Parallel water entry: Experimental investigations of hydrophobic/hydrophilic spheres

When a solid object impacts an undisturbed, free water surface and penetrates the water, it initiates a complex, nonlinear, and transient flow event involving the interaction between the solid, liquid, and gas phases. This phenomenon, commonly referred to as “water entry” in classical mechanics, encompasses a wide range of dynamic behaviors and events. Among these are air cavity formation, evolution, and closure, as well as splash crown and dome formation, water jet generation, and the occurrence of cavity ripples and shedding. These phenomena can significantly impact the dynamics of the object's movement and induce changes in its kinematic behavior.^1–8 For over a century, engineers and researchers have dedicated significant efforts to studying the topic of water entry, driven by its broad spectrum of applications. Some notable applications include the following:

1. Industrial processes: Water entry plays a crucial role in spraying, painting, printing, fluid atomization, coatings, and surface cleaning applications.^9–13 

2. Naval engineering: Water entry phenomena are significant in areas like ship launching, slamming (impact of a ship hull with waves), and the development of naval weapons.^13–21 

3. Aerospace and aeronautical engineering: Water entry studies are relevant to seaplane and spacecraft landing, air-to-water weapons, and they offer insight into the behavior of bombs during water impact.^22–26 

4. Sports engineering: Water entry research contributes to the technical analysis of oars in rowing and athletes in diving.^12,26,27

5. Observations in nature: Water entry phenomena have been observed in various natural scenarios, such as lizards that can walk on water surfaces, seabirds that dive into the water, and beetles that walk upside down underneath the water surface.^1,25,28–31

With the emergence and expansion of high-speed camera technology, the study and investigation of water entry phenomena have entered a new phase, enabling the examination of objects with diverse geometries and under distinctive conditions. This technological advancement has significantly enhanced the recording, accuracy, and analysis of related phenomena.³² It is noteworthy that a majority of the studies presented and reported thus far have primarily focused on the water entry of spherical bodies,^{4,9,13,16,17,25,33–36} since such projectiles are capable of generating and exhibiting a wide range of interesting phenomena during water entry, including various main cavity closure regimes. Furthermore, their axisymmetric geometry allows for the formation of nearly axisymmetric cavities, jets, and splashes. As a result, conducting tests with a single high-speed camera is generally sufficient. Additionally, spherical bodies tend to experience minimal rotation and deviation in their trajectory. Nevertheless, several researchers have also focused on investigating water entry phenomena involving objects with non-spherical geometries. Examples of such objects include conical and wedge-shaped projectiles,^37–43 flat plates,^1,14,44–48 vertical cylindrical and slender bodies,^8,32,49–52 bodies with asymmetric nose shapes,⁵³ and thick and thin hollow objects.^54–58 Moreover, studies have explored the water entry behavior of objects with more complex geometries, such as 3D-printed birds and teardrop-shaped objects.^31,59,60 It is important to note that apart from geometric shapes, researchers are also interested in various other factors. These include the material and surface properties of the projectile, the physical properties of the fluid, the angle and rotation rate of the projectiles when they enter the water, the free-water surface deformations due to forced entry and exit of bodies, and even scenarios involving microgravity.^{2–4,8,12,26,61–67}

A review of the existing literature highlights that the field of water entry research has primarily concentrated on examining single projectile impacts on the water surface. However, the investigation of parallel (side-by-side) or tandem (consecutive) water entry involving multiple objects remains relatively unexplored. This topic can be regarded as a highly unresolved research subject within the field. It is important to note that this problem differs from the extensively studied scenario of fluid passing over multiple bodies, which typically focuses on one-phase fluid flow phenomena.^68–74 The parallel or simultaneous entry of multiple projectiles can give rise to interactions that significantly influence the development of air cavities, splash curtains, and jets, which in turn can impact the movement and dynamics of the objects involved. The phenomenon of parallel water entry has diverse applications, including synchronized diving in Olympic competitions, parallel diving behaviors of seabirds, side-by-side entry of airborne projectiles into the water, water model experiments simulating molten iron bath desulphurization processes to predict penetration depth and residence time of agents, inkjet printing technology, simultaneous entry and exit of adjacent oars in rowing, extreme waves affecting offshore wind turbine platforms and offshore vessels or structures installed side-by-side, and the consideration of loads on catamarans' twin hulls, among others.

As mentioned, the number of published studies focusing on multiple-object water entry is limited, but the topic has received quite some attention in recent years. In 2010, Wang and Wang⁷⁵ conducted a numerical analysis on the parallel synchronous water entry of cylinders to investigate the impact of various parameters, such as cylinder radius, the distance between cylinders, and their velocities, on the hydrodynamic forces. In 2014, Ern and Brosse⁷⁶ conducted a study examining the interaction of two identical disks falling side-by-side into salty water, varying the aspect ratios. They observed that when the lateral distance between the disks' centers was less than $4.5$ diameters, the projectiles repelled each other and that the repulsion kinematics was also dependent on the aspect ratio. Yousefnezhad and Zeraatgar⁷⁷ utilized the boundary element method (potential theory) to analyze the parallel water entry of two wedges at a constant vertical velocity. They proposed a regression formula for determining the maximum pressure coefficient on the projectile surface for different wedge angles. In 2018, Yu⁷⁸ investigated the cavity evolution and kinetics of two bodies entering the water in tandem, employing both numerical and experimental techniques. Jiaxing et al.⁷⁹ conducted an empirical study on the water entry process of two parallel cylinders. They compared the cavity characteristics of single and double cylinders at various Froude numbers. Their findings revealed that the entire cavity exhibited good mirror symmetry, whereas each cylinder's cavity displayed significant asymmetry. Furthermore, their results demonstrated that for low-impact Froude numbers (⁠ $Fr I$⁠), the cavity closure followed a characteristic pinching-off pattern, with the pinching point moving upward as $Fr I$ increased. When $Fr I$ reached a critical value, the closure transitioned to a surface-seal pattern, and the closure point moved downward with increasing $Fr I$⁠.

