Experimental investigation on the characteristics of the shock wave emitted by the cavitation bubble near the air bubble

Abstract

This study explores the mitigation of cavitation damage in hydraulic engineering through air entrainment. The primary aim is to experimentally analyze the shock wave characteristics emitted by cavitation bubbles adjacent to air bubbles affixed to a tube nozzle. The schlieren optical system is utilized to visualize the shock wave, while a hydrophone measures its pressure. Experiments are conducted on cavitation bubbles induced by the spark-generated method in the vicinity of air bubbles, varying the dimensionless distances and sizes of the air bubbles. The results indicate that (1) The introduction of an air bubble noticeably changes the morphology, kinematic behavior, and shock wave features of the cavitation bubble. (2) Four distinct shock wave patterns are identified based on the quantity and shape of the shock wave, with variations in the cavitation bubble's collapsing behavior and shock wave characteristics across different patterns. (3) The dimensionless distance γ and size δ exert significant influence on the shock wave's quantity, pressure peak, shape, and energy. With γ decreases or δ increases, the shock wave quantity increases while the shock wave intensity decreases. This investigation of the interaction between cavitation bubbles and air bubbles is essential for elucidating the mechanism through which air entrainment mitigates cavitation damage.

1. Introduction

Cavitation is a hydrodynamic phenomenon including a gas–liquid two-phase, which is prevalent in hydraulic machinery [1], [2], marine engineering [3], [4], underwater explosion [5], [6], and medical application [7]. This phenomenon is characterized by the phase transition process, where the fluid undergoes vaporization when its local pressure falls below the saturated vapor pressure, leading to the formation of cavitation bubbles. The formation of these bubbles triggers oscillation and subsequent collapse, leading to the release of significant energy. This collapse generates shock waves reaching peak pressures of several hundred megapascals (MPa)[8], [9], along with microjets exceeding speeds of 100 m per second (m/s)[10], [11]. Hence, structural noise [12], vibration [13], and damage [14] are caused when the cavitation bubble collapses in the vicinity of different boundaries, and this phenomenon is known as cavitation erosion. Two primary mechanisms responsible for cavitation erosion are the shock wave and high-speed jet generated during bubble collapse. In the field of hydraulic engineering, air entrainment has proven to be an effective method for mitigating cavitation damage [15]. A large amount of experimental results have revealed that air entrainment protects underwater structures (such as dam [16] and tunnel [17]) and metal machinery [18], [19] from cavitation erosion. The cavitation phenomenon is eliminated with the increase of air concentration, and when the air concentration reaches a certain value, it could completely attenuate cavitation erosion [20]. Scholars have proposed that the reduction in cavitation damage through air entrainment can be attributed to the entrained air's attenuation effect on the shock wave and high-speed jet produced during cavitation bubble collapse [21], [22]. This research on reducing cavitation damage through air entrainment takes a micro-level approach, specifically examining the interaction between two kinds of bubbles.

The progression of a cavitation bubble over time contains expansion, contraction, and rebound stages, and the collapse phenomenon occurs at the end of the contraction stage. The shock wave generated during the collapse is a phenomenon of great interest and has been extensively studied due to its high-pressure peak and rapid propagation speed [23], [24]. Researchers have employed experimental devices such as hydrophones and optical systems to investigate shock wave mechanisms under different boundary conditions. Considering the simple phenomenon that the cavitation bubble in the open space. Matula et al. [25] utilized a needle hydrophone to examine the acoustic emissions from a single cavitation bubble in a open space, revealing four shock wave pulses associated with one main collapse pulse and three rebound pulses. Huang et al. [26]. conducted experiments and numerical simulations to explore acoustic waves generated by a single bubble in the open space, identifying compression waves, rarefaction waves, and shock waves during the bubble's expansion, contraction, and collapse stages, respectively. In order to study the influence of the fluid viscosity on the shock wave emission and cavitation bubble dynamics, Brujan [27] employed high-speed photography and acoustic measurements to study a single cavitation bubble in polymer solutions, observing notable reductions in maximum shock wave pressure and extended bubble oscillation times in the presence of polymer additives.

Furthermore, researchers have investigated the interaction between cavitation bubbles and various boundaries. Vogel et al. [28] comprehensively studied laser-induced cavitation bubble dynamics near solid boundaries using optical and acoustic methods, highlighting the dominant role of shock wave energy in spherical bubble collapse, while it plays a minor part in non-spherical bubble collapse. Using time-resolved shadowgraph and hydrophone pressure measurement, Supponen et al. [29] studied the shock wave emission from a non-spherical bubble when the bubble is subjected to a force given by the free surface, rigid boundary and gravity, respectively. They defined an anisotropy parameter and developed a predictive framework for the pressure peak and energy of shock waves. Additionally, the cavitation bubble dynamics near multiple boundaries has been studied. Huang et al. [30] employed the schlieren method to investigate bubble dynamics between free surfaces and rigid boundaries. Their findings revealed intriguing behaviors of shock waves, such as the transformation into rarefaction waves when reflected by free surfaces, as opposed to retaining their characteristics when reflected by rigid boundaries. The cavitation bubble near the elastic boundary has also been investigated due to the compliant materials have the application in cavitation erosion reduction [31], [32]. Brujan et al. [33] conducted a detailed investigation of a laser-induced cavitation bubble near an elastic boundary, showcasing that the influence of the boundary results in a distinctive mushroom-shaped bubble configuration during the collapse phase. Intriguingly, the bubble was observed to undergo a splitting process during this phase, accompanied by the generation of two sets of shock waves. Recently, the application of shock waves in ice breaking has attracted much attention. Cui et al. [34] utilized the shadowgraph method to examine the influence of dual collapsing cavitation bubbles on ice fracturing. Their study revealed diverse bubble shapes, including splitting, counter-jets, and asymmetric toroidal formations. The experiment is carried out by changing bubble–bubble and bubble-boundary distances, which shows that the shock wave is crucial to the fracturing of the ice.

Recent years have witnessed an increased focus on exploring the cavitation bubbles dynamics in the vicinity of air bubbles, driven by the potential of air bubbles to reduce cavitation-induced damage. The presence of air bubbles significantly impacts the cavitation bubble behavior, influencing factors such as collapse direction and subsequent shock wave generation. Xu et al. [35] delineated three potential collapse trajectories for the cavitation bubble concerning the adjacent air bubble: towards, both towards and away from, and away from the air bubble. Moreover, the collapse direction of the cavitation bubble is determined by the distance between the two bubbles and the roundness of the air bubble. Goh et al. [36] conducted investigations on the jet orientation of a cavitation bubble in the vicinity of a solid boundary attached to an air bubble. Their findings postulate that the inter-bubble distance is a determining factor for the jet formation within the cavitation bubble, with the ratio of the cavitation bubble collapse time to the air bubble oscillation time being instrumental in defining the jet direction. Additionally, the size of the air bubble was found to impact the collapse trajectory, with Xu et al. [37] observing that the cavitation bubble tends to collapse towards smaller air bubbles, whilst veering away from larger air bubbles. Their experimental work also reported instances of cavitation bubbles merging with air bubbles, resulting in non-spherically shaped merged entities. In subsequent investigations, Luo et al. [21] utilized pressure measurements to analyze the amalgamation of cavitation and air bubbles, noting a marked reduction in the peak collapse pressure value subsequent to the emergence of a cavitation bubble characterized by its gaseous composition. Li et al. [38] expanded on this area of study, examining the effect of multiple air bubbles on the cavitation bubble dynamics. Their experimental setup included arrangements of two air bubbles on one side, a singular air bubble on each side, and a quartet of air bubbles encircling the cavitation bubble. The findings revealed that the number of air bubbles influenced the cavitation bubble's collapse direction, oscillation time, and collapse pressure.

