
==== Front
Ultrason Sonochem
Ultrason Sonochem
Ultrasonics Sonochemistry
1350-4177
1873-2828
Elsevier

S1350-4177(24)00264-5
10.1016/j.ultsonch.2024.107016
107016
Original Research Article
Interaction between cavitation bubbles and plastrons on superhydrophobic surfaces
Huang Caisheng
He Xiaolong xiaolonghescu@scu.edu.cn
⁎
Zhang Jianmin zhangjianmin@scu.edu.cn
⁎
State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China
⁎ Corresponding authors. xiaolonghescu@scu.edu.cnzhangjianmin@scu.edu.cn
08 8 2024
10 2024
08 8 2024
109 1070166 6 2024
21 7 2024
5 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The interaction between cavitation bubbles and plastrons on superhydrophobic surfaces was investigated using a low-voltage discharge device and high-speed photography techniques. The plastron adhered to the superhydrophobic surface acts as a liquid–gas interface, giving the boundary the ability to repel cavitation bubbles. The direction of bubble collapse is determined by the vector synthesis of the Bjerknes repulsive force from the plastron and the Bjerknes attractive force from the rigid wall when the bubble collapses for the first time. Various collapse behaviors were observed, including bubbles moving away from the plastron, bubbles orienting towards the plastron, and bubbles splitting into sub-bubbles in opposite directions. During the subsequent evolution of the bubbles, the expansion of the plastron led to the reversal of the downward jet or reduced the impact velocity of the jet. Seven jet patterns were identified based on the evolution of the cavitation bubble. Starting from the impact velocity of the jet, three jet patterns, namely, the jet away from the plastron (JA), the funnel-shaped jet away from the plastron (JAF), and the funnel-shaped jet away from the plastron with vortex shedding (JAFV), were found to have a weaker effect on the boundary. Three criteria for the design of plastrons on superhydrophobic surfaces were established: VP>0.25Vmax, HP>0.55Rmax, DP>1.2Rmax. Passive pulsation of the plastron in response to the cavitation bubble exhibited similar behaviors across seven jet patterns except for the JAF pattern: torus-shaped, dish-shaped, and skirt-shaped. The dimensionless wall distance, volume ratio, and plastron morphology parameters were identified as significant factors influencing the interaction between cavitation bubbles and the plastron.

Keywords

Cavitation bubble
Plastron
Superhydrophobic surface
Jet pattern
==== Body
pmc1 Introduction

Cavitation, characterized by the growth and collapse of bubbles due to a significant reduction in liquid pressure, has attracted the attention of researchers for centuries due to its enigmatic nature and potential dangers [1]. Its practical applications span a wide range of fields, from civilian engineering, such as ultrasonic cleaning [2], materials processing [3], and pharmaceutical treatments [4], to aerospace engineering [5], nuclear engineering [6], and military applications [7]. The primary physical mechanism underlying the widespread applications involves the collapse of cavitation bubbles near a rigid wall. The presence of a rigid wall impedes the radial flow of the surrounding fluid around the cavitation bubble. This results in pressure gradients that cause the cavitation bubble to collapse towards the rigid wall, generating high-pressure shockwaves, exothermic reactions, and microjets [8], [9]. It is important to note that in hydraulic and ocean engineering, the collapse of cavitation bubbles often results in damage to flow boundaries, such as spillway surfaces and propeller blades. This not only reduces the efficiency of energy utilization but also poses a threat to the safe operation of engineering structures [10], [11], [12], [13]. Therefore, many researchers and engineers have begun to explore methods to mitigate cavitation.

Different boundary forms determine the motion modes of cavitation bubbles. In contrast to microjets directed towards the boundary generated by bubbles collapsing near a rigid wall, the effect of a free surface on cavitation bubbles is quite different. Jets are repelled by the free surface and diverge away from it [14], [15], indicating that the significant energy released upon bubble collapse does not directly contact the boundary, thus reducing erosion. Zhang et al. [16] systematically examined jet morphology, jet tip velocity, and the migration of the bubble centroid under the influence of the free surface. The results indicated that the closer the initial position of the cavitation bubble is to the free surface, the faster the downward migration speed of the bubble upon collapse. In addition to studying the motion of cavitation bubbles near an open free surface, Cui et al. [17] conducted experimental research on bubble dynamics under a horizontal rigid plate with a hole. Although most free surfaces were replaced with a rigid wall, the remaining free surfaces at the hole still caused microjets to move away from the boundary. The impact of flexible boundaries on cavitation bubbles lies between that of rigid walls and free surfaces. Sieber et al. [18] generated single laser-induced cavitation bubbles near agarose hydrogels with varying elasticities to investigate the effects of boundary elasticity and the distance between the bubble and the elastic surface on bubble dynamics. Close to flexible boundaries, the sidewall of the bubble shrinks faster than the top boundary, resulting in a distinct necking phenomenon in a mushroom-shaped bubble. Eventually, the bubble collapses, generating microjets directed towards and away from the boundary, consistent with previous experimental findings [19], [20].

Due to the ability of free surfaces to keep cavitation bubbles away from boundaries, numerous studies have begun exploring the introduction of liquid–gas interfaces near rigid walls to mitigate cavitation. Smith and Mesler [21] discussed the interaction between spark cavitation bubbles and nearby air bubbles, finding that air bubbles located at rigid boundaries can prevent surface damage. Goh et al. [22] investigated the influence of air bubbles adhered to rigid walls on the jet orientation of a collapsing bubble and identified the ratio of the oscillating bubble oscillation time and the wall-attached bubble oscillation time (T′) as a crucial factor in determining the jet direction, in addition to the dimensionless stand-off distance, γ. Furthermore, Wang et al. [23] summarized five motion modes of wall-attached air bubbles under the influence of cavitation bubbles and found that when the dimensionless volume ratio V′<3, the air bubble can be considered a free surface. Meanwhile, Wang et al. [24] also considered the interaction between an air bubble adhered to a flexible structure and cavitation bubbles. Tan et al. [25] investigated the effect of an air bubble adhered to the tube nozzle on the collapsing behavior of a cavitation bubble. The results of shadowgraph experiments indicated that the shockwave generated upon collapse significantly weakened after the cavitation bubble merged with the air bubble. Sun et al. [26] and Yin et al. [27] respectively analyzed the influence of a rigid wall with a gas-entrapping hole on the collapse behavior of cavitation bubbles from experimental and numerical perspectives. As the cavitation bubble changed its migration direction, the oscillation period decreased, thereby reducing the likelihood of cavitation damage occurring on the wall. Altering surface microstructures to trap gas at the solid–liquid interface is also a common approach [28], [29], [30]. Gonzalez-Avila et al. [28] designed a biomimetic gas-trapping microtextured surface. Wei et al. [29], [30] considered the gas-trapping capability of annular groove arrays and a porous medium copper plate with hydrophobic modification, as well as their influence on the direction of cavitation bubble collapse. Experimental results consistently affirm the beneficial effect of introducing the liquid–gas interface in mitigating cavitation.

