Experimental study on influence of particle shape on shockwave from collapse of cavitation bubble

Abstract

The bubble dynamics under the influence of particles is an unavoidable issue in many cavitation applications, with a fundamental aspect being the shockwave affected by particles during bubble collapse. In our experiments, the method of spark-induced bubbles was used, while a high-speed camera and a piezoresistive pressure sensor were utilized to investigate how particle shape affects the evolution of shockwaves. Through the high-speed photography, we found that the presence of the particle altered the consistency of the liquid medium around the bubble, which result in the emitting of water hammer shockwave and implosion shockwave respectively during the collapse of the bubble. This stratification effect was closely related to the bubble-particle relative distance φ and particle shape δ. Specifically, when the bubble-particle relative distance φ < 1.34 $e^{-0.10\delta }$, particles disrupted the medium consistency around the bubbles and led to a nonspherical collapse and the consequent stratification of the shockwave. By measuring the stratified shockwave intensity affected by different particle shapes, we found that the stratified shockwave intensity experienced varying degrees of attenuation. Furthermore, as the particle shape δ increased, the attenuation of the particle on shockwave intensity gradually reduced. These new findings hold significant theoretical implications for elucidating cavitation erosion mechanisms in liquid–solid two-phase flows and applications and prevention strategies in liquid–solid two-phase cavitation fields.

1. Introduction

The interactions between the two in the bubble-particle system occur throughout the lifetime of bubbles, which occur in numerous cavitation scenarios such as biomedical treatment [1], [2], hydraulic machinery [3], [4], [5], ultrasonic cleaning [6], [7], [8], [9] and drug delivery [10], [11], etc. The mechanisms underlying the influence of particle on microjet and shockwave forms the theoretical basis for these application scenarios.

Since the pioneering establishment of the ideal spherical cavitation bubble model by Rayleigh [12], numerous scientists have conducted extended research on bubble dynamics, encompassing phenomena like microjets during bubble collapse. Some researchers have investigated the bubbles collapse near various boundaries through experimental methods and have made numerous discoveries, such as the evolution of jets near wall [13], [14], [15], the development of microjets under the influence of a free surface [16], mechanism of microjets in cavitation erosion [17], [18], the formation mechanisms of microjets and counterjets [19], microjets under different gravity [20], the development of microjets affected by a wall and a free surface [21]. Another group of scientists, by combining numerical simulations with their research, have also achieved significant outcomes, such as the development of microjets near wall surfaces [22], [23], [24], the impact pressure characteristics of microjets on the wall [25], scale laws between different types of cavitation bubbles [26], unified studies on microjets [27], the characteristics of microjets near a wall in compressible mediums [28], unified studies on different classical bubble equations [29]. The above research results on the microjet from near-wall bubble collapse have laid a foundation for the mechanism of shockwave generation affected by particles.

The aforesaid researches findings elucidate bubble dynamics from the perspective of microjets. However, there are few researches on the phenomenon of shockwave. Ohl et al. [30] analyzed the asymmetric collapse of individual bubbles within a bubble cluster and the associated emission of shockwaves in an ultrasonic field. Vogel et al. [31] investigated the energy configuration of shockwave and cavitation bubble during the evolution of bubbles. Akhatov et al. [32] used the numerical simulation method to study different-sized cavitation bubbles and they found that the bubble radii and shockwave intensities obtained from calculations were agreed with experimental data. Additionally, the research found the significant influence of the concentration of non-condensable gas in the computational model on the dynamics of bubbles. Sankin et al. [33] investigated the nonspherical shape of bubble collapse and the shockwaves emission. Wang et al. [34] found, in a specific case study, that the impact pressure on the wall resulting from the collapse shockwave was higher than the impact pressure from the microjet, and it affected a larger wall area. Supponen et al. [35], [36] investigated the shockwaves during the evolution of weakly deformed bubble and nonspherical bubbles through experiments. Supponen et al. [37], through experimental research, discovered the relationship between jet direction and secondary compression waves. Luo et al. [38] studied the stratification phenomenon of shockwaves affected by an air bubble and established the relationship of the characteristic parameters of the air bubble on the intensity of the shockwave.

Based on these two main types of bubble dynamics research, scientists have found that different types of boundaries around the cavitation bubble exert a certain degree of influence on the evolution of the cavitation bubble, such as elastic boundaries [39], [40], [41], closed spherical free surfaces [42], curved rigid surfaces [43], [44] and floating structures [45]. Meanwhile, these boundaries exhibit different responses under the effect of cavitations bubble. Poulain et al. [46] experimentally found that the moving velocity of particle was related to bubble-particle distance. Three types of jet pattern from bubble collapse under particle action [47], important influencing factors in bubble-particle interaction (i.e., bubble-particle relative size and relative density) [48], a classification of bubble-particle interaction (“weak,” “intermediate,” and “strong”) [49] and the cavitation bubbles induce particles motion, and conversely, the motion of particles exerts an influence on the bubbles themselves [50] were discovered. Lv et al. [51] experimentally investigated particle-bubble interactions and analyzed the dynamics of particle motion qualitatively and quantitatively under the influence of bubbles. Wu et al. [52] conducted experiments about the interaction between bubbles and freely settling particles. They established a model to describe the particle-bubble interaction. Wu et al. [53] experimentally investigated the near-wall bubble-particle interaction. They combined numerical simulation results with theoretical analysis to determine the limiting conditions of bounce and impact during the interaction processes. Zevnik and Dular [54] conducted numerical simulations using the finite volume method to study the bubble evolution near particle and they found that the microjet impact load on the surface of the particle is influenced by the bubble-particle size ratio and distance. Hu et al. [55] numerically simulated the collapse shockwave and microjet dynamics of a cavitation bubble in the vicinity of two spherical particles, and defined four typical characteristics of the bubble collapse. Wang et al. [56] conducted an experimental and theoretical study to comprehensively investigated the dynamics of laser-induced cavitation bubbles near fixed spherical particles. It uncovered the applicability range of the Kelvin impulse theory for bubbles in proximity to spherical particles and established a predictive model for bubble motion. Zheng et al. [57] established a theoretical model for effectively predicting the motion characteristics of a cavitation bubble in the vicinity of two spherical particles by combining experimental and numerical investigation methods. Ren et al. [58] found through experimentation that bubbles can lift metal particles, and further employed theoretical methods to describe the force that lifts the particles. Teran et al. [59] conducted an experimental and numerical study on the interaction between cavitation bubble and particle near a wall, and their results showed that the size and density of the particle and the relative position of the particles and the cavitation bubbles had a significant effect on the maximum velocity of the particles. Zhang et al. [60], [61] conducted an experimental study on the interaction between near-wall particle and cavitation bubble, and found the effects of bubble-wall distance, bubble-particle distance, and particle size on the behavior of bubbles collapse. Yin et al. [62] conducted a numerical study on the interaction between near-wall particle and cavitation bubble, and found that the particle size, the relative bubble-particle–wall position play a key role in the variation of the wall pressure peak resulting from bubble collapse, and the main source of the wall pressure peak at different bubble-wall distances was demonstrated. In the above studies concerning the bubble-particle interactions, the focus has mainly been on the relative movement of particles caused by cavitation bubbles. Few research has been conducted on the significant dynamic phenomenon of shockwaves under the influence of particles.