For the shock wave emitted by the cavitation bubble near the air bubble, many scholars have conducted detailed experiments through hydrophone and optical systems. Tan et al. [22] employed the schlieren method to investigate the collapse behavior of a spark-induced cavitation bubble adjacent to an air bubble, observing a decrease in the intensity of the shock wave and the collapse velocity of the cavitation bubble concomitant with a reduction in the inter-bubble distance. Furthermore, upon the fusion of the two kinds of bubbles, schlieren images indicate that the shock wave manifests as subtle waves surrounding the cavitation bubble. Similarly, Li et al. [39] utilized a hydrophone to assess the collapse strength of the merged bubble, noting a progressive weakening of the collapse sound pressure and maximal contraction speed with an increase in the air bubble radius. Luo et al. [40] conducted research using the shadowgraph method and a pressure measurement system to examine how the presence of an air bubble affects the shock wave generated when a cavitation bubble collapses. The high-speed imagery and pressure measurements revealed the critical roles played by the inter-bubble distance and air bubble size in influencing the stratification of the shock wave, in addition to affecting the maximum pressure and the energy contained within the shock wave.

Based on a review of the cited literature, researchers have extensively investigated shock wave emission from cavitation bubbles in various scenarios, encompassing free-field conditions, proximity to different boundaries, and the interactions between cavitation and air bubbles. However, the research on the shock wave emitted by the cavitation bubble near the air bubble is insufficient. Some researchers have employed shadowgraph and schlieren methods to capture shock waves. However, there is a notable gap in the comprehensive analysis of shock wave characteristics, including their quantity and shape. In addition, the schlieren method has been identified as possessing superior sensitivity for capturing minute details of the shock wave as compared to the shadowgraph method [41]. Although hydrophones have been employed to gauge the pressure of shock waves, their current usage has been limited to discussing the intensity of shock waves during air bubble collapse at different distances from the cavitation bubbles. There is a lack of systematic exploration of the acoustic variations in the collapse period of cavitation bubbles near air bubbles, and no comprehensive elucidation of the functional relationship between shock wave intensity and dimensionless distance has been provided. Therefore, this study aims to provide a thorough comparative analysis of the dynamic features of cavitation bubbles in both free-field and near-air bubble conditions, with a particular emphasis on assessing how the presence of the air bubble affects the morphology and intensity of shock waves. Furthermore, this experiment utilizes the schlieren method and a hydrophone to examine the attributes of the shock wave emitted by the cavitation bubble when it is in proximity to an air bubble. Through the qualitative analysis of the schlieren images and the quantitative analysis of the pressure data, the paragraph discusses the influence of the distance between the two bubbles on the quantity, shape, pressure, and energy of the shock wave, as well as the impact of air bubble size. In addition, the relationship between the quantity and intensity of shock waves is further explored. The objectives of this work are twofold: (1) to clarify how air bubbles influence the dynamic behavior of cavitation bubbles, and (2) to investigate the characteristics of shock waves across a broad range of dimensionless parameters.

2. Experimental setup

2.1. Bubble generation

The experimental apparatus' schematic diagram is presented in Fig. 1. The experiment involved the creation of the cavitation bubble just above the air bubble, achieved through bubble generator. The methodology employed to generate the cavitation bubble is the spark-charged technique, built by Zhang et al. [42] and further developed into a schlieren observation platform by Huang et al. [30]. The bubble generator is comprised of a 220 V AC power source and a $6.6\times {10}^{-3}$ F Capacitive element, with the capacitor's terminals connected to copper wires fitted with positive and negative electrodes, respectively. The procedure begins by charging the capacitor from the power source, regulating the charging voltage to 600 V for each trial. The positive and negative electrodes are then brought into contact to form a connection point, at which point the capacitor discharges, leading to the creation of the cavitation bubble through the release of intense heat from the electrodes. Experimental repetitions have determined that the range of the cavitation bubble's radius is 17.5 ± 2.5 mm, with its initial center consistently positioned at the connection point. Consequently, the accurate determination of the cavitation bubble's original placement is feasible.

Fig. 1.

The schematic diagram of the experimental setup.

In Fig. 1, the cavitation bubble is formed within a 0.5 × 0.5 × 1 m transparent water container, part of the bubble generator assembly. Exceeding 15 times the largest diameter of the bubble, the spatial separation between the transparent water boundaries and the central point of the cavitation bubble eradicates any minor boundary effects on the bubble's conduct. The spatial separation between the transparent water boundaries and the cavitation bubble's central point exceeds 15 times the bubble's maximum radius, thereby negating the negligible boundary effects on the behavior of the bubble. The bubble generator, oscilloscope, and high-speed camera are controlled by the synchronization unit. The bubble generator's manual discharge prompts the simultaneous triggering of the oscilloscope and high-speed camera, allowing the hydrophone to record pressure data and the high-speed camera to capture the transient bubble's temporal evolution.

2.2. Air bubble system

The experimental setup consists of an air bubble system, as depicted in Fig. 1, which comprises an air pump, a throttle valve, and an air tube. The air pump generates the air bubble, with its production rate controlled by the throttle valve. It has been observed that the air bubble tends to rise more readily when the air tube's airflow velocity is marginally increased. When the air bubble rises, the original structure of the air bubble undergoes significant alterations due to surface tension, varying considerably between experiments. Additionally, the velocity at which the air bubble ascends differs, posing a challenge to the experimental design. In order to mitigate these variables, the experiment is structured to keep the air bubble affixed to the tube's spout while generating the cavitation bubble. This approach ensures consistency in the original form and speed of the air bubble across all trials, thereby eliminating the confounding effects of the air bubble's shape and speed. Additionally, adjusting the throttle valve regulates the air tube's flow rate to nearly zero, causing both the air bubble and the gas inside the tube to remain motionless at the start of the experiment. The pressure at tube's spout is assumed to be equal to the hydrostatic pressure at that particular depth.

2.3. Schlieren optical system

In addition, the schlieren optical system, delineated in Fig. 1, encompasses the entirety of the light path section and is employed for the purpose of capturing the shock waves emanating from the cavitation bubble. The schlieren optical system is comprised of a lighting element, bulging lens, gap, flat mirrors, schlieren mirrors, and a schlieren cut. After passing through the calculation section, which is characterized by a non-uniform refractive index resulting from variable flow field density, the parallel rays are converged by plane mirror 2 and then directed through the schlieren cut. This process results in a schlieren image exhibiting gradients of light and shade, with the contrast of the image being indicative of the distribution of spatial changes in the light's refractive index's first derivative within the measured flow field. Furthermore, the light refractive index is directly proportional to the flow field density, implying that any disturbances altering the fluid density concurrently affect the fluid's light refractive index. Consequently, the schlieren optical system visualizes the shock waves, which are then captured by a high-frame-rate camera operating at 27,000 frames per second, with an exposure time of 0.39 μs per frame. The specific camera models are shown in Table. 1.