It is important to note that, while the previously mentioned methods for introducing the liquid–gas interface are novel and provide valuable references, their practical significance is limited. Under flow conditions, the attachment of air bubbles to common hydrophilic surfaces (such as concrete and metal) is random and difficult to control. The collapse of cavitation bubbles is primarily assessed in terms of their interaction with rigid walls. Moreover, the complex design of the microstructure on rigid surfaces results in a significant increase in engineering construction costs, making it feasible only under specific conditions and for limited applications. In hydraulic engineering, aeration is the most frequently used method for introducing the liquid–gas interface [31], [32], [33]. Li et al. [34], [35] described the mechanism of aeration to mitigate cavitation as the merging-redirection-extension-shielding effect of air bubbles. The greater quantity, smaller size, and more uniform distribution of air bubbles lead to a more pronounced suppression effect on the collapse of cavitation bubbles. However, many existing hydraulic structures lack aeration facilities or have inadequate aeration facilities to meet operational requirements due to their age. As a result, engineers are now considering the application of protective coatings on hydraulic concrete surfaces. One type is surface film-forming coatings [36], while the other type is penetrating film-forming coatings [37], [38]. The use of penetrating film-forming coatings imparts superhydrophobic characteristics to the surface of hydraulic concrete, effectively enhancing its resistance to chemical corrosion.

Superhydrophobic surfaces have the ability to naturally form a liquid–gas interface (hereinafter referred to as a plastron) in water and effectively capture air bubbles within the water [39], [40], [41]. However, the cavitation mechanism on superhydrophobic surfaces remains unclear. Previous studies often considered the presence of the plastron as a supplement to the initial gas nuclei for cavitation [42], [43], [44], [45], potentially leading to more severe cavitation. Limited research has examined the impact of the plastron on cavitation bubbles on superhydrophobic surfaces. Jasikova et al. [46] established the relationship between the plastron thickness on various wetting surfaces and the Reynolds number. They also explored the interaction between a single cavitation bubble and hydrophobic surfaces, revealing a mutual attraction between the plastron and the cavitation bubble. However, there is a lack of quantitative analysis on the influence of the plastron on the evolution of cavitation bubbles.

This study investigates the interaction between cavitation bubbles and the plastron on a superhydrophobic surface, using a low-voltage discharge device and high-speed photography techniques. Seven different jet patterns of cavitation bubbles are identified and analyzed. Starting from the impact velocity of collapsing jets, the influence of the volume ratio of the plastron to the cavitation bubble, as well as the morphological parameters of the plastron and the dimensionless wall distance on the jet patterns are discussed. Three criteria are proposed under weak jet impact zone. Furthermore, the coupled response of the plastron when subjected to an oscillating cavitation bubble is also investigated.

2 Experimental setup and method

2.1 Preparation and characterization of samples

Using a 5-mm-thick polymethyl methacrylate (PMMA) plate as the substrate, a fluorosilane polymer dissolved in butyl acetate at a concentration of 0.8 g/ml was sprayed onto the surface of the substrate using a spray gun. The sample was then allowed to dry at room temperature for 24 h, resulting in a superhydrophobic surface. Optical images and the three-dimensional morphology of the sample were obtained using a three-dimensional optical scanning profiler (Keyence, VR-5000), as shown in Fig. 1. The low surface energy fluorosilane polymer formed random micro-nano hierarchical structures on the PMMA plate, with an average surface roughness of 21.5 µm for the samples. As shown in Fig. 1(c), the thickness of the superhydrophobic coating ranges from 38.6 to 65.4 µm. The water contact angle (WCA) of the samples was measured at 25 ± 1 °C using an optical contact angle measurement instrument (Kruss, DSA25). A 4 µL water droplet was deposited onto the sample surface, and the WCA was determined using the Young–Laplace approximation. To ensure the reliability of the results, five different positions on the same surface were selected for measurement during the process, and the measurements were averaged. As shown in Fig. 2(a), the WCA of the PMMA plate as the substrate was 81.53 ± 1.8°, demonstrating hydrophilicity and serving as the control group in subsequent experiments as a rigid wall boundary condition. The WCA of the superhydrophobic surface was measured to be 153.0 ± 2.1°, as shown in Fig. 2(b).Fig. 1 Surface morphology of superhydrophobic sample: (a) optical image; (b) three-dimensional morphology; (c) typical cross-sectional image of the superhydrophobic specimen.

Fig. 2 Water contact angle measurements: (a) PMMA plate; (b) superhydrophobic plate.

2.2 Experimental device

A schematic diagram of the experimental setup is illustrated in Fig. 3(a). It includes a low-voltage discharge device, a high-speed photography system, and a plastron control device. The experiments were conducted in a glass water tank measuring 15 × 15 × 20 cm3, filled with deionized water up to a depth of 15 cm. The temperature was maintained at 25 ± 1 °C.Fig. 3 Schematic diagram of experimental setup: (a) experimental system; (b) schematic description of the radius of the cavitation bubble and initial plastron shape parameters; (c) definition of characteristic positions in the evolution process of the cavitation bubble and plastron; (d) actual morphology of the plastron in water.

The low-voltage discharge device utilized to generate cavitation bubbles consists of a regulated DC power supply, a 2 kΩ resistor, and two parallel 4700 μF capacitors [47]. The DC power supply converts 220 V AC electricity into the required output DC voltage to charge the capacitors. Once the capacitors are fully charged, the discharge circuit is activated, causing the stored electrical energy in the capacitors to converge at the intersection of two copper wires with a diameter of 0.15 mm. A short circuit then occurs at the discharge electrode submerged in water, instantly heating and vaporizing the surrounding water at the electrode intersection, thus generating a cavitation bubble. The initial center of the cavitation bubble always remains at the intersection of the copper wire electrodes. This allows for precise control of the initial position of the cavitation bubble through the use of a three-axis displacement platform. The experiments yielded cavitation bubbles with maximum diameters ranging from 4.29 to 6.72 mm, with the diameter of the copper wire being only 2.2 % to 3.5 % of the maximum bubble radius. The influence of the electrodes on bubble dynamics can be neglected [19], [48]. Additionally, given that the distances from the electrode intersection to the tank wall and free surface are both greater than 10 times the maximum bubble radius, the effects of the tank boundaries and free surface on the evolution of cavitation bubbles can be disregarded [49], [50].

Using a high-speed camera, the evolution process of cavitation bubbles is recorded at a resolution of 256 × 256, capturing at a rate of 13,000 frames/s. A non-strobing LED light source is positioned on the opposite side of the water tank, as shown in Fig. 3(a), providing sufficient brightness for the high-speed camera. To assist in capturing the events, frosted glass is placed between the light source and the camera to diffuse the light. Throughout the experiment, the centers of the light source, electrode intersection, and camera lens are aligned on the same horizontal line.

Superhydrophobic surfaces can naturally form a plastron [41], [51], [52], but controlling its thickness quantitatively is challenging. Moreover, the diameter of spark-generated bubbles (on a millimeter scale) tends to be larger than those produced under realistic conditions (on a micrometer scale). To address this issue, a plastron control device was developed. Air is injected at a low rate using a syringe pump through a needle attached to the backside of the superhydrophobic plate, allowing for regulation of the gas volume of the plastron. Once air is accumulated, the plastron can move freely on the superhydrophobic surface underwater, reducing the impact of substrate perforations. The underwater morphology of the plastron is depicted in Fig. 3(d). Its main component is air supplied by a syringe pump. During the experiment, the volume of the plastron is controlled between 150.78 and 200.48 µL, which is half the volume of the plastron when it detaches from the superhydrophobic surface. Within this range, the plastron exhibits good morphological similarity as the gas volume increases. After the gas supply is completed, the internal and external pressures of the plastron are in a balanced state, equivalent to the hydrostatic pressure at the boundary.