From the statements above, it becomes evident that existing research has predominantly focused on microjets, with less attention devoted to investigating the influence of particle shape on collapse shockwaves. However, in various cavitation application scenarios, such as shockwave lithotripsy [63], and ultrasonic cleaning [6] etc., the influence of different shapes of particle on the shockwave from the collapse of the cavitation bubble remains an inevitable issue. Therefore, this paper experimentally investigates the evolution of shape and intensity of collapse shockwaves influenced by different shapes of particle. The aim is to delve deeper into the cavitation erosion mechanism within the water–sand two-phase flow.

2. Experimental equipment and method

In our experiments, a comprehensive experimental setup was designed to accomplish the experimental objectives of observing shockwaves from the collapse of the cavitation bubble under the influence of the particle and measuring the impact pressure of these shockwaves on the wall. The setup included a generator for spark-induced bubbles, a high-speed photography system for observation, and a piezoresistive pressure sensor for measurement.

The method of spark-induced bubble was used for generating cavitation bubbles. This method offered precise control over the bubble location and size [41]. The bubble radii were in the range of 8.55 ± 0.43 mm. In the experiments of this paper, we repeated the experiments five times for each experimental condition to obtain accurate bubble size. In order to reduce the effect of the electrodes on bubble dynamics, a slender copper wire with a radius of 0.08–0.09 mm (significantly smaller than the maximum bubble radius, accounting for about 1 % of the bubble radius) was chosen as the discharge electrodes [64]. The electrodes were securely affixed to a holder within the water tank.

A transparent glass tank measuring 30 cm × 20 cm × 35 cm was selected for the experiment to avoid shockwave reflection from the tank walls or the free surface during the first bubble collapse. The electrodes contact point was situated in the centre of the tank, and the deionized water depth was about 30 cm. In our experiments, the distance between the bubble center and the boundaries is greater than ten times the maximum bubble radius [65]. Throughout the experiments, the ambient temperature was set to approximately 22 °C, while the ambient pressure was maintained at 95.5 kPa.

The cavitation bubbles generated through underwater low-voltage discharge exhibit an extremely short lifetime (millisecond scale) during their evolution. Consequently, the examination of bubble dynamics affected by the particle necessitated the utilization of a high-speed camera, which has a maximum acquisition rate of one million frames per second (Photron Inc. Fastcam SA-Z). Given the diminutive dimensions of the cavitation bubbles (millimetre scale) created by the cavitation bubble generation system, the camera must be equipped with a macro lens (NIKKOR 85 mm). A point light source was introduced to supplement illumination during the experiments, as shown in Fig. 1(a).

Fig. 1.

Experimental setup and parameter description: (a) Experimental setup (Top view); (b) Description of the spatial position parameter of bubble and particle.

During the evolution of the shockwave, the water density at the shock front was different from that in surrounding regions. In light of this, the shadowgraph method was adopted to capture the shockwaves in this experimental setup. The principle was to use a concave mirror to redirect the light, which permeates the photographed area. This method leverages the fluctuations in light transmission intensity resulting from changing water density due to the spread of shockwaves. In this study, a shooting speed of 180,000 fps and an exposure time of 0.25 μs were selected. The image resolution is set to 384 pixels × 128 pixels, with each pixel measuring approximately 0.2439 mm pixel⁻¹.

The rapid evolution of shockwaves, outpacing the speed of sound in water [66], necessitated careful consideration of pressure transducer characteristics, including bandwidth, rise time, and dimensions, within experimental setups. Before the experiment, we tested the collapse shockwave of underwater low-voltage bubble and found the time span of single waveform was about 4 μs – 30 μs, and the corresponding frequency was 33.3 KHz – 250 KHz. A piezoresistive pressure transducer (Test Electronics Information Co. Ltd, CY400) was suitable for this study following a series of tests. This pressure sensor exhibited a maximum response frequency of 500 KHz and a rise time of 0.2 μs. Its circular sensing surface has a radius of 1.50 mm, and it can measure physical processes with a maximum pressure of less than 20 MPa (the accuracy is 0.25 % of the maximum range). The sensor operated at a sampling frequency of 10 MHz. Ahead of the experiments, the pressure sensor was embedded into the wall, aligning its sensing surface as closely as possible with the wall. Throughout the experimental proceedings, the bubble centre remained level with the sensing area centre of the pressure sensor, as shown in Fig. 1(b).

The primary aim of this paper was to investigate the influence of particle shape while maintaining consistent volume and density on the collapse shockwaves. 3D printing technology was employed to fabricate particles with diverse shapes to achieve this experimental goal. These particles were designed to possess a smooth surface and a uniform resin composition, resulting in an average density of 1.149 g / cm³. In the experiment, a slender nylon rope with a diameter of 0.05 mm was utilized to suspend the particle. The particle centre was aligned with the contact point of the electrodes to ensure precision, as shown in Fig. 1(b). Eight distinct particle shapes were selected for examination, considering the ratio of the horizontal semi-axis to the vertical semi-axis of the ellipsoidal particles. Each particle exhibited varying ratios of horizontal to vertical semi-axes, thereby mimicking diverse sand grain shapes within the same volume. The specific physical characteristics of these particles were outlined in Table 1.

Table 1.

Physical parameters of particles.