Wang and Lyu⁷ conducted an experimental analysis of the cavity dynamics and kinetics of two spheres impacting the water surface side-by-side. The surfaces of the spheres had a static contact angle ranging from $80°±10°$⁠, indicating a hydrophilic nature. The impact velocity chosen for the experiments was above $14 m s − 1$⁠, which exceeds the threshold velocity, above which cavity formation occurs for hydrophilic spheres. They investigated the influence of time intervals between releases and the lateral distances on the water entry process. Their findings revealed that at a lateral distance between the spheres of more than $5.5$ diameters, both the cavity characteristics and kinetics of the spheres resembled those observed in single water entry scenarios. However, when considering the combined effect of release time intervals and lateral distances, they observed intriguing phenomena, such as the expansion and contraction of the walls of two adjacent cavities, the formation of an M-shaped splash, and more. For moderate time intervals and lateral distances of less than 1.5 diameters, one sphere was horizontally attracted to the other, resulting in a successive water entry pattern. In contrast, in cases of synchronous water entry, repulsion between the two spheres was observed. These phenomena were explained by the relative strength of expulsion and pressure forces acting on the spheres.

In 2020, Yun et al.⁸⁰ conducted an experimental study to investigate the hydrodynamic features associated with oblique water entry of two tandem spheres. Their research revealed the formation of a suction region within the cavity created by the front sphere. This suction region resulted in the acceleration of the rear sphere, and in some cases, even led to a collision between the two spheres. In a separate experimental study, Yun et al.⁸¹ investigated vertical tandem water entry of two spheres. They observed that when the rear sphere impacted the upward water jet generated by the front sphere, a significant amount of kinematic energy was transferred to the flow. As a result, a considerably larger cavity formed behind the rear sphere compared to the case of a single-sphere water entry.

Rabbi et al.⁸² conducted an experimental study where they investigated the tandem water entry of two spheres. Their primary focus was to measure the forces exerted on the trailing sphere. They observed that a reduction in impact deceleration occurred for the rear sphere, which was influenced by the cavity dynamics generated by the front sphere and the relative timing of the trailing sphere's impact. To categorize the phenomena arising from these combined effects, they introduced a non-dimensional number called Matryoshka number. Four main regimes were identified: on-cavity, inside-cavity, on/inside-bowl, and on-Worthington jet modes. The results revealed that in the first, second, and fourth regimes, the impact force on the trailing sphere could decrease by up to $78%$⁠. However, in the third regime, a surprising increase in over $400%$ in the force was observed in the worst-case scenario.

Rabbi⁸³ experimentally examined the simultaneous side-by-side water entry of hydrophobic spheres of the same size. He observed that the air cavities formed by the spheres are not influenced when the lateral center-to-center distance (⁠ $Δ x c$⁠) is higher than $4.0 D s$ (here, $D s$ is the sphere's diameter). In this study, for $Δ x c<4.0 D s$⁠, two bowl-shaped asymmetric cavities were observed, which prevented a clean pinch-off; instead, two long air tubes formed (string shedding). The pinching time of both cavities monotonically decreased with increasing sphere distance approaching $t pinch=2.06 D s / 2 g$⁠, i.e., the relation for predicting the pinch-off time of single sphere water entry. Rabbi also proposed a potential flow model to explain experimental observations (e.g., the dynamic interaction between the cavities' walls). Furthermore, he showed that as $Fr I$ increases, the pinching times decrease and confirmed that two tilted Worthington jets are created after pinch off, which was most noticeable for smaller $Δ x c$⁠. For very small lateral distances, a combined upper cavity was observed, resulting in the formation of a single chaotic and merged Worthington jet.

In 2021, Lyu et al.⁸⁴ conducted experimental investigations to examine the impact of the time interval between two tandem spheres entering the water on cavity evolution and the motion characteristics of the spheres. Their findings revealed that the time interval between the spheres influences the pattern of cavity closure for the rear sphere. Lu et al.⁸⁵ examined high-speed parallel projectile's water entry (from $280$ to $400 m s − 1$⁠) numerically. At such impact velocities, their results show that the cavity distribution is nonuniform due to the strong interference between the projectiles. In 2022, Lu et al.⁸⁶ numerically investigated the asynchronous parallel water entry of two slender cylindrical projectiles. They reported that, at first, the air entrainment cavity of the first projectile is squeezed, and the cavity of the second one is expanded, and eventually, both cavities merged and gradually stretched. In 2022, Lyu et al.⁸⁷ experimentally and numerically analyzed the different modes of water entry for the trailing sphere in a successive water entry scenario of two spheres. Regarding the impact Froude number, three typical modes were introduced: (1) Steady mode: when $Fr I>33.0$⁠, the rear sphere falls into the opening cavity of the front sphere and then enters the water, (2) Transition mode: for $25.2< Fr I≤33.0$⁠, the rear sphere first contacts the splash formed by the front sphere and then enters the water, and (3) Perturbation mode: for $3.4< Fr I≤25.2$⁠, the second sphere enters the water after hitting the Worthington jet of the primary sphere. In 2023, Wang et al.⁸⁸ conducted a study using a 3D numerical volume-of-fluid (VOF) approach and a six-degree of freedom solver to simulate the water entry of two spheres positioned side-by-side at varying lateral distances and time intervals. They validated their approach with the experimental results of Wang and Lyu.⁷ The findings of their study indicated that the time interval between the entries of the spheres does have an impact on the cavity dynamics, influencing the expansion and contraction processes. However, they noted that the influence of the time interval diminishes significantly when the lateral distance between the two spheres is increased.