2.3 Experimental parameters

The parameters of the cavitation bubble and plastron are shown in Fig. 3(b). The maximum radius of the cavitation bubble is defined as(1) Rmax=S/π

where S represents the cross-sectional area of the cavitation bubble at its maximum volume Vmax during the first oscillation stage.

H0 represents the distance from the intersection of the electrodes to the superhydrophobic surface. DP, HP, and VP represent the horizontal spread diameter, maximum vertical height, and gas volume of the undisturbed plastron at the initial moment, respectively. Utilizing these parameters, four dimensionless quantities are defined: the dimensionless wall distance γ, the volume ratio V′, the dimensionless plastron height α, and the dimensionless plastron spread diameter β.(2) γ=H0Rmax

(3) V′=VPVmax

(4) α=HPRmax

(5) β=DPRmax

To provide a detailed illustration of the evolution process of cavitation bubbles and plastrons, Fig. 3(c) defines three characteristic distances. H1 and H2 represent the perpendicular distances from the top and bottom points of the cavitation bubble to the boundary, respectively. H3 represents the perpendicular distance from the highest point of the plastron to the boundary. The reference plane for calculating characteristic distances is the superhydrophobic surface. At the initial moment of the experiment, H3 is equal to HP.

3 Results and discussion

3.1 Collapse characteristics of cavitation bubble near rigid wall

Cavitation bubbles collapsing near a rigid wall generate microjets directed towards the boundary (termed as JT pattern), accompanied by the release of intense shock wave causing material damage. Previous studies [8], [53] have shown that when γ>1.0, the damage to the material exhibits a distinct annular distribution, closely associated with the secondary collapse of cavitation bubbles at the boundary. Fig. 4 depicts the evolution process of a cavitation bubble after its first oscillation near a rigid wall. The first row of the image sequence shows the second oscillation of the bubble at the boundary, while the second row illustrates the vortex evolution that forms after the secondary collapse of the cavitation bubble. As shown in Fig. 4(a) and movie S1, the bubble collapsed for the first time at γ=1.47 and Rmax=5.51mm, generating a microjet directed towards the boundary. However, the jet did not immediately impact the rigid wall but continued to move downwards until t=1.846ms. At t=2.077ms, the bubble reached its maximum volume within the second oscillation stage, primarily contracting downwards. Around t=2.462ms, a ring-shaped bubble forms, with the inner side of the ring developing faster than the outer side, generating a vortex ring containing dispersed small cavitation bubbles. This vortex ring, known as a wall vortex, develops tangentially along the wall. Fig. 5 shows the evolution of the horizontal spreading diameter dwall of the ring bubble. At 1 ms after the second collapse, the diameter of the vortex ring has reached 13.12 mm, nearly twice the horizontal spreading diameter at the time of the cavitation bubble's second collapse.Fig. 4 Two typical evolution patterns of the cavitation bubble collapsing near a rigid wall: (a) wall vortex (Rmax=5.51mm,γ=1.47); (b) free vortex (Rmax=6.02mm,γ=1.19). The first row of image sequences corresponds to the second oscillation of the bubble, while the second row depicts the evolution of the toroidal bubble. Time in the upper left corner begins from the moment of cavitation bubble excitation.

Fig. 5 The horizontal spreading diameter evolution of annular bubble after the secondary collapse of cavitation bubbles. The initial moment corresponds to the time of the secondary collapse of cavitation bubbles, with wall vortex conditions at t=2.462ms (Fig. 4a) and free vortex conditions at t=2.769ms (Fig. 4b).

When the dimensionless wall distance γ<1.3, the vortex formed after the secondary collapse of the cavitation bubble will exhibit another typical pattern [54], [55], as shown in Fig. 4(b) and movie S2. This is referred to as the free vortex (γ=1.19, Rmax=6.02mm). The microjet generated during the first collapse directly impacts the rigid wall. Subsequently, during the bubble shrinkage stage, an inward flow parallel to the rigid wall becomes dominant. At t=2.846ms, the collapse of the annular bubble begins at the outer periphery, gradually detaching from the substrate. This creates a circular vortex perpendicular to the substrate, which enters the liquid. During the vortex ring detachment process, the spreading diameter dfree of the remaining bubble on the rigid wall remains almost unchanged, as shown in Fig. 5. Considering the damage pattern of the material [8], [53], the erosion caused by the collapse of cavitation bubbles on the rigid wall primarily results from the impact of microjets on the boundary during the first collapse and the pressure generated during the collapse when the annular bubble comes into direct contact with the rigid boundary during the second oscillation stage. Reducing the impact of both factors on the boundary will help mitigate cavitation damage.

3.2 Interaction between cavitation bubble and plastron under different patterns

Previous studies [56], [57], [58], [59] have shown that a rigid wall exerts a Bjerknes attractive force on a cavitation bubble, while the liquid–gas interface repels the bubble, causing it to move away. When the centroid of the bubble coincides with the centroid of the plastron spread on the superhydrophobic surface, both located perpendicular to the rigid wall, as shown in Fig. 3(b), the direction of the collapsing bubble is primarily determined by the vector synthesis of these two Bjerknes forces. During the first collapse of the cavitation bubble, three distinct types of motion perpendicular to the rigid wall were observed in experiments: movement away, towards, and bidirectional. Simultaneously, changes in the dimensionless wall distance γ resulted in different evolution patterns for these three jet directions. For cavitation bubbles moving away from the boundary, as γ decreases, the jet patterns observed were categorized as follows: jet away from the plastron (JA), and funnel-shaped jet away from the plastron (JAF). In the case of cavitation bubbles moving towards the boundary, only one jet pattern was observed under the experimental conditions of this study: jet towards the plastron followed by a reversal (JTR). For cavitation bubbles moving bidirectionally, as γ decreases, the jet patterns observed were summarized as follows: jet towards and away from the plastron followed by downward jet reversal (JTRA), jet towards and away from the plastron with a predominance of jets moving away (JAT), jet towards and away from the plastron with a predominance of jets moving towards (JTA), and funnel-shaped jet moving away from the plastron accompanied by vortex shedding (JAFV). Further details regarding the main jet patterns utilized in this study are outlined in Table 1. The distinctive characteristics of cavitation bubbles and the plastron under the various patterns will be elaborated on and visually represented in the subsequent sections.Table 1 Detailed information regarding the main jet patterns.

