Experimental Research on the Electric Spark Bubble Load Characteristics under the Oblique 45 Degree Curved Surface Boundary

Abstract

In order to study the influence of the pressure load generated during the pulsation of cavitation bubbles on the oblique 45-degree curved surface boundary. The curved surface boundaries have different curvatures. This study also designs a multi-angle bracket to make the oblique boundary oblique 45 degrees. This research uses high-voltage electric discharge to generate electric spark bubbles, which act as cavitation bubbles. When the explosion source is at different distances from the oblique 45-degree boundary, a high-speed camera is used to capture the pulsation process of electric spark bubbles. A pressure sensor is used to measure the pulsating load of the electric spark bubbles on the oblique 45-degree boundary during the pulsation process. In this study, we use the dimensionless parameter ζ to represent the curvature of the oblique 45-degree boundary. The dimensionless parameter γ is used to represent the shortest distance between the explosion source and the oblique 45-degree boundary. It is found through experiments that the oscillation characteristics and pulsating load of electric spark bubbles will be affected by ζ and γ. This study summarized six types of unique bubble pulse shapes from 44 groups of experiments. They are “mushroom shape without jet”, “mound shape with jet”, “jellyfish shape with jet”, “oval shape with jet”, “drop shape without jet”, and “spherical shape without jet”. In this paper, the ABAQUS/Explicit is used to simulate the ring-shaped bubble that is not clearly captured by the high-speed camera. Finally, the study summarizes the shock wave load generated during the explosion, the first pulsation load, and the second pulsation load of electric spark bubbles, and finds an obvious overall trend of change.

1. Introduction

It is very important to study the cavitation bubbles’ load characteristics to nearby boundary structures. To study the cavitation bubbles load characteristics to the 45-degree curved surface boundary structure is one of the important areas of research. This research is helpful to the anti-riot design of the bottom of the ship. Most researchers use experimental methods to study the pressure load of cavitation bubbles on the nearby boundary.

In terms of the generation of cavitation bubbles, Lauterborn et al. [1] used laser equipment to generate cavitation bubbles. Lindau et al. [2], Shaw et al. [3], Zhao et al. [4], and Tomita, Y. et al. [5] used the same method. Cui et al. [6] used high-voltage electric discharge equipment to generate electric spark bubbles. C. GL. et al. [7], Fong et al. [8], Gong et al. [9], Luo et al. [10], Ma et al. [11], Zhang et al. [12], and Ma et al. [13] used the same method. Klaseboer et al. [14] used the same method. Reich et al. [15] proposed the bubble dynamics of the ablation bubble through X-ray imaging technology. Barcikowski et al. [16] extended the ultrasonic sonochemistry and pulsed laser ablation in liquid techniques to investigate the formation and collapse of cavitation bubbles. Brujan et al. [17] conducted the interaction of the laser-induced cavitation bubble with elastic boundary through the experimental investigation. Tomita et al. [18] studied the laser-induced cavitation bubbles near the curved rigid boundary through the experimental technology, and a boundary integral method was proposed to explain the liquid jet behavior. Also, De et al. [19] used optical emission spectroscopy technology, a fast shadowgraph technology laser scattering technology, to explain the influence of the cavitation dynamic on the ablation of bulk and wire-shaped targets.

In the measurement of the pressure load generated by cavitation bubbles, Tong et al. [20] studied the effect of different distance parameters. Through research, they preliminarily summarized the trend of bubble load with increasing distance parameters. Shima et al. [21] and Tomita et al. [22] measured underwater laser bubble loads through experiments, and preliminarily studied the damage of underwater laser bubble loads to surrounding structures. Wang et al. [23] measured the bubble load of cavitation bubbles by PVDF. Jin-Lei et al. [24] used a pressure sensor to measure the bubble load produced by a micro-explosive. Jayaprakash et al. [25] used a pressure sensor to measure the wall pressure load generated by the spark bubble in the vertical direction. Cui et al. [26], Yao et al. [27,28], and Ma et al. [13] used Hopkinson rods to measure spark bubble loads and micro-explosive bubble loads. Through previous research results, it is known that the explosion distance is an important cause of damage to surrounding structures due to the cavitation bubble load. This conclusion is applicable to all explosion sources used in the current research, such as laser bubbles, electric spark bubbles, and micro-explosive bubbles.

In terms of the numerical simulation of underwater bubbles, Klaseboer et al. [14], Wang et al. [29], and Jin et al. [30,31] used the boundary element method. Li et al. [32,33,34,35,36] used BEM to simulate the bubble pulsation process caused by cavitation bubbles. Liu et al. [37] used FTM to simulate the cavitation bubble pulsation process near the boundary and its pressure load. Hsiao et al. [38] solved the finite difference problem of the compressibility of cavitation bubbles. Tian et al. [39] used the finite element method to simulate the bubble pulsation evolution process near the boundary structure.

To sum up, these studies all involve the pulsation characteristics and load characteristics of cavitation bubbles, but rarely give their changing laws. Therefore, this study intends to carry out a series of experimental studies, combined with numerical methods to describe the pulsation process of electric spark bubbles near the oblique 45 degree curved boundary and the relationship between the pressure load generated by the boundary curvature and the burst distance.

There are three highlights in this study:

①

We use a measuring system that can adjust the angle and replace the target plate.

In this way, the purpose of measuring the curved surface boundary structure with different curvatures under the cavitation bubbles load is realized.

②

On this basis, the change law of bubble dynamics and load characteristics near the oblique 45 degree boundary with different curvatures is discussed.

③

We analyze the change law of bubble shape under different curvature parameters ζ and different blasting distance parameters γ under the premise that the center line of the boundary structure and the vertical direction are at an angle of 45 degrees.

2. Experimental Setup

2.1. The Boundary Structure Model

The boundary structure was composed of three oblique 45 degree hemispherical structures with different radii and one oblique 45 degree flat plate structure. They were all made of aluminum alloy. The radii of the hemispherical structure were 25, 50, and 75 mm. The flat plate structure had a radius of 125 mm and a thickness of 10 mm. We drilled M10 × 1.0 threaded holes in the center of the hemispherical structure and used the flat plate structure to install the pressure sensor.

