Drop impact onto a moving substrate: Aerodynamic rebound

International Journal of Multiphase Flow 184 (2025) 105113 Available online 22 December 2024 0301-9322/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow Research paper Drop impact onto a moving substrate: Aerodynamic rebound Bastian Stumpfa, Samaneh Abdi Qezeljeh a, Reda Kamal a,b, Fabien Dezitter c, Alessandro Martuffod, Ilia V. Roisman a, Jeanette Hussong a,∗ aTechnische Universität Darmstadt, Institute of Fluid Mechanics and Aerodynamics, Darmstadt, 64289, Germany bPolitecnico di Milano, Department of Aerospace Science and Technology, Milano, 20156, Italy cAirbus Helicopters S.A.S., Aéroport International Marseille Provence, Marignane, 13725, France dAirbus Helicopters, Aeromechanics & Performance, Marignane, 13725, France ARTICLE INFO Dataset link:https://zenodo.org/doi/10.5281/ zenodo.12684793 Keywords: Drop impact Multiphase flow Drop deposition and rebound Boundary layer Wetting and icing Moving substrate ABSTRACT The dynamics of droplets approaching fast moving surfaces of high surface-tangential velocities is relevant to numerous technical applications, such as icing phenomena in aviation. Due to the substrate motion a boundary layer is formed which interacts with impacting droplets. In the present study, the transition from drop impact and splashing to boundary layer induced drop rebound is investigated for varying drop diameters, drop and plate velocity, as well as impact angles. It is found that this transition is strongly influenced by the degree of drop deformation that is induced by aerodynamic forces acting on the drop when it enters the boundary layer. Based on these considerations, a threshold model is obtained that describes the transition from splash to aerodynamic rebound. It is shown that the model is valid for a laminar and a turbulent boundary layer agreeing well with own and existing experimental data. 1. Introduction The interaction of a drop with a planar surface that exhibits a fast motion tangential to its interface is relevant to numerous technical applications. One example is icing in aviation caused by small supercooled drops in the atmosphere that impact onto the aircraft surface or onto rotating helicopter blades (Cao and Chen,2010;Cao et al.,2018). Another example is droplet based cooling of fast rotating components, such as rotor windings in electric machines. In electrical machines, the interaction of the drop with the moving surface needs to be fully understood to be able to reliably predict the cooling heat flux (Lim and Kim,2014;Liu et al.,2019). For this, it is of particular interest to know whether the droplets are in direct contact with the substrate and if so, what the impact outcome will be. For the icing, the resulting ice layer changes the aerodynamic properties of the aircraft, which leads to decreased efficiency and might lead to hazardous flight conditions (Lee et al.,1984;Kind et al.,1998;Cao et al.,2015;Yamazaki et al.,2021). Also, the ice accretion rate depends, among other phenomena, on the impact outcome. Generally, a drop impact can be subdivided into drop splash, deposition, or rebound. For reliable modeling of technically relevant processes, it is important to determine the deposited mass ratio and the parameters of the secondary spray formed by rebound and splash since this secondary spray could again impact onto surfaces. The impact of drops onto stationary wetted and dry surfaces has been studied extensively (Yarin,2006;Moreira et al.,2010;Josserand ∗Corresponding author. E-mail address: [email protected] (J. Hussong). and Thoroddsen,2016;Yarin et al.,2017). The outcome of drop impact, it is bouncing (Sprittles,2024), deposition and splash (Tropea and Marengo,1999) is mainly determined by the impact parameters, the drop velocity 𝑈𝑑 ,𝑛 normal to the substrate, and the drop diameter 𝐷and the material properties of the liquid, including the kinematic viscosity 𝜈dr op, the density 𝜌dr op and the