No.	Horizontal semi-axis A (mm)	Vertical semi-axis B (mm)	Particle shape δ (A / B)	Density (g / cm3)	Volume (cm3)
1	10.00	10.00	1.00	1.151	4.189
2	13.10	8.74	1.50	1.149	4.189
3	15.87	7.94	2.00	1.149	4.189
4	18.42	7.37	2.50	1.147	4.189
5	20.80	6.93	3.00	1.148	4.189
6	23.05	6.59	3.50	1.147	4.189
7	25.20	6.30	4.00	1.148	4.189
8	27.26	6.06	4.50	1.154	4.189

Upon capturing the bubble collapse through camera recordings, the subsequent step involved extracting key parameters of the bubble, such as its radius and centre position, through image processing techniques. The equivalent radius R of the cavitation bubble was calculated using image greyscaling, represented by the radius of the circle that has the same pixel points as the bubble (as shown in Fig. 1(b)). The maximum bubble radius was also determined, denoted as R_max. The first position of the electrode crossing point was taken as the bubble centre. In this paper, the minimum distance from the cavitation bubble centre to the particle surface was represented by L, and the distance from the bubble centre to the wall was denoted as D (as shown in Fig. 1(b)). A and B, respectively, denoted the horizontal and vertical semi-axes of the ellipsoidal particle (as shown in Fig. 1(b)). For dimensionless treatment, we adopted φ, δ, and γ to denote the relative distance of the cavitation bubble-particle (L / R_max, hereinafter referred to as the bubble-particle distance), the shape of the particle (A / B, hereinafter referred to as the particle shape), and the relative distance of the cavitation bubble-wall surface (D / R_max, hereinafter referred to as the bubble-wall distance), respectively. In addition, we used P and E to denote the impact pressure on the wall surface and the collapse shockwave energy, respectively. P_max denoted the corresponding pressure peak. P'_max and E' denoted the pressure peak and energy of the shockwave impacting the wall at the first collapse of the bubble in the absence of particles with the same conditions, respectively.

3. The influence of the particle on shape of shockwave

In order to investigate the influence of the particle on the collapse shockwaves in unbounded field, the following sections will analyze the influence of φ and δ on the shape of the shockwave from the bubble collapse respectively.

3.1. The influence of bubble-particle distance on the shape of the shockwave

Fig. 2 shows the influence of the particle on the shape of the collapse shockwave when φ is varied.

Fig. 2.

Influence of bubble-particle distance on the shape of shockwave (The stratified structures shown in Fig. 2 have been marked with red arrows): (a) R_max = 8.645 mm, φ = 0; (b) R_max = 8.421 mm, φ = 0.505; (c) R_max = 8.953 mm, φ = 0.712; (d) R_max = 8.488 mm, φ = 1.001; (e) R_max = 8.386 mm, φ = 1.115; (f) R_max = 8.677 mm, φ = 1.224; (g) R_max = 8.737 mm, φ = 1.459; (h) R_max = 8.670 mm, φ = 1.961.

In Fig. 2(a), φ is equal to 0. The bubble expanded to its maximum volume at t = 1.311 ms, and the left side of the bubble was wrapped entirely around the particle surface. When t = 2.278 ms, the bubble volume hit a minimum, and an apparent mushroom head shape can be observed, as shown in Fig. 2(a₃). The left side of the cavitation bubble was stretched into a mushroom neck shape, and remained adsorbed upon the particle surface during collapsing processes. Same phenomenon appeared in another research by Ma et al. [43]. In this scenario, the bubble collapsed toward the opposite direction of the particle, and fractured at the middle of “mushroom neck.” This fracturing phenomenon also emerged in the research carried out by Tomita et al. [44]. A faint layer of shockwave was then emitted at the middle of “mushroom neck”, while a barely visible layer of water hammer shockwave was emitted at the “mushroom head.” In a high-precision numerical simulation investigating the bubble-sphere interaction, carried out by Li et al. [50], the form of microjet development at the bubble collapse affected by particle was simulated. This simulation captured the flow velocity field and pressure field of the surrounding water when the bubble-sphere distance was relatively close. These results revealed that the cavitation bubble generated a toroidal high-pressure zone at the mushroom head, which continuously elongated the mushroom neck. Eventually, the toroidal jet at the mushroom head impacted the external boundary of bubble. This development of the toroidal microjet dispersed the jet energy, resulting in a weaker water hammer shockwave. Consequently, capturing the trace of the water hammer shockwave in Fig. 2(a) was challenging in this study.

In Fig. 2(b), φ increased to 0.505, R_max was 8.421 mm. The right side of the bubble experienced collapsed firstly, which led to the emission of a single-layer shockwave (as shown in Fig. 2(b₄). Subsequently, the left side of the bubble collapsed and emitted a shockwave (as shown in Fig. 2(b₆)). Throughout the bubble collapse, two layers shockwaves became apparent, indicating a stratification phenomenon. In Fig. 2(c), φ increased to 0.712 and R_max was 8.953 mm, similar to the bubble collapse in Fig. 2(b), the cavitation bubble took on a droplet shape. It was divided into two parts during its collapse. The difference was that the left side of the bubble constituted a more significant portion of the bubble volume compared to Fig. 2(b). Similarly, stratified shockwaves were observed (as shown in Fig. 2(c₅)).

In Fig. 2(d), φ = 1.001 and R_max = 8.488 mm, the left side of the bubble was not adhered to the particle surface when its volume reached a minimum. However, the bubble collapse was also divided into two parts, and the minimum volume during collapse was further reduced compared to that shown in Fig. 2(c), stratified shockwaves appeared. In Fig. 2(e), φ further increased to 1.115, the image showed a reduced degree of stratification in the shockwaves. Additionally, the distance between the stratified shockwaves narrowed compared to the earlier sets of experiments.

In Fig. 2(f-h), the particles were positioned farther away from the cavitation bubble, φ were 1.224, 1.459, and 1.961, respectively. The bubbles collapsed with a spherical shape, emitting a single-layer shockwave. The shockwave shape displayed a regular spherical form, reminiscent of the shockwave during the bubble collapse observed in the study by Ohl et al. [30], where the cavitation bubble remained unaffected by particle. In Fig. 2(f-h), the shockwave experienced reflection upon reaching the particle, as shown in Fig. 2 (f₆, g₅, h₆).

Analyzing the evolution of the collapse shockwave while varying the bubble-particle distance, under a constant particle shape δ, a distinct correlation between shockwave stratification and the bubble-particle distance φ becomes evident. Smaller bubble-particle distances led to the shockwave stratification. As φ increased, the stratification extent of shockwave gradually decreased, eventually stratification disappeared.

The presence of the particle disrupted the medium consistency around the bubble, causing an asymmetrical collapsing shape and even potential tearing when φ was small. This asymmetry in collapsing shape was responsible for the observed stratification of the shockwave. As φ increased, the influence of particle on the bubble collapse diminished, which led to a restoration of a spherical collapse shape resembling that in unbounded field. The outward-emitting shockwave during the collapse was primarily attributed to the implosion shockwave triggered by the bubble rebound. Consequently, the shockwave stratification also disappeared.

3.2. The influence of particle shape on the shape of the shockwave

Fig. 3 shows the influence of particle shape δ on the collapse shockwave shape while maintaining a constant bubble-particle distance φ. To explore whether the same pattern of influence persisted for particle shape δ between 0 and 1.0, given that the experiment focused on particle shapes δ ≥ 1.0, we rotated the particle with a shape δ = 4.5 by 90° to hang vertically in the water, the particle shape δ = 0.22.

