Numerical Investigation of Cavitation Bubble Jet Dynamics near a Spherical Particle

: Synergistic interaction between cavitation bubbles and particles is critical for the operational performance of hydro turbines. The jet dynamics near the wall have been extensively investigated; however, the jet dynamics near the particles are not clear. In the present paper, the bubble jet dynamics near a spherical particle are numerically investigated based on a compressible two-phase ﬂow solver considering the effects of heat transfer and mass transfer between the phases. Furthermore, the effect of the distance between the particle and the initial position of the bubble on the jet characteristics is analyzed in detail. Based on the simulations, three typical cases (i.e., jet during the rebound stage, jet pointing towards the particle, two jets facing each other) of jet behavior are categorized together with the range of dimensionless parameters. As the distance between the particle and the bubble increases, the three cases of jet impacts occur in the rebound stage, in the ﬁrst period, and in the transition from the ﬁrst period to the rebound stage, respectively.


Introduction
Hydraulic components (e.g., nozzle or blades) often suffer from the synergistic interaction effects of particle erosion and cavitation damage during hydromechanical system operation [1][2][3]. An in-depth exploration of particle-cavitation bubble interactions is essential to understand the phenomenon and reduce the damage caused by such synergistic effects [4][5][6]. In this paper, the interaction between particle and cavitation bubble is numerically investigated using the open-source software OpenFOAM [7].
The jet during bubble collapsing has been widely and intensively investigated as one of the important phenomena during cavitation bubble collapse near structures (e.g., rigid walls, surfaces, and particles, etc.). Bai et al. [8] and Luo et al. [9] observed microjets during cavitation bubble collapse near a rigid wall by using high-speed photography. Reuter and Ohl [10] found that needle-like jets with an average velocity of 850 m/s were produced when cavitation bubbles collapsed close to the wall. Lechner et al. [11,12] numerically investigated the formation of a jet near a rigid wall and identified two types of axial jets. They determined the range of the standoff distance for the two types of jets and the transition interval between them. Furthermore, Bußmann et al. [13] revealed three types of jet (needle, mixed, and regular jets) formation mechanisms during cavitation bubble collapsing near a wall. They found that the needle jet produced pressure pulses on the wall that exceeded the pressure caused by the bubble collapse. Li et al. [14] found that the jet impact and high pressure inside the bubble are responsible for the double-peak structure of the wall pressure profile at a specific standoff distance based on the boundary integral method. Liu et al. [15] found that the jet induces a first peak at the center of the wall and a second peak at high pressure inside the bubble, based on the front tracking method. Based on numerical schlieren (which also visualize turbulent structures [16,17]), Tian et al. [18] state that the shock wave causes the first pressure peak, and the jet causes the second pressure peak. Sun et al. [19] reported that the rigid wall with a gas entrapping hole can reduce the pressure of the jet on the wall. During the rebound stage, Vogel et al. [20] observed the counter jet phenomenon in laser-induced cavitation bubble experiments. Yin et al. [21] observed the generation of a counter jet during the rebound stage through numerical simulations. Zhang [22] found that at standoff distances of 1~2, the "counter jet" may be part of the bubble spiral. Recently, Zhang et al. [23] experimented and simulated that the maximum height of the counter jet increases and then decreases with the characteristic distance. However, the bubble jet phenomenon still requires further exploration due to its complexity in terms of the jet characteristics and the jet formation mechanisms.
Specifically, the research on bubble jets near particles will be given briefly. For jet dynamics near a spherical particle, Zhang et al. [24] conducted an experimental investigation on a laser-induced cavitation bubble with the aid of high-speed photography. The identification criteria of three representative cavitation bubble collapse shapes were given quantitatively. Lv et al. [25] performed research on the dynamics of a cavitation bubble near a suspended spherical particle and found that jets are one of the factors contributing to the particle motion. Poulain et al. [26] conducted experimental investigations on particle motion induced by a cavitation bubble and found that the bubble is repulsive to the particle during growth and attractive to the particle during the collapse. According to the analytical model, the velocity of the particle depends on the distance from the bubble. Recently, Wang et al. [27] predicted the direction and strength of the jet near the particle based on the Weiss theorem and Kelvin impulse. Zevnik and Dular [28] numerically investigated the interaction of a cavitation bubble with a spherical particle, demonstrating the variation of the jet shape with increasing distance between the cavitation bubble and the particle. However, the formation mechanism, evolution, and influencing parameters of the bubble jet near the particle are still not clear.