The model that simulates the oblique 45 degree curved surface boundary structure is shown in Figure 1. In order to better explain the internal structure of the experimental equipment, Figure 1 uses a sectional view to show the relationship between the explosion source and the boundary. The center line of the hemispherical structure and the flat plate structure is at an angle of 45 degrees from the vertical. The pressure sensor is arranged at the center of the hemispherical structure and the flat plate structure. The distance between the explosion source and the measuring surface of the sensor is represented by d, as shown in Figure 1. The d in Figure 1 is measured manually, and the unit is mm. $R_{max}$ is the maximum radius of the bubble, in millimeters. The measuring surface of the sensor is the closest point to the explosion source in the oblique 45 degree curved surface boundary structure. In the following, it is called the bottom end of the boundary.

The measuring surface radius of the 37,311 pressure sensor is 0.03 m, as shown in Figure 2. According to the formula $S=\pi r^2$, the area of the sensor measuring surface is 0.002826 m². The radii of the three hemispherical structures are 25, 50, and 75 mm. According to the formula $S=\pi r^2$, the surface areas of the three hemispherical structures were 0.19625, 0.785, and 1.76625 m². Therefore, the ratio of the measuring surface of the sensor to the surface area of the three hemisphere structures was 1.4%, 0.3%, and 0.1%, respectively. Because the proportion is relatively small, it is still assumed in the experiment that the combination of the sensor and the hemisphere creates a complete hemispherical structure. The details are shown in Figure 3.

2.2. Occurrence and Measurement of Electric Spark Bubbles

The experimental equipment for simulating the effect of cavitation bubble pulsating load on an oblique 45-degree curved surface boundary is shown in Figure 4. This equipment was based on the experiment of Ma et al. [13] to improve the experimental equipment and optimize the experimental circuit. We added a replaceable boundary interface and a multi-angle mounting base. This experiment used a 10 pF electricity storage box with a discharge voltage of 400 V. The electricity storage box was connected to a thin copper tube through a high-voltage line. During discharge, high temperature and burning occurred at the intersection of thin copper tubes. This produced electric spark bubbles. Turangan et al. [40] research shows: “In the experimental study, there are mainly three kinds of technique to generate bubble, that are charge explosion technique, laser focusing technique and spark discharge technique”. Zhang et al. [41] used electric spark bubbles for experimental investigation of the interaction between a pulsating bubble and a rigid cylinder. And achieved good experimental results. So, this experiment uses electric spark bubbles to simulate cavitation bubbles. The pulsating load generated by the electric spark bubbles was measured by a 37,311 pressure sensor. The pulsation process of the electric spark bubble was captured by a Phantom VEO 710 high-speed camera. Phantom VEO-710 high-speed camera is produced by AMETEK in the United States.

A multi-angle mounting base was applied to the experimental setup to measure the electric spark bubble pulse load on an oblique 45-degree boundary. One can see the electric spark bubble dynamics near the oblique 45-degree boundary. A waterproof tube was placed in the angle-fixed bottom axis. One can adjust the angle-fixed bottom axis so that the angle measurer points to 45 degrees. In this way, the center line of the waterproof tube was at 45 degrees from the vertical because the center line of the waterproof tube coincides with the center line of the boundary structure. The multi-angle mounting base is shown in Figure 5.

In this experiment, the Phantom VEO 710 high-speed camera shot electric spark bubbles at a speed of 24,800 frames/s. At this time, the image rate of the high-speed camera was 512 × 512. The exposure speed of the camera was 2.41 μs. The time for the spark bubble to end in the second pulse was about 10.343 ms. In other words, during this period, the high-speed camera took approximately 248 photos. In order to avoid the LED lightbulb image appearing in the pictures taken, in this experiment, frosted glass was installed between the lamp and the experimental water tank. This experiment used a CY-YD-205piezoelectric pressure sensor, abbreviated as 37311. The pressure measurement range was 0‒100 MPa, and the overload capacity was 120%. The sensor is shown in Figure 6. The spark bubble generator and curved surface boundary structure model were placed in the experimental water tank. The experimental water tank was a glass cube with a side length of 1 m. The experimental water tank was fixed on the bracket to prevent it from moving during high-voltage discharge.

Both the electric spark bubble generator and the pressure measuring surface of the curved surface boundary structure model were placed at the center of the experimental water tank. The maximum diameter of electric spark bubbles generated in the free field was 60 mm. Because of the large volume of the experimental water tank, the possible boundary effect of the water tank boundary could be ignored. The diameter of the electrode used in the experiment was 0.5 mm, and its influence could be ignored compared to the bubble volume. Therefore, in this experiment, only the boundary structure could affect the bubble dynamics. The exposure speed of the camera was 2.41 μs. The time for the electric spark bubble to end in the second pulse was about 10.343 ms. The exposure speed of the camera was much smaller than the spark bubble pulsation time. The error was small enough. The focal length of the high-speed camera was at the intersection of the electrodes and the midpoint of the curved surface boundary structure. In this way, the whole process of the effect of the electric spark bubble pulsation process on the oblique 45-degree curved surface boundary structure could be captured better.

2.3. Nondimensionalization

In order to better discuss the experimental results, we used the bubble maximum radius ($R_{max}$), boundary structure radius (r), and explosion distance (d) to dimensionlessly deal with the boundary structure curvature parameter (ζ) and explosion distance parameter (γ). See Equations (1) and (2) for details [42,43]:

$$\zeta =R_{max}/r$$

(1)

$$\gamma =d/R_{max}$$

(2)

The discharge voltage, the discharge electrode and the depth of the explosion source will affect the maximum radius of the cavitation bubbles formed. In order to increase the repeatability of the experiment, the influence of ζ and γ on cavitation bubbles is explored. Therefore, in this experiment, the discharge voltage, the discharge electrode and the depth of the explosion source are all fixed.