Based on the literature review, it is evident that the manner in which projectiles enter the water (whether individually, in parallel, tandem, etc.) can have a strong effect on descent trajectories, cavity dynamics, fluid-structural integrity, and hydrodynamic forces. For instance, when a single sphere enters the water, the pinch-off time remains relatively constant across a wide range of release heights. However, in the case of parallel entry, the lateral distance between the objects and their surface wettability can have a significant impact on the pinch-off time. Therefore, the objective of this study is to experimentally investigate parallel water entry for spheres with identical size and material properties. It aims to examine the influence of different coating layers (hydrophobic and hydrophilic) and the lateral center-to-center distance between the spheres. The study focuses on analyzing the fundamental patterns of cavity formation, evolution, pinching, and the kinetics of projectiles. To effectively capture and analyze the dynamics of the water entry process, this study employs a high-speed photography system and an image processing technique. The paper is structured as follows: Sec. II provides a description of the experimental apparatuses and test procedures. Section III presents the qualitative outcomes, including the stages of cavity formation, evolution, and collapse as well as splash crown and Worthington jet behavior. Section IV presents the main quantitative results, which encompass the dimensions of the cavities and the dynamics of the projectiles. Finally, Sec. V concludes the paper by summarizing the main findings.

Figure 1 displays a schematic depiction of the water entry test rig and the associated coordinate system. The projectiles utilized in the experiment are steel spheres with a diameter of $D s=20 mm$⁠, a density of $ρ s≈7783 kg m − 3$⁠, and a mass of $m s≈0.0326 kg$⁠. The original spheres have a smooth surface with a static contact angle of approximately $85$°, indicating a hydrophilic behavior. In order to create hydrophobic spheres, some of the spheres are coated with a layer of ULTRA EVER DRY^® TOP Coat produced by UltraTech International, Inc. This coating significantly increases the contact angle to around $165–170$°, resulting in a superhydrophobic surface. In this paper, the abbreviations SHP and HPI refer to the spheres coated with the hydrophobic layer (superhydrophobic) and the uncoated spheres (hydrophilic), respectively. For testing purposes, an acrylic water tank with a cross-sectional area of $50×50 cm 2$ and a depth of $150 cm$ is utilized. The tank is filled with water, with a water depth of approximately $130 cm$⁠. The experimental setup utilizes demineralized water, which possesses the following properties: a temperature range of $T w≈23–25 °C$⁠, a density of $ρ w≈998 kg m − 3$⁠, a dynamic viscosity of $μ w≈8.91× 10 − 4 Pa s$⁠, and a surface tension of $σ w≈72.0 mN m − 1$⁠.

The release mechanism consists of a machined metallic plate with a row of small holes, an electromagnet, and a pushing spring. The plate is hinged to a fixed rod, and when it is attached to the electromagnet, it remains in a completely horizontal position. The dimension of the plate and the location of the holes have been designed so that the spheres impact the water surface in the center of the tank. The small holes are meticulously drilled to ensure specific center-to-center distances of the spheres (⁠ $l d$⁠), ranging from $1.0 D s$ to $5.0 D s$ in intervals of $0.5 D s$⁠. Upon deactivation of the electromagnet, the compressed spring forcefully propels the plate downward. This downward movement is fast enough to ensure that the spheres resting on the holes are simultaneously released. To ensure simultaneous impact and to validate the absence of sphere rotation caused by the release mechanism, the impact events of the two spheres were carefully examined using high-speed imaging. The entire release mechanism is mounted on an adjustable horizontal beam. This beam can be vertically moved using a motorized system, allowing for precise adjustment of the desired release height (⁠ $H R$⁠) above the water surface. When the spheres are released from the plate, they impact the water surface at a speed of $U I= 2 g H R$⁠, where the acceleration due to gravity is represented by $g=9.81 m s − 2$⁠. Therefore, for the range of release heights of $H R=15–95 cm$ used in these experiments, the impact velocities can be approximated as $1.72≤ U I≤4.32 m s − 1$⁠. Each simultaneous release test is conducted three times, and the data obtained from each test are averaged for analysis. This approach ensures reproducibility and accuracy in the measurements. Table I presents a summary of the key dimensional and dimensionless parameters required for the analysis.

To record the water entry processes, two high-speed cameras are employed. The first camera, a Photron FASTCam-SA3, operates at a frame rate of $2000$ fps and possesses an image resolution of $1024×1024$⁠. It is specifically utilized for close-up capturing of the impact, splash formation, and cavity development. The second camera, a Kron Technologies CHRONOS-2.1HD, operates at a frame rate of $1512$ fps and has an image resolution of $1280×1024$⁠. Its purpose is to capture wide-angle images from a distance to record the spheres' trajectories up to a water depth of approximately $50 cm$⁠. To achieve the desired shots, the following lenses are used: “Nikkor-35 mm-f/1.1,” “MicroNikkor-55 mm-f/2.8” for the close-up images, and “Nikkor-24–85 mm-f/2.8–4” for the wide-angle images. An LED light panel with sufficient contrast is mounted on the backside of the water tank. This ensures that the lighting conditions are optimal, resulting in high-quality images with clear contrast and detail resolution.