surface tension of the drop 𝜎dr op. Correspondingly, the outcome is governed by the Reynolds number and Weber number, Re=𝑈𝑑 ,𝑛𝐷dr op∕𝜈dr op and We=𝜌dr op𝑈2 𝑑 ,𝑛𝐷dr op∕𝜎dr op, respectively. If the drop impacts onto a wetted substrate also the dimensionless film thickness, scaled by the drop diameter 𝐷, becomes an influencing parameter. Wall-normal drop impact onto a stationary and planar surface leads to a radially expanding flow in a thin lamella. The outcome is defined by the interaction of this flow with the outer wall film. If the inertial effects are dominant in comparison with the viscous and capillary forces, this interaction leads to the emergence of a corona-like liquid jet (Yarin and Weiss,1995;Roisman and Tropea, 2002). The corona splash is then caused by the instability of the Taylor rim (Taylor,1959;Roisman,2010;Agbaglah et al.,2013;Wang and Bourouiba,2021). Drop impact onto a solid dry wall is influenced also by the conditions at the substrate surface, its morphology, and wetting properties (Mundo et al.,1995;Riboux and Gordillo,2014;Roisman et al., 2015;Josserand and Thoroddsen,2016). The evolution of the drop https://doi.org/10.1016/j.ijmultiphaseflow.2024.105113 Received 13 July 2024; Received in revised form 8 October 2024; Accepted 16 December 2024 International Journal of Multiphase Flow 184 (2025) 105113 2 B. Stumpf et al. diameter is determined by the dynamics of the Taylor rim formed at the edge of the spreading lamella (Roisman et al.,2002). However, the outcome of drop impact depends significantly on the aerodynamic effects in the surrounding gas. It is known that the conditions leading to corona emergence and thus to the corona splash are influenced by the properties of the surrounding gas (Xu et al.,2005). Recently, the main mechanisms of the corona emergence associated with the dynamics of the gas flow in a spreading wedge has been considered and explained (Riboux and Gordillo,2014). Drop bouncing from a dry or even liquid surfaces occurs for impacts with relatively low Weber number, We ∼(1). It is attributed to a very thin air layer between the substrate and the drop (de Ruiter et al., 2014). For the drop impact onto moving substrates (Mundo et al.,1995) showed that for the splash deposition limit only the normal component of the drop impact velocity is relevant if the tangential substrate velocity is smaller or comparable with the drop impact velocity. Most studies that investigated the interaction of a drop with a moving substrate focused on the hydrodynamics of spreading on wetting and non-wetting substrates (Almohammadi and Amirfazli,2017;Moghtadernejad et al., 2021). Only a few studies investigated parameter ranges where the inertia of the gas boundary layer is sufficiently high to play a significant role in the drop dynamics. A complete aerodynamic rebound of droplets due to the airflow in the boundary layer was first observed and analyzed in Povarov et al. (1976). In this study, the threshold velocity 𝑈⋆ plat e of the substrate, corresponding to the inception of the drop rebound, is related to the thickness 𝛿of the viscous boundary layer in the gas flow. 𝑈⋆ plat e∼𝑈𝑑 ,𝑛√𝜌dr op 𝜌gas 𝐷dr op 𝛿.(1) The same relation was confirmed by Gauthier et al. (2016,2018) for laminar boundary layers. In the present study, we show that the threshold velocity (1) at which droplet rebound can be observed is valid for both laminar and turbulent boundary layers. However, the coefficient of proportionality in (1) changes significantly. This means that not only the boundary layer thickness but also the velocity profile plays a significant role in the bouncing phenomenon. 