Category	Jet pattern	Rmax(mm)	Vmax(µL)	γ	V'	α	β	
Away	JA	4.76	179.49	1.85	0.40	0.7	1.33	
JAF	5.49	189.46	1.14	0.27	0.55	1.25	
Bidirectional	JTRA	6.08	189.15	1.94	0.20	0.52	1.09	
JAT	5.88	160.36	1.73	0.19	0.47	1.12	
JTA	6.40	186.38	1.27	0.17	0.5	1.30	
JAFV	5.13	186.58	1.47	0.33	0.61	1.31	
Towards	JTR	6.06	154.97	1.85	0.17	0.46	1.08	

3.2.1 Bubble developing away from the boundary upon the first collapse

(a) JA: jet away from the plastron

Fig. 6 and movie S3 illustrates the typical evolution process of a cavitation bubble and plastron under the JA pattern with γ=1.85 and V′=0.40. The protrusion in the first image represents the undisturbed plastron. During the expansion phase of the first oscillation stage, the upper half of the bubble remained spherical, while the lower half expanded in a conical shape until t=0.462ms, resulting in an overall balloon shape. As depicted in Fig. 7(a), the lowest point of the bubble is also closest to the plastron at t=0.462ms. At t=0.539ms, when the bubble reaches its maximum volume, the bottom surface starts to flatten, becoming increasingly apparent during the bubble collapse process. This results in the formation of a liquid jet with a width approximately equal to the diameter of the bubble, as shown in Fig. 6. During the collapse process of the first oscillation stage, no liquid jet penetrating from the top of the bubble was observed. The bubble exhibited a bowl shape when it reached its minimum volume at t=1.000ms. However, in the subsequent second oscillation stage, a secondary jet impinged on the top of the rebounding bubble, causing it to deform into a toroidal shape while creating a protrusion at the top of the bubble, termed as a “jet torus”. At t=1.539ms, the bubble reached its minimum volume during the second oscillation, after which the toroidal bubble is separated by the jet torus and migrates upwards, with the flow direction of the vortex ring as shown in Fig. 6. Fig. 7(a) illustrates the temporal evolution of characteristic distances and the centroid of the bubble. As discussed earlier, during the first oscillation stage, the development of the upper and lower surfaces of the bubble was not synchronous due to the influence of the plastron. Following the onset of the shrinkage phase in the first oscillation stage, the speed at which the bubble centroid moved away from the boundary significantly increased, consistently progressing in the direction away from the boundary.Fig. 6 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JA pattern (Rmax=4.76mm,γ=1.85,V′=0.40). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 Results under the JA pattern (Rmax=4.76mm,γ=1.85,V′=0.40): (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

On the other hand, under the influence of cavitation bubbles evolving in the JA pattern, three simple and intriguing plastron morphologies were observed: torus-shaped, dish-shaped, and skirt-shaped, as shown in Fig. 6. As the bubble expands gradually in a balloon-like manner, the horizontal spread diameter and maximum vertical height of the plastron continuously decrease. At the same time, a downward liquid jet appear in the center of the plastron [60]. At t=0.462ms, when the plastron reaches its minimum volume, it takes the form of a torus-shaped bubble. In fact, the change in the plastron volume is closely related to the internal and external pressures. Previous studies [61], [62] have shown that during the expansion of the cavitation bubble, the fluid pressure near the rigid boundary first increases and then decreases. The expansion of the cavitation bubble compresses the volume of the plastron, causing the internal pressure to rise while the pressure of the external fluid gradually decreases. The moment of pressure balance between the inside and outside of the plastron occurs when the plastron volume is compressed to its minimum (t=0.462ms). Subsequently, when the internal pressure of the plastron exceeds the external fluid pressure, it begins to expand, driving the fluid at the top to flow upwards. This is also the reason for the flattened bottom of the bubble at t=0.539ms. It is worth noting that during the expansion of the cavitation bubble, the pressure peaks of the fluid near the boundaries are located just on either side of the plastron, which limits the horizontal expansion of the plastron during its volume rebound process. However, due to the slip characteristics of gases on superhydrophobic surfaces [63], which significantly reduce the apparent friction factor, the edge of the plastron can easily expand outward, ultimately leading to step-like surface instability at the edge of the plastron. During the subsequent expansion of the plastron, the surface instability at the edge gradually diminishes. At t=1.000ms, slight curvature is observed only at locations on either side of the plastron, with the plastron resembling a dish, and its volume reaches its maximum.

During the second oscillation of the cavitation bubble, the volume of the plastron enters another shrinkage phase. Unlike the first shrinkage process, the upper surface of the plastron contracts downward in a ring-shaped manner under the effect of the shock waves released by the cavitation bubble collapse [64], [65]. At t=1.615ms, when the toroidal bubble is punctured by the jet torus, the volume of the plastron is at its minimum, appearing as a skirt. Fig. 7(a) illustrates the evolution process of the highest point of the plastron. It can be observed that during the first oscillation of the bubble, the movement of the highest point of the plastron is synchronous with the lowest point of the bubble, as the plastron is able to freely move on the superhydrophobic surface. However, in the second oscillation stage, the bubble gradually moves away from the boundary, resulting in a delayed impact of the bubble on the plastron. It is worth noting that although a downward liquid jet appears within the plastron during the expansion stage of the cavitation bubble, no significant damage was observed on the superhydrophobic surface after the experiment. Under the JA pattern, the superhydrophobic surface was effectively protected.

From the perspective of vector synthesis [66], [67], [68], the JA pattern is primarily attributed to the plastron’s greater repulsion towards the bubble compared to the attraction exerted by the rigid wall, as illustrated in Fig. 7(b). As the volume ratio V′ increases, the plastron more closely resembles the free surface, thereby exerting a stronger repulsive force on cavitation bubbles. Moreover, as the dimensionless wall distance γ decreases, the attraction and repulsion of the wall and plastron on cavitation bubbles become more pronounced.

(b) JAF: funnel-shaped jet away from the plastron

When γ decreases, bubbles still form far from the boundary during the collapse process. However, the jetting behavior of the bubble and its coupled response with the plastron are significantly different from the JA pattern. Fig. 8 and movie S4 illustrates another pattern of bubble movement away from the boundary, referred to as the JAF pattern, for γ=1.14 and V′=0.27. In this pattern, the jet takes on a funnel shape away from the plastron. During the growth phase of the first oscillation stage, a balloon-like expansion similar to the JA pattern is observed. At t=0.385ms, the plastron is compressed into a torus shape under the downward liquid jet, reaching its minimum volume. At this moment, the lowest point of the cavitation bubble nearly touches the superhydrophobic surface, as shown in Fig. 9(a). However, from the subsequent independent development of the bubble and the plastron, there still exists a thin layer of liquid between them. Notably, decreasing γ enhances the interaction between the bubble and the plastron. In the JA pattern, approximately 86 % of the time it takes for the bubble to reach its maximum radius coincides with the plastron volume shrinking to its minimum. In contrast, in the JAF pattern, the plastron volume reaches its minimum in only about 42 % of the time it takes for the bubble to expand to its maximum radius. This indicates that as γ decreases, the cavitation bubble affects the pressure balance inside and outside the plastron earlier. Compared to the JA pattern, the rebound of the plastron volume is advanced under the JAF pattern. As shown in Fig. 8, at t=0.539ms, a distinct stepped surface instability is observed at the edge of the plastron. During the subsequent expansion of the bubble, its bottom flattens, and a fine jet with a diameter of 0.74 mm begins to develop upward from the bottom of the bubble at t=0.385ms, reaching the top of the bubble by t=0.923ms. During the subsequent shrinkage of the bubble, the thin jet penetrates the top surface, forming a jet torus. Meanwhile, the upward thin jet transforms the bubble into a funnel shape, accelerating the liquid between the bubble and plastron, resulting in a sharp conical shape of the plastron. By t=1.539ms, the highest point of the plastron nearly contacts the bottom of the bubble, leading to the first collapse of the bubble and violent shrinkage of the plastron surface towards the superhydrophobic surface due to inertia. By t=2.000ms, the plastron volume reaches its minimum. Unlike the skirt shape observed in the JA pattern, the disturbance caused by intense surface shrinkage in the JAF pattern results in a multi-layered cauliflower shape of the plastron. The subsequent plastron oscillation displays a more chaotic state, with some microbubbles detaching and forming a bubble cloud due to the distortion of the gas–liquid interface.Fig. 8 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JAF pattern (Rmax=5.49mm,γ=1.14,V′=0.27). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9 Results under the JAF pattern (Rmax=5.49mm,γ=1.14,V′=0.27), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