In this paper, jet dynamics near a particle are investigated based on the finite volume method (FVM) and the volume of fluid (VOF). Three jet behaviors were identified based on the process of jet formation and development at different particle-bubble distances. The rest of the paper is organized as follows: Section 2 introduces the numerical method and implementation. Section 3 presents the physical, boundary, and initial conditions as well as the mesh independence for the simulations. Further, numerical validation is carried out through our experimental results. Section 4 reveals the formation mechanism and evolution of the jet based on the three cases observed. Section 5 quantitatively analyses the jet characteristics in terms of jet velocities. Section 6 gives some concluding remarks.

Governing Equations
In this paper, a compressible two-phase flow model, which takes into account the compressibility and viscosity of the liquid, thermodynamic effects, phase changes, and surface tension at the interface, is employed to simulate the jet phenomenon near particles. For simplicity, the effect of gravity is neglected. The VOF is employed to capture the liquid-vapor interface. Here, the continuity equation is where .
m is the mass transfer rate.
. m + is the condensation rate of the vapor phase.
. m − is the vaporization rate of the liquid phase. ρ is the mixture density. ρ l and ρ v are the density of liquid and vapor, respectively. α l and α v are the volume fraction of liquid and vapor, respectively. U is the mixture velocity. U l and U v are the velocity of liquid and vapor, respectively. The Schnerr-Sauer cavitation model [29] derived from the Rayleigh-Plesset equation [30] is introduced to solve the mass transfer rate, . m. The Schnerr-Sauer cavitation model has been validated to have high confidence in predicting jets [21,31]. In the Schnerr-Sauer cavitation model, the condensation rate and evaporation rate are: where n is the number of nuclei per cubic meter. p sat is the saturated vapor pressure. The volume fraction of the nucleation site α Nuc is where d Nuc is the nucleation site diameter.
To capture liquid-vapor interface, the volume fraction is solved with the transport equation [21]: where ∇ · (α l α v U r ) is introduced to ensure a sharp liquid-vapor interface [32,33]. U r is the relative velocity between the liquid phase and the vapor phase [34]. The momentum equation is [21]: where µ = α l µ l + α v µ v is the dynamic viscosity. µ l and µ v are the dynamic viscosity of liquid and vapor, respectively. I is the unit tensor. σ is the surface tension coefficient. In the last term of Equation (10), α l is obtained from the volume fraction α l by smoothing it over a finite region around the interface using the Lafaurie filter [34,35].
The energy equation is [21] ∂ρT ∂t + ∇ · (ρUT) + α l C p,l + α v where T is the temperature. C p,l and C p,v are the heat capacity of liquid and vapor, respectively. λ l and λ v are the thermal conductivity of liquid and vapor, respectively. K is kinematic energy. The liquid is considered as an ideal fluid, and the equation of state (EOS) [36] is where, when T = 0, the density of the liquid ρ 0 = 1000 kg/m 3 in this study, The vapor phase is assumed to be an ideal gas, and the EOS is where R v = 284.75 J/(kg·K) denotes the vapor constant.