Table 1.

Physical picture and main performance parameters of high-speed camera.

In Fig. 2(a), the schlieren optical system visualizes the shock wave, while the hydrophone is positioned at the periphery of the image. The hydrophone is set at a distance of 54 mm beyond the midpoint of the cavitation bubble, which is used for measuring the pressure of the shock wave. The hydrophone used in this study is manufactured by Müller-Platte, and its sensitive material is Piezoelectric PVDF. The pressure measurement range of the hydrophone is −10 to 150 MPa, which is suitable for the measurement scale of the shock wave emitted by the cavitation bubble in water. The hydrophone has a small sensitive diameter of less than 0.5 mm and an extremely short rise time of about 50 ns. The maximum frequency bandwidth of the hydrophone is 11 MHz, which is capable of a detailed sampling of the shock wave. The specific technical data of Müller-Platte’s PVDF-type hydrophones are shown in Table. 2. After converting the pressure data recorded by the hydrophone into electrical voltage, the oscilloscope records the information. The electrical signal can be converted into a pressure signal following the sensitivity of 12.8 mV/1MPa. Finally, the analysis and unit conversion calculation are performed on the pressure data, displayed in Fig. 2(b).

Fig. 2.

(a) Typical schlieren image of the shock wave. (b) Typical hydrophone pressure signal of the shock wave emitted at the collapse stage of the cavitation bubble.

Table 2.

The technical data of Müller-Platte’s PVDF-type hydrophones.

Pressure range	−100 to 1500 bar (-10 to 150 MPa)
Sensitive element	Piezoelectric PVDF
Rise time	50 ns
Sensitive diameter	≪ 0.5 mm
Sensitivity	About 0.3 pC/bar or 1 mV/bar incl. 2 m cable
Bandwidth	0.3 – 11 MHz ± 3.0 dB
Temperature range	Max. 60 °C

2.4. Dimensionless parameters

Both the spatial distance from the cavitation bubble to the air bubble and the air bubble's size play a pivotal role in shaping the dynamics of the cavitation bubble. As shown in Fig. 3, the utmost diameter of the cavitation bubble is established as

$$R_{\max }=\sqrt{S_C/\pi }$$ (1)

Where parameter S_C denotes the largest surface of the cavitation bubble as extracted from the experimental images.

Fig. 3.

The schematic diagram of the parameters.

At the start of the experiment, the intentional attachment of the air bubble to the tube's outlet positions the initial distance between the two types of bubbles to be the same as the length from the center of the cavitation bubble to the air tube's nozzle (L). The definition of γ, the dimensionless distance between the two types of bubbles, is established as

$$\gamma =\frac{L}{R_{\max }}$$ (2)

Variations in the inner diameter of the air tube (D) directly affect the size of the air bubble. In the present experiment, ranging from 4 mm to 10 mm, the inner diameter of the air tube is noted. To provide a quantitative description of the air bubble's size, the maximum radius of the air bubble is established as

$$r_{\max }=\sqrt{S_A/\pi }$$ (3)

Where parameter S_A represents the largest region of the air bubble measured from experiments. Therefore, the dimensionless size between the two types of bubbles δ is established as

$$\delta =\frac{r_{\max }}{R_{\max }}$$ (4)

Furthermore, the experiment is meticulously designed to encompass a wide spectrum of dimensionless distances γ and dimensionless sizes δ, by varying parameters L and r_max respectively.

3. Results and discussion

3.1. Comparison of the dynamic characteristics of the cavitation bubble in both open space and near the air bubble

3.1.1. The temporal evolutions of the cavitation bubble in the open space and near the air bubble

As illustrated in Fig. 4(a), the temporal sequence of a spherical cavitation bubble's oscillation in an open field is partitioned into three discernible periods: expansion (t = 01.850 ms), contraction (t = 2.5903.293 ms), and rebound (t = 3.3304.440 ms). In the expansion period, the cavitation bubble originates and subsequently expands to its utmost size, reaching a radius of R_max = 17.2 mm. Concurrently, the electrode discharge triggers the generation of a high-temp, high-pressure plasma, resulting in a plasma shock wave at t = 0.037 ms. During the shrink stage, the cavitation bubble contracts to the minimum size at t = 3.293 ms, exhibiting an almost spherical shape. In rebound phase, the cavitation bubble undergoes collapse followed by expansion. The non-condensable gas within the cavitation bubble counteracts the external fluidic pressure at t = 3.330 ms, releasing a substantial amount of energy that concludes with the creation of a shock wave. This event is observable in the schlieren image as “bright and dark arcs,” with the shock wave exhibiting spherical symmetry. In an open space, the cavitation bubble's collapse generates a solitary strong shock wave. At t = 3.700 ∼ 4.440 ms, the interaction between the primary shock wave and the water tank boundaries causes the generation of reflected shock waves. This collision also induces vibrations in the tank boundary, resulting in the creation of extra waves. The repeated reflections ultimately render the image background chaotic. Throughout the rebound phase, the central locus of the cavitation bubble remains relatively constant, with negligible alterations in its morphology or volume.

Fig. 4.

The comparison of the temporal evolutions of the cavitation bubble in the open space and near the air bubble. (a)Temporal evolution of the cavitation bubble in the open space. (b) Temporal evolution of the cavitation bubble near the air bubble for γ = 1.48 and δ = 0.76.

Fig. 4(b) illustrates the time-based development of the cavitation bubble in proximity to the air bubble, characterized by a dimensionless distance γ = 1.48 and a dimensionless size δ = 0.76. In the expansion phase (t = 0 ∼ 1.480 ms), the cavitation bubble reaches its utmost size, measuring R_max = 17.2 mm. Simultaneously, at t = 0.037 ms, the electrodes discharging causes the formation of a plasma shock wave. During the contraction phase (t = 2.220 ∼ 3.108 ms), the cavitation bubble diminishes to its smallest size, the nearby air bubble reaches its utmost size. This differential in volumetric changes between the two bubbles imparts a conical shape to the cavitation bubble, characterized by a flat base juxtaposed with a spherical apex. This differential in volumetric changes between the two bubble imparts a conical shape to the cavitation bubble, characterized by a horizontal foundation and rounded peak. During rebound stage (t = 3.145 ∼ 4.070 ms), the cavitation bubble crumbles and bounces back. At t = 3.145 ms, multiple shock waves are released by the breakdown of the cavitation bubble adjacent to the air bubble, which is quite different from the case of the open space in the amount of the shock wave at 3.330 ms. Furthermore, the shape of the shock waves is not completely spherical symmetry. While t = 3.330 ∼ 4.070 ms, the emergence of reflected shock waves results in a disordered background in the image. During the entire rebound process, the morphology and volume of the cavitation bubble undergo significant changes, leading to complex motion of the bubble. Initially, the cavitation bubble undergoes secondary expansion along two opposite directions, one oriented towards the air bubble and the other directed in the opposite direction, forming a “plus sign” shape that gradually intensifies over time. Until t = 3.700 ms, the lower jet breaks through the cavitation bubble, initiating the collapse of the lower portion of the cavitation bubble, which then contracts towards the upper end. Simultaneously, the cavitation bubble continues to expand in the direction opposite to the air bubble.