Fig. 3.

Influence of particle shape on the shape of shockwave (The stratified structures shown in Fig. 3 have been marked with red arrows): (a) R_max = 8.233 mm, φ = 0.774, δ = 0.22; (b) R_max = 8.953 mm, φ = 0.712, δ = 1.00; (c) R_max = 8.333 mm, φ = 0.765, δ = 1.50; (d) R_max = 8.799 mm, φ = 0.724, δ = 2.00; (e) R_max = 8.741 mm, φ = 0.729, δ = 2.50; (f) R_max = 8.967 mm, φ = 0.711, δ = 3.00; (g) R_max = 8.658 mm, φ = 0.736, δ = 3.50; (h) R_max = 8.987 mm, φ = 0.709, δ = 4.00; (i) R_max = 8.861 mm, φ = 0.719, δ = 4.50.

In Fig. 3(a), δ is equal to 0.22, the bubble shape was cylindrical at t = 1.944 ms. At this stage, the side of the bubble far away from the particle collapsed earlier, and no evident shockwave was captured in the image. Subsequently, the side of the bubble near the particle collapsed towards the particle and emitted stratified shockwaves (as shown in Fig. 3(a₆)). In Fig. 3(b), the bubble collapse resembled that in Fig. 3(a), with a key distinction being the emergence of a single-layer collapse shockwave of the side of the cavitation bubble oppose to the particle, as shown in Fig. 3(b₃). A visibly stratified structure of the collapse shockwave became evident in Fig. 3(b₆). Notably, the depiction in Fig. 3(b₆) offered a considerably clearer visualization of the shockwave compared to Fig. 3(a₆).

Simiar to Fig. 3(a, b), Fig. 3(c, d) showed stratified collapse shockwaves. The difference was that the bubble minimum volume during the first collapse decreased as the particle shape δ increased in Fig. 3(c, d). The adsorption area of the particle to the cavitation bubble during collapsing processes was also decreased. Consequently, the colour of the collapse shockwave deepened, as shown in Fig. 3(c₅, d₅). Interestingly, an obvious deformation of the shockwave was observed as it passed the particle surface (Fig. 3(c₅, d₅)). Particularly, the colour of the shockwave on the left side of the particle was apparently lighter than in other regions (Fig. 3(c₆)).

In Fig. 3(e-i), the values of δ were 2.50, 3.00, 3.50, 4.00, and 4.50, respectively. As the cavitation bubbles contracted to their minimum radii, they separated from the particle and their minimum volumes during the collapses decreased further than that in Fig. 3(d). In the images, all collapse shockwaves exhibited a stratified structure. Notably, the extent of stratification progressively decreased with increasing particle shape δ. An reflection phenomenon occurred as the shockwave crossed the particle (as shown in Fig. 3(i₅)).

When the bubble was near the particle initially, the contact of bubbles with particles led to differential contraction speeds on both sides of the bubble. Consequently, the bubble collapsed with a nonspherical shape. With increased δ, the contact area of the bubble with the particle during the collapse gradually decreased. As a result, the influence of particle on the bubble collapse weakened. The collapsing shape of the cavitation bubble gradually converged towards a spherical shape, accompanied by a reduction in the degree of stratification within the collapse shockwave.

3.3. The critical condition of the stratification effect

According to the above experiments, it becomes evident that specific conditions for the bubble-particle distance φ and particle shape δ lead to a stratification effect on the shockwave. We systematically analyze the experimental results, focusing on the influences of φ and δ on the shockwave, determination based on whether the shockwave exhibits stratification or not. Critical conditions for the stratification of the shockwave were derived under the premise of consistent particle volume, as shown in Fig. 4, the value of φ ranges from 0 to 2.23, and the value of δ within a range of 0.22 to 4.5.

Fig. 4.

Critical conditions for stratification of shockwaves by particles.

From Fig. 4, it is obvious that, for a constant particle shape δ, the shockwave evolution initiates with stratification and gradually transitions to an unstratified state as the bubble-particle distance φ increases. Similarly, when the bubble-particle distance φ remains constant, altering the particle shape from spherical to ellipsoidal corresponds to an analogous shift from stratified to unstratified shockwave evolution. Furthermore, Fig. 4 indicates that when φ < 1.34 $e^{-0.10\delta }$, the influence of particle on the bubble collapsing shape results in stratification of the shockwave. Conversely, when φ > 1.34 $e^{-0.10\delta }$, the particle affects the bubble collapsing shape slightly, and the shockwave remains unstratified.

Additionally, Fig. 4 demonstrates that the critical bubble-particle distance φ, inducing the stratification effect on the shockwave, decreases as particle shape δ increases. This trend suggests that with an infinitely large δ, the influence of particle on the shockwave shape becomes negligible, the bubble collapsing form is similar to a free-field spherical collapse. The trend of the critical bubble-particle distance φ curve within the range of δ = 0 – 0.22 and the selection of the fitting function form will be discussed in Section 5.

Numerical simulations of the bubble collapse near a wall were conducted by Beig [67]. The findings indicated that the water hammer shockwave and implosion shockwave converge into a single-layer shockwave before impacting on the wall as relative bubble-wall distance exceeds 2.0. Moreover, the water hammer shockwave and implosion shockwave became virtually indistinguishable as relative bubble-wall distance surpasses 4.0. The critical cavitation bubble-wall relative distance of 2.0 for shockwave stratification affected by the wall is indicated by the red square in Fig. 4. The particle affects the shockwave shape slightly than that of the wall. Consequently, the curve depicting the variation of the critical bubble-particle distance φ should consistently remain below the straight line representing φ = 2.0.

Through extensive experimentation, we have established the critical condition for the stratification of collapse shockwaves affected by particles, i.e., 1.34 $e^{-0.10\delta }$. By considering the physical property governing the dispersion of the medium surrounding the bubble, it becomes evident that the presence of particles disrupts the medium consistency because of the differing compressibilities of particles and water. This inconsistency in the medium around the bubble lastly causes asymmetrical collapsing shapes, in contrast to the spherical collapse in unbound area. Analyzing the generation mechanisms of implosion shockwaves and water hammer shockwaves reveals that when φ > 1.34 $e^{-0.10\delta }$, with larger bubble-particle distance φ, the particles affect the bubble collapsing shape less. The collapse shockwave is primarily attributed to the implosion shockwave generated by the bubble rebound. Conversely, when φ < 1.34 $e^{-0.10\delta }$, the presence of particles affects the bubble development and shape during the collapse. And the impact of the microjet on the opposing bubble surface arise to a water hammer shockwave [50]. Subsequent bubble collapse and rebound led to the emergence of an implosion shockwave. This stratification phenomenon is also observed in bubble collapse near walls. Lindau and Lauterborn [15] confirmed the formation mechanism of the stratification of the collapse shockwave by studying shockwaves of bubbles near a wall. Similarly, numerical simulations on the collapse of a near-wall bubble by Beig [67] further validate this formation mechanism.