As shown in Fig. 9(a), after the first collapse, the bubble undergoes a brief expansion process. The lowest point of the bubble migrates towards the boundary and approaches the plastron at t=1.769ms. Subsequently, the bubble moves away from the superhydrophobic surface. Fig. 9(a) also illustrates the temporal evolution of the bubble's centroid. Compared to the JA pattern, in the JAF pattern, both the Bjerknes attraction and Bjerknes repulsion forces increase due to the decrease in γ, while the reduction in volume ratio V′ decreases the Bjerknes repulsion force. Assuming that the increase in attraction and repulsion forces due to the decrease in γ is the same, the jet in the JAF pattern should be weaker. However, the JAF pattern exhibits a stronger jet, which indicates that the decrease in γ has a more significant effect on the increase of Bjerknes repulsion, ultimately resulting in a funnel-shaped jet (Fig. 9(b)). In summary, under both JA and JAF patterns, the migration and collapse of the cavitation bubble occur in regions outside the boundary, preventing direct contact between the bubble and the boundary and effectively protecting the material surfaces.

3.2.2 Bubble splitting into sub-bubbles in opposite directions upon the first collapse

(a) JTRA: jet towards and away from the plastron followed by downward jet reversal.

Fig. 10 and movie S5 illustrates a typical evolutionary pattern of collapsing jets when the Bjerknes attractive force from the boundary slightly exceeds the Bjerknes repulsive force from the plastron. During the second oscillation stage, the jet directed towards the boundary experiences a reversal in direction and migrates away from the plastron. This jet pattern is referred to as the JTRA pattern. Fig. 10 illustrates the detailed motion process of a cavitation bubble and plastron under the JTRA pattern (γ=1.94, V′=0.20). Within the first oscillation stage of the bubble, compared to the JA pattern, the effect of the plastron’s pulsation on the cavitation bubble has been weakened with the decrease of the volume ratio V′ under the JTRA pattern. The balloon shape mentioned before does not occur. Even at t=0.692ms, when the pressure inside the plastron is higher than the external fluid pressure, and the volume begins to rebound, the cavitation bubble continues to grow spherically. The bottom of the cavitation bubble just becomes slightly flattened at the end of the growth stage due to the expansion of the plastron. At t=1.615ms, the plastron expands to its maximum volume, while the bubble is still in the radial shrink stage of the first oscillation stage. As shown in Fig. 11(a), before the bubble reaches its minimum volume at t=1.769ms within the first oscillation stage, the maximum height of the plastron hardly changes. The shrinkage of the plastron mainly involves a reduction in the horizontal spread diameter, resulting in the bubble assuming a peapod shape. Shrinkage velocities on both sides of the bubble exceed those at the top and bottom. Annular flow towards the center of the bubble results in bubble splitting and the formation of two jets in opposite directions perpendicular to the equatorial plane of the bubble [69].Fig. 10 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JTRA pattern (Rmax=6.08mm,γ=1.94,V′=0.20). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11 Results under the JTRA pattern (Rmax=6.08mm,γ=1.94,V′=0.20), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

Examining the development of subsequent jets, it is apparent that the jet generated from the lower part of the bubble is stronger than the jet in the opposite direction. This indicates that there is a greater attraction at the boundary than there is repulsion from the plastron [69]. Generally, the jet moving towards the boundary will ultimately collide with the plastron at a certain speed. However, in the JTRA pattern, it is interesting to observe that the jet towards the plastron experiences a reversal before contacting the plastron, as shown in Fig. 10. At t=1.8462.154ms, the lower bubble experiences a second oscillation. The volume of the bubble reaches its maximum at t=2.000ms, while the downward jet approaches its nearest position to the plastron (Fig. 11(a)). At this moment, the plastron is once again compressed to its minimum, taking on a skirt-like shape. As the bubble shrinks to its smallest volume, the plastron continues to expand, accelerating the upward flow of liquid between the plastron and the bubble. As shown in Fig. 11(a), the lower bubble, upon reaching its maximum volume, demonstrates a continuous shift of its centroid towards a direction away from the boundary. The reversal of the jet’s direction is primarily attributed to the expansion of the plastron during the bubble shrinkage stage. The phase difference Δθ between the cavitation bubble and the plastron during the second oscillation period of the cavitation bubble as follows:(6) Δθ=2πΔtTosc,2

where Δt represents the time difference when the cavitation bubble and the plastron reach their minimum volumes during the second cycle, and Tosc,2 is the second oscillation period of the cavitation bubble. In the JTRA and JTR patterns, the phase difference between the cavitation bubble and the plastron is 0.8π. Similar to previous studies [70], [71], when bubbles oscillate out of phase, meaning their expansion and collapse stages do not coincide, the collapse of the bubbles will form jets directed away from each other. It is worth noting that at t=2.385ms, the presence of thin jets within the plastron is noticeable. This indicates that during the first oscillation, the spherical expansion of the cavitation bubble still caused downward liquid jets within the plastron [22], [23], [67]. At t=0.539ms, the plastron appeared torus-shaped.

(b) JAT: jet towards and away from the plastron, predominantly jet away.

Fig. 12 and movie S6 depicts the evolution of a cavitation bubble with γ=1.73 and V′=0.19, which also serves as a typical case of the JAT pattern. Following the first collapse, jets emerge towards and away from the plastron, with the latter being more prominent. Within the JAT pattern, the first oscillation of the bubble resembles that of the JTRA pattern. During the growth stage, the bubble maintains a predominantly spherical shape. Due to the expansion of the plastron, the bottom of the bubble flattens at t=0.385ms, a phenomenon that becomes more pronounced during the shrinkage stage. The plastron reaches its maximum volume prior to the collapse of the bubble (t=1.462ms). As the bubble contracts radially, the plastron primarily contracts horizontally, creating high-pressure zones on either side of the bubble at t=1.615ms, with the inward annular flow generating two opposing jets [69].Fig. 12 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JAT pattern (Rmax=5.88mm,γ=1.73,V′=0.19). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The main difference between the JAT and JTRA patterns is most evident in the evolution of the jet after the first collapse. As shown in Fig. 12, first, the jet moving away from the plastron dominates in the two opposing jets; second, the jet moving towards the plastron does not undergo reversal but eventually merges with the plastron. At t=1.615ms, the annular flow generates jets in opposite directions, but the bubble does not split into two sub-bubbles. Instead, necking occurs, completing the division during the shrinkage stage of the second oscillation, while the plastron volume also reaches a minimum (t=1.923ms), taking on a skirt shape. In terms of subsequent jet development, as shown in Fig. 13(a), the lower bubble does not undergo a significant volume oscillation process, suggesting that the downward jet moves towards the plastron only under the influence of inertia. Meanwhile, the rebound of the plastron propels the liquid to flow upwards, reducing the speed of the downward jet. When the lower bubble reaches its lowest point and contacts the plastron at t=2.539ms, its velocity is only 2.94 m/s, close to the expansion velocity of the plastron at the top, which is 3.92 m/s. After the plastron merges with the lower bubble completely, the top of the plastron continues to move upward until t=2.769ms when the plastron volume reaches its maximum. This indicates that under the JAT pattern, although the downward jet is directed towards the plastron and merges with it eventually, its impact on the boundary is negligible. The impact from the lower bubble is consumed through the oscillation of the plastron, offering effective protection for the superhydrophobic surface.Fig. 13 Results under the JAT pattern (Rmax=5.88mm,γ=1.73,V′=0.19), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