2. Experimental setup The experimental setup, schematically shown in Fig. 1(a), consists of the rotating plate, the drop generation system, and the imaging system. The plate is made of carbon fiber-reinforced plastic (CFRP) and has a radius of 𝑅= 90 mm. It is attached to a brushless DC motor that can accelerate the plate to an angular velocity of 𝛺≤17.000 rpm (revolutions per minute). The velocity is controlled, monitored, and logged by an in-house LabView script. To capture the drop impact, a high-speed camera is used in the imaging system. Depending on the experiment the utilized models are either Photron SA-X2 or Phantom T3610. A high-speed LED (Constellation 120E) and a diffuser plate provide uniform background illumination, and the imaging system can reach frame rates of 100,000 fps. The spatial resolution in the experiments varies in the range of 9 μm to 15 μm per pixel. The drop generator is a commercial mono-disperse drop chain generator from FMP Technology GmbH which will be referred to as FMP generator in the following. It generates a mono-disperse stream of droplets by inducing a Rayleigh–Plateau instability onto a liquid jet using a piezoelectric actuator (Brenn and Tropea,1996). The droplet size can be varied in the range 80 μm≤𝐷≤500 μm by either changing the size of the outlet orifice of the FMP or by altering the excitation frequency. The radial coordinate of impact is 88 mm but can fluctuate approximately ±1 mm. The absolute velocity of the droplet is set by controlling the pressure in the pressurized tank and can be varied in the range 5.5m/s ≤𝑈𝑑 ,abs ≤12 m/s. The velocity component 𝑈𝑑 ,𝑛 of the drop normal to the plate can be altered by altering the impact angle 10◦≤𝛽≤40◦of the droplet or 𝑈𝑑 ,abs. In Fig. 1(b) and Fig. 1(c), the geometry of the impact is illustrated. The normal and horizontal velocity components 𝑈𝑑 ,𝑛 and 𝑈𝑑 ,ℎ can be obtained directly from the high-speed recordings, as will be explained below. A consequence of the inclined impact is that the droplet has a velocity component in circumferential direction 𝑈𝑑 ,𝜑 =𝑈𝑑 ,ℎcos𝛼where 𝛼is a constant offset angle of 7.1◦, relative to the camera axis. To account for this the relative velocity in circumferential direction 𝑈𝜑,r el is considered the characteristic velocity which can be formed as 𝑈𝜑,r el =𝑈𝜑,plat e−𝑈𝑑 ,𝜑,(2) where 𝑈𝜑,plat erepresents the circumferential velocity of the plate. The plate Reynolds number, Reaer o=𝑟2𝜔∕𝜈𝑎, with 𝜔being the angular velocity of the plate and 𝜈𝑎the kinematic viscosity of air, varies in the range 2.5 × 105<Reaer o<8.7 × 105. The lower limit is chosen to avoid the boundary layer’s transitional regime, while the upper limit aligns with typical values for helicopter blades and aircraft wings. The corresponding plate velocity at the impact location is in the range 40 m∕s < 𝑈𝜑,plat e<160 m∕s. At the upper end of this range, weak compressibility effects may be expected as the Mach number approaches approximately 0.5. Experimental results by Theodorsen and Regier (1944) for Mach numbers up to 0.62 indicate that the theoretical predictions by von Kármán (1946), which are based on the assumption of incompressibility, exhibit strong agreement with the experimental data for the moment coefficient, which in turn is derived from the boundary layer profile. Furthermore, the boundary layer profile shows a steep gradient near the wall, confining any compressibility effects to a small fluid region. Consequently, within the range investigated, compressibility effects are negligibly small and are therefore not considered in the present study. Finally the boundary layer is fully turbulent, as the laminar-toturbulent transition for the plate occurs in the range 1.8 × 105<Reaer o< 3.5 × 105, as shown in Shevchuk (2009). 