To further compare the dynamic characteristics of the cavitation bubble in both the open field and in close proximity to the air bubble, Fig. 5 displays the kinematic behavior in these two scenarios. The kinematic parameters under investigation encompass the variations in bubble size, speed, and acceleration. The bubble size is ascertained through the quantification of the schlieren image, while the bubble's velocity and acceleration are derived from the first and second derivatives of the bubble radius, respectively. Since the dimensionless bubble size δ and dimensionless distance γ both affect the radius, velocity, and acceleration of cavitation bubbles, we opted for the most classical air bubble influence model, MSW. In this model, the shock wave intensity of cavitation bubble collapse sharply decreases, and the effect of air bubbles on cavitation bubbles is maximal. The experimental images represent the combined results of multiple identical conditions, and the results have all undergone the following dimensionless processing:

$$R^{\ast }=R/R_{\mathit{MAX}}{V}^{\ast }=V\ast T/R_{\mathit{MAX}}{A}^{\ast }=A\ast T\ast T/R_{\mathit{MAX}}$$

$R_{\mathit{MAX}}$ represents the maximum radius of the cavitation bubble, and T is the collapse period of the cavitation bubble.

Fig. 5.

The comparison of the kinematic behaviors of the cavitation bubble in the open space and near the air bubble.

According to Fig. 5, the maximum bubble radiuses in both the open space and proximity to the air bubble exhibit nearly identical measurements. However, the time to expand to the maximum radius is different, which is 1.480 ms and 1.850 ms for the air bubble and open space, respectively. The discrepancy is attributed to the air bubble experiencing a more substantial initial acceleration, leading to a higher expansion velocity compared to the open space. During the shrink stage, the cavitation bubble adjacent to the air bubble contracts until reaching its minimum radius at t = 3.108 ms, while the cavitation bubble in the open space maintains a radius of 10 mm. Similarly, this occurs because the air bubble exhibits greater acceleration and velocity compared to the open space. Therefore, the acceleration and velocity of the cavitation bubble in both the expansion and shrink phases are escalated due to the impact of the nearby air bubble. In addition, either in the open space or near the air bubble, the acceleration decreases to a minimal value of about 5 × 103 m/s2 and the velocity reduces to a zero value when the cavitation bubble attains its utmost size. Subsequently, as the cavitation bubble contracts, the acceleration increases to a value of about 30 × 103 m/s2, and the velocity rises to a value of about 20 m/s. This indicates that the cavitation bubble undergoes deceleration during expansion and accelerates during shrinkage. Based on the preceding analysis, it is concluded that while the magnitudes of the cavitation bubble's acceleration and velocity are impacted by the air bubble, the trends in the bubble's radius, velocity, and acceleration remain unaffected by the presence of the air bubble.

3.1.2. The characteristics of the collapse shock wave in the open space and near the air bubble

Analysis of Fig. 4 leads to the conclusion that the shock wave characteristics exhibit significant disparities between the open space and the vicinity of the air bubble. Hence, shock wave pressures in these two cases recorded by the hydrophone are presented. As shown in Fig. 6, the pressure data detected by the hydrophone exhibits two peak values (a and b). High-speed camera images reveal that during the initial generation of cavitation bubbles by electrical sparks, the plasma impact generates substantial energy, covering the entire photo with intense light. At this moment, it precisely corresponds to the maximum point (a) in the hydrophone pressure data. The collapse of the cavitation bubble also generates a strong shock wave, aligning with the peak point (b) in the hydrophone pressure data. By comparing the hydrophone data with the timing of photos captured by the high-speed camera, it can be confirmed that these two points precisely correspond to the plasma shock wave and the collapse shock wave. The pressure of the peak point a is 7.969 MPa, and the pressure of the peak point b is 3.000 MPa. According to the previous study, the following equation accurately describes the correlation between the shock wave's pressure (p_s) and its velocity (u_s) [24]:

$$p_s=c_1{\rho }_0{u}_s({10}^{(u_s-c_0)/c_2}-1)+p_{\infty }$$ (5)

Where ρ₀ is the water density before compression by the shock wave (ρ₀ = 1000 kg/m³), c₀ denotes the speed of sound within the water, and p_∞ is the hydrostatic pressure (p_∞ ≈ 101325 Pa). In addition, c₁ and c₂ are the constants, corresponding to 5190 m/s and 25306 m/s. The initial velocity of the shock wave reaches 4000 m/s, and the initial pressure of the shock wave is as large as 10GPa [43], [44]. However, the velocity and pressure of the shock wave attenuate dramatically over time, and for the spark-induced bubble, the velocity of the shock wave decreases rapidly to the sound speed in the water within 0.01 ms [26]. In the present experiment, the distance between the hydrophone and the center of the cavitation bubble is 54 mm, so the estimate for the velocity of the shock wave u_s is around 1500 m/s when the shock wave arrives at the position of the hydrophone. According to the images at t = 3.330 ∼ 3.367 ms in Fig. 4(a), the velocity of the shock wave u_s is calculated as 1505 m/s by image processing. Therefore, the pressure of the shock wave p_s calculated by Eq. (5) is 3.656 MPa, which is close to the value of the collapse shock wave measured by the hydrophone. This shows that the pressure of the shock wave decays violently from the shock wave center to the pressure measuring point.

Fig. 6.

The pressure in the open space and near the air bubble recorded by the hydrophone. (a) The pressure in the open space recorded by the hydrophone. (b) The pressure near the air bubble for γ = 1.48 and δ = 0.76 recorded by the hydrophone.

According to Fig. 6(b), the plasma shock wave’s pressure in close proximity to the air bubble registers at 7.890 MPa, a value akin to the pressure observed in the plasma shock wave within the open space. In fact, since the voltage for charging is nearly identical (600 ± 20 V), there are only minor differences in the plasma shock wave’s pressure in each experiment. However, the collapse shock wave’s pressures in open space and near the air bubble have a great difference. The collapse shock wave's maximum pressure in the vicinity of the air bubble reaches 0.688 MPa, markedly less than the 3.000 MPa pressure in an open space. This indicates that the air bubble has the ability to significantly attenuate the shock wave’s pressure emitted by the cavitation bubble. Besides, there is a single strong shock wave in an open space, while the collapse of the cavitation bubble in proximity to the air bubble generates multiple shock waves. It is assumed that with an increasing number of shock waves, the energy of the shock wave becomes more dispersed, resulting in the collapse shock wave’s pressure drop. Moreover, comparing the collapse shock wave’s pressure curves under the two cases, there exists more pressure fluctuation near the air bubble.