Numerical Implementation
In this paper, the governing equations are discretized using the FVM and solved iteratively by using the PISO algorithm. During the simulation, the minimum time step is automatically adjusted through a maximum Courant number of 0.3, but the maximum step does not exceed 1 × 10 −7 s. Tables 1 and 2 list the numerical schemes and solvers employed for the simulations in Section 2.1. Table 3 lists the property parameters of the liquid phases, vapor phases, and cavitation model in the simulation.   Figure 1 illustrates the schematic of the computational domain. In Figure 1a, to save the computational resources, the model is axisymmetric, and the X-axis is the symmetry axis. The right endpoint of the particle is located at the origin of the Cartesian coordinates, and the cavitation bubble is located on the right side of the particle (Figure 1b). The computational domain size is 60 R max (R max represents the maximum radius of the cavitation bubble) to ensure that the size of the computational domain does not affect the simulation results. To accurately capture the liquid-vapor interface, the region near the bubble is refined locally (green region in Figure 1b). The average node distance on Line 1 and Line 2 ( Figure 1b) is employed as a quantitative criterion for the degree of refinement. Figure 2 illustrates the mesh of the computational domain and the mesh near the particle.  Figure 1a is the computational domain and the green region in Figure 1b is the refinement region. In this research, the dimensionless parameters γ and t* are defined as follows:  Figure 1a is the computa domain and the green region in Figure 1b is the refinement region. In this research, the dimensionless parameters γ and t * are defined as follows:

Numerical Validations
where the particle radius p R =1.00 mm in this paper. l denotes the center distanc tween the particle and the bubble at the initial moment of the simulation or experi  To simulate the growth and collapse process of the cavitation bubble, the initial radius of the bubble is 0.2 mm. The initial pressure and temperature inside the bubble are 23 MPa and 593.3 K, respectively. The pressure and temperature of the liquid are 101,325 Pa and 293.3 K. Initially, the liquid and vapor are still. In the simulation, the particle surface is the wall. The boundary conditions for pressure, velocity, and temperature are fixed pressure flux, no-slip, and zero gradient. For the outlet, the pressure is the wave transmissive flow condition, and the velocity and temperature conditions are both zero gradients.
To mitigate the effect of mesh size on the accuracy of the liquid-vapor interface and to reduce the computational cost [37], the results for eight different mesh sizes were compared. Figure 3 illustrates the effect of mesh size on R max . Obviously, R max gradually converges with decreasing mesh size. The absolute error of 0.5% on results for 3.2 µm and 2.4 µm mesh. The optimal mesh has 1042 and 1500 nodes in Line 1 and Line 2 ( Figure 1), respectively, and the initial cavitation bubble with a radius of 0.2 mm contains more than 200,000 mesh. Therefore, the mesh size of 3.2 µm (the total number of mesh is 2.81 million) is adopted to satisfy both accuracy and efficiency. Figure 4 depicts experimental and simulated cavitation bubble shape evolution over time at γ = 0.78. In the experiment, the particle was fixed at the tip of the needle, and the tail of the needle was connected to a three-dimensional movable platform. The particle was completely submerged in a tank filled with water. The platform adjusted the particle to be near the focusing point of the laser. A digital delay generator controlling the laser generates a cavitation bubble near the particle. At same time, a high-speed camera (900,000 fps) is controlled to record the shapes of the cavitation bubble [24]. During the experiment, the maximum radius of the cavitation bubble was 1.15 mm.
to reduce the computational cost [37], the results for eight different mesh sizes were compared. Figure 3 illustrates the effect of mesh size on max R . Obviously, max R gradually converges with decreasing mesh size. The absolute error of 0.5% on results for 3.2 µm and 2.4 µm mesh. The optimal mesh has 1042 and 1500 nodes in Line 1 and Line 2 ( Figure 1), respectively, and the initial cavitation bubble with a radius of 0.2 mm contains more than 200,000 mesh. Therefore, the mesh size of 3.2 µm (the total number of mesh is 2.81 million) is adopted to satisfy both accuracy and efficiency.  In the experiment, the particle was fixed at the tip of the needle, and the tail of the needle was connected to a three-dimensional movable platform. The particle was completely submerged in a tank filled with water. The platform adjusted the particle to be near the focusing point of the laser. A digital delay generator controlling the laser generates a cavitation bubble near the particle. At same time, a high-speed camera (900,000 fps) is controlled to record the shapes of the cavitation bubble [24]. During the experiment, the maximum radius of the cavitation bubble was 1.15 mm.