2.1. Image processing The experiments are analyzed in a custom MATLAB script, utilizing the MATLAB image processing toolbox, where the drop diameter as well as the horizontal and vertical velocity components of the drop before impact are measured. For this, first, a moving background subtraction with a delta of 5 frames followed by a binarization is performed. From the resulting binary images, individual objects i.e. droplets, their centroid, and diameter can be evaluated. A simple particle tracking algorithm based on the nearest neighbor principle is then used to recognize the droplets in consecutive frames and thus determine the trajectories. To make the evaluation more robust against false detection, only trajectories with objects that are recognized and allocated in 10 consecutive frames are considered. A median drop diameter is determined for each trajectory using images of the drop in multiple consecutive frames before impact. To address potential bias due to outliers, drop images with diameters that deviate more than two standard deviations from the mean are considered outliers and are excluded. Eventually, mean values of the diameter and the respective horizontal and vertical velocity components are formed using all trajectories in an experiment. Furthermore, cases in which the droplet diameter or the droplet impact velocity scatter by more than 20% are discarded from the evaluation. The mean relative standard deviation for all cases is 𝜎𝐷= 4.49% for the drop diameter, 𝜎𝑈𝑑 𝑛= 4.1% in the normal direction and 𝜎𝑈𝜑,rel = 0.25% for the relative plate velocity. 3. Results of drop impact and rebound When consecutive, mono-disperse droplets interact with a rotating plate, various phenomena can be observed depending on the impact parameters. In Fig. 2, a time series of impacting droplets of constant diameter and velocity is shown at three different plate velocities. For International Journal of Multiphase Flow 184 (2025) 105113 3 B. Stumpf et al. Fig. 1. Schematic representation of the experimental setup (a) side view (b) top view and (c) front view. the lowest plate velocity, 𝑈𝜑,r el = 60 m/s (see Fig. 2(a)) a strong interaction of the droplet with the plate can be observed. During the interaction, liquid adheres to the fast-moving plate and gets dragged along, while in parallel, ligaments form that atomize into secondary droplets and are carried away by the airflow in the boundary layer. The liquid that adheres to the plate will reach the impact location after one full revolution of the rotating plate where consecutive droplets may impact onto the now wetted surface. In this case, splashing is enhanced. Studies on single drop impact onto thin liquid films have shown that very thin films with dimensionless film thicknesses  ℎ < 0.02 enhance splashing significantly (Geppert,2019;Zhu et al.,2021; Stumpf et al.,2023). In contrast, for the highest plate velocities, 𝑈𝜑,𝑟𝑒𝑙 = 140 m/s (see Fig. 2(c)) no splash and also no residual can be observed anymore, as droplets undergo a complete aerodynamic rebound, preventing drop-plate contact. At medium plate velocities, 𝑈𝜑,𝑟𝑒𝑙 = 80 m/s (see Fig. 2(b)) splashing can only be observed occasionally and in a less pronounced form. Droplets partially rebound while it is evident in the enlarged section at 𝑡= 0.1ms that a small amount of the droplet resides on the plate. There are two reasons why the splash might only occur for some droplets: •The turbulent character of the airflow leads to small fluctuations of the impact location in the radial direction that result in variations of the relative velocity at impact. The latter’s are caused by the radial boundary layer of the disk and are within the depth of field of the camera leading to a possible error in the impact position of ± 2 mm. In consequence, when the plate velocity is close to the critical velocity for rebound, some of the drops will exceed it and only some drops will have contact with the plate. •As depicted in Fig. 2(b) (see instant 0.01 ms), the interaction between the droplet and the plate results in the formation of small patches of residual liquid. At instants 0.10 ms and 0.11 ms it can be observed that such patches interact with a subsequent droplet after one revolution, resulting in a splash. It is evident that such patches potentially influence the outcome of subsequent drop impacts. However, these patches cannot be observed in the aerodynamic rebound regime. Thus, for increasing plate velocity the transition from splash/contact to aerodynamic rebound is continuous. For the following analysis, three categories of interaction are defined. If the outcome of the impact results in a splash then the experiment will be classified as ‘‘splash/contact’’. If splash can only be observed for some of the droplets then the experiment will be categorized as ‘‘transition’’. If no splash can be observed throughout the experiment and the droplets are completely repelled from the plate, we conclude that there is no more contact and thus no liquid residual on the plate. This regime is defined as ‘‘aerodynamic rebound’’. 