An in-house MATLAB script is employed for image processing to analyze various parameters in this study. The script is specifically developed to evaluate parameters such as the trajectory of the projectiles (⁠ $z s t$ in $m$⁠), pinching time (⁠ $t p$ in $ms$⁠), pinching depth (⁠ $L p$ in $m$⁠), and cavity sizes. The script automatically analyzes the individual images of a water entry event and increases the contrast of the images to clearly identify the spheres and cavities. For impact scenarios where a clear distinction of the two spheres and the cavities is possible, the images are subsequently subdivided into distinct image regions to evaluate areas, centroid positions, and a bounding box for each object (connected component, i.e., sphere and attached cavity). The images are converted into black and white images, and continuous regions are identified to obtain the positions of the spheres and the cavities. In cases where the separation ratios of the spheres are low, individualized, not fully automated techniques are used to evaluate the sphere's positions.

In the realm of water entry, a range of events and phenomena can take place both above and below the water surface. These include the formation, evolution, and closure (pinch-off) of air entrainment cavities, the creation and sealing of splash crowns/domes, the generation of Worthington water jets, the presence of cavity wall waves/ripples, and the shedding of cavities, among others. The objective of this section is to provide a comprehensive analysis of qualitative and visual data related to the simultaneous impact of two spheres on the water surface. It explores specifically in detail the impact of the lateral center-to-center distance between the spheres and the surface wettability on the observed water entry phenomena. It is important to note that in this study, the term “symmetric” refers specifically to axisymmetry around the vertical axis of the spheres, as the photography and captured phenomena are two-dimensional. This should not be confused with symmetry with respect to the midplane between the spheres, which will be referred to as “mirrored.”

For the smallest separation ratio (⁠ $l d/ D s=1.0$⁠), indicating that the two spheres are in direct contact with each other, significant distortions are observed in the shape of the cavities. After the spheres in contact with each other impact the water surface, a large merged air cavity is formed. This merged cavity is particularly evident in the initial time frames. However, as the spheres descend further into the water, they exert repelling forces onto each other, resulting in an ongoing increase in the separation distance between the spheres. Consequently, two distinct cavities form, each attached to its respective sphere. Although this repelling force and the subsequent separation also occur for other values of $l d/ D s$⁠, this is particularly pronounced in the case of $l d/ D s=1.0$⁠. Figure 3(b) provides insight into the phenomenon of spheres moving apart employing an aerodynamic analogy. The figure illustrates that the shape of the sphere and the attached cavity resembles an asymmetric hydrofoil with one flat and one curved profile. The accelerated flow on the curved surface generates a low-pressure zone on the left side of the sphere (suction side) and the flow along the flat side creates a high-pressure zone on its right side (pressure side). This configuration creates a repelling force that acts from the pressure side toward the suction side, causing the spheres to move away from each other and increase their separation distance. An additional intriguing phenomenon, which is specifically observed in the case of $l d/ D s=1.0$⁠, is the occurrence and the progression of the cavity-shedding string (air-tube) over time. To illustrate this phenomenon, Fig. 4 presents a selection of close-up images of the water entry event shown in the last row of Fig. 2. These images span from time $t=77$ to $97 ms$⁠, with intervals of $Δt=5$ ms. Notably, around $t=87 ms$⁠, the air tubes come into contact with each other at a location close to the lower air cavities attached to the spheres. Subsequently, a third air tube forms between the existing strings. These three air tubes largely maintain their continuity until the time $t=97 ms$⁠. It should be pointed out that in the work of Rabbi,⁸³ only a single merged air string for a specific state of $l d/ D s=1.0$ was observed and the presence of two or three separate air tubes was not mentioned. This discrepancy in the observations may be attributed to the higher interval of $15 ms$ between the presented images in Rabbi's study compared to the finer time resolution of 5 ms used in Fig. 4 of the present study. Another possible interpretation can refer to the difference in the experimental settings. The sphere's diameters and release heights used in this study lead to slender and long cavities attached to the spheres with a high aspect ratio, whereas the images provided in Rabbi's work show cavities with a lower aspect ratio, which are similar to a bluff body. Consequently, in the latter case, a more unsteady and turbulent wake is produced, which significantly distorts the strings and impedes the development of clearly identifiable separate strings. The more streamlined cavity shapes of this study experience a low degree of boundary layer separation and the strings emerging from the bubbles experience lower turbulent distortions, which allows for the formation of a third air tube further upward. It is worth noting that further comprehensive studies, including numerical simulations, are necessary to delve deeper into this phenomenon and gain a thorough understanding of its intricate details.

Here, to conclude the analysis of the splash phenomenon, Fig. 8 presents the events above the water surface for a case in the surface seal regime. This figure showcases the chronophotography of the splash and Worthington jet, capturing the simultaneous water entry of two hydrophobic spheres released from a height of 65 cm. This investigation encompasses three distinct scenarios, characterized by $l d/ D s$ ratios of $5.0, 2.5$⁠, and $1.0$⁠. The illustrations show the emergence of a splash dome and the subsequent interaction of the Worthington jet with this dome. The initial splash dome for $l d/ D s=5.0$ and 2.5 is asymmetric. The subsequent surface sealing events produce splashes, which are not located in the center of the dome. After the surface sealing, a merged splash dome rises in the $l d/ D s=2.5$ scenario. The interaction of the Worthington jet with the splash dome produces complicated disruptions of the splash dome, which do not allow for a comprehensive analysis of the events and the Worthington jet angles in meticulous detail. The scenario $l d/ D s=1.0$ produces a central jet pair, as has been observed in the deep-seal regime discussed before. This jet pair persists during the surface sealing event and the subsequent rise of the merged splash dome, until it is finally completely distorted by a strong central Worthington jet (not shown in the figure).