For the simulation, a circular domain with a radius of 0.2 mm, whose center is located on the symmetry axis, was marked on the right side of the particle ( Figure 1). This circular domain corresponds to the cavitation bubble produced in the experiment. The growth and collapse of the bubble in the experiment was simulated by setting the pressure and temperature of the fluid inside the circle higher than the surrounding.
By comparing the experimental and simulated images, the shapes of the cavitation bubble obtained through the simulation agreed well with the experimental results. There is only a small discrepancy between the experiment and the simulation at the end of the first period. A quantitative comparison with the experiment is shown in Figure 5. d R indicates the distance between the right end point of the cavitation bubble and the origin. Since the experiment captures the side of the bubble, while the simulation shows the crosssection of the cavitation bubble, the simulation results are smaller than the experiment at the end of the bubble collapse. By comparing the absolute errors of the Rd obtained from experiments and simulations, the largest absolute error occurs at t * = 0.26, which is less than 4%. From the above comparative analysis, the numerical model is reliable.  For the simulation, a circular domain with a radius of 0.2 mm, whose center is located on the symmetry axis, was marked on the right side of the particle (Figure 1). This circular domain corresponds to the cavitation bubble produced in the experiment. The growth and collapse of the bubble in the experiment was simulated by setting the pressure and temperature of the fluid inside the circle higher than the surrounding.
By comparing the experimental and simulated images, the shapes of the cavitation bubble obtained through the simulation agreed well with the experimental results. There is only a small discrepancy between the experiment and the simulation at the end of the first period. A quantitative comparison with the experiment is shown in Figure 5. R d indicates the distance between the right end point of the cavitation bubble and the origin. Since the experiment captures the side of the bubble, while the simulation shows the cross-section of the cavitation bubble, the simulation results are smaller than the experiment at the end of the bubble collapse. By comparing the absolute errors of the R d obtained from experiments and simulations, the largest absolute error occurs at t* = 0.26, which is less than 4%. From the above comparative analysis, the numerical model is reliable.

Typical Jet Dynamics
In this section, the jet phenomena of the cavitation bubble at different dimensionless distances (0.25 ≤ γ ≤ 1.60) are investigated in detail. Three typical cases are defined for jet behaviors.
In Case 1, the distance between the cavitation bubble and the particle is very small, and no jet is produced during the first period. A weak jet is formed during the rebound stage.
In Case 2, the distance between the cavitation bubble and the particle is medium. A directed jet towards the particle is produced.
In Case 3, the distance between the cavitation bubble and the particle is large and two jets facing each other are produced.
To clarify the three types of jet phenomena, γ = 0.25, 0.70, and 1.10 were selected, which correspond to Case 1, Case 2, and Case 3 respectively. Figure 6 illustrates a schematic of the particle and bubble positions for the three γ. Table 4 lists the parameter ranges and main characteristics of the three cases. Table 4. Main characteristics of the three representative cases.

Ranges of γ Main Characteristics Example
Case 1 0.20 ≤ γ ≤ 0.45 No jet produced in the first period, but jet appeared in the rebound stage In Case 3, the distance between the cavitation bubble and the particle is large and two jets facing each other are produced.