As illustrated in Fig. 6, there are periodic pressure fluctuations during the expansion, shrink and rebound stages. Therefore, the pressure fluctuation frequency spectrum is presented based on the Fast Fourier Transform (FFT) method, as shown in Fig. 7. Whether it is the expansion and shrink stages or the rebound stage, the primary frequency of the pressure is approximately 3000 Hz, whether in the open space or in close proximity to the air bubble. This observation underscores the persistent value of the wave frequency emitted by the cavitation bubble. The dominant frequency of the wave does not change with the variation of boundary conditions and time evolution.

Fig. 7.

(a) Pressure frequency spectrum in the open space and near the air bubble during the expansion and shrink stages. (b) Pressure frequency spectrum in the open space and near the air bubble during the rebound stage.

The analysis from Fig. 6 indicates that the existence of the air bubble significantly diminishes the collapsing shock wave’s peak pressure. As a result, there is a need for a comprehensive investigation into how the presence of the air bubble affects the collapse power of the cavitation bubble. To facilitate this inquiry, it is imperative to quantitatively assess the energy associated with the collapse shock wave. The shock wave's energy, denoted as E_S and derived by Vogel et al [45], is calculated as follows:

$$E_S=\frac{4\pi r^2}{\rho c}\int P^{2}dt$$ (6)

Where ρ denotes the water density (ρ = 1000 kg/m³), c denotes the velocity of sound within the water (c = 1500 m/s), and r denotes the space between the origin of the shock wave and the location of pressure measurement (r = 0.054 m in present experiment). The energy is obtained by integrating the pressure curve along with the collapse shock wave's entire time range. By Eq. (6), the collapse shock wave’s energy in the open space and adjacent to the air bubble are 0.395 J and 0.020 J, respectively. This indicates that the air bubble decreases the collapse shock wave’s energy by 95 %, which exerts a substantial influence in diminishing the collapse potency of the cavitation bubble.

The content above reflects the impact of the air bubble on the shock wave’s characteristics emitted by the cavitation bubble. Whether the air bubble is present or not, there remains a variation in the ratio between the shock wave's energy at the measuring point and that of the cavitation bubble. The cavitation bubble’s energy, which is called the potential energy, varies gradually with the time evolution. The bubble potential energy transforms into the flow field’s motion energy and the shock wave’s energy during the entire evolution process. The cavitation bubble’s potential energy E_B is obtained from the following equation [46]:

$$E_B=\frac{4\pi }{3}{R}^3(P_{\infty }-P_B)$$ (7)

Where R represents the cavitation bubble’s radius, P_∞ represents environmental pressure, and P_B represents internal pressure of the cavitation bubble. The (P_∞ - P_B) represents the pressure variance between the interior and exterior of the cavitation bubble. However, the environmental pressure P_∞ and internal pressure within the cavitation bubble P_B fluctuate throughout the evolution process, presenting challenges in the measurement of P_B. Therefore, the pressures P_∞ and P_B are not acquired separately. The (P_∞ - P_B) is obtained by the Rayleigh-Plesset [47] equation:

$$P_B-P_{\infty }=\rho (R\ddot{R}+\frac{3}{2}{\dot{R}}^2+\frac{4\mu }{R}\dot{R}+\frac{2\sigma }{\rho R})$$ (8)

Where ρ represents water’s density (ρ = 1000 kg/m³), μ represents the dynamic viscosity of the water (μ = 1 × 10^-3Pa∙s), σ denotes the surface tension (σ = 0.0728 N/m), and $R,\dot{R},\stackrel{⃛}{R}$ represent the cavitation bubble's radius, speed, and acceleration, respectively.

Utilizing Eqs. (7), (8) along with the information from Fig. 5, the cavitation bubble’s potential energy is computed both in the open space and in the vicinity of the air bubble, as displayed in Fig. 8. The maximum potential energy occurs at the moment of the cavitation bubble’s largest diameter. The open space exhibits a maximum potential energy of 3.676 J, while near the air bubble, it peaks at 4.334 J. The energy of the shock wave, at a measuring point 54 mm from its emission center, is 0.395 J in the open space and 0.020 J near the air bubble. Therefore, the shock wave energy occupies 10.75 % and 0.46 % of the cavitation bubble energy in open space and near the air bubble, respectively. This demonstrates the considerable reduction in shock wave energy emitted by the cavitation bubble due to the presence of the air bubble.

Fig. 8.

The variation of the radius and potential energy of the cavitation bubble in the open space and near the air bubble.

3.2. The characteristics of the shock wave emitted by the cavitation bubble near the air bubble at different patterns

3.2.1. Global shock wave patterns under various dimensionless parameters

From Section 3.1, the presence of the air bubble significantly alters the cavitation bubble's morphology and kinematic behavior of the cavitation bubble, as well as influencing the quantity, shape, pressure peak, and the resultant shock wave’s energy. Therefore, the shock wave’s characteristics engendered by the cavitation bubble in proximity to the air bubble deserve further study. The non-dimensional range γ and non-dimensional size δ are critical parameters affecting the shock wave’s characteristics. In the current experimental setup, a comprehensive range of γ values from 1.0 to 3.0, and δ values from 0.2 to 1.0 were explored. Notably, distinctive phenomena pertaining to the shock wave witnessed in connection with the shock wave as the cavitation bubble collapsed. As shown in Fig. 9, the experimental outcomes have been classified into four distinct patterns based on the quantity and morphology of the shock wave, namely: single strong shock wave (SSSW), several shock waves (SSW), multiple shock waves (MSW), and merged weak shock waves (MWSW). For the SSSW pattern, the shock wave in this pattern resembles the shock wave in the open space, which speculates that the shock wave’s intensity is relatively strong. The shock wave is singular in quantity, and it exhibits a high degree of spherical symmetry in shape. For the SSW pattern, there is an increase in the shock wave’s quantity, resulting in a reduction in the spherical symmetry compared to the previous pattern. For the MSW pattern, the shock wave’s quantity is maximal, and the shape is not completely spherical symmetry. For the MWSW pattern, the air bubble combines with cavitation bubble, thereby significantly attenuating the shock wave’s intensity to the extent that it becomes nearly imperceptible. The resultant shock waves manifest as few slight waves to the cavitation bubble’s right, devoid of spherical symmetry.

Fig. 9.

Overview of the shock wave under various dimensionless parameters, four classical phenomena are observed, namely single strong shock wave (SSSW), several shock waves (SSW), multiple shock waves (MSW), and merged weak shock waves (MWSW).