Fig. 13(b) shows a schematic diagram illustrating the vector synthesis for the cavitation bubble, plastron, and rigid wall. Unlike the JTRA pattern, where the boundary attracts the bubble slightly more than the plastron repels it, the repulsive effect of the plastron on the bubble strengthens as γ decreases. In the JAT pattern, the Bjerknes repulsive force of the plastron on the bubble is slightly greater than the Bjerknes attractive force of the boundary on the bubble. Consequently, when the bubbles collapse to form jets in the opposite direction, the jet away from the boundary dominates.

(c) JTA: jet towards and away from the plastron, predominantly jet towards.

With a decrease in γ, the repulsion of the plastron on the cavitation bubble becomes more pronounced during the first oscillation stage. Fig. 14 and movie S7 illustrates the typical evolution process of the bubble under the JTA pattern when γ=1.27 and V′=0.17. During the first oscillation stage, the growth of the bubble is almost consistent with the JAF pattern, maintaining a balloon shape until t=0.385ms when the volume of the plastron reaches a minimum. Subsequently, the plastron begins to expand, causing the bottom of the bubble to flatten and generating a thin jet from the bottom to the top inside the bubble. It is worth noting that when the bubble expands to its maximum volume at t=1.077ms, the development of the thin jets differs from the JAF pattern. Instead of affecting the top surface of the expanding bubble, the jets penetrate the bubble’s surface to form a jet torus during the shrinkage process of the bubble (t=1.385ms). During the development of the thin jet, there was no distinct conical shape observed in the plastron. Only minor oscillations were noted on the top surface of the plastron, suggesting that the thin jet generated under the JTA pattern was much weaker compared to the JAF pattern. Similar to the JTRA and JAT patterns, the volume of the plastron reached its maximum earlier than the moment of bubble collapse at t=1.769ms. The horizontal shrinkage of the plastron primarily caused annular flow on the equatorial plane of the bubble (t=1.846ms). As the bubble collapses, jets form directed towards and away from the plastron.Fig. 14 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JTA pattern (Rmax=6.40mm,γ=1.27,V′=0.17). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

After a brief oscillation, the bubble splits into two sub-bubbles at t=2.154ms. Similar to the JAT pattern, there is no significant volume oscillation as the lower bubble migrates towards the plastron. Prior to contact with the expanding plastron (t=2.231ms), the jet velocity of the lower bubble is 7.91 m/s, slightly higher than that of the JAT pattern but still significantly lower than the jet velocity generated when bubbles collapse near a rigid wall. The evolution after the merge of the lower bubble with the plastron (t=2.231-3.231ms) are also illustrated in Fig. 14. The lower bubble fully merges with the plastron at t=2.615ms, halting its expansion in the vertical direction. Expansion predominantly occurs horizontally until the plastron reaches its maximum volume at t=2.923ms. Subsequently, the shrinkage of the plastron is mainly observed in the horizontal direction. This indicates that the impact of the downward jet suppresses the vertical development of the plastron. Under the JTA pattern, the bubble has a greater impact on the plastron compared to the JAT pattern.

Fig. 15(b) illustrates the vector synthesis of a cavitation bubble, plastron, and rigid wall under the JTA pattern. As γ decreases, the repulsive effect of the plastron on the bubble increases. As previously mentioned, the magnitude of the Bjerknes repulsive force increases by a greater amount than the Bjerknes attractive force of the wall on the bubble. However, when compared to the JAT pattern (Fig. 12), the volume ratio V′ also decreases, which weakens the Bjerknes repulsive force of the plastron on the cavitation bubble. This, in turn, results in a slightly greater Bjerknes attractive force of the wall compared to the Bjerknes repulsive force of the plastron on the bubble. Upon collapse, the bubble splits into two jets in opposite directions, with the jet towards the plastron being dominant.Fig. 15 Results under the JTA pattern (Rmax=6.40mm,γ=1.27,V′=0.17), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

(d) JAFV: funnel-shaped jet away from the plastron accompanied by vortex shedding.

In addition to the three bidirectional jet patterns discussed above, another interesting jet pattern was observed during the experimental process, which resembles a weakened version of the JTA pattern. After the first collapse of the bubble, a funnel-shaped jet forms away from the plastron, accompanied by vortex shedding towards the plastron. This jet behavior is referred to as the JAFV pattern. Fig. 16 and movie S8 illustrates the typical evolution process of the JAFV pattern (γ=1.47,V′=0.33). During the first oscillation stage of the bubble, the JAFV pattern appears almost identical to the JTA pattern, with premature shrinkage of the plastron leading to the formation of annular flow around the bubble (t=1.539ms). It is worth noting that when the plastron reaches its maximum volume (t=1.462ms), pulsating phenomena observed in the JTA pattern are absent on the top surface of the plastron. This indicates that during the shrinkage process of the bubble, the interaction between the bubble and the plastron is weaker in the JAFV pattern, causing annular flow to occur below the equatorial plane of the bubble. Subsequent to the first collapse of the bubble at t=1.615ms, a vortex ring detaches from the bottom of the bubble and moves towards the plastron. In contrast to the JTA pattern, the volume of the bubble is mainly concentrated in the upper half, away from the plastron in the JAFV pattern. During the migration of the vortex ring towards the plastron, similar to the JTA pattern, significant oscillations in the volume of the vortex ring are not observed. As shown in Fig. 17(a), the migration speed of the lower bubble slows as the plastron expands, and at t=2.615ms, when the lower bubble contacts the plastron, its impact velocity is only 2.74 m/s. The smaller volume and lower velocity of the lower bubble causes it to have less momentum than the JAT pattern, and the plastron remains in the expansion stage even after completely merging with the lower bubble. It can be said that the impact of the downward jet on the plastron in the JAFV pattern is less significant than in the JAT pattern.Fig. 16 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JAFV pattern (Rmax=5.13mm,γ=1.47,V′=0.33). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 17 Results under the JAFV pattern (Rmax=5.13mm,γ=1.47,V′=0.33), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

In terms of vector synthesis (Fig. 17(b)), the JAFV pattern exhibits a notable increase in the volume ratio of the plastron to bubble compared to the JTA pattern. The plastron exerts a stronger repulsive force on the bubble, causing the annular flow plane to shift towards the plastron’s direction upon bubble collapse. The volume of the lower bubble decreases, ultimately detaching in the form of a vortex ring.