4. The influence of the particle on intensity of shockwave

By analyzing the influencing factors of the stratification effect on the shockwave in a free field, we found the critical conditions for the stratification effect. The intensities of stratified shockwaves vary under the influence of particles requires further investigation. Hence, this section focuses on delineating the patterns in shockwave intensity variation affected by the particle. Comprehensive numerical simulations by Beig [67] demonstrated that collapse shockwaves do not exhibit stratification as the relative bubble-wall distance exceeds 2.0. Consequently, this section introduces the wall to acquire the variation trend of the shockwave intensity under the influence of the particle. In order to exclude the wall-induced effect on the shockwave, we have chosen the bubble-wall distance to be three times the maximum bubble radius in our experiments.

4.1. The influence of bubble-particle distance on the pressure peak of the shockwave

Fig. 5 shows a series of images illustrating the evolution process of the shockwave under the influence of particle shape δ = 1.50 near a wall when the bubble-particle distance φ changes. The right black area in Fig. 5 represents a rigid wall equipped with a pressure sensor.

Fig. 5.

Influence of bubble-particle distance near the wall on the shape of shockwave (The walls located in the right of each image): (a) R_max = 8.494 mm, γ = 3.002; (b) R_max = 8.905 mm, φ = 0, γ = 2.864; (c) R_max = 8.223 mm, φ = 0.517, γ = 3.102; (d) R_max = 8.122 mm, φ = 0.785, γ = 3.140; (e) R_max = 8.161 mm, φ = 1.042, γ = 3.125; (f) R_max = 8.126 mm, φ = 1.151, γ = 3.138; (g) R_max = 8.662 mm, φ = 1.227, γ = 2.943; (h) R_max = 8.387 mm, φ = 1.520, γ = 3.040; (i) R_max = 8.473 mm, φ = 2.006, γ = 3.010.

Fig. 5(a) showed the evolution of the collapse shockwave near a wall in the absence of particle. During this stage, the bubble collapsing form resembled the spherical collapse in a free field. It emitted a regular single-layer spherical shockwave outward, as shown in Fig. 5(a₅). Notably, when the bubble-wall distance was three times the maximum bubble radius, the influence of the wall on the shape of the shockwave was negligible. This phenomenon aligns with findings by Beig [67] from numerical simulations, which indicated that stratification was absent in the collapse shockwave when the relative bubble-wall distance was larger than 2.0.

In Fig. 5(b), φ = 0, the side of the bubble close to the particle gradually extended into a mushroom neck shape and remained affixed to the particle surface. As the toroidal jet impacted the external boundary of the bubble [50], the “mushroom neck” fractured from the middle. The smaller side of the bubble, near the particle, collapsed earlier, emitting a layer of shockwave (as shown in Fig. 5(b₄)). Subsequently, the toroidal jet impacted the outside of the bubble, and a water hammer shockwave can be observed in Fig. 5(b₅).

In Fig. 5(c), φ increased to 0.517. The side of the bubble near the wall collapsed firstly, emitting a single-layer spherical shockwave. Subsequently, the side of the bubble contacting the particle collapsed, with the impact on the particle becoming apparent in the image. In Fig. 5(d), where φ = 0.785, there was a difference from Fig. 5(c). In this case, the collapsing shape of the left side of the bubble contacted the particle surface was flatter and more elongated. It emitted stratified collapse shockwaves. However, the shockwave trace was less evident in Fig. 5(d₆).

In Fig. 5(e), φ = 1.042, R_max = 8.161 mm. The bubble detached from the particle as its volume reached a minimum, and the minimum volume decreased compared with the above three groups of experiments. The shockwave had an obvious stratified structure, and its trace was clearly visible in contrast to Fig. 5(b-d), as shown in Fig. 5(e₄). In Fig. 5(f), φ was further increased to 1.151, and the shockwave stratification almost disappeared, as shown in Fig. 5(f₄).

In Fig. 5(g-i), the bubble collapsing form resembled the spherical collapse in unbound area, and a regular single-layer spherical shockwave was emitted in the collapse, with a darker color of the shock front and an obvious shockwave image. The shockwave reflection occurred when it contacted the particle surface, as shown in Fig. 5(g₆, h₆, i₆).

Regarding the shockwave intensity after stratification, we need to analyze it with the impact pressure collected by the pressure sensor. The processes of the shockwaves impacting on the wall in Fig. 5 are shown in Fig. 6.

Fig. 6.

The process of the shockwaves impacting on the wall under different bubble-particle distance. (Within the images, the above images depict the impact processes of the shockwave on the wall during the corresponding experiments in Fig. 5. The red dashed line signifies the sensor position, all shockwaves were indicated by white arrows.)

In Fig. 5(a), the bubble underwent a spherical collapse, emitting a regular single-layer spherical shockwave. As a result, only a single pressure peak can be observed in Fig. 6(a), where the peak value of the impact pressure was 5.038 MPa. For the expansion or collapse of cavitation bubbles, the shockwaves generated are primarily due to the implosion of the bubble or the water hammer of microjets. The underlying reason for this is the compressibility of the liquid. Thus, based on the conservation of momentum of the shock front, the relationship between the shockwave velocity of the bubble $u_s$ and its impact pressure $p_s$ was shown in the following equation [68]:

$$p_s=c_1{\rho }_0{u}_s\left({10}^{\frac{u_s-c_0}{c_2}}-1\right)+p_{\infty }$$ (1)

where ${\rho }_0$ denotes the density of the water before it is compressed by the shockwave, $c_0$ is the speed of sound in the water (22 °C, c₀ ≈ 1482.9 m/s), c₁ = 5190 m/s, c₂ = 25306 m/s, and $p_{\infty }$ is the hydrostatic pressure. Through Fig. 5(a₃, a₆), we can calculate the shockwave velocity is about 1490.125 m/s. Therefore, we can estimate the shockwave pressure is about 5.181 MPa, and this value is closer to the experimental data obtained from the measurements in this paper.

In Fig. 5(b), the first collapse of the “mushroom neck” of the bubble produced the first pressure peak in Fig. 6(b), reaching a peak value of 0.833 MPa. Subsequently, the toroidal jet at the “mushroom head” impacted the external boundary of the cavitation bubble [50], resulting in a water hammer shockwave. This phenomenon was represented by the second pressure peak in Fig. 6(b), with a peak value of 0.885 MPa.