Experimental observations indicate a higher prevalence of the SSSW pattern when the γ value is the largest and the δ value is the smallest. This implies a substantial separation between the two kinds of bubbles, coupled with the diminutive size of the air bubble, which collectively result in a shock wave pattern akin to that observed in the open space. With the decrease of γ and increase of δ, the shock wave pattern is changed. The transition between the SSSW pattern and the SSW pattern occurs at δ = 0.8γ-1.1. In addition, the division between the SSW pattern and the MSW pattern appears at δ = 1.65γ-2.1, and the demarcation between the MSW pattern and the MWSW pattern is δ = 2.8γ-3.3. In the MWSW pattern, which is observed at the smallest γ and largest δ values, the merging of the two kinds of bubble is evident, thereby emphasizing the profound influence of the air bubble on the emitted shock wave. This indicates that the evolution forms of shock waves are influenced by the combined effects of dimensionless distance γ and dimensionless bubble size δ. When the dimensionless distance is sufficiently large, and the dimensionless bubble size is small, the impact of the air bubble on the collapse of the cavity is weak. Thus, the cavity collapse can be approximated as an open space, resulting in only one high-energy shock wave, namely the SSSW mode. As the dimensionless distance decreases and the dimensionless bubble size increases (γ decreases and δ increases), the air bubble begins to affect the collapse of the cavity. The collapse of the cavity produces a stronger shock wave, which propagates to the surroundings. After passing through the air bubble, the lower shock wave reflects multiple weak reflected waves from the air bubble, increasing the number of shock waves, defining the SSW mode. As γ further decreases and δ increases, during the cavity contraction process, the air bubble gradually approaches the cavity. According to Ma et al [48]. (https://doi.org/10.1007/s42241-019–0056-7), during cavity contraction, compression waves are released, and these waves continue as compression waves after reflecting off the air bubble. This process reduces the pressure gradient around the cavity, resulting in a decrease in the intensity of the shock wave. When the cavity contracts to a minimum, it releases a shock wave of high intensity, where the compression waves interact with the rebounded compression waves, dispersing into numerous small waves, known as the MSW mode. When γ is at its maximum and δ is at its minimum, during cavity collapse, the air bubble and the cavitation bubble have already merged. The energy released by the cavitation bubble is neutralized by the air bubble, and no shock wave is generated. Further sections will elaborate on the specifics of the cavitation bubble's collapsing behavior and the distinctive attributes of the shock wave within each pattern.

3.2.2. Collapsing behavior of the cavitation bubble near the air bubble at four shock wave patterns

The generation of shock waves occurs upon the collapse of the cavitation bubble, hence the cavitation bubble’s collapsing behavior at four shock wave patterns is investigated, as shown in Fig. 10. A typical example of the SSSW pattern for γ = 2.23 and δ = 0.36 is displayed in Fig. 10(a). At t = 3.552 ms, a single strong shock wave is induced by cavitation bubble collapse, and the shock wave’s shape is highly spherical symmetry, which is comparable to the shock wave in the open space. During the entire rebound process, the rebounded bubble oscillates in one position, and its morphology and volume do not change significantly. Due to both the sizable separation between the two types of bubbles and the air bubble's small dimensions, the air bubble’s influence on the cavitation bubble is small, which leads to the cavitation bubble’s dynamics in the SSSW pattern being similar to that in the open space.

Fig. 10.

The collapsing behavior of the cavitation bubble near the air bubble at four shock wave patterns. (a) SSSW pattern for γ = 2.23 and δ = 0.36. (b) SSW pattern for γ = 1.76 and δ = 0.61. (c) MSW pattern for γ = 1.54 and δ = 0.75. (d) MWSW pattern for γ = 1.37 and δ = 0.90.

Fig. 10(b) illustrates the cavitation bubble’s temporal development when the value of γ decreases to 1.76 and δ increases to 0.61. At t = 2.997 ms, the cavitation bubble undergoes collapse, emitting multiple shock waves, exhibiting an increased quantity compared to the SSSW pattern. Furthermore, the shock wave’s spherical symmetry in the SSW pattern is not as good as that in the SSSW pattern by comparing the two collapsing images. During the rebound stage (t = 2.997 ∼ 3.700 ms), the cavitation bubble changes greatly in shape and volume, and it expels a jet in the opposite direction of the air bubble. From the comparison of the SSW pattern and SSSW pattern, it is found that the cavitation bubble's dynamics are distinctly affected by both the separation between the two bubbles and the size of the air bubble, prominently influencing the shock wave’s traits and the rebounded bubble’s path.

In the instance where the formation of the cavitation bubble occurs near the air bubble., characterized by γ = 1.54 and δ = 0.75, the results are depicted in Fig. 10(c). At t = 3.293 ms, the generation of the shock waves is observed around the cavitation bubble. At t = 3.330 ms, multiple shock waves are observed throughout the whole image. For shock waves, the MSW pattern boasts the highest quantity. Furthermore, the shock wave’s spherical symmetry in the MSW pattern is further reduced by comparing it with the two previous patterns. During the rebound stage, the cavitation bubble’s movement differs significantly from that in the SSW pattern. Within the time frame of t = 3.330 ∼ 3.626 ms, the cavitation bubble split, producing two axial jets moving in opposite directions. In the vicinity of the air bubble, the bottom stream tilts downwards, while the top stream recoils upwards. Moving to the time interval at t = 3.922 ∼ 4.440 ms, the lower jet undergoes a reversal in its trajectory, eventually coalescing with the upper jet to culminate in a unified bubble ascending upwards.

A classic representation of the Merged Weak Shock Waves (MWSW) pattern, typified by the smallest γ = 1.37 and the largest δ = 0.90, is illustrated in Fig. 10(d). During t = 2.701–––2.775 ms, the fusion of the cavitation and air bubbles is conspicuous. At t = 2.738 ms, the emergence of slight waves encircling the cavitation bubble is noted, though their intensity is so low that they are nearly imperceptible. Moreover, the deviation from spherical symmetry in the shock wave's morphology is apparent. From t = 2.960–––3.700 ms, the rebound phase is marked by the cavitation bubble slowly moving away from the air bubble. At the time point t = 2.960 ms, the cavitation bubble starts to separate from the air bubble, there is a noticeable lack of jet formation by the cavitation bubble. Furthermore, in the interval t = 3.330–––3.700 ms, the schlieren images do not capture any reflected shock waves. The analysis indicates that the fusion of the cavitation and air bubbles has the dual effect of reducing jet velocity and substantially diminishing the shock wave’s intensity.

According to the description of the four shock wave patterns, it is found that the non-dimensional distance γ and non-dimensional size δ significantly influence the cavitation bubble’s collapsing behavior and the shock wave’s characteristics. Specifically, the shock wave’s quantity and shape vary greatly with the parameters γ and δ. When the cavitation bubble remains unmerged with the air bubble, a decrease in γ and an increase in δ lead to an elevation in the shock wave’s quantity, accompanied by a reduction in the spherical symmetry. Upon the fusion of the cavitation bubble and the air bubble, it generates non-spherically symmetric slight shock waves. This section discusses the combined influence of the parameters γ and δ on the shock wave’s quantity and shape. In subsequent sections, the analysis extends to the influence of individual parameters on the shock wave, particularly focusing on shock wave intensity.