In the experimental environment of the oblique 45-degree curved surface boundary structure model, the bubbles were affected by the boundary effect of the boundary structure. Therefore, $R_{max}$ was chosen to be obtained by the 400 V discharge in the free field. At this time, the boundary structure was removed. Twenty groups of free-field 400 V discharge electric spark bubble experiments were carried out in the experimental water tank to ensure that the gravity and buoyancy of the bubbles in the free field and non-free field experiments were the same. Therefore, the space position of the spark bubble generator in the free field experiment was the same as that in the non-free field experiment. The results show that the bubble radius fluctuated between 57.7 and 61.9 mm. After calculation, the average value was 60 mm. The maximum error of the average radius was 3.8%. Therefore, suppose $R_{max}$ = 60 mm. In this study, the radius of the flat plate structure was considered to be infinite (∞) based on the above analysis. The variables in this study are shown in

Table 1. Correspondence between ζ and r.

r (mm)	∞	25	50	75
ζ	0	0.4	0.6	1.2

and

Table 2. Correspondence between γ and d.

d (mm)	10	15	20	25	30	35	40	45	50	55	60
γ	0.33	0.50	0.67	0.83	1.00	1.17	1.33	1.50	1.67	1.83	2.00

.

3. Finite Element Method ABAQUS/Explicit

This article we use ABAQUS/Explicit to simulate cavitation bubbles. Because ABAQUS/Explicit is suitable for short-term, transient, and dynamic analysis, such as impact and explosion problems. During the simulation, we used Euler’s formula in ABAQUS/Explicit. A Eulerian solver is very suitable for dealing with the calculation of large deformation problems, such as the movement of liquid and gas. This is because the mesh nodes of the Euler body are always fixed, only the material flows inside the mesh, without excessive deformation of the element. The Euler element can contain multiple materials at the same time, and the Eulerian solver will automatically consider the contact between different materials.

This article uses spark bubbles to simulate cavitation bubbles. Gong SW et al. [44] compared and analyzed the evolution characteristics of the three bubbles of explosive, laser and electric spark, and proposed the similar relationship between different bubble formation mechanisms. In the studies of Gong SW et al. [44], it has been proved that electric spark bubbles are the same as cavitation bubbles in the pulsation shape and pressure load. The numerical method used in this paper is to simulate cavitation bubbles. So numerical methods can also simulate spark bubbles. Although we use the empirical formula of cavitation bubbles to derive the initial conditions of electric spark bubbles, this article is still studying electric spark bubbles.

The numerical simulation in this paper was based on experimental conditions, using Euler components to simulate the entire water tank, so the materials in the Euler body were air, water, and explosion source. Lagrangian components were used for flat and curved boundaries. All the material properties mentioned above are consistent with the experiment.

In ABAQUS simulation calculations, to simulate the mechanical properties of fluids, it was necessary to use the equation of state to determine the material density, temperature, energy, pressure, and other parameters. There were three media in the Euler domain in this study: water, air, and explosion source. The equations of state are explained below.

3.1. Equation of State for Air Medium

We used the ideal gas Equations (3) and (4) of state to describe the air medium:

$$p+p_a=(\gamma -1) \rho E_m$$

(3)

$$E_m=C_v\left(\theta -{\theta }^Z\right)$$

(4)

In these formulae, $p_a$ is the ambient pressure, $\gamma$ is the specific heat ratio, $E_m$ is the initial specific energy at initial temperature,$\rho$ is the air density, $\theta$ is the current temperature, ${\theta }^Z$ is absolute zero, and $C_v$ is the specific heat.

3.2. Equation of State of Water Medium

The $U_s$‒$U_p$ equation of state (EOS) model in ABAQUS/Explicit can simulate incompressible viscous or inviscid laminar flow controlled by Navier‒Stokes equations of motion. We used Hugoniot as a reference curve. The choice of water, shear modulus, and viscosity should be as small as possible to avoid an excessive response. EOS assumes that pressure is related to density and internal energy per unit mass. The most common form of the Mie‒Grüneisen equation is

$$p-p_H={\Gamma }_{\rho }\left(E_m-E_H\right)$$

(5)

where $E_m$ is the internal energy per unit mass; $p_H$ and $E_H$ are the Hugoniot pressure and specific energy, respectively; Γ is the Grüneisen ratio; and $\Gamma ={\Gamma }_0\frac{{\rho }_0}{\rho }$ (${\Gamma }_0$ is the material constant and ${\rho }_0$ is the reference density).

The relationship between energy $E_H$ and pressure $p_H$ in the Hugoniot equation is as follows:

$$E_H=\frac{p_H\eta }{2{\rho }_0}$$

(6)

The pressure $p_H$ in Hugoniot is only a function of density. η is the volumetric compressive strain.

$$\eta =1-\frac{{\rho }_0}{\rho }$$

(7)

Substituting these variables into Equation (5), we get

$$p=p_H\left(1-\frac{{\Gamma }_0\eta }{2}\right)+{\Gamma }_0{\rho }_0{E}_m$$

(8)

The common fit of Hugoniot data is

$$p_H=\frac{{\rho }_0{c}_0^2\eta }{{\left(1-s\eta \right)}^2}$$

(9)

Substituting this into Equation (4), we can get the linear $U_s–U_p$ Hugoniot equation:

$$p=\frac{{\rho }_0{c}_0^2\eta }{{\left(1-s\eta \right)}^2}\left(1-\frac{{\Gamma }_0\eta }{2}\right)+{\Gamma }_0{\rho }_0{E}_m$$

(10)

where $c_0$ and s define the linear relationship between the linear impact velocity $U_s$ and the particle velocity $U_p$, as shown below:

$$U_s=c_0+s{U}_p$$

(11)

${\rho }_0{c}_0^2$η is the elastic bulk modulus with a smaller strain.

3.3. Equation of State of Explosion Source

There are many kinds of state equations for explosion source, among which the most widely used is the JWL state equation. The explosive material in the simulation is the explosion source; the explosion products produced by the explosion use the JWL equation of state, as follows:

$$P=A\left(1-\frac{\omega \eta }{R_1}\right)e^{-\frac{R_1}{\eta }}+B\left(1-\frac{\omega \eta }{R_2}\right)e^{-\frac{R_2}{\eta }}+\omega \eta {\rho }_{0}e$$

(12)

In this formula, $\eta$ is the ratio of the density of the explosive product to the density of the initial explosive, ($\eta =\rho /{\rho }_0$), $A,B,R_1,R_2,\mathrm{and}\omega$ are constants related to the type of explosive, and e is the internal energy per unit mass of the explosive.

