/water interface), while the excess pressure forces built inside the film/rivulet heads were found to become much smaller in comparison to other forces, the aerodynamic drag forces acting on the film/rivulet heads were found to play a dominant role in conquering the restraining surface tension forces to drive the film/rivulet flow running back over the test plate.

This research work is partially supported by the National Science Foundation (NSF) under Award Nos. CMMI-1824840 and CBET-1916380 and Iowa Energy Center for Wind Turbine Icing Study under the IEC Competitive Grant No. 312350.

The authors have no conflicts to disclose.

K.Z. contributed to model design and manufacture, wind tunnel experiments, data acquisition and processing, writing the original draft. H.H. contributed to the development of methodology, formal analysis, writing—review and editing, funding acquisition, and supervision.

The data that support the findings of this study are available from the corresponding author upon reasonable request.