In Fig. 6(c), there were two peak values, measuring 0.999 MPa and 1.680 MPa, respectively. These corresponded to the collapse shockwaves of the two separated portions of the bubble in Fig. 5(c). The first pressure peak was generated by the collapse of the near-wall side of the cavitation bubble, while the second peak corresponded to the collapse of the side of the cavitation bubble in contact with the particle.

Similarly, in Fig. 5(d), the first pressure peak was resulted from the collapse of the side of the bubble close to the wall, with a peak value of 0.656 MPa. However, the inter-wave distance between the stratified collapse shockwaves of the side of the bubble near the particle in Fig. 5(d) was larger, which can be recognized through the pressure sensor, manifesting as the second and third pressure peaks in Fig. 6(d), with peak values of 0.773 MPa and 0.797 MPa, respectively. It's notable that these pressure peaks were smaller than those in Fig. 5(c).

In Fig. 6(e), a secondary pressure peak and a main pressure peak were evident, with peak values of 0.750 MPa and 2.850 MPa, respectively. These pressure peaks corresponded to the two-layer shockwaves shown in Fig. 5(e₄), where one layer was faint, and the other was obvious. When compared to Fig. 6(b-d), there was a noticeable increase in the pressure peak. In Fig. 6(f), the secondary pressure peak almost disappeared due to the closer proximity of the two-layer collapse shockwaves. Instead, it appeared as a single pressure peak with a peak value of 3.225 MPa, which was larger than that in Fig. 6(e).

In Fig. 6(g-i), it was evident that the pressure process exhibited only a single main pressure peak, with peak values of 4.702 MPa, 4.772 MPa, and 4.967 MPa, respectively. These peak pressure values of the shockwaves progressively approached the pressure peak value of 5.038 MPa observed when no particle was present during the bubble collapse. As φ increased to around 1.25, the attenuating of particles on the pressure peak of the shockwave was significantly decreased. According to the discussion of the pressure of shockwaves, it can be inferred that at smaller bubble-particle distances φ, particles have an attenuating effect on the shockwave intensity.

In the preceding discussion, we analyzed the influence of φ on the shape and intensity of the collapse shockwave. To provide a more intuitive understanding of how both φ and δ affect the shockwave intensity, we have plotted the P-φ relationship under different particle shapes δ into a single image, as shown in Fig. 7.

Fig. 7.

Relationship between bubble-particle distance and relative pressure peak of shockwaves.

From Fig. 7, it was evident that under the influence of different particle shapes δ, the relative pressure peak of the shockwave, represented by P_max / P^'_max, exhibited varying degrees of reduction. At φ = 0: The presence of the particle had the most significant effect on the shockwave, resulting in the most significant decrease in P_max / P^'_max. At φ = 0.50: The influence of the particle on the P_max / P^'_max weakened, which led to an increase in P_max / P^'_max. At φ = 0.75: A decrease in P_max / P^'_max was observed. In Fig. 5(d), the side of the bubble near the particle had a more elongated collapsing shape compared to Fig. 5(c), which led to weaker stratified collapse shockwaves. This phenomenon contributed to the decrease of the relative pressure peak. As φ continued to increase: The attenuation effect of particles on the shockwave decreased, causing the relative pressure peak P_max / P^'_max to increase gradually.

When φ increased to a value between 1.25 and 1.50, the influence of the particle on P_max / P^'_max became almost negligible. When the values of δ were 0.22 and 1.00, the critical bubble-particle distance φ for the attenuation effect of the particle on P_max / P^'_max fell within the range of 1.25–1.50. For δ of 2.00, 2.50, and 3.00, the critical bubble-particle distance φ for the attenuation effect of the particle on P_max / P^'_max was in the range of 1.10–1.25. For δ of 3.50 and 4.00, the critical bubble-particle distance φ for the attenuation effect of the particle on P_max / P^'_max lay in the range of 1.00–1.10. When δ was 4.50, the critical bubble-particle distance φ fell within the range of 0.75–1.00. It's obvious that the critical bubble-particle distance φ for the attenuation effect of the particle on P_max / P^'_max is closely linked to the critical bubble-particle distance φ for the stratification effect of the shockwave under the influence of particles, as shown in Fig. 4.

Before reaching the minimum critical bubble-particle distance φ, while keeping φ constant, P_max / P^'_max demonstrates an ascending trend with increasing δ. It is worth noting that within Fig. 7, when φ = 0, P_max / P^'_max of δ = 0.22 is larger than that of δ = 1.00, considering that it may be due to the experimental error caused by δ = 0.22 not being placed completely vertically in our experiments.

It is noteworthy that the trend of the critical φ for the variation of P_max / P^'_max reflected in Fig. 7 is highly consistent with the trend of the change of the stratification effect of the shockwave in Fig. 4. Accordingly, utilizing the critical bubble-particle distance φ = 1.34 of the stratification effect when δ = 0 (further elaborated in the Section 5) as a boundary line. We define the region where φ < 1.34 as the “affected area.” Within this domain, the influence of the particles weakens the relative pressure peak. Conversely, the region with φ > 1.34 is defined as the “unaffected area.” Here, the influence of particles on P_max / P^'_max is negligible. Consequently, the pressure peak of the shockwave resembled that of a cavitation bubble without influence of particles.

The bubble collapsing intensity near wall is influenced by the dimensionless bubble-wall distance. In general, when the cavitation bubble is closer to the wall, the pressure tends to be greater. Interestingly, the surface of a particle can also be treated as a curved, movable wall. In this analogy, the particle shape δ can be likened to the curvature of this “curved wall.” Existing research, such as the study by Ma et al. [43], indicates that when the relative bubble-wall distance is constant, an increase in the curvature of the wall (corresponding to an increase in particle shape δ) leads to a higher shockwave load. This finding aligns with the trend shown in Fig. 7.

4.2. The influence of particle shape on the pressure peak of the shockwave

Fig. 8 shows the images of the influence of particle shape δ on the shape of the collapse shockwave near the wall under same φ.

Fig. 8.

Influence of particle shape near the wall on the shape of shockwave (The walls located in the right of each image): (a) R_max = 8.590 mm, φ = 0.742, δ = 0.22, γ = 2.969; (b) R_max = 8.401 mm, φ = 0.759, δ = 1.00, γ = 3.035; (c) R_max = 8.393 mm, φ = 0.759, δ = 1.50, γ = 3.038; (d) R_max= 8.148 mm, φ = 0.782, δ = 2.00, γ = 3.130; (e) R_max = 8.390 mm, φ = 0.759, δ = 2.50, γ = 3.039; (f) R_max = 8.334 mm, φ = 0.765, δ = 3.00, γ = 3.060; (g) R_max = 8.129 mm, φ = 0.784, δ = 3.50, γ = 3.137; (h) R_max = 8.238 mm, φ = 0.773, δ = 4.00, γ = 3.095; (i) R_max = 8.466 mm, φ = 0.753, δ = 4.50, γ = 3.012.