4. Experimental Results

It was found through the experiments that the bubble shape change and the load change near the oblique 45-degree boundary mainly depended on ζ and γ. When both are variables, the situation becomes extremely complicated. Therefore, we explored the law of bubble pulsation when another variable changes, while controlling one variable. The specific experimental numbers are shown in

Table 3. Experimental cases.

No.	ζ	γ	Type		No.	ζ	γ	Type
1	0	0.33	I	Mushroom shape without jet	23	0.6	0.33	I	Mushroom shape without jet
2		0.50	II	Mound shape with jet	24		0.50	IV	Oval shape with jet
3		0.67			25		0.67
4		0.83			26		0.83
5		1.00	III	Jellyfish shape with jet	27		1.00	V	Drop shape without jet
6		1.17			28		1.17
7		1.33			29		1.33
8		1.50			30		1.50
9		1.67	VI	Spherical shape without jet	31		1.67	VI	Spherical shape without jet
10		1.83			32		1.83
11		2.00			33		2.00
12	0.4	0.33	I	Mushroom shape without jet	34	1.2	0.33	I	Mushroom shape without jet
13		0.50	IV	Oval shape with jet	35		0.50
14		0.67			36		0.67	IV	Oval shape with jet
15		0.83			37		0.83
16		1.00	V	Drop shape without jet	38		1.00
17		1.17			39		1.17	V	Drop shape without jet
18		1.33			40		1.33
19		1.50			41		1.50
20		1.67	VI	Spherical shape without jet	42		1.67	VI	Spherical shape without jet
21		1.83			43		1.83
22		2.00			44		2.00

. We found that different ζ and γ combine with each other, and a total of six different bubble pulsation forms can occur. See Table 3 for details.

4.1. Effect of Different ζ and γ on Bubble Dynamics

4.1.1. Type I: Mushroom Shape without Jet

Type I bubbles specifically refer to bubbles in the case of 0 ≤ ζ ≤ 0.6, γ = 0.33, and ζ = 1.2, 0.33 ≤ γ ≤ 0.50. For type I, we choose one γ of each ζ to discuss. The details are shown in Figure 7. To facilitate the comparison of bubble dynamics in different situations, in Section 4.1, the time scale was standardized as the moment the explosion starts. We defined the start time of the explosion as 0 ms. Other moments were adjusted accordingly.

Figure 7(1) shows the bubble pulsation process when ζ = 0 and γ = 0.33. At this time, we used a flat plate structure. When t = 0 ms, a dazzling light was emitted from the intersection of the electrodes, which was the beginning of the high-voltage discharge. Since the explosion source was close to the flat plate structure (10 mm), the expansion of the bubble in the first cycle was completed along the surface of the flat plate structure. At this time, the bubble presented a hemispherical shape, as shown by t = 2.791 ms. In the contraction process of the first cycle of the bubble, the bubble began to appear mushroom-shaped. When t = 4.958 ms, the shape of a mushroom head could be seen. However, at this time the mushroom neck was still difficult to observe. The bubble collapsed at t = 6.583 ms. This marked the end of the first pulse cycle of the bubble. Except for the microbubble cloud attached to the plate during the second pulsation of the bubbles, no other special phenomena existed. The high-speed camera did not capture the jet phenomenon during the entire bubble pulsation process. When t = 0 ms, a shock wave was generated. When t = 7.499 ms, the first pulsation load of bubbles was generated. When t = 6.583 ms, the second pulsating load of bubbles was generated.

Figure 7(2) shows the bubble pulsation process when ζ = 0.4 and γ = 0.33. At this time, we used a hemispherical structure with a radius of 75 mm. The same was true for ζ = 0 and γ = 0.33: the bubble appeared hemispherical in the expansion phase of the first cycle, as shown at t = 4.958 ms. During the contraction of the bubble in the first cycle, its mushroom shape was more obvious than when ζ = 0. Part of the mushroom neck appeared at t = 4.958 ms. Although the bubble was also in the form of a microbubble cloud during the second cycle of pulsation, its volume was larger than when ζ = 0. This may be why the adsorption force of Bjerknes gradually decreased as the boundary area shrank. The results from the high-speed camera showed that there was no obvious jet flow during the bubble pulse. When t = 0 ms, a shock wave was generated. When t = 6.416 ms, the first pulsation load of bubbles was generated. When t = 9.707 ms, the second pulsating load of bubbles was generated.

Figure 7(3) shows the bubble pulsation process when ζ = 0.6 and γ = 0.33. At this time, we used a hemispherical structure with a radius of 50 mm. Compared with ζ = 0.4, the mushroom shape of the bubble in the first cycle of contraction was more obvious. When t = 5.5 ms, the head of the mushroom is rounder and the neck was longer. In the expansion phase of the second bubble pulse cycle, an obvious strawberry-shaped microbubble cloud was produced. The strawberry-shaped microbubble cloud expanded and contracted to the lower left. This may have been the result of the combined action of the boundary Bjerknes force and gravity. The microbubble cloud with ζ = 0.6 was larger than that with ζ = 0.4, which may be related to the decrease of the contact surface of the boundary structure and the weakening of Bjerknes adsorption force. The results of the high-speed camera showed that there was no obvious jet flow during the bubble pulse. When t = 0 ms, a shock wave was generated. When t = 5.958 ms, the first pulsation load of bubbles was generated. When t = 9.416 ms, the second pulsating load of bubbles was generated.