3.2.3 Bubble developing towards the boundary upon the first collapse

At larger dimensionless wall distances γ and smaller volume ratios V′, a jet towards the plastron is generated upon the first collapse of the bubble. However, during subsequent evolution, the direction of the jet reverses, and the bubble moves away from the plastron. This jet behavior is referred to as the JTR pattern. Fig. 18 and movie S9 illustrates a typical case of the JTR pattern (γ=1.85,V′=0.17). Unlike the six jet patterns mentioned earlier, the first oscillation of the bubble under the JTR pattern is hardly influenced by the oscillation of the plastron. Even after the plastron starts rebounding at t=0.462ms, the bubble remains spherical. At t=1.846ms, the bubble collapses, followed by a jet towards the plastron, and the plastron enters a new shrinkage phase. Similar to the JTRA pattern, during the second cycle of the cavitation bubble, the plastron and the cavitation bubble oscillate out of phase (phase difference Δθ=0.8π), causing the jet towards the plastron to reverse. When the bubble volume reaches its maximum during the second oscillation, the plastron volume is compressed to a minimum (t=2.077ms). The expansion of the plastron propels the fluid upward, and during the shrinkage phase, the centroid of the bubble gradually moves away from the plastron (Fig. 19(a)). It is worth noting that at t=2.154ms, it appears that the plastron merges with the jet. However, this is merely a trace of small bubbles generated during the shrinkage of the bubble. Fig. 19(a) illustrates the motion of the lowest point of the bubble and the highest point of the plastron, showing no apparent correlation in the first oscillation stage. The larger γ and smaller V′ weaken the repulsive effect of the plastron on the cavitation bubble, emphasizing the attraction of the wall to the bubble. The result of vector synthesis leads to the formation of a jet towards the plastron when the bubble collapses for the first time (Fig. 19(b)).Fig. 18 Typical evolution process of the cavitation bubble near the plastron on a superhydrophobic surface under the JTR pattern (Rmax=6.06mm,γ=1.85,V′=0.17). The contour of the plastron volume at its extremum is marked with a solid red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 19 Results under the JTR pattern (Rmax=6.06mm,γ=1.85,V′=0.17), (a) temporal evolution of characteristic distances H1, H2, H3, and the bubble centroid; (b) vector synthesis schematic of cavitation bubble, plastron, and rigid wall. The magnitude of the forces is qualitatively reflected in the length of the arrows, according to the jet morphology at the moment of collapse.

3.3 Criterion for plastron design

Fig. 20 compared the impact of the presence or absence of the plastron on the dimensionless bubble oscillation period. The dimensionless oscillation period of the bubble is defined as:(7) tosc∗=Tosc,11.83Rmaxρ/(P∞-Pv)

where Tosc,1 denotes the first oscillation period of the cavitation bubble, P∞=95kPa is the ambient pressure, Pv=3.17kPa is the vapor pressure of water at 25 °C, and ρ is the density of water. As shown in Fig. 20, the rigid wall extends the collapse time of the bubble, while the free surface shortens the oscillation period of the bubble. The current experimental results are generally consistent with those reported in the literature [16], [72]. Notably, among all jet patterns, the presence of the plastron reduces the contraction speed of the cavitation bubble, correspondingly decreasing the intensity of bubble collapse and reducing the likelihood of damage to the solid wall.Fig. 20 Evolutions of the non-dimensional bubble period with γ. The results of Zhang et al. [16], Yang et al. [72]. are included for comparison.

On the other hand, as described in Section 3.1, the damage caused by cavitation bubbles to a rigid wall primarily stems from the impact of the jet during the first collapse and the pressure released upon the collapse of the annular bubble. The presence of the plastron prevents the annular collapse of the bubble on a superhydrophobic surface across all seven jet patterns. To assess the impact of the downward jet on the boundary, Fig. 21 illustrates the relationship between jet impact velocity and γ, focusing on the jet pattern with a downward jet and comparing the impact velocity when bubbles collapse near the rigid wall (JT pattern). v is defined as the time-averaged velocity from time t1 to time t2. t2 refers to the frame immediately prior to the jet’s contact with the plastron or the rigid wall, while t1 corresponds to the frame before t2. Ljet represents the vertical distance traveled by the lowest point of the lower bubble between t1 and t2. It is worth noting that there is a possibility of contact between the jet and the plastron under the JTRA and JTR patterns when the lower bubble reaches its maximum volume. The velocity of the bubble at its closest position to the boundary is similar to that in the JT pattern, but the jet direction then reverses, moving away from the boundary. As shown in Fig. 21, in the JT pattern, γ decreases from 2.13 to 1.01, and the impact velocity of the bubble increases from 3.92 m/s to 31.36 m/s. However, for the JAT, JTA, and JAFV jet patterns where the lower bubble merges with the plastron, the first collapse of the cavitation bubble is primarily characterized by radial contraction on both sides of the bubble, which results in lower initial jet velocities towards the boundary compared to the JT pattern. At the same time, the plastron is in its expansion phase, pushing the upper fluid upwards and intensifying the fluid's hindrance to the jet towards the plastron. Consequently, the impact velocities in the JAT, JTA, and JAFV patterns remain relatively low. As γ decreases from 1.78 to 1.16, the impact velocity of the jet does not increase significantly, rising only from 3.77 m/s to 7.91 m/s. This is mainly because as the water layer between the jet and the plastron becomes thinner, the buoyancy effect of the plastron on the jet becomes more pronounced. Notably, the impact velocity is highest in the JTA pattern, similar between the JAT and JAFV patterns, and in the JAFV pattern, the lower bubble takes the form of a small vortex ring, resulting in a weaker impact on the boundary.Fig. 21 Impact velocity of the downward jet as a function of γ.

The influence of dimensionless wall distance γ and the volume ratio V′ on the jet patterns is illustrated in Fig. 22. Based on the analysis of jet patterns in Section 3.2 and the distribution of the jet impact velocities in Fig. 21, five regions are identified based on the varying effect on the boundary, ranging from weak to strong. First, when γ>1.51 and V′>0.25, the evolution of bubbles is predominantly in the JA pattern, with the jet moving away from the plastron, exerting minimal influence on the boundary. The jet morphology under JAF and JAFV patterns is similar and falls within the same region (γ<1.51,V′>0.25). The main body of the bubble forms a funnel-shaped jet developing away from the plastron. The plastron undergoes conical shrinkage under the influence of the funnel-shaped bubble (JAF pattern) or merging with a low-speed vortex ring (JAFV pattern). The plastron experiences slightly greater influence than in the JA pattern. As shown in Fig. 22, two regions are delineated within the range of γ>1.51 and V′<0.25. For larger values of γ and smaller values of V′, the evolution of the cavitation bubble mainly occurs in two patterns: JTR and JTRA. The jet towards the plastron undergoes a reversal in direction during the second oscillation. It is noteworthy that in certain events, a downward jet was observed to impact the plastron at high velocities before the direction reversal occurred. In comparison, the impact of the downward jet on the boundary in JTR and JTRA patterns remains lower than that in the JAT pattern, where the jet impacts the plastron and eventually merges with it. When γ<1.51 and V′<0.25, under the JTA pattern, the jet towards the boundary does not reverse but impacts the plastron at a higher velocity with the potential of causing boundary damage.Fig. 22 Jet patterns for the cavitation bubble near the plastron on a superhydrophobic surface in terms of the dimensionless wall distance γ and the volume ratio V′.