To clarify the three types of jet phenomena, γ = 0.25, 0.70, and 1.10 were selected, which correspond to Case 1, Case 2, and Case 3 respectively. Figure 6 illustrates a schematic of the particle and bubble positions for the three γ. Table 4 lists the parameter ranges and main characteristics of the three cases.  Figure 7 illustrates the bubble shapes for Case 1 during the growth stage, the collapse stage, and the rebound stage of the cavitation bubble. Due to the small distance between the particle and the bubble, in the preliminary growth stage, the left side of the bubble contacted the particle, and the right side remained nearly hemispherical without significant influence from the particle (Figures 7a-7d). When the bubble reaches its maximum volume (Figure 7d), a part of the particle is swallowed by the bubble, and then the bubble enters the collapse stage. As shown in Figures 7e and 7f, in the preliminary collapse stage, the right side of the bubble remains approximately hemispherical, but the left side of the bubble contacts the particle consistently and forms a neck. Since the bubble shrinks faster in the vertical direction than in the horizontal direction, the bubble has an elongated shape in the late stage of collapse (Figure 7g). At the moment demonstrated in Figure 7h, the cavitation bubble reaches its minimum volume for the first time. Obviously, during the first period (0.000 ≤ * t ≤ 1.000), there is no jet produced. During the rebound phase ( * t > 1.000), the right-side bubble shrinks along the vertical direction and detaches from the main bubble body, and then the detached bubble disappears quickly (Figure 7i-k). During this process, a jet directed towards the particle is generated, and the jet subsequently reaches the particle surface (Figure 7l).    Figure 7 illustrates the bubble shapes for Case 1 during the growth stage, the collapse stage, and the rebound stage of the cavitation bubble. Due to the small distance between the particle and the bubble, in the preliminary growth stage, the left side of the bubble contacted the particle, and the right side remained nearly hemispherical without significant influence from the particle (Figure 7a-d). When the bubble reaches its maximum volume (Figure 7d), a part of the particle is swallowed by the bubble, and then the bubble enters the collapse stage. As shown in Figure 7e,f, in the preliminary collapse stage, the right side of the bubble remains approximately hemispherical, but the left side of the bubble contacts the particle consistently and forms a neck. Since the bubble shrinks faster in the vertical direction than in the horizontal direction, the bubble has an elongated shape in the late stage of collapse (Figure 7g). At the moment demonstrated in Figure 7h, the cavitation bubble reaches its minimum volume for the first time. Obviously, during the first period (0.000 ≤ t * ≤ 1.000), there is no jet produced. During the rebound phase (t * > 1.000), the right-side bubble shrinks along the vertical direction and detaches from the main bubble body, and then the detached bubble disappears quickly (Figure 7i-k). During this process, a jet directed towards the particle is generated, and the jet subsequently reaches the particle surface ( Figure 7l). Figure 8 illustrates the velocity and pressure distributions during the collapse and rebound stage for Case 1. The upper subfigure is the pressure distribution and the lower one is the pressure distribution. In addition, the direction and magnitude of the flow are indicated with arrows on the velocity distributions. For the convenience of demonstration of the physical processes, and the scaling of the arrows varies in each subfigure. As shown in Figure 8a, at the end of growth, the right side of the bubble remains in growth while the left side begins to contract. In Figure 8b, the local high-pressure liquid near the neck continues to act on the bubble, causing the neck to contract at a much greater rate than the other locations. When the necks meet (Figure 8d), the high-pressure fluid (4.91 MPa) at the meeting location causes the right part of the bubble to detach and form a jet directed towards the particle. Figure 9 demonstrates the variations of the bubble shape during the first period for Case 2. During the growth stage (Figure 9a-c), the left side of the bubble grows along the particle surface, and the right side of the bubble keeps an approximate hemispherical shape. In addition, a liquid film exists between the bubble and the particle and gradually thins during the growth stage of the bubble (Figure 9b-d). As shown in Figure 9e-h, with the collapse of the bubble, the neck gradually shrinks towards the symmetry axis (resulting in a decrease in the neck radius) and simultaneously develops to the right (resulting in a widening of the neck). In Figure 9e, a directional jet directed towards the particle is formed and subsequently pierces the bubble (Figure 9j). Afterwards, the cavitation bubble travelled along the particle surface and reached the minimum volume (Figure 9l). Figure 10 illustrates the velocity and pressure distributions during the collapse stage for Case 2. In Figure 10a,b, the adverse pressure gradient deflects the outward flow at the liquid film, which gradually flows towards the bubble and causes the formation of a neck in the bubble close to the particle (see Figure 9e). And the contraction velocity of the neck was significantly smaller than that of the right side of the bubble. With the bubble collapse, a high-pressure region covering half of the bubble appears on the right side of the bubble (Figure 10c), and the pressure increases closer to the right end of the bubble. In Figure 10d, the high-pressure region concentrated at the right end of the bubble accelerates the collapse of the right side of the bubble, consequently forming an obvious jet (Figure 10e). In Figure 10g, the jet impacted the particle surface and the instantaneous pressure on the particle surface reached 58.76 MPa, which is a relatively high value causing damage.  