The chronophotography presented in the last row of Fig. 9 illustrates the phenomena observed during the water entry of two in-contact spheres. The right sphere is hydrophilic, and according to expectations, the formation of an air entrainment cavity is deemed impossible at the given impact velocity. The images in the first and second rows of Fig. 9 support this notion, as no cavity is observed behind the hydrophilic sphere. However, in the third row, it becomes apparent that when the spheres are in close contact; i.e., at a separation ratio $l d/ D s=1.0$⁠, the air from the main cavity formed by the hydrophobic left sphere partly encompasses the right hydrophilic sphere and, for the release height of the presented images, a smaller cavity remains attached at the hydrophilic right sphere. This can be observed in the images at $t=23$ and $38 ms$⁠. After a few milliseconds (⁠ $t=53 ms$⁠), this newly formed cavity separates from the main cavity and travels downward along with its respective sphere. This separation process is accompanied again by the formation of a narrow air tube, as can be observed in the image at $t=68 ms$⁠. At this stage, the cavity behind the left sphere takes on a shape resembling a long circular pipe, with a diameter determined by the leading sphere. This air pipe then pinches off, which results in the formation of a second narrow air tube, as can be observed at $t=83 ms$⁠. Interestingly, in this scenario, the previous observation of the hydrophilic sphere moving slower than the hydrophobic sphere is no longer true. Instead, the hydrophilic sphere now moves slightly faster. It is presumed that this behavior change can be attributed to four factors. First, the tear-shaped air cavity behind the right sphere suppresses the formation of ring-like vortices, resulting in a reduced drag on the combined sphere-cavity system. Second, differences in the separation location of the water flow on the sphere surface can lead to different pressure distributions on the sphere surface. Third, the hydrophilic sphere experiences a lower upward buoyancy force compared to the hydrophobic sphere due to its smaller cavity volume. Finally, the hydrophilic sphere travels a longer distance with a detached cavity from the water surface, minimizing the influence of surface tension forces between the meniscus and the sphere. In contrast, the hydrophobic sphere maintains an attached cavity that remains connected to the water surface for a significant duration, leading to a stronger impact of the mentioned surface tension force on its movement. It is important to note that the latter two factors will have a relatively small impact compared to the first two factors. It is worth noting that the occurrence of this double cavity regime depends on the impact velocity and is observed in scenarios where $H R≥55 cm$⁠. Conversely, in cases where $H R≤45 cm$⁠, a single cavity regime can be observed, similar to the scenarios with larger separation ratios described above, however without the occurrence of a second pinch-off event but a rather highly disturbed cavity. Upon analyzing the frame-by-frame images, it becomes evident that immediately after the impact, the cavity generated by the left hydrophobic sphere partially covers the adjacent hydrophilic sphere. Subsequently, the two spheres move slightly apart, resulting in water flow entering the gap between them. In cases where the sphere velocity is higher (i.e., $H R≥55 cm$⁠), the water flow entering the gap fails to completely cut off the air attached to the sphere, despite its hydrophilic surface. This implies that the water flow within the gap follows a more direct upward path, cutting through the entrained air at a location further away from the sphere's surface, leading to a cavity pinch-off event and an air cavity that remains attached to the hydrophilic sphere.

The regime presented and discussed in the second row of images in Fig. 9, where a strong influence of the hydrophilic sphere on the air-entrainment cavity of the hydrophobic sphere is observed, which results in the formation of a second pinching point, is explored further in the following. The effect of impact velocity is investigated and presented in Fig. 10(b). This figure highlights the interaction between the hydrophilic sphere and the cavity for four different release heights: $H R$ =  $15$⁠, $35$⁠, $45$⁠, and $95 cm$⁠. In all the images, the separation ratio of the spheres is $l d/ D s=2.0$⁠. It is evident that, as the impact velocity increases, the strength of the vortices shed by the right sphere and their influence on the opposing cavity become more pronounced. This intensifies the deformation of the cavity shape and increases the likelihood of the formation of a second pinching point. A relationship between the minimum release height and the separation ratio regarding the occurrence of a second pinch-off event is shown in Table II. The data demonstrate that a second pinch-off is more likely to occur at lower separation ratios. When the hydrophilic sphere is located closer to the hydrophobic one, even the shedding of weaker vorticity can have a strong enough impact on the neighboring cavity to produce a second pinching event.

Section III A investigates the events beneath the water surface. As it is also valuable to examine the phenomena occurring above, this aspect is addressed in Fig. 11 (Multimedia views) and provides a visual representation of the splash crown formation during the simultaneous water entry of a hydrophobic and a hydrophilic sphere. Both spheres are released from a height of $H R=25$ cm, resulting in initial values of the impact Weber, Reynolds, and Froude numbers being $1360$⁠, $49 670$⁠, and $5.0$⁠, respectively. The sequence of images is captured at regular intervals of $Δt=15$ ms, starting from $t=10 ms$ after the impact and continuing until $85 ms$⁠. This figure highlights two scenarios. The left column corresponds to the scenario where $l d/ D s=2.5$⁠, and the right column corresponds to the scenario where $l d/ D s=1.0$⁠. In the case of the separation ratio of $2.5$⁠, it becomes evident that both the hydrophobic and hydrophilic spheres exhibit behaviors similar to an isolated water entry scenario. Specifically, for the hydrophobic sphere, an approximately symmetric splash crown is observed. Conversely, the hydrophilic sphere on the right generates a single vertical water jet as the water film layer surrounding its surface is displaced, which aligns with its behavior in an independent water entry context. It should be noted that the hydrophobic sphere eventually creates an inclined Worthington jet, which indicates a slightly asymmetric cavity shape at pinch-off. In the case of a 1.0 separation ratio, similar to the events discussed for two hydrophobic spheres, the formation of the jet-pair is observed. However, this jet pair is not located in the center of the splash curtain and rises in an oblique direction. In addition to the formation of this jet-pair, another interesting phenomenon is observed. A transverse jet (instead of a vertical jet) emerges from the side of the splash crown facing the hydrophilic sphere. This jet is probably created due to the same movement of the water layer around the hydrophilic surface. The interaction with the splash crown of the left sphere does not permit a vertical jet development. Such a transverse jet can not only be observed in the deep seal regime presented in Fig. 11 but also in a surface seal regime. In the latter case, the jet emerges directly from the water surface and becomes weaker with increasing release height.