Figure 7(4) shows the bubble pulsation process when ζ = 1.2 and γ = 0.33. At this time, we used a hemispherical structure with a radius of 25 mm. Compared with ζ = 0.6, the mushroom shape of the bubble in the first cycle of contraction was more obvious. When t = 5.083 ms, the head of the mushroom was smoother and rounder, and the neck was longer. In the expansion phase of the second bubble pulsation cycle, the bubble jet and bubble separation phenomenon obviously occurred in the lower right. There were two reasons for this phenomenon: ① The contact surface of the boundary structure was reduced, which reduced the Bjerknes adsorption force; ② the load generated when the bubble collapsed produced a reaction force on the bubble, which prompted the remaining bubbles to move in the opposite direction of the boundary structure. The results of the high-speed camera showed that there was no obvious jet flow during the bubble pulse. When t = 0 ms, a shock wave was generated. When t = 5.5 ms, the first pulsation load of bubbles was generated. When t = 8.208 ms, the second pulsating load of bubbles was generated.

In summary, the existence conditions of type I bubbles are 0 ≤ ζ ≤ 0.6, γ = 0.33, and ζ = 1.2, 0.33 ≤ γ ≤ 0.50. What they have in common is: ① The bubble appears hemispherical in the expansion stage of the first cycle. This is because the explosion source is very close to the boundary structure. ② The bubble will appear mushroom-shaped in the contraction phase of the first cycle. The larger the ζ, the more obvious the mushroom shape. This is because, as ζ increases, the Bjerknes adsorption force at the center of the boundary structure is greater than that at other positions. This makes the mushroom neck appear. This change of bubbles is also described in the study by Shima et al. [44]. They have predicted necking as well as jet formation, depending on the stick-slip motion at the boundary and many reports have since then experimentally verified the effects. The differences are: With the increase in ζ, the volume of the microbubble cloud generated by the bubble in the second cycle expansion stage will become larger and larger, until the jet and bubble separation occur in the opposite direction. This is because, as ζ increases, the Bjerknes adsorption force produced by the edge of the boundary structure gradually decreases. Therefore, the Bjerknes adsorption force ($F_B$) gradually decreases, while the pressure ($F_c$) generated by the first collapse of the bubble remains unchanged. This causes their composition of forces ($F_{cf}$) direction to gradually increase, and the direction points 45 degrees to the lower right. The details are shown in Figure 8.

4.1.2. Type II: Mound Shape with Jet

Type II bubbles specifically refer to bubbles in the case of ζ = 0, 0.50 ≤ γ ≤ 0.83. For type II, we choose ζ = 0, γ = 0.83 to represent this type for discussion. The detailed information is shown in Figure 9.

It can be seen from Figure 9 that, when t = 0 ms, the high-voltage electric discharge began to explode underwater. At this time, a dazzling light was generated at the intersection of the electrodes. The bubbles expanded into a hemispherical shape in the first cycle. At t = 3.499 ms, the bubble reached its maximum volume. In the contraction process of the first cycle of the bubble, the bubble took on a mound shape, accompanied by a jet. When t = 6.249 ms, the bubbles were in the shape of a mound. When t = 6.499 ms, the bubble jet began to form. When t = 6.624 ms, the bubble jet penetrated the other end of the bubble. When t = 7.041 ms, the bubble completed the collapse of the first pulse cycle. At t = 7.999 ms and t = 10.998 ms, the bubbles in the expansion and contraction phases of the second cycle were in the form of microbubble clouds. When t = 11.457 ms, the bubble completed the collapse of the second pulse cycle.

The use of the ABAQUS/Explicit simulation results to simulate the first pulsation period of the bubble is shown in Figure 10(2). When t = 0 ms, bubbles started to form. When t = 3.499 ms, the artificial bubble expanded to the maximum. At this time, the simulated bubble became a hemisphere. t = 6.249 ms, the simulated bubble shrank into a mound shape. When t = 6.499 ms, the simulated bubble started to produce a jet. When t = 6.624 ms, the jet passed through the bubble and reached the other side of the bubble. When t = 7.041 ms, the first pulsation period of the simulated bubble collapsed, and the generation of annular bubbles could be seen.

4.1.3. Type III: Jellyfish Shape with Jet

Type III bubbles specifically refer to bubbles in the case of ζ = 0, 1.00 ≤ γ ≤ 1.50. For type III, we choose ζ = 0, γ = 1.17 to represent this type for discussion. The detailed information is shown in Figure 11.

It can be seen from Figure 11 that when t = 0 ms, electric spark bubbles began to form. The bubble expanded in a spherical shape during the first pulsation cycle. This was different from types I and II. This was caused by the increase in explosion distance. At t = 4.166 ms, the bubble reached its maximum volume. During the contraction process of the first cycle of the bubble, the bubble was in the shape of a jellyfish, accompanied by a jet, as shown at 6.874, 6.999, and 7.082 ms. When t = 6.874 ms, the jet began to form. When t = 6.999 ms, the jet reached and passed through the other end of the bubble. When t = 7.082 ms, the bubble jet reached the boundary of the flat plate. When t = 7.332 ms, the bubble collapsed for the first time. The bubbles pulsated in the form of microbubble clouds in the second cycle, as shown at 8.040 and 10.457 ms. When t = 11.165 ms, the bubble collapsed for the second time.

Using the ABAQUS/Explicit simulation results to simulate the first pulsation period of the bubble is shown in Figure 12(2). When t = 0 ms, bubbles started to form. When t = 2.90 ms, the simulated bubble expanded to the maximum. At this time, the simulated bubble became spherical. When t = 5.65 ms, the simulated bubble shrank into a jellyfish shape. When t = 5.65 ms, the simulated bubble started to produce a jet. When t = 5.75 ms, the jet was already very obvious. When t = 5.80 ms, the jet passed through the bubble and reached the other side. When t = 5.85 ms, the first pulsation period of the simulated bubble collapsed, and the generation of annular bubbles could be seen.

4.1.4. Type IV: Oval Shape with Jet

Type IV bubbles specifically refer to bubbles in the case of 0.4 ≤ ζ ≤ 0.6, 0.50 ≤ γ ≤ 0.83, and ζ = 1.2, 0.67 ≤ γ ≤ 1.00. For type IV, we choose one γ of each ζ to discuss. The details are shown in Figure 13.