Figure 9 demonstrates the variations of the bubble shape during the first period fo Case 2. During the growth stage (Figure 9a-c), the left side of the bubble grows along th particle surface, and the right side of the bubble keeps an approximate hemispherica thins during the growth stage of the bubble (Figure 9b-d). As shown in Figure 9e-h, with the collapse of the bubble, the neck gradually shrinks towards the symmetry axis (resulting in a decrease in the neck radius) and simultaneously develops to the right (resulting in a widening of the neck). In Figure 9e, a directional jet directed towards the particle is formed and subsequently pierces the bubble (Figure 9j). Afterwards, the cavitation bubble travelled along the particle surface and reached the minimum volume (Figure 9l).  Figure 10 illustrates the velocity and pressure distributions during the collapse stage for Case 2. In Figure 10a,b, the adverse pressure gradient deflects the outward flow at the liquid film, which gradually flows towards the bubble and causes the formation of a neck in the bubble close to the particle (see Figure 9e). And the contraction velocity of the neck was significantly smaller than that of the right side of the bubble. With the bubble collapse, a high-pressure region covering half of the bubble appears on the right side of the bubble (Figure 10c), and the pressure increases closer to the right end of the bubble. In Figure 10d, the high-pressure region concentrated at the right end of the bubble accelerates the collapse of the right side of the bubble, consequently forming an obvious jet (Figure 10e). In Figure 10g, the jet impacted the particle surface and the instantaneous pressure on the particle surface reached 58.76 MPa, which is a relatively high value causing damage. The collapse stage of the bubble with jet formation, development and impact the particle. Grey, white, and black areas in each subfigure indicate particle, liquid, and bubble, respectively. The subfigure size is 2.6 mm × 2.6 mm (width × height). γ = 0.70, R max = 1.15 mm. Figure 11 demonstrates the variations of the bubble shape during the first period for Case 3. In this case, the cavitation bubble is farther away from the particle and the influence of the particle on the bubble is weakened. As a result, during the growth stage (Figure 11a-c), the bubble remains spherical. During the primary stage of bubble collapse (Figure 11d-f), the left side of the bubble gradually contracted in the direction away from the particle, but the shrinking velocity was smaller than the right side, and consequently, the bubble gradually evolved from a spherical shape to a droplet shape. In Figure 11g, a depression appears on the left side of the bubble, which then gradually develops into a jet directed away from the particle (Figure 11g-j). At the same time, the right end of the bubble flattens ( Figure 11i) and subsequently develops into a jet pointing towards the particle (Figure 11j). And the jet develops more rapidly compared to the jet moving away from the particle. In Figure 11k, two jets with opposite directions penetrate the bubble and the bubble becomes a toroidal bubble. In Figure 11l, due to the impact of the two oppositely directed jets, the flow direction at the encounter location is deflected to move inside the bubble.  Figure 11 demonstrates the variations of the bubble shape during the first period Case 3. In this case, the cavitation bubble is farther away from the particle and the in ence of the particle on the bubble is weakened. As a result, during the growth stage ( Fig   Figure 10. Pressure distributions (upper part of each subfigure) and distributions (lower part of each subfigure) at different dimensionless times t * for Case 2. (a,b) The bubble begins to collapse from maximum volume. (c-h) The high pressure on the right side of the bubble drives the formation of the jet and its movement toward the particle. Grey and white areas in each subfigure indicate particle and bubble, respectively. The arrows indicate the magnitude and direction of the liquid velocity, but the length of the arrows varies between each subfigure. The subfigure size is 2.6 mm × 2.6 mm (width × height). γ = 0.70, R max = 1.15 mm. ble flattens (Figure 11i) and subsequently develops into a jet pointing towards the particle (Figure 11j). And the jet develops more rapidly compared to the jet moving away from the particle. In Figure 11k, two jets with opposite directions penetrate the bubble and the bubble becomes a toroidal bubble. In Figure 11l, due to the impact of the two oppositely directed jets, the flow direction at the encounter location is deflected to move inside the bubble.