3.1. Observed impact outcomes To characterize the outcome of the drop impact onto a rotating plate the normal impact velocity 𝑈𝑑 ,𝑛 as well as the relative plate velocity 𝑈𝜑,r el and the drop diameter are systematically varied. In Fig. 3(a) the outcome of impact is shown for the relative tangential and normal impact velocity. It becomes apparent that for higher 𝑈𝑑 ,𝑛 also higher 𝑈𝜑,𝑟𝑒𝑙 are necessary to achieve aerodynamic rebound. A straight line could separate the aerodynamic rebound from the transition and splash in the plot. In Fig. 3(b) the outcomes are shown for a larger drop diameter of 𝐷≈ 452 μm. In comparison to the 𝐷≈ 91.3 μm case for the larger drops, higher plate velocities are necessary to achieve an aerodynamic rebound. It becomes evident that the higher the inertia of the drop the higher also the inertia of the airflow needs to be to reflect the droplet from the plate. In preceding studies (Povarov et al.,1976;Gauthier et al.,2018) threshold parameters, describing the onset of aerodynamic rebound and based on the relation of drop and plate inertia are formulated in (1) assuming a deceleration of the droplet throughout the boundary layer and possible trajectory deflection due to the aerodynamic drag. This assumption is examined in Fig. 4. In Fig. 4(a) the temporal evolution of the distance 𝑦of the center of mass of the drop to the plate and its velocity are exemplified for one experiment in the aerodynamic rebound regime. The instant 𝑡= 0is defined as the time when the undeformed (spherical) drop with an undisturbed trajectory would touch the plate. In Fig. 4(b) the evolution of the corresponding velocity is shown. The estimated thickness of the turbulent boundary layer thickness in this case is 𝛿𝑡≈ 4.5 mm. It can be seen that the center of mass of the droplet is not decelerated until it is in the close vicinity of the plate surface (𝑦≈ 100 μm). This suggests that rather than the deceleration of the whole droplet, the deformation of the droplet is governing the rebound. 3.2. Boundary layer induced drop deformation: governing scales The dynamics of a drop moving through an airflow is determined by a balance of the stresses in the airflow and in the deforming drop. The main aerodynamic stresses include 𝑝gas ∼𝜌gas𝑈2 gas,(3) 𝑝unst eady ∼𝜌gas𝐷dr op d𝑈gas d𝑡∼𝜌gas𝐷dr op 𝜕 𝑈gas 𝜕 𝑧𝑈d,abs,(4) International Journal of Multiphase Flow 184 (2025) 105113 4 B. Stumpf et al. Fig. 2. Snapshots of a stream of monodisperse drops impacting on a plate at different plate velocities 𝑈𝜑,r el. The droplet of interest is highlighted with a green overlay. (a) splash occurring at 𝑈𝜑,r el = 60 m∕s, (b) transition at 𝑈𝜑,r el = 80 m/s, (c) aerodynamic rebound at 𝑈𝜑,r el = 140 m/s. Impact parameters are 𝐷≈ 230 μm, 𝑈𝑑 ,𝑛 ≈ 2m/s, 𝛽= 19◦. The corresponding videos are available in the supplementary material. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 3. Velocity maps showing the drop impact outcomes ‘‘splash’’, ‘‘transition’’ and ‘‘rebound’’ for the plate velocity 𝑈𝜑,r el and the normal drop impact velocity 𝑈d,nfor (a) 𝐷≈ 91.3 μm, (b) 𝐷≈ 452 μm. The error bars show the root mean square error (RMSE) of the corresponding velocities of different drops in one experiment. where 𝑈gas is the relative gas velocity. It can be shown that if the gas velocity is much higher than the drop velocity, 𝑈gas ≫ 𝑈d,abs, the value of 𝑝unst eady is negligibly small in comparison with 𝑝gas. However, this is not the case when the thickness of the boundary layer 𝛿is much smaller than the initial diameter of the drop, 𝐷dr op. The stresses in the drop are defined by the drop deformation velocity 𝑈def . The pressure terms in the drop, arising from inertial effects International Journal of Multiphase Flow 184 (2025) 105113 5 B. Stumpf et al. Fig. 4. Evaluation of the drop trajectory (a) normal distance to the plate as a function of time. (b) Velocity in normal direction as a function of time. Error bars are the RMSE between the trajectories in the experiment. Impact conditions: 𝑈𝜑,r el = 120 m∕s,𝐷= 218 μm,𝑈𝑑 ,𝑛 = 2.1 m∕s,𝛽= 19.25◦. due to the deformation flow, are 𝑝dr op ∼𝜌dr op𝑈2 def , 𝑝dr op,unst eady ∼𝜌dr op𝐷dr op d𝑈def d𝑡(5) The drop deformation velocity outside the boundary layer is negligible, given that the aerodynamic pressure associated with the air relative velocity is considerably smaller than the capillary pressure of the drop. Upon entering the boundary layer, the relative gas velocity increases, resulting in the deformation of the drop. For the modeling of this stage of drop motion, it is sufficient to consider only the unsteady pressure term. The validity of this assumption will be examined later, once the deformation velocity has been determined. Consequently, the balance of pressure at the drop interface yields: 𝜌gas𝑈2 gas =𝜌dr op𝐷dr op d𝑈def d𝑡.(6) The relative gas velocity profile in the boundary layer can be represented in the form 𝑈gas =𝑈plat e𝑓(𝜉) −𝑈dcos 𝛽 , 𝜉=𝑧 𝛿,(7) where 𝑓(𝜉)is a dimensionless function of the similarity variable 𝜉, and 𝛿is the thickness of the boundary layer. The solution of the differential Eq. (6) is obtained in terms of the variable 𝜉using the transformation of the variable d𝑡=𝛿d𝜉∕𝑈dr op,𝑛 𝑈def (𝜉) = 𝛿 𝜌gas𝑈2 plat e 𝐷dr op𝜌dr op𝑈d,𝑛 ∫𝜉∗ 𝜉 [𝑓(𝜉) −𝑘]2d𝜉 ,(8) 𝑘=𝑈dcos 𝛽 𝑈plat e , 𝑈d,𝑛 =𝑈dsin 𝛽 ,(9) where 𝜉∗is the dimensionless variable at which the relative gas velocity equals zero. The aerodynamic rebound, defined as the onset of the upward movement of the droplet without contact with the wall, is driven by the aerodynamic forces experienced by the droplet within the boundary layer. These forces, in turn, cause the droplet to deform. The droplet rebound occurs if the deformation velocity at the wall (𝜉= 0) is equal to the normal component of the droplet impact velocity 𝑈d,𝑛. This condition can now be written with the help of (8) in the form 𝐵=𝐵r ebound where 𝐵= 𝛿 𝜌gas𝑈2 plat e 𝐷dr op𝜌dr op𝑈2 d,𝑛 ∫𝜉∗ 0 [𝑓(𝜉) −𝑘]2d𝜉 .(10) 𝐵r ebound being an empirical constant of order of unity. It is interesting that the scaling (10) has a form similar to that defined in (1), obtained in Gauthier et al. (2016,2018) for laminar airflow from different considerations. In this study, the scaling is applied to the turbulent and laminar flows, generated by the rotating disk. 3.3. Solution for turbulent boundary layer The regime of the flow around a rotating disk is determined by the aerodynamic Reynolds number Reaer o. As mentioned earlier, in this study the flow is turbulent for all the experimental conditions. An approximate solution for the turbulent boundary layer (Shevchuk and Khalatov,1997) is chosen for the consideration of drop deformation. In this solution, the integral method is combined with the fitting of the existing experimental data from Cham and Head (1969), Itoh and Hasegawa (1994), Littell and Eaton (1994). The profile of the thickness of the boundary layer is expressed as 𝛿≈ 0.48𝑟Re−1∕6 aer o(11) In our experiments, the value of 𝛿is of the order of 1 mm. The azimuthal velocity component of the gas flow in the laboratory reference frame is assumed in the form 𝑢𝜑=𝑓(𝜉)with 𝑓(𝜉) ≈ 1 −𝜉1∕9, 𝜉=𝑧 𝛿.