Figure 13(1) shows the bubble pulsation process when ζ = 0.4 and γ = 0.67. At this time, we used a hemispherical structure with a radius of 75 mm. When t = 0 ms, a dazzling light was emitted from the intersection of the electrodes, which was the beginning of high-voltage discharge. A shock wave was generated. When the bubble expanded to the maximum volume state in the first period, the bubble appeared hemispherical, as shown by t = 2.958 ms. During the contraction of the bubble’s first cycle, the bubble began to take on an oval shape. Obvious oval bubbles could be seen at 4.333 and 5.124 ms. At t = 5.791 ms, a clear jet directed at the wall could be seen. The bubble collapsed at t = 6.082 ms and generated the first pulsating load. This marked the end of the first pulse period of the bubble. In the pulsation phase of the second cycle, the bubbles presented a microbubble cloud state, as shown at 6.707 and 8.790 ms. The bubble collapsed at t = 9.415 ms and generated a second pulsating load. This marked the end of the second pulse period of the bubble.

Figure 13(2) shows the bubble pulsation process when ζ = 0.6 and γ = 0.67. At this time, we used a hemispherical structure with a radius of 50 mm. Compared with ζ = 0.4, the oval shape of the bubble was more obvious in the first contraction cycle. A more obvious oval shape could be seen at 4.666 and 5.291 ms. When t = 5.916 ms, the jet directed at the wall could be seen. The second pulse period of the bubble was similar to ζ = 0.4. It also took the form of a microbubble cloud. When t = 0 ms, a shock wave was generated. When t = 6.166 ms, the first pulsating load of the bubble was generated. When t = 9.832 ms, the second pulsating load of bubbles was generated.

Figure 13(3) shows the bubble pulsation process when ζ = 1.2 and γ = 0.33. At this time, we used a hemispherical structure with a radius of 25 mm. Compared with ζ = 0.6, the oval shape of the bubble was more obvious in the first contraction cycle. When t = 5.333 ms, the head of the oval bubble became smoother and rounder. The neck of the oval bubble was more slender than ζ = 0.4 and ζ = 0.6. When t = 5.833 ms, the jet from the tail of the oval bubble to the wall could be seen clearly. In the expansion phase of the second bubble pulsation cycle, a strawberry-shaped microbubble cloud was clearly seen to the left. The strawberry-shaped microbubble cloud expanded to the left. This phenomenon may have been caused by the combined action of Bjerknes’s adsorption force, collapse load, gravity, and buoyancy. When t = 0 ms, a shock wave was generated. When t = 5.916 ms, the first pulsating load of bubbles was generated. When t = 9.541 ms, the second pulsating load of bubbles was generated.

The ABAQUS/Explicit simulation results were used to simulate the first pulse period of the bubble, as shown in Figure 14(2), (4), and (6). One can see from the figure the jet flow and the state of the annular bubble when the bubble collapsed. Figure 14(2) simulates the bubble pulsation process when ζ = 0.4 and γ = 0.67. When t = 5.55 ms, the jet can be seen. When t = 5.70 ms, one can see that the bubbles become ring-shaped. When t = 5.50 ms, the jet can also be seen in Figure 14(4) and (6). When t = 5.60 ms, ring-shaped bubbles can also be seen in Figure 14(4) and (6).

4.1.5. Type V: Drop Shape without Jet

Type V bubbles specifically refer to bubbles in the case of 0.4 ≤ ζ ≤ 0.6, 1.00 ≤ γ ≤ 1.50, and ζ = 1.2, 1.17 ≤ γ ≤ 1.50. For type V, we choose one γ of each ζ to discuss. The details are shown in Figure 15.

Figure 15(1) shows the bubble pulsation process when ζ = 0.4 and γ = 1.17. At this time, we used a hemispherical structure with a radius of 75 mm. When t = 0 ms, a dazzling light was emitted from the intersection of the electrodes, which was the beginning of high-voltage discharge. A shock wave was generated. When the bubble expanded to the maximum volume state in the first time period, the bubble took on a spherical shape, as shown by t = 3.166 ms. During the first period of bubble contraction, the bubble began to appear drop-shaped. From 5.041 ms, clear drop-shaped bubbles could be seen. From t = 5.666 ms and t = 5.708 ms, a clear jet directed towards the wall could be seen. The bubble collapsed at t = 5.749 ms and produced the first pulsating load. This marked the end of the first pulse period of the bubble. In the pulsation phase of the second cycle, the bubbles formed a strawberry-shaped microbubble cloud facing the wall, as shown at 7.499 and 9.582 ms. The bubble collapsed at t = 10.249 ms and generated a second pulsating load. This marked the end of the second pulse period of the bubble.

Figure 15(2) shows the bubble pulsation process when ζ = 0.6 and γ = 1.17. At this time, we used a hemispherical structure with a radius of 50 mm. Compared with ζ = 0.4, the drop shape of the bubble was more pronounced in the first contraction cycle. From 5.167 ms, more obvious drop-shaped bubbles could be seen. When t=5.625 ms and t=5.708 ms, jets directed at the wall could be seen. The second pulse period of the bubble was similar to ζ = 0.4. It also formed a strawberry-shaped cloud of microbubbles facing the wall. When t = 0 ms, a shock wave was generated. When t = 5.750 ms, the first pulsating load of the bubble was generated. When t = 9.916 ms, the second pulsating load of bubbles was generated.

Figure 15(3) shows the bubble pulsation process when ζ = 1.2 and γ = 0.33. At this time, we used a hemispherical structure with a radius of 25 mm. Compared with ζ = 0.6, the drop shape of the bubble was more obvious in the first contraction cycle. When t = 5.208 ms, the head of the drop-shaped bubble became smoother and rounder. The head of the drop-shaped bubble was sharper than ζ = 0.4 and ζ = 0.6. When t = 5.833 ms, the jet from the tail of the drop-shaped bubble to the wall could be seen clearly. In the expansion phase of the second bubble pulse cycle, a strawberry-shaped double microbubble cloud was clearly visible. Two strawberry-shaped microbubble clouds expanded to the lower left and the surface of the hemispherical structure, respectively. This phenomenon is unique: ① the strawberry-like microbubble cloud extending to the surface of the hemispherical structure may have been the result of the Bjerknes adsorption force; ② the strawberry-shaped microbubble cloud that expanded to the lower left may have been caused by the combined action of Bjerknes’s adsorption force, collapse load, gravity, and buoyancy. When t = 0 ms, a shock wave would be generated. When t = 5.916 ms, the first pulsating load of the bubble would be generated. When t = 9.416 ms, the second pulsating load of the bubble would be generated.