In Fig. 8(a, b), the values of δ were 0.22 and 1.00, respectively. During the collapsing processes, the side of the bubble near the wall collapsed earlier, emitting weak stratified shockwaves, as shown in Fig. 8(a₃, b₃), and then the side of the bubble in contact with the particle surface collapsed toward the particle, emitting weak stratified shockwaves, as shown in Fig. 8(a₆, b₆). In Fig. 8(c), δ increased to 1.50, the bubble collapse was still divided into two parts, and the stratification of the shockwave still existed. As δ increased to 2.00, as shown in Fig. 8(d), stratified shockwaves were emitted when the side of the bubble contacted the particle collapsed.

In Fig. 8(e-h), it become evident that, under the constant φ condition, the shockwave stratification consistently persisted as δ increased. However, a gradual reduction in the degree of stratification was observable in the high-speed images. As the particle shape δ increased to 4.50, the stratification phenomenon of the shockwave nearly disappeared, as shown in Fig. 8(i).

Fig. 9 shows the pressure processes corresponding to the shockwaves impacting the wall in Fig. 8. As shown in Fig. 8 (a₃), the side of the bubble near the wall collapsed earlier, emitting a single-layer shockwave towards the water, forming the first weak pressure peak in Fig. 9(a), with a peak value of 0.441 MPa. While the other side of the cavitation bubble impacted the particle after that, forming stratified shockwaves as shown in Fig. 8(a₆), but the distance between waves in the stratified structure was minimal, which was challenging to be recognized when the shockwave impacted on the wall. So, it was shown as the second weak pressure peak with a peak value of 0.447 MPa in Fig. 9(a).

Fig. 9.

The process of the shockwave impacting on the wall under different particle shapes. (Within the images, the above images depict the impact processes of the shockwave on the wall during the corresponding experiments in Fig. 8. The red dashed line signifies the sensor position, all shockwaves were indicated by white arrows.)

Fig. 9(b) showed two weak pressure peaks with 0.562 MPa and 0.427 MPa, respectively. The first pressure peak caused by the collapse of the near-wall side of the bubble, and the second pressure peak was formed by the collapse of the side of the cavitation bubble in contact with the particle (Fig. 8(b₅)). In Fig. 8(c₆), there was a stratification of the shockwave, as shown in Fig. 9(c) ,as the second and third pressure peaks due to the large distance between the stratified waves. The first pressure peak resulting from the collapse of the near-wall side of the bubble, and the three peaks were 0.781 MPa, 0.653 MPa, and 1.016 MPa, respectively.

In Fig. 8(d), the bubble collapse was divided into two parts, the pressure peak value of the collapse shockwave of the near-wall side of the bubble was 0.822 MPa, and the pressure peak value of the collapse shockwave of the side of the bubble in contact with the particle was 1.395 MPa (as shown in Fig. 9(d)). In Fig. 8(e), the pressure peak value of the collapse shockwave of the near-wall side of the bubble was 0.976 MPa, and the pressure peak value of the collapse shockwave of the side of the bubble in contact with the particle was 1.698 MPa (as shown in Fig. 9(e)).

In Fig. 9 (f-i), where a secondary pressure peak and a main pressure peak appeared in the pressure process of each experiment, with the secondary pressure peak at the front and the main pressure peak at the back. From Fig. 9 (f-i), it can be observed that the secondary pressure peaks of all four sets of experiments were maintained near 1 MPa, and the main pressure peaks increased sequentially, with the main peak values of 2.292 MPa, 2.568 MPa, 2.702 MPa, and 2.737 MPa, respectively. In addition, it was evident that the time interval between the secondary peaks and the main peaks of the pressure processes decreased with the particle shape δ, which was in line with the results of Fig. 8(f-i), i.e., the distance between the two layers of shockwaves corresponding to the main and secondary pressure peaks decreased gradually with the increase of particle shape δ.

It can be observed that when the bubble-particle distance φ was constant, the pressure peak of the shockwave increased with the particle shape δ, combining the shockwave pressure peaks from several sets of experiments.

Fig. 10 shows a scatter plot illustrating the variation of the relative pressure peak P_max / P^'_max as a function of particle shape δ under different bubble-particle distances φ.

Fig. 10.

Relationship between particle shape and relative pressure peak of shockwaves.

Fig. 10 shows that when the bubble-particle distance φ falls within the range of 0 to 1.10, the relative pressure peak of the shockwave, represented by P_max / P^'_max, consistently increases with increasing particle shapes δ. At a bubble-particle distance φ of 1.25, P_max / P^'_max continues to rise as δ increases from 0.22 to 1.50. Subsequently, as δ continues to increase beyond 1.50, P_max / P^'_max is almost unchanged, maintaining a consistently high level. When the values of the bubble-particle distance φ were increased to 1.50 and 2.00, the influence of particles on P_max / P^'_max becomes nearly negligible, causing P_max / P^'_max to exhibit minor fluctuations while remaining close to 1.00. Furthermore, when holding the particle shape δ constant, the trend of P_max / P^'_max variation with changing bubble-particle distance φ aligns with the pattern shown in Fig. 7.

After analyzing the data in Fig. 10, it becomes apparent that when φ remains constant, the attenuating effect of particles on P_max / P^'_max decrease with the increasing δ. It can be inferred that as the particle shape δ approaches infinity, the influence of particles on P_max / P^'_max almost disappears by observing the trend in Fig. 7. The eight data sets converge and align along the same horizontal line, with P_max / P^'_max close to a value of 1 infinitely, which resembles the trend shown in Fig. 4, where the critical φ for the stratification effect of the shockwave tends towards 0 as δ approaches infinity.

4.3. The influence of particles on the energy of the shockwave

The above analysis primarily focuses on the features of the pressure peak of shockwaves. To comprehensively assess the intensity of the shockwave under the influence of the particle, the energy of the shockwave is further evaluated. The quantification of collapse shockwave energy stemming can be achieved through an assessment of the pressure distribution, as demonstrated in previous studies [35], [69], [70]:

$$E=\frac{4\pi s^2}{{\rho }_0{c}_0}\int P^{2}dt.$$ (2)

In Eq. (2), s denotes the distance between the centre of shockwave emission and the measurement surface, and ρ₀ and c₀ denote the density of the water and the speed of sound in the water, respectively (22 °C, c₀ ≈ 1482.9 m·s⁻¹). Complete waveform of the shockwave is guaranteed in the choice of the integration boundary [36]. Based on Eq. (2), we assessed the alteration in relative energy E / E' of the collapse shockwave from the first bubble cycle affected by different particle shapes δ in Fig. 7, and the results are shown in Fig. 11.