In contrast to Sec. III, the primary focus of this section is to conduct a comprehensive analysis of quantitative data concerning the conducted experiments. The main objective is to examine in detail the influence of both the lateral center-to-center distance between the spheres at impact and the surface wettability pairing on various aspects, including the spheres' kinetics, the cavities' pinch-off time and depth, or the angles of Worthington jets. The data utilized in this analysis are obtained through the application of image processing techniques, allowing for a more precise and detailed investigation of the observed phenomena. The main steps of the employed algorithms are outlined in Sec. II. All experimental tests have been conducted with a minimum of three replications. The following results represent the average values of these replications. The square root of the variance, representing the deviation between the measured data and the average value, is used to quantify the uncertainty. The “error bar” symbol is employed to visually represent this uncertainty. However, to maintain clarity in the graphs, the uncertainty is explicitly emphasized in only one figure, namely, Fig. 16, while the remaining figures present the average values without error bars to avoid confusion.

The findings of the study demonstrate a substantial influence of the lateral center-to-center distance between the spheres and their interaction during parallel water entry on the values of pinch-off time (⁠ $t p$⁠) and pinch-off depth (⁠ $L p$⁠) of the cavities formed. Figures 12(a) and 12(b) showcase the variation of the dimensionless pinch-off depth (⁠ $L p/ D s$⁠) and of the dimensionless pinch-off time (⁠ $t p U I/ D s$⁠) in relation to the separation ratio (⁠ $l d/ D s$⁠), respectively. An analysis of Fig. 12(a) shows that regardless of release height and surface wettability pairing, an increase in the separation ratio results in a reduction of the pinch-off depth. However, with increasing the separation ratio, the influence of the neighboring sphere on the pinch-off depth gradually diminishes, and the graphs tend to level off horizontally. These curves are expected to approach the same pinch-off depth observed in a single water entry scenario, particularly for very high separation ratios. It is worth noting that the inclination of the curves for the SHP-SHP scenario does not reach zero at $l d/ D s=5.0$⁠, which concurs with the observation of a slight asymmetry in the cavities even at that separation ratio, as described in Sec. III. Furthermore, it is evident that for a given release height, the cavity closure of a hydrophobic sphere takes place at a shallower depth when it is in proximity to a neighboring hydrophilic sphere. This effect becomes less pronounced at higher release heights, so that for the largest release height considered in the current experiments (i.e., $H R=95 cm$⁠), the results for closure depth become nearly identical for both the SHP-SHP and SHP-HPI configurations.

The graphs presented in Fig. 12(b) corroborate the findings of Fig. 12(a), indicating that an increase in the separation ratio leads to a decrease in the pinch-off time. This observation holds true regardless of the release height and surface wettability pairing. Moreover, it is evident that for high separation ratios, the pinch-off time stabilizes at a relatively constant value, aligning with the pinch-off time observed in a single-sphere water entry scenario. Once again, it is worth reiterating that the curves for the SHP-SHP scenario do not become fully horizontal since a slight influence of the neighboring sphere and the cavities remains even at $l d/ D s=5.0$⁠. Figure 12(b) also includes three horizontal green lines that represent the semi-analytical dimensionless pinch-off time for single-sphere water entry, calculated using the formula proposed by Duclaux et al.,³ i.e., $t p , ∞=2.06 D s / 2 g$⁠, for the cavity pinching time in an undisturbed deep-sealing regime (refer to the next paragraphs for an explanation of the notation chosen here). The comparison between these lines and the test results from this research provides valuable insight. First, it serves to validate the reliability of the release mechanism employed in the experiments, as the close alignment between the single sphere limit and the test results for large separation ratios suggests that the spheres were released without rotation or other distortions. Second, the comparison demonstrates that this relationship remains valid for single water entry of superhydrophobic spheres.

As mentioned in Sec. III A, the splash crown does not form a dome-like shape at low release heights. The absence of a dome-like shape provides the ideal opportunity to capture the Worthington jet since it does not crash into a splash curtain. Furthermore, in Fig. 5, it is observed that decreasing the lateral center-to-center distance in the SHP-SHP configuration results in an increase in the inclination angle of the Worthington jets (⁠ $θ$⁠). To quantify this observation, Fig. 14(a) illustrates the variation of $θ$ as a function of time for two release heights (⁠ $H R=15$ and $25 cm$⁠) and three separation values of $l d/ D s=3.0, 4.0$⁠, and $5.0$⁠. The angle $θ$ is measured manually as indicated in Fig. 14(a), and the provided values are calculated by averaging the measured values of the left and right Worthington jets. For separation values below a certain threshold (⁠ $l d/ D s≤2.0$⁠), the upper bowl-shaped cavities merge, resulting in the formation of a single chaotic Worthington jet instead of two separate jets (see Fig. 5). This merging makes it challenging to measure the inclination angle of the individual jets. One notable observation is the increasing angle of the jet over time, with a tendency to become more vertical. This behavior can be attributed to the reduction in the dimensions of the upper cavities and the reduction in the velocity of the water close to the surface as time progresses. As the upper cavities become smaller, their influence on each other decreases. This, together with the lower velocities in the water phase, leads to a more symmetrical shape. Consequently, the jets tend to align more vertically. Another observation is the correlation between the release height and the angles of the jets. It is observed that as the release height increases and the spheres impact the water surface at higher speeds, the jet trajectory becomes more oblique. This tilt can be attributed to the stronger asymmetry of the upper cavity resulting from the higher impact speeds. Finally, as expected, when the lateral center-to-center distance increases and the configuration tends more toward single water entry, the angle of the Worthington jets increases and the jets tend to be more vertical. This issue is also displayed using combined images in Fig. 14(b) for $H R=25 cm$ in eight selected time-frames. It is worth noting that Rabbi⁸³ also reported a similar trend in the variation of jet angles with respect to $l d/ D s$ and release height. However, the development of the tilt angles over time is not addressed and the specific times at which the angles are evaluated are not specified in his study.