3.2.3. The influence of the dimensionless distance on the characteristics of the shock wave

Fig. 11 displays temporal changes in the cavitation bubble near the air bubble at different γ and the same δ (about 0.60), which aims to investigate the distance influence on the shock wave characteristics. In case (a), γ = 2.49 and δ = 0.58, it is evident that this case is the SSSW pattern. The cavitation bubble spherically expands and shrinks, and a single strong shock wave is observed at t = 3.219 ms. Furthermore, the shock wave shape is highly spherical symmetry. Clearly, case (b) is the SSW pattern with γ = 1.76 and δ = 0.60, several shock waves are induced by the cavitation bubble collapse at t = 3.700 ms, while the shock wave shape is relatively spherical symmetry. In the comparison between case (a) and case (b), a substantial decrease in the γ value does not bring about significant changes in the shock wave, and its shape continues to exhibit spherical symmetry. It is indicated that the shock wave characteristics exhibit a subtle difference when γ surpasses 1.76. In case (c), γ = 1.52 and δ = 0.63, this case is similar to the MSW pattern. At t = 3.478 ms, multiple shock waves are generated, which is quite different from cases (a) and (b) in the shock waves number. Moreover, the shock wave shape in the MSW pattern is not completely spherical symmetry. Comparing case (b) and case (c), a little difference of γ causes the dramatical variation in the quantity of the shock wave, which illustrates that the turning point that caused this drastic change occurs at 1.5 ∼ 1.76. Finally, in case(d), γ = 1.39 and δ = 0.60, apparently this case is the MWSW pattern. The small distance leads to the fusion between the two kinds of bubbles, which further causes an extreme reduction in the shock wave intensity. During the period t = 3.774 ms, some weak waves appear around the cavitation bubble, displaying a shape that deviates from spherical symmetry. From the above analysis, when the δ is constant and with the decrease of γ, the shock wave quantity is increased and the shock wave spherical symmetry is decreased before the fusion between the two kinds of bubbles. As γ further decreases, the shock wave generated by the fusion of the two kinds of bubbles appears as slight waves.

Fig. 11.

The time evolution of the cavitation bubble near the air bubble at different γ and the same δ: case (a) γ = 2.49 and δ = 0.58; case (b) γ = 1.76 and δ = 0.60; case (c) γ = 1.52 and δ = 0.63; case (d) γ = 1.39 and δ = 0.60.

The quantity and shape of the shock wave are significantly influenced by γ, but the impact of γ on the shock wave characteristics is not explicitly addressed. Hence, the analysis based on the pressure data is carried out. The pressure data recorded by the hydrophone that corresponds to all the cases in Fig. 11 is presented in Fig. 12. In case (a), γ = 2.49 and δ = 0.58, the shock wave’s maximum pressure during collapse is 1.808 MPa. Furthermore, the shock wave’s energy is calculated as 0.138 J by applying Eq. (6). Compared to the pressure in the open space in Fig. 6(a), the quantity and shape of the shock wave in case (a) are quite similar to that in the open space, whereas there is a large difference in the shock wave’s maximum pressure. This illustrates that air bubble is helpful to reduce the shock wave intensity. Within the context of case (b), a decrease in γ corresponds to an elevation in the shock wave's quantity, resulting in a maximum pressure and shock wave energy of 1.637 MPa and 0.114 J, respectively. With the contrast of case (a) and case (b), it is concluded that the reduction of the distance between the two bubbles leads to a rise in shock wave quantity, which further causes a reduction in shock wave’s maximum pressure. In case (c), the shock wave reaches its maximum quantity when γ decreases to 1.52. Furthermore, the shock wave achieves its maximum pressure, reaching 1.250 MPa, and its maximum energy, amounting to 0.057 J. Comparing the three cases (a), (b), and (c), it is found that the shock wave's intensity decreases swiftly when the shock wave’s quantity dramatically increases. In case (d), γ = 1.39 and δ = 0.60, the presence of multiple slight waves emitted by the merged bubble suggests an extremely low intensity of the waves. Accordingly, there is no pressure peak in the pressure data at the collapse stage. This observation supports the hypothesis that the combination of the two kinds of bubbles significantly weakens their collapse, approaching nullity. It is plausible that this phenomenon is a result of condensable air entering the interior of the cavitation bubble, leading to an increase in its minimum volume and the absorption of collapse energy. In summary, the analysis of the pressure data unequivocally confirms the inverse relationship between γ and both the utmost pressure and shockwave intensity. Additionally, the intensity of the shock wave witnesses a significant attenuation after the bubbles merge.

Fig. 12.

The pressure data recorded by the hydrophone at different γ and the same δ: case (a) γ = 2.49 and δ = 0.58; case (b) γ = 1.76 and δ = 0.60; case (c) γ = 1.52 and δ = 0.63; case (d) γ = 1.39 and δ = 0.60.

3.2.4. The influence of the dimensionless size on the characteristics of the shock wave

The focus in this section is on examining how the size of the air bubble affects the shock wave characteristics. The time evolution of the cavitation bubble in proximity to the air bubble at different δ and the same γ (about 1.75) is presented in Fig. 13. In case (a), δ = 0.49 and γ = 1.75, this case is similar to the SSSW pattern. At t = 3.293 ms, the cavitation bubble produces a solitary and intense shock wave. In case (b), δ = 0.61 and γ = 1.76, it is obvious that this case is the SSW pattern. Several shock waves are induced at t = 2.997 ms, and a discernible arc was observed between the bubbles as the shock waves reflected. At t = 3.034 ms, the propagation of the reflected waves is observed throughout the whole image. Apparently, case (c) is also the SSW pattern with δ = 0.72 and γ = 1.73, and several shock waves are released by the cavitation bubble collapse at t = 3.367 ms. Similarly, the reflected wave’s propagation can be seen at t = 3.404 ms. However, with the contrast of frames b₆ and c₆, it is found that there are more waves in frame c₆. This observation suggests an increase in the shock wave’s quantity with the raising of δ. Hence, the analysis indicates that the shock wave’s quantity is heavily dependent on the air bubble size. Specifically, as the size of the air bubble increases, so does the number of shock waves.

Fig. 13.

The time evolution of the cavitation bubble near the air bubble at different δ and the same γ: case (a) δ = 0.49 and γ = 1.75; case (b) δ = 0.61 and γ = 1.76; case (c) δ = 0.72 and γ = 1.73.

Fig. 14 shows the pressure data recorded by the hydrophone corresponding to all the cases in Fig. 13. In case (a), there is a single shock wave emitted by the cavitation bubble when δ is the smallest. In addition, utmost pressure and shockwave intensity are 1.934 MPa and 0.179 J, respectively. In case (b), an increase of δ leads to a raising in the shock wave number, which further reduces the pressure peak of the shock wave to 1.363 MPa. Moreover, shockwave intensity is calculated as 0.084 J. As the value of δ steadily rises, the shock wave's quantity in case (c) becomes the largest among the three cases. However, utmost pressure and shockwave intensity in this case are the smallest, which are 1.262 MPa and 0.077 J, respectively. Comparing case (b) and case (c), it is worth noting that the two cases belong to the same SSW pattern, while the increase in δ corresponds to a reduction in the shock wave intensity. Therefore, from the analysis of the pressure data, it is concluded that the collapse strength of the cavitation bubble can be attenuated by the air bubble size. Specifically, utmost pressure and shockwave intensity are reduced with the increase of δ.

Fig. 14.

The pressure data recorded by the hydrophone at different δ and the same γ: case (a) δ = 0.49 and γ = 1.75; case (b) δ = 0.61 and γ = 1.76; case (c) δ = 0.72 and γ = 1.73.