Fig. 11.

Relationship between bubble-particle distance and relative energy of shockwaves.

Fig. 11 shows that the overall trend of the relative energy E / E' increases with the bubble-particle distance φ, followed by a tendency to stabilize with increasing bubble-particle distance φ, with a peak at φ = 2.0. This pattern of E / E' variation aligns closely with the trend of P_max / P^'_max and φ, as shown in Fig. 7. Moreover, for smaller φ, the relative shockwave energy increases with δ under constant φ, which concurs with the trend of P_max / P^'_max and δ shown in Fig. 7.

In Fig. 11, E / E' is obviously reduced in different degrees under the influence of different particle shapes δ, compared to the scenario without particles. The bubble-particle interaction leads to displacement of the particle, partially transferring the bubble energy to the particle [46], [47], [48], [49], [50], [51], [52], [53], [54], [71]. The largest reduction in E / E' occurs when δ are 0.22 and 1.00. As the particle shape δ increases, the degree of reduction in shockwave relative energy E / E' decreases. At δ = 4.50, the reduction in shockwave relative energy E / E' reaches a minimum. Additionally, Fig. 11 shows that when the particle shape δ is equal to or less than 1.00, variations in particle shape δ have a relatively minor influence on the shockwave relative energy E / E'.

5. Discussion

Regarding the critical condition of the stratification effect of the shockwave in Fig. 4, when the particle shape δ changes trend between 0 and 0.22, we have carried out two assumes: (1) when the particle shape δ tends to 0 infinitely, the particle approximates to an infinitely long thin line, the influence of particle on the bubble collapse is negligible, the bubble collapse approximates to spherical collapse in unbound field, and the critical bubble-particle distance φ = 0, the critical condition curve of the stratification effect of the shockwave falls to zero at δ = 0. (2) Even if δ gradually decreases to 0, there is still an influence of particle on the shape of the collapse shockwave, and the critical φ for the stratification effect of the shockwave increases with the decrease of δ and intersects with a point on the y-axis. Based on these two assumes, we make further discussion.

In Fig. 12(a), the bubble collapse in a free field resulted in a single-layer regular spherical shockwave emission. In Fig. 12(b), a horizontal placement of a 0.5 mm diameter steel wire was used to simulate an infinite particle shape δ, with φ = 0.223, the collapse shockwave displayed a regular single-layer spherical structure similar to that in Fig. 12(a). These experimental findings implied that when the value of particle shape δ tends to infinity, the bubble collapse form tended towards a purely spherical collapse and the influence of particle on the shockwave was negligible, which established a critical bubble-particle distance φ of 0.

Fig. 12.

Evolution of the shockwave with different boundary conditions: (a) free field, R_max = 10.233 mm; (b) R_max = 9.375mm, φ = 0.223; (c) R_max = 10.118mm, φ = 0.978; (d) wall, R_max = 9.155 mm, γ = 2.003.

In Fig. 12(c), by placing the wire vertically to emulate a particle with δ tending towards 0 and with φ = 0.927, the bubble volume hit a minimum at t = 2.300 ms, followed by the emergence of a stratified shockwave from its collapse. This empirical observation supported the conclusion that even as the particle shape δ approached 0, it still had some influence on the shockwave shape, which confirmed the validity of the second assumption.

To further investigate the critical bubble-particle distance φ when particle shape δ = 0, we included a set of experiments involving the bubble-wall interaction, as shown in Fig. 12(d). With γ = 2.003, the proximity of γ to the critical bubble-wall relative distance of 2.0 (obtained from numerical simulations by Beig [67]) was evident. The stratified shockwaves were observed, as shown in Fig. 12(d₄). Nevertheless, the degree of stratification in this case was minimal. Since the particle were influenced by the action of the cavitation bubble [46], [47], [48], [49], [50], [51], [52], [53], [54], [72] and possess curved surfaces [43], [44], its influence on the shockwave shape was less compared to that of a wall. Consequently, the critical bubble-particle distance φ of the stratification effect on the shockwave, when δ = 0, was likely to be lower than the critical bubble-wall relative distance of 2.0 derived from numerical simulations by Beig [67].

Based on the above analysis, we set the critical condition functional form as follows:

$${\phi =ae}^{b\delta }$$ (3)

In Eq. (3), a needs to be a positive number less than 2.0, and b needs to be a negative number to ensure that the critical value φ satisfies the condition of being less than 2.0 when δ tends to 0 and the critical value φ = 0 when δ tends to infinity. After the experiment in Section 3, it can be determined that a and b in Eq. (3) are 1.34 and −0.10, respectively, which satisfy the conditions that a is less than 2.0 and b is negative. Therefore, Eq. (3) is finally:

$$\phi =1.34e^{-0.10\delta }$$ (4)

Calculation according to Eq. (4) shows that the critical bubble-particle distance φ = 1.34 for the stratification effect of the shockwave when the particle shape δ = 0.

6. Conclusion

In this paper, we experimentally study the influence of the particle shape on the shape and intensity of collapse shockwaves of bubbles. The following major conclusions are drawn:

(1) The particle near bubble induces a stratification effect on the collapse shockwave of the bubble, and the influence is closely related to φ and δ. When φ is large (i.e., φ > 1.34 $e^{-0.10\delta }$), the particle exerts minimal effect on the bubble collapse, and the collapsing form of the bubble resembles the spherical collapse and the single-layer shockwave is primarily attributed to the bubble implosion. When φ is small (i.e., φ < 1.34 $e^{-0.10\delta }$), the presence of the particle disrupts the medium consistency around bubble, which leads to the irregular bubble collapsing shape. This disruption produces spatial and temporal delays in the microjet-induced water hammer shockwave and the bubble rebound-induced implosion shockwave, culminating in shockwave stratification.

(2) Shockwave intensity variations closely correlate with the stratification effect. The presence of particles resulted in different degrees of attenuation in the intensity of the shockwave. In addition, the degree of attenuation of shockwave intensity is influenced by particle shape δ and bubble-particle distance φ. The attenuating effect of particles on shockwave intensity decreases with increasing particle shape δ. As φ increases, the attenuation of particles on shockwave intensity decreases.

In this paper, the influences of particle shape on the shockwave evolution are investigated experimentally. It reveals the microscopic mechanisms about the reduction of the cavitation erosion by particles under the cavitation pattern of shockwaves in sediment-laden water. These new findings hold theoretical implications that contribute to understanding cavitation erosion mechanisms in sediment-laden flows [73] and enhancing ultrasonic cleaning efficiency, etc. [6], [7].

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.