As discussed in Sec. III A and supported by Fig. 3(b), it is observed that when two hydrophobic spheres impact the water at a low separation ratio, they exert horizontal forces on each other, resulting in an increase in the horizontal center-to-center distance between the spheres (⁠ $Δx$⁠). To quantify this phenomenon, the dimensionless parameter $Δx/ l d$ is plotted as a function of descent time (measured from the moment of impact, milliseconds) in Fig. 15. The release height is $H R=65 cm$⁠, and separation ratios are $l d/ D s=1.0$⁠, $1.5$⁠, $2.0$⁠, $3.0$⁠, and $5.0$⁠. The figure demonstrates that for a separation ratio of $l d/ D s=5.0$⁠, the spheres exhibit a slight tendency to move closer to each other over time. However, for lower separation ratios, an increase in the distance between the spheres is observed. This effect becomes more pronounced as the separation ratio decreases. In the case of the minimum separation ratio, the distance between the centers has more than doubled in size within the recorded trajectory range. The findings also indicate the presence of a threshold separation ratio of approximately $4.5$⁠. At values above this threshold, the spheres tend to attract each other, whereas, below it, they move farther apart.

During the water entry process, the downward motion of the projectile is caused by the combined effects of gravity and kinetic inertia. Meanwhile, the motion is opposed by resistance forces, including the hydrodynamic drag force, buoyancy force, and forces due to surface tension. The dynamic behavior of the projectile during water entry is governed by the intricate interplay between these driving and resistance forces. However, as mentioned before, in the case of parallel water entry, specifically in the SHP-HPI configuration, additional factors come into play. The formation and shedding of ring-like vortices in the wake of the hydrophilic sphere have a significant impact, not only on the neighboring cavity but also on the kinetics of the spheres themselves. In Fig. 16, the kinetics of the SHP-HPI spheres released from a height of 95 cm (positioned at separation ratios $l d/ D s=2.0$⁠, $2.5$⁠, $3.5$⁠, and $4.5$⁠) is presented as an illustrative example. The figure includes the spheres' vertical position over time $z s t (m)$⁠, the velocity $V s t= z ̇ s$ (⁠ $m s − 1$⁠), the acceleration $a s t= z ¨ s$(⁠ $m s − 2$⁠), and the total hydrodynamic force coefficient $C F , s t$⁠. As mentioned in the first paragraph of Sec. IV, error bars are specifically illustrated for the case of $l d/ D s=3.5$⁠. Highlighting the error bars provides a visual representation of the uncertainty associated with the measurements in that particular case representing the statistical contribution to the observed deviations. Systematic measurement errors, which are hard to quantify, e.g., due to camera positioning, lens distortion, etc., are not included and would lead to higher error bars. The results, especially those based on the second-time derivative of the vertical position data $z s t$⁠, exhibit a certain waviness, which can be attributed to small errors in $z s t$⁠. Based on the figure and the plotted graphs, several significant findings can be pointed out.

Figures 16(a) and 16(b) indicate that in the range of these experiments, the spheres' separation ratio has a minimal effect on their vertical trajectory and descent velocity. However, it is noticeable that the hydrophilic sphere (HPI) experiences a slower descent and reaches its terminal velocity (around 2  $.0 m s − 1$⁠) at a shallower depth compared to the hydrophobic sphere. Furthermore, details regarding these findings have been discussed in Sec. III B. The cases shown in Fig. 16 represent the cavity regimes shown in the first two rows of Fig. 9. The influence of the neighboring sphere on the cavity dynamics plays a minor role in the kinetics of steel spheres with a high solid-to-liquid density ratio. The low effect of the separation ratio indicates that the vertical velocity and vertical acceleration of the two spheres are similar to that observed in single water entry events.