4.1.6. Type VI: Spherical Shape without Jet

Type VI bubbles specifically refer to bubbles in the case of 0.4 ≤ ζ ≤ 0.6, 1.00 ≤ γ ≤ 1.50, and ζ = 1.2, 1.17 ≤ γ ≤ 1.50. For type VI, we choose one γ of each ζ to discuss. The details are shown in Figure 16.

Figure 16(1) shows the bubble pulsation process when ζ = 0 and γ = 2.00. At this time, we used a flat plate structure. When t = 0 ms, a dazzling light was emitted from the intersection of the electrodes, which is the beginning of high-voltage discharge. Since the distance between the explosion source and the flat plate structure was 60 mm, the expansion process of the bubble in the first cycle did not cause it to come into contact with the surface of the flat plate structure. At this time, the bubbles were spherical, as shown by t = 3.375 ms. During the contraction of the bubble in the first cycle, the bubble was also spherical. The bubble collapsed at t = 5.958 ms. This marked the end of the first pulse period of the bubble. The bubbles expanded and contracted in the form of strawberry-shaped microbubble clouds during the pulsation of the second cycle, as shown at 7.750 and 9.541 ms. The strawberry-shaped microbubbles cloud faced 45 degrees to the upper left. During the pulsation of the second cycle of the bubble, there was a jet toward the flat plate structure. When t = 9.791 ms, the second pulsating load of bubbles was generated.

Figure 16(2) shows the bubble pulsation process when ζ = 0.4 and γ = 2.00. At this time, we used a hemispherical structure with a radius of 75 mm. Same as ζ = 0 and γ = 2.00, the bubble was spherical in the expansion phase of the first cycle, as shown by t = 4.416 ms. During the bubble-shrinking process of the first cycle, the bubble also shrank in a spherical shape, as shown by t = 6.624 ms. Much like ζ = 0 and γ = 2.00, the pulsation process of the bubbles in the second cycle formed a strawberry-shaped microbubble cloud, as shown at 8.874 and 10.832 ms. The difference was that the direction of expansion of the strawberry-shaped microbubble cloud was not the upper left 45 degrees, but the upper left 20 degrees. This may be why the Bjerknes adsorption force gradually decreased as the boundary area shrank. During the pulsation of the second cycle of the bubble, there was a jet 20 degrees to the upper left. When t = 0 ms, a shock wave was generated. When t = 7.249 ms, the first pulsating load of bubbles was generated. When t = 11.249 ms, the second pulsating load of bubbles was generated.

Figure 16(3) shows the bubble pulsation process when ζ = 0.6 and γ = 2.00. At this time, we used a hemispherical structure with a radius of 50 mm. The similarities to ζ = 0.4 were: ① the bubble pulsated in a spherical shape in the first cycle, as shown at 3.291 and 5.624 ms; ② the bubble pulsed with a strawberry shape in the second cycle, as shown at 7.457 and 9.124 ms. The difference to ζ = 0.4 was: The expansion direction of the strawberry-shaped microbubble cloud and the jet direction of the second cycle of the bubble were not the upper left 20 degrees, but the lower left 20 degrees, as shown at 7.457 and 9.124 ms. This may have been the result of the combined action of the boundary Bjerknes force and gravity. When t = 0 ms, a shock wave was generated. When t = 6.207 ms, the first pulsating load of the bubble was generated. When t = 9.499 ms, the second pulsating load of bubbles was generated.

Figure 16(4) shows the bubble pulsation process when ζ = 1.2 and γ = 2.00. At this time, we used a hemispherical structure with a radius of 25 mm. The similarities to ζ = 0.6 were: ① the bubble pulsated in a spherical shape in the first cycle, as shown at 3.208 and 5.750 ms; ② the bubble pulsed with a strawberry shape in the second cycle, as shown at 7.708 and 9.291 ms. The difference to ζ = 0.6 was: the expansion direction of the strawberry-shaped microbubble cloud and the jet direction of the second cycle of the bubble was not the lower left 20 degrees, but the lower left 65 degrees, as shown at 7.708 and 9.291 ms. This may have been the result of the combined effect of the Bjerknes force and gravity on the bubble as the Bjerknes attractive force at the boundary decreases. When t = 0 ms, a shock wave was generated. When t = 6.208 ms, the first pulsating load of bubbles was generated. When t = 9.666 ms, the second pulsating load of bubbles was generated.

4.2. Effect of Different ζ and γ on Bubble Pulse Load

In this study, pressure sensors were used to measure the pressure loads generated by electric spark bubbles. The pressure load generated when γ = 0.33 and 0 ≤ ζ ≤ 1.2 is shown in Figure 17. Four pressure load peaks are marked in Figure 17(1). The time taken by the pressure sensor to obtain them was 0.902, 7.501, 7.515, and 11.33 ms, respectively. At t = 0.916 ms, a dazzling light was generated at the intersection of the electrodes, and spark bubbles began to form. Therefore, the pressure load measured by the pressure sensor at 0.902 ms was a shock wave load, and its load value was 7.556 MPa. When t = 7.499 ms, the bubble collapsed for the first time. In the pictures taken by the high-speed camera, no obvious jet phenomenon was found in the bubbles during the first pulsation stage. Therefore, it can only be assumed that one of the pressure loads measured by the pressure sensor at 7.501 and 7.515 ms was the bubble collapse load, and the other was the jet load. Based on past experience, we conclude that the pressure loads measured by the pressure sensor at 7.501 and 7.515 ms were the jet load and bubble collapse load, respectively. Their pressure load values were 40.47 MPa and 34.98 MPa, respectively. When t = 9.748 ms, the bubble shrank to the minimum volume, and the bubble collapsed for the second time. The pressure load measured by the pressure sensor at 11.33 ms was the bubble collapse load, and its load value was 11.33 MPa.