(12) The power-law approximation was first proposed by von Kármán (1921) using an analogy with the turbulent flow in a round pipe and on a flat plate. While in the original work, the exponent is 1/7, from a comparison with the experimental result a better agreement is found with a value of 1/9, (Shevchuk,2009). For more detailed information on the turbulent boundary layer approximation, the interested reader is referred to comprehensive reviews (Kobayashi,1994;Crespo del Arco et al.,2005;Shevchuk,2009;Lingwood and Henrik Alfredsson,2015; Alfredsson et al.,2023). Now, the coordinate 𝜉∗corresponding to the position at which the relative gas velocity equals zero is 𝜉∗= (1 −𝑘)9.(13) Now the expression for the integral in the right-hand side of (10) can be derived explicitly and the dimensionless rebound threshold parameter 𝐵for turbulent flow can be expressed in the form 𝐵= 𝛿 𝜌gas𝑈2 plat e(1 −𝑘)11 55𝐷dr op𝜌dr op𝑈2 d,𝑛 .(14) Expression for the threshold conditions for rebound (14) has been developed using the theory which neglects completely the surface tension effects. These effects can be taken into account by considering the aerodynamic Weber number Weaer o= 𝜌gas𝐷dr op𝑈2 plat e 𝜎(15) In Fig. 5the outcome map of drop impact onto a rotating disk is shown in terms of the 𝐵number and the Weber number Weaer ofor turbulent air flow. International Journal of Multiphase Flow 184 (2025) 105113 6 B. Stumpf et al. Fig. 5. Outcome map of drop impact onto a rotating disk in terms of the dimensionless parameter 𝐵, defined in (10), and Weaer o, defined in (15). The experimental parameters correspond to the turbulent flow in the airflow around the disk. Fig. 6. Dependence of the rebound threshold parameter 𝐵r ebound on the geometrical parameter 𝛿∕𝐷dr op for the experimental data from Gauthier et al. (2018) with range of oil drop diameters 1.4< 𝐷dr op <3.1 mm and from Povarov et al. (1976) with the range of water drop diameters 0.3< 𝐷dr op <4 mm. It becomes evident that 𝐵r ebound is well suited to describe the onset of the aerodynamic rebound regime. Furthermore, a slight dependence of 𝐵r ebound on Weaer obecomes apparent at smaller values of the Weber number Weaer o<10. As expected, this dependence is minor at high values of Weaer ofor which 𝐵r ebound ≈ 0.5,f or Weaer o>10.(16) It is important to note that the fitted value of the parameter 𝐵r ebound is of the order of unity. This is an important indicator of the validity of the model in which the main physical factors are taken into account. At this stage the validity range of parameters of the solution for the deformation velocity (8) can be examined. The model used in this study is valid if the ratio of the steady and unsteady pressures, expressed in (5), satisfies the condition 𝑝dr op∕𝑝dr op,unst eady ≪1. The upper bound for this ratio is estimated using (8): 𝑝dr op 𝑝dr op,unst eady ≈𝛿2 𝐷2 dr op 𝜌gas𝑈2 plat e 𝜌dr op𝑈2 dr op [∫1 0 𝑓(𝜉)2d𝜉]2 .(17) In the example 𝑈plat e= 140 m∕s,𝑈dr op = 1 m∕s,𝐷dr op = 0.5𝛿, the estimation (17) yields 𝑝dr op∕𝑝dr op,unst eady ≈ 0.03. The experimental parameters in this study satisfy the condition 𝑝dr op∕𝑝dr op,unst eady ≪1; thus the assumptions in this model are justified. 3.4. Solution for laminar boundary layer, normal impact (𝑘= 0) In the case of a laminar boundary layer a similarity solution for the velocity distribution has been developed by von Kármán (1921) and solved by Cochran (1934). The boundary layer thickness, defined as the normal distance to the plate from a point where the air velocity has reached 1% of the plate velocity, is expressed (Schlichting and Gersten, 2016) in the form 𝛿= 5.5√𝜈gas∕𝜔, (18) where 𝜔is the angular plate velocity, 𝜈gas is the gas kinematic viscosity. The evolution of the air velocity can be obtained by numerically solving the coupled ordinary differential equation system from von Kármán (1921). Correspondingly, the three components of the gas velocity can be expressed in terms of the similarity variable 𝜉=𝑧∕𝛿. The numerical integration of the expression on the right-hand side of (10), using the velocity profile 𝑓(𝜉)for laminar flow, yields 𝐵= 0.12𝛿 𝜌gas𝑈2 plat e 𝐷dr op𝜌dr op𝑈2 d,𝑛 ,laminar f low, 𝑘= 0.