In general, when the volume ratio V′>0.25, the plastron exerts a Bjerknes repulsive force on bubbles that is far greater than the Bjerknes attractive force from the wall. Consequently, bubbles tend to move away from the rigid wall. Even under the JAFV pattern, the influence of the downward vortex ring on the plastron is limited. Therefore, the range of V′>0.25 (equivalent to VP>0.25Vmax) is referred to as the weak jet impact zone. As mentioned earlier, the presence of the plastron effectively reduces the likelihood of cavitation damage occurring on the wall. By increasing the gas volume of the plastron, the impact of the collapsing jet on the boundary can be further weakened. VP>0.25Vmax serves as one of the criteria for the design of the plastron on a superhydrophobic surface.

Furthermore, an analysis was conducted on the morphological parameters of the plastron (such as dimensionless plastron height α and dimensionless plastron spread diameter β) and their influence on the distribution of jet patterns. As shown in Fig. 23(a), the weak jet impact zone appears when α>0.55. This leads to the establishment of the second criterion for plastron design: HP>0.55Rmax, indicating that a higher maximum vertical height of the undisturbed plastron reduces the likelihood of boundary failure. On the other hand, as depicted in Fig. 23(b), the weak jet impact zone primarily occurs at β>1.2. Therefore, considering the horizontal spread diameter of the plastron, the third criterion for plastron design is established: DP>1.2Rmax. A larger horizontal spread diameter of the plastron enhances the repulsive effect on bubbles, thus minimizing the impact on the boundary during bubble collapse.Fig. 23 Jet patterns for the cavitation bubble near the plastron on a superhydrophobic surface in terms of the dimensionless wall distance γ and the morphological parameters of the plastron: (a) dimensionless plastron height α, (b) dimensionless plastron spread diameter β.

4 Conclusions

On a superhydrophobic surface, the plastron acts as the liquid–gas interface, influencing the flow field and pressure distribution during the evolution of a cavitation bubble. This study explores the interaction between the cavitation bubble and the plastron by varying the size of the cavitation bubble, the volume of the plastron, and the dimensionless distances between the bubble and the plastron as well as between the bubble and the wall (1.04<γ<2.43,0.13<V′<0.59). The following conclusions are drawn from this study:(1) When a cavitation bubble collapses for the first time near the plastron on a superhydrophobic surface, the bubble exhibits three types of motion: moving away from the plastron, splitting into sub-bubbles in opposite directions, and moving towards the plastron. Seven interesting jet patterns are defined based on the subsequent evolution of the jets: jet away from the plastron (JA); funnel-shaped jet away from the plastron (JAF); jet towards and away from the plastron, followed by downward jet reversal (JTRA); jet towards and away from the plastron, predominantly jet away (JAT); jet towards and away from the plastron, predominantly jet towards (JTA); funnel-shaped jet away from the plastron accompanied by vortex shedding (JAFV); and jet towards the plastron followed by reversal (JTR).

(2) The direction of motion of the bubble during its first collapse is determined by the vector synthesis of Bjerknes forces. When the Bjerknes repulsive force from the plastron on the bubble exceeds significantly the Bjerknes attractive force from the wall, the bubble primarily displays JA and JAF patterns. If the Bjerknes repulsive force from the plastron approaches the Bjerknes attractive force from the wall, the bubble splits into two daughter bubbles, with the larger one forming in the direction of the stronger force (JTRA, JAT, JTA, and JAFV patterns). In cases where the Bjerknes repulsive force from the plastron is less than the Bjerknes attractive force from the wall, the bubble moves towards the plastron (JTR pattern). The repulsive and attractive forces acting on the bubble are mainly influenced by the dimensionless wall distance γ and the volume ratio V′. As γ decreases, the increase in the Bjerknes repulsive force from the plastron on the bubble becomes greater than the increase in the Bjerknes attractive force from the wall. Furthermore, an increase in the volume ratio V′ enhances the Bjerknes repulsive force from the plastron on the bubble.

(3) During the second oscillation stage, the bubble and the plastron oscillate out of phase, leading to the reversal of the jet directed towards the plastron. In the JTRA and JTR patterns, as the lower bubble expands to its maximum volume, the plastron is compressed to its minimum. Subsequently, the expansion of the plastron pushes the fluid upwards, causing the shrinking bubble to move away from the boundary. In contrast, for the JAT, JTA, and JAFV patterns, there is no observable oscillation process of the lower bubble. The expansion of the plastron in these cases slows down the speed of the lower bubble towards the boundary.

(4) Compared to the cavitation bubble collapsing near a rigid wall, the presence of the plastron on a superhydrophobic surface extends the bubble collapse time. Additionally, it prevents the collapse of annular bubbles at the boundary and significantly reduces the impact velocity of the downward jet, thereby considerably reducing the possibility of cavitation damage to the boundary. Among the seven jet patterns, the downward jet impacts the plastron at higher velocities under the JAT pattern, resulting in the greatest influence on the boundary. In comparison, the JA, JAFV, and JAF patterns have a smaller impact on the plastron. Based on this, three design criteria for achieving weak jet influence on the plastron are established: VP>0.25Vmax, HP>0.55Rmax, DP>1.2Rmax.

(5) Under the influence of oscillating cavitation bubbles, the evolution of the plastron in seven jet patterns generally follows a similar process. When the plastron is first compressed to its minimum, it takes on a torus-like shape. During the rebound process, the pressure distribution outside the plastron and the attraction of the superhydrophobic surface to the plastron cause the edges of the plastron to exhibit a stepped surface instability. When the volume reaches its maximum, the plastron resembles a dish-like shape. Subsequently, as the cavitation bubble collapses, the release of shock waves causes the plastron to once again compress to its minimum, resulting in a skirt-like shape. The stronger interaction between the bottom surface of the bubble and the top of the plastron leads to the observation of different shapes under the JAF pattern, including conical and cauliflower-like structures.

This study not only elucidates the positive significance of the plastron on superhydrophobic surfaces in resisting cavitation erosion but also provides theoretical support for the design of superhydrophobic surfaces resistant to cavitation. It is worth noting that during the experiments, the plastron was observed to detach from the superhydrophobic surface after multiple oscillations. Of genuine concern is whether the coupled response of the plastron on the superhydrophobic surface under oscillating cavitation bubbles will lead to a decline in the surface’s ability to trap air, which requires further research.

CRediT authorship contribution statement

Caisheng Huang: Writing – original draft, Methodology, Investigation, Conceptualization, Writing – review & editing. Xiaolong He: Writing – review & editing, Supervision, Investigation, Formal analysis. Jianmin Zhang: Writing – review & editing, Supervision, Investigation, Funding acquisition, 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.

Appendix A Supplementary data

The following are the Supplementary data to this article:Supplementary video 1

Supplementary video 2

Supplementary video 3

Supplementary video 4

Supplementary video 5

Supplementary video 6

Supplementary video 7

Supplementary video 8

Supplementary video 9

Acknowledgements

The authors gratefully acknowledge the support by the 10.13039/501100001809 National Natural Science Foundation of China (grant number: U22A20236 ).

Appendix A Supplementary data to this article can be found online at https://doi.org/10.1016/j.ultsonch.2024.107016.
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