Three pressure load peaks are marked in Figure 17(2). The time taken by the pressure sensor to obtain them was 0.458, 6.846 and 10.18 ms, respectively. When t = 0.458 ms, a dazzling light was generated at the intersection of the electrodes, and electric spark bubbles began to form. Therefore, the pressure load measured by the pressure sensor at 0.458 ms was a shock wave load, and its load value was 8.35 MPa. When t = 6.874 ms, the bubble collapsed for the first time. The pressure load measured by the pressure sensor at 6.846 ms was the bubble collapse load, and its load value was 50.81 MPa. When t = 10.18 ms, the bubble shrank to the minimum volume, and the bubble collapsed for the second time. The pressure load measured by the pressure sensor at 10.18 ms was the bubble collapse load, and its load value was 16.04 MPa.

Four pressure load peaks are marked in Figure 17(3). The time taken by the pressure sensor to obtain them was 0.982, 6.969, 10.45, and 10.46 ms, respectively. When t = 0.999 ms, a dazzling light was generated at the intersection of the electrodes, and spark bubbles began to form. Therefore, the pressure load measured by the pressure sensor at 0.982 ms was the shock wave load, and its load value was 9.13 MPa. When t = 6.957 ms, the bubble shrank to its minimum volume, and the bubble collapsed for the first time. The pressure load measured by the pressure sensor at 6.969 ms was the bubble collapse load, and its load value was 67.25 MPa. At t = 10.415 ms, the bubble collapsed for the second time. In the pictures taken by the high-speed camera, no obvious jet phenomenon was found in the bubbles during the second pulsation stage. Based on past experience, we conclude that the pressure loads measured by the pressure sensor at 10.45 and 10.46 ms were the jet load and bubble collapse load, respectively. Their pressure load values were 33.24 and 17.76 MPa, respectively.

Five pressure load peaks are marked in Figure 17(4). The time taken by the pressure sensor to obtain them was 0.466, 5.986, 6.009, 8.693, and 8.714 ms, respectively. When t = 0.499 ms, a dazzling light was produced at the intersection of the electrodes, and spark bubbles began to form. Therefore, the pressure load measured by the pressure sensor at 0.466 ms was a shock wave load, and its load value was 10.22 MPa. When t = 5.999 ms, the bubble collapsed for the first time. In the pictures taken by the high-speed camera, no obvious jet phenomenon was seen in the bubbles during the first pulsation stage. Based on past experience, we conclude that the pressure loads measured by the pressure sensor at 5.986 and 6.009 ms were the jet load and bubble collapse load, respectively. Their pressure load values were 134.8 and 54.85 MPa, respectively. From the FEM simulation of the bubble pulsation process, when ζ = 1.2 and γ = 0.33, there was a jet in the first contraction of the bubble pulsation process. The details are shown in Figure 18b. Because 134.8 MPa exceeded the range of the pressure sensor (100 MPa), the measured value was an abnormal value, and we did not include it in the analysis. When t = 8.707 ms, the bubble collapsed for the second time. In the pictures taken by the high-speed camera, no obvious jet phenomenon was seen in the bubbles during the second pulsation stage. Therefore, we conclude that one of the pressure loads measured by the pressure sensor at 8.693 and 8.714 ms was the bubble collapse load, and the other was the bubble jet load. Their pressure load values were 43.55 and 24.12 MPa, respectively.

The bubble collapse load and jet load were caused by the bubble pulsation process. In this study, we only focused on the peak load generated during the pulsation process, because it has the greatest effect in terms of damage to the oblique 45-degree curved surface boundary structure. It can be preliminarily seen from Figure 16 that, with the increase in ζ, the shock wave load, the first cycle pulsation load of the bubble and the second cycle pulsation load of the bubble showed an increasing trend.

In order to further verify that, with the increase in ζ, the shock wave load, the first cycle pulsation load of the bubble, and the second cycle pulsation load of the bubble show an increasing trend, we continued to use pressure sensors to measure the load generated by electric spark bubbles to the oblique 45-degree curved surface boundary under different combinations of ζ and γ. We drew the measurement results as a curve, as shown in Figure 19 and Figure 20. It can be seen from Figure 19 and Figure 20 that, with the increase in γ, the peak load of shock wave and the pulsation peak load of the first and second cycles of the bubble gradually decreased. With the increase in ζ, the peak load of shock wave and the pulsation peak load of the first and second cycles of the bubble gradually increased. Appendix A provides the shock wave peak load measured by the pressure sensor when 0 ≤ ζ ≤ 1.2 and 0.33 ≤ γ ≤ 2.00, and the pulsation peak load data of the first and second cycles of the bubble.

5. Conclusions

In this study, a simple structure with an angle of 45 degrees to the vertical was used to simulate the oblique 45-degree curved surface boundary structure. We used high-voltage electric discharge to simulate cavitation bubbles. High-speed cameras and pressure sensors were used to study the effect of pressure loads generated during the pulsation of cavitation bubbles on the oblique 45-degree boundary. The conclusions of this study are as follows:

(1)

The centerline of the boundary structure was determined, assuming that it is at an angle of 45 degrees to the vertical. When γ was the same, the load generated by the cavitation bubbles increased with the increase in ζ.

(2)

When ζ was the same, the load generated by the cavitation bubbles became larger as γ became larger.

(3)

The bubble shape was roughly divided into six categories: “mushroom shape without jet”, “mound shape with jet”, “jellyfish shape with jet”, “oval shape with jet”, “drop shape without jet”, and “spherical shape without jet”.

The main purpose of this study was to expand the research in the field of cavitation bubble load measurement. We hope that our research can make a contribution to the anti-riot design of a ship’s bottom.

6. Patents

The experimental equipment for this research has been patented. Patent name: Experimental device for multi-angle electric spark bubble wall pressure load on curved surface. Patent number: CN201911298040.8.