Cases Ranges of γ
Main Characteristics Example Case 1 0.20 ≤ γ ≤ 0.45 No jet produced in the first period, but jet appeared in the rebound stage Figure 7 Case 2 0.50 ≤ γ ≤ 1.00 Directed jet towards particle   Figure 12a, the low-pressure liquid at the film flows towards the particle, while the liquid on the right side of the bubble is already flowing towards the bubble. As a result, the contraction of the bubble near the particle lagged behind the bubble away Figure 11. The variations of the bubble shape with dimensionless time t * for Case 3. (a-c) The growth stage of the bubble. (d-l) The collapse stage of the bubble with the formation and development of two jets. Grey, white, and black areas in each subfigure indicate particle, liquid, and bubble, respectively. The subfigure size is 2.6 mm × 2.4 mm (width × height). γ = 1.10, R max = 1.15 mm. Figure 12 illustrates the velocity and pressure distributions during the collapse stage for Case 3. In Figure 12a, the low-pressure liquid at the film flows towards the particle, while the liquid on the right side of the bubble is already flowing towards the bubble. As a result, the contraction of the bubble near the particle lagged behind the bubble away from the particle, and the bubble was gradually elongated. In Figure 11b,c, the pressure gradient of the liquid around the bubble gradually increases, resulting in the bubble close to the particle starting to shrink towards the bubble's center. In Figure 12d, the high-pressure liquid in the depression to the left of the bubble drives the formation of the jet away from the particle. In Figure 12e, the high pressure at the tail of the jet drives the jet and the bubble towards the particle.
Furthermore, in Figure 12e, the liquid near the tail of the left jet begins to move away from the cavitation bubble. Thereby, a vortex is formed in the liquid from the tail of the left jet to the tail of the jet. In Figure 12e-h, as the cavitation bubble and vortex move towards the particle, the maximum velocity of the vortex gradually decreases due to the conservation of momentum. In the aforementioned process, the vortex radius gradually becomes larger as the vortex volume and momentum spread in the liquid. from the cavitation bubble. Thereby, a vortex is formed in the liquid from the ta left jet to the tail of the jet. In Figure 12e-h, as the cavitation bubble and vorte towards the particle, the maximum velocity of the vortex gradually decreases du conservation of momentum. In the aforementioned process, the vortex radius g becomes larger as the vortex volume and momentum spread in the liquid.   Figure 13 illustrates the variations of jet velocity over time for the three cases. The jet velocity is represented by the velocity at the right endpoint of the bubble over the symmetry axis. In the same subfigure, the jets corresponding to different γ have similar behaviors. In addition, a positive velocity indicates a direction towards the particle and a negative one indicates a direction away from the particle. negative one indicates a direction away from the particle.