3.3. Analysis of the intensity of the shock wave under different dimensionless parameters

The above content elucidates how the dimensionless distance γ and dimensionless size δ impact various characteristics of the shock wave, such as quantity, shape, pressure peak, and energy. Furthermore, this section will analyze the correlation between the shock wave intensity specifically utmost pressure and shockwave intensity and dimensionless parameters (γ and δ). The hydrophone records typical pressure data, as depicted in Fig. 15. The pressure peaks labeled 'a' and 'b' specifically denote the plasma and collapse, respectively, and are enlarged to present detailed information. The plasma shock wave magnitude implies the input energy, which affects the collapse shock wave magnitude. Hence, the parameter λ is utilized to describe the shock wave peak pressure, which is defined as:

$$\lambda =\frac{P_{\mathit{CSW}}}{P_{\mathit{PSW}}}$$ (9)

Where P_CSW denotes the shock wave peak pressure during the collapse stage, and P_PSW denotes the plasma shock wave peak pressure. Similarly, the parameter η is adopted to describe the shock wave energy intensity, which is defined as:

$$\eta =\frac{E_{\mathit{CSW}}}{E_{\mathit{PSW}}}$$ (10)

Where E_CSW denotes the collapse shock wave energy, and E_PSW denotes the plasma shock wave energy. Obviously, the larger values of the λ and η represent the higher intensity in the shock wave.

Fig. 15.

Typical pressure data for a cavitation bubble near the air bubble recorded by the hydrophone.

According to the experimental data, the connection linking the pressure peak intensity λ and the dimensionless parameter γ/δ is shown in Fig. 16. A power function curve(Luo et al., 2021) (dashed line) is fitted with the three shock wave patterns as follows:Fig. 17.

$$\lambda =0.21{(\frac{\gamma }{\delta }-1.78)}^{0.28}$$ (11)

Fig. 16.

The relationship between the pressure peak intensity of the shock wave λ and the dimensionless parameter γ/δ.

Fig. 17.

The relationship between the energy intensity of the shock wave η and the dimensionless parameter γ/δ.

The correlation coefficient R² value in Eq. (11) is 0.60, allowing for the estimation of the pressure peak intensity when γ and δ are known. The fitting curve reveals that as the dimensionless parameter γ/δ increases, the pressure peak intensity (λ) also shows a corresponding increase. The γ/δ in MSW pattern, SSW pattern, and SSSW pattern approximately occurs in the range of 2 ∼ 3, 3 ∼ 4, and larger than 4, respectively. Additionally, as γ/δ increases, the shock wave patterns transition sequentially through MSW, SSW, and SSSW. It is worth mentioning that the variation of λ from the MSW pattern to the SSW pattern is sharp, while from the SSW pattern to the SSSW pattern is smooth. This indicates that the highest-pressure peak exists in the SSSW pattern, which corresponds to the analysis in Section 3.2. Moreover, the shock wave peak pressure is greatly reduced when multiple shock waves are generated. It is found that either an increase of γ or a decrease of δ leads to a reduction in the shock wave quantity. This phenomenon may be attributed to the concentration of collapse energy released by the cavitation bubble, potentially contributing to the high-pressure peak. Besides, when the fusion between the two kinds of bubbles, the pressure peak intensity λ is almost zero and remains constant regardless of changes in γ/δ. The γ/δ in the MWSW pattern approximately appears in the range of 1 ∼ 2. Furthermore, when the cavitation bubble is in the open space as an example in Fig. 6 (a), the shock wave pressure during the collapse stage is 3.000 MPa, and the plasma shock wave pressure is 7.969 MPa. It is observed that, regardless of the specific conditions involving a cavitation bubble near an air bubble, the pressure peak intensity λ consistently registers lower values than those observed in open space. This illuminates that the presence of the air bubble leads to a reduction in the pressure peak intensity.

According to the pressure data recorded by the hydrophone and energy calculation Eq. (6), the connection linking the shock wave energy intensity η and the dimensionless parameter γ/δ is shown in Fig. 16. Similarly, a power function curve (dashed line) is fitted with the three shock wave patterns as follows:

$$\eta =0.03{(\frac{\gamma }{\delta }-1.55)}^{0.64}$$ (12)

The correlation coefficient R² value in Eq. (12) is 0.72. The fitting curve analysis reveals that the energy intensity (η) demonstrates an increase as the value of γ/δ increases. Overall, the change in η with the increase of γ/δ is smooth. The elevation of γ/δ leads to a sequential transition of shock wave patterns, progressing through MSW, SSW, and SSSW patterns. This demonstrates that the energy of the cavitation bubble reaches its peak when a single shock wave is generated. In addition, the energy intensity η in the MWSW pattern is nearly zero. Taking the open space in Fig. 6 (a) as an example, the collapse shock wave energy is 0.395 J, and the plasma shock wave energy is calculated as 2.647 J. Therefore, the energy intensity η in the open space is 0.15. Similarly, it is found that the η in the open space is larger than the η when the air bubble exists.

4. Conclusions

During the above experiment, the shock wave characteristics emitted by the cavitation bubble close to the air bubble adhering to the tube nozzle are investigated by changing the dimensionless distance γ and dimensionless size δ. The cavitation bubble is produced using the spark-induced method, while precise control is exerted to ensure the attachment of the air bubble to the tube nozzle. The schlieren optical system is employed for capturing the shock wave resulting from the cavitation bubble during the collapse stage, while a hydrophone is utilized to measure the shock wave pressure. The results suggest a substantial impact of the air bubble on the dynamic of the cavitation bubble, and the shock wave characteristics including quantity, shape, pressure peak, and energy are greatly influenced by the two dimensionless parameters γ and δ. The main observations include:

• (1)The cavitation bubble characteristics in the open space and near the air bubble have great differences. A single shock wave is produced in the open space, while the shock wave quantity is increased near the air bubble, and the intensity of the shock waves is significantly weakened. The morphology and dynamic characteristics of cavitation bubbles also undergo noticeable changes.
• (2)According to the quantity and shape of the shock wave, four shock wave patterns are categorized, namely SSSW, SSW, MSW, and MWSW patterns. The four shock wave patterns are closely related to the two dimensionless parameters γ and δ.
• (3)Variations of substantial significance are noted in the collapsing characteristics of the cavitation bubble and the shock wave's intensity within the four identified patterns. The shock wave intensity decreases sequentially across the three patterns. In the MWSW mode, the two bubbles first merge, then separate, and eventually move away at a slower speed. The shock wave intensity in this pattern is markedly weak, with the pressure peak and energy nearing zero.
• (4)The dimensionless distance γ and dimensionless size δ exert a substantial impact on the shock wave's characteristics, including its quantity, shape, pressure peak, and energy. When the cavitation bubble does not merge with the air bubble, a decrease in γ or an increase in δ results in an increased quantity of shock waves and a decreased spherical symmetry. When the two kinds of bubbles merge, the shock wave intensity is sharply attenuated.
• (5)Utmost pressure and shockwave intensity are a power function of γ/δ. By establishing the relevant functions, it becomes possible to anticipate the shock wave intensity when the cavitation bubble collapses based on γ/δ.

CRediT authorship contribution statement

Jin Zhu: Writing – original draft, Data curation, Conceptualization. Mindi Zhang: Writing – review & editing, Validation, Software, Methodology, Conceptualization. Zhenkun Tan: Methodology, Investigation. Lei Han: Resources, Methodology. Biao Huang: Writing – review & editing, Validation, Supervision.

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.