Figure 16(c) showcases the vertical acceleration experienced by both spheres. Figure 16(d) presents the total hydrodynamic force coefficient calculated using Eq. (1). As anticipated, the deceleration of both spheres decreases over time, tending toward zero. However, it is notable that the hydrophilic sphere exhibits a higher deceleration in the first part of its descent compared to the hydrophobic counterpart. Consequently, the deceleration of the hydrophilic sphere reaches zero earlier, and it nearly reaches its terminal velocity within the measurement range of the experiments. The hydrophobic sphere still experiences significant deceleration at the end of the measurement range. The results for acceleration and the calculated force coefficient tend to be similar to those published by Truscott et al.⁹² for steel spheres with a solid-to-liquid density ratio of about 7.8. In the hydrophobic case, the inertia is so high that the dynamic effects of cavity pinching can hardly be detected in the deceleration curves. Similar to the observations reported by Truscott et al., the hydrophilic sphere exhibits higher initial deceleration and consequently higher resistive forces. The main reasons have been analyzed in detail by Truscott et al. Here, a brief summary regarding the main force components is provided, which differs slightly in the approach regarding the hydrophilic case, where the work of Mansoor et al.⁹³ is considered. The deceleration rate is primarily influenced by the forces of added mass (⁠ $F a$⁠), buoyancy (⁠ $F b$⁠), unsteady vortex shedding (⁠ $F I$⁠), and the unsteady pressure fluctuations due to the expansion and collapse of cavities (⁠ $F c$⁠). The added mass can be estimated using the equation $m a= C m ρ w V c$⁠, where $V c$ represents the combined volume of the sphere and the attached cavity, and $C m$ is the added mass coefficient. The value of $C m$ depends on the flow pattern around the sphere and the attached cavity. For spheres moving without an attached cavity, $C m$ is typically $0.5$⁠, while for a streamlined combination of sphere and cavity, a mean value of $0.672$ is assumed over the entire descent. $V c$ for a single descending sphere is generally lower than for a sphere traveling with an attached cavity. Consequently, it can be argued that $F a , SHP> F a , HPI$⁠. The total buoyancy force also depends on $V c$ (see Mansoor et al.⁹³ for pre- and post-pinch-off expressions), which proves that $F b , SHP> F b , HPI$⁠. For hydrophobic spheres, the attached cavity suppresses vortex shedding to a large extent, resulting in $F I , SHP≅0$⁠, however $F c , SHP>0$ (especially at and after pinch-off). In contrast, for a non-cavity-forming hydrophilic sphere, the kinetic energy of the shed vorticity leads to $F I , HPI≫0$⁠, while the absence of a cavity results in $F c , HPI≅0$⁠. The main conclusion is consequently that the higher deceleration observed for the hydrophilic sphere can be attributed to the effect of $F I , HPI$⁠, which describes the momentum transfer from the sphere motion to the unsteady vortex wake.

The experimental investigation conducted in this study focuses on the vertical parallel water entry of two spheres with identical geometric and material properties but differing surface wettability (hydrophilic or hydrophobic). In these tests, two different scenarios are emphasized: hydrophobic–hydrophobic (abbreviated as SHP-SHP) and hydrophobic–hydrophilic (abbreviated as SHP-HPI) side-by-side water entry. The spheres are simultaneously released from varying heights (ranging from $15$ to $95 cm$⁠), resulting in a range of impact velocities of approximately $1.71$– $4.32$m s⁻¹. The range for Bond, Froude, Weber, and Reynolds numbers listed in Table I indicates that the experimental tests were conducted within the deep and surface seal regimes. The lateral center-to-center distances between the spheres are also varied, ranging from $1.0$ to $5.0$ times the sphere's diameter. To analyze the water entry process, a high-speed photographing system is employed, and image processing techniques are utilized. The study examines various aspects, including the behavior of the air-entrainment cavity (shape, closure patterns, and air tubes), water flow features (Worthington jets and splashes), and sphere kinetics (position, velocity, acceleration, and forces). The parallel water entry of the two spheres in both wettability scenarios leads to asymmetry in the flow field surrounding the cavities, impacting various cavity parameters such as the pinching time, Worthington jet angle, the formation of air tubes between the upper and lower cavities, and the horizontal distance between the spheres during descent, among others. By decreasing the lateral center-to-center distances between the two spheres, these influences are further accentuated, revealing the intricate relationship between the spheres' positioning and the resulting water entry dynamics. This experimental study expands the understanding of water entry by investigating configurations that have not been tested thus far, in particular superhydrophobic coating in parallel water entry and pairing of superhydrophobic and hydrophilic spheres. Some specific and unique findings can be highlighted as follows:

• SHP-SHP pairing:

  • When the separation ratio is 1.0, the air tubes come into contact with one another near the lower air cavities, which are attached to the spheres. Subsequently, a third air tube forms between the already existing structures.

  • When the separation ratio is 1.0, an interesting observation is made regarding the planar jet formed by the merging of the two splash crowns. It concentrates and splits into two distinct roughly cylindrical jets, referred to as “jet-pair,” located at the boundaries of the splash crown.

  • The study presents a regression formula for the pinch-off time of the parallel water entry of two hydrophobic spheres, which is compared to experiments by a previous study and a previously proposed theoretical formulation based on the potential flow model.

• SHP-HPI pairing: Three main regimes regarding the cavity dynamics can be identified.

  • When separation ratios exceed a certain value, the influence of the neighboring spheres is negligible.

  • If the hydrophilic sphere comes closer to the hydrophobic sphere, its vortex wake disturbs the cavity behind the hydrophobic sphere, which can result in a second pinch-off event. The sphere's kinetics are hardly influenced by this disturbance and are akin to that of individual water entry events of hydrophobic or hydrophilic spheres, respectively.

  • At a separation ratio of 1.0 and high impact velocities, a smaller cavity detaches from the larger cavity behind the SHP-sphere and attaches to the HPI-sphere, moving along with it.

The authors would like to acknowledge Johannes Kepler University Linz for supporting this study. This project has received funding from the EU-Horizon 2020 Marie Skłodowska-Curie Individual Fellowship Program under the name of WE-EXPERTH (Grant No. 101022112), covering the period from 2022 to 2024. The authors would also like to express their appreciation to Professor Brenden Epps⁸⁹ for publishing his Matlab code, “smoothspline.m version 5.0,” under a GNU general public license, which was utilized to determine the best smoothing spline fit for estimating the descent velocity and acceleration of the spheres.

The authors have no conflicts to disclose.

Pooria Akbarzadeh: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Michael Krieger: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal). Dominik Hofer: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal). Maria Thumfart: Methodology (equal); Resources (equal). Philipp Gittler: Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon request. Movie files of this set of experiments will be made available on the Zenodo-platform (zenodo.org) and linked to the DOI of this article.