(19) The dependence of the rebound threshold parameter 𝐵r ebound, computed using Eq. (19), on the geometric parameter 𝛿∕𝐷dr op, as well as the values of 𝐵r ebound, computed with the help of (14), is shown in Fig. 6 using the experimental data from Gauthier et al. (2018) for silicon oil drops (𝜈= 100 mm2/s, 𝜌= 960 kg/m3,𝜎= 21 mN/m) and from Povarov et al. (1976) for water. Moreover, the results (Povarov et al.,1976) for 𝐵r ebound in the turbulent regime are also in the same order of magnitude, indicating a clear improvement of the present model, which accounts for the velocity profile in the air boundary layer. The scatter of the data for 𝐵r ebound indicates that the parameter 𝐵 is rather sensitive to the variations of the impact parameters. We can identify a monotonic but weak growth of 𝐵r ebound for higher values of 𝛿∕𝐷dr op. Nevertheless, the threshold values of 𝐵are in the range 0.18 < 𝐵r ebound <0.8. 4. Conclusion In this study, drop impact onto a dry rotating plate is experimentally investigated using a high-speed video system. Various types of impact outcomes have been observed for water drops in the diameter range of 80 μm< 𝐷 <500 μm and relative tangential velocities of 40 m∕s < 𝑈𝜑,r el <160 m∕s. These outcomes include drop complete or partial rebound, splash and deposition. The threshold conditions for the drop full rebound are determined from experiments and are modeled theoretically. In the model, the drop deformation velocity caused by the aerodynamic pressure in the near-wall boundary layer of the airflow is estimated from the balance of the pressure at the drop surface. The corresponding dimensionless parameter 𝐵has been formulated, which is expressed not only in terms of the impact parameters and densities of liquid and gas but also on the entire velocity profile in the boundary layer. Since the inertial effects in the gas flow and deforming drop are dominant, the dependence of the threshold value of the parameter 𝐵r ebound, corresponding to the rebound threshold, on the aerodynamic Weber number, is rather weak. The threshold values of 𝐵=𝐵r ebound are rather close for laminar and turbulent flow regimes which indicates that the main physical factors are considered in the model. Furthermore, a weak dependence of 𝐵r ebound on the geometrical parameter 𝛿∕𝐷dr op is identified. International Journal of Multiphase Flow 184 (2025) 105113 7 B. Stumpf et al. CRediT authorship contribution statement Bastian Stumpf: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation. Samaneh Abdi Qezeljeh: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation. Reda Kamal: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation. Fabien Dezitter: Supervision, Funding acquisition. Alessandro Martuffo: Supervision, Conceptualization. Ilia V. Roisman: Writing – review & editing, Supervision, Funding acquisition, Formal analysis, Conceptualization. Jeanette Hussong: Writing – review & editing, Supervision, Funding acquisition, Formal analysis, Conceptualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The work of Samaneh Abdi Qezeljeh is partially supported by the joint DFG/FWF Collaborative Research Centre CREATOR (DFG: Project-ID 492661287/TRR 361; FWF: 10.55776/F90) at TU Darmstadt, TU Graz and JKU Linz. Furthermore, this research project is funded by the German Federal Ministry for Economic Affairs and Energy (BMWI) within the framework concept ‘‘LuFo VI’’, subproject ‘‘NANNY’’. This project has received funding from the European Union’s Horizon Europe research and innovation program under the Marie Skłodowska-Curie grant agreement No 101072551 (TRACES). The authors gratefully acknowledge the valuable preliminary work done by Dr.-Ing. Mark Gloerfeld and the assistance in the experiments of Osaid Ur Rahman Siddiqui. Data availability We have made the data and the videos available at the following link: https://zenodo.org/doi/10.5281/zenodo.12684793. References Agbaglah, G., Josserand, C., Zaleski, S., 2013. Longitudinal instability of a liquid rim. Phys. 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