Quantitative Analysis of Jet Characteristics
In Case 1, the jet velocity first increases gradually due to the high-pressure liquid effect on the right side of the bubble. Subsequently, the contraction of the neck towards the symmetry axis inhibits the development of the jet, resulting in a decrease in jet velocity. After the neck meets, the high-pressure liquid at the meeting location causes a sudden increase in the jet velocity. Subsequently, the jet velocity gradually decreases due to energy conservation. Particularly for γ = 0.25, the right end point of the bubble starts to move away from the particle before the neck meets, and therefore the velocity gradually increases. In addition, the jet velocity gradually increases with increasing γ. In Case 2 and Case 3, the jet velocity exhibits the same trend over time. The jet velocity first gradually increases and then decreases. With the increase of γ, the jet velocity first increases gradually, then the jet velocity reaches a maximum at γ = 0.80, and then it decreases.  In Case 1, the jet velocity first increases gradually due to the high-pressure liquid effect on the right side of the bubble. Subsequently, the contraction of the neck towards the symmetry axis inhibits the development of the jet, resulting in a decrease in jet velocity. After the neck meets, the high-pressure liquid at the meeting location causes a sudden increase in the jet velocity. Subsequently, the jet velocity gradually decreases due to energy conservation. Particularly for γ = 0.25, the right end point of the bubble starts to move away from the particle before the neck meets, and therefore the velocity gradually increases. In addition, the jet velocity gradually increases with increasing γ. In Case 2 and Case 3, the jet velocity exhibits the same trend over time. The jet velocity first gradually increases and then decreases. With the increase of γ, the jet velocity first increases gradually, then the jet velocity reaches a maximum at γ = 0.80, and then it decreases. Figure 14 illustrates the effect of γ on the jet characteristics when the jet impacts or punctures the bubble. v jet denotes the velocity of the jet when it punctures the bubble. ∆t * = (t jet − t c )/t c denotes the dimensionless jet impact time difference, and t jet denotes the time at jet impact. ∆t * < 0 indicates that the jet impact occurs before the end of the first cycle, and ∆t * > 0 indicates that the jet impact occurs after the end of the first period. V * = V/V max denotes the dimensionless cavitation bubble volume at jet impact. V denotes the bubble volume at jet impact. V max indicates the maximum volume of the bubble. L jet * = L jet /R max denotes the dimensionless jet length at jet impact. L jet denotes the jet length at jet impact. d * = d/R max denotes the dimensionless bubble center displacement, and d is the displacement of the bubble center compared to the initial position of the bubble. d > 0 indicates the bubble is moving towards the particle, d < 0 indicates that the bubble is moving away from the particle.
increases, indicating that the bubble has entered the rebound stage before the jet impacts. And with γ increasing, the difference increases between the jet impact time and the collapse time. As γ increases, the influence of the particle on the bubble gradually decreases, and the bubble wall close to the particle gradually moves away from the particle; therefore the jet length and the displacement of the bubble center gradually decrease. When γ > 1.4, the jet length increases with γ due to the fact that the jet impact occurs in the rebound phase.  For Case 1 (0.25 ≤ γ ≤ 0.45), the jet impact velocity gradually increases with γ, and the effect on the particles is increased. ∆t* > 0 indicates that the jet impact occurs in the rebound stage. As γ increases, ∆t * , V * , and L jet * decrease, and the jet impact gradually occurs simultaneously with the collapse of the bubble.
In Case 2 (0.5 ≤ γ ≤ 1.0), as γ increases, v jet , ∆t * , and V * exhibit a consistent trend, with all increasing and reaching a maximum at γ = 0.8, then subsequently decreasing with further increases in γ. As γ increases, the initial position of the bubble gradually moves away from the particle, but the left side of the bubble keeps contacting the particle, and therefore the jet length and center displacement gradually increase.
For Case 3 (1.05 ≤ γ ≤ 1.6), ∆t * decreases with increasing γ at 1.05 ≤ γ ≤ 1.2. In particular, ∆t * ≈ 0 when γ = 1.2, indicating that the cavitation bubble reaches the minimum volume during the first period when the jet impacts. When γ > 1.2, ∆t * > 0 and gradually increases, indicating that the bubble has entered the rebound stage before the jet impacts. And with γ increasing, the difference increases between the jet impact time and the collapse time. As γ increases, the influence of the particle on the bubble gradually decreases, and the bubble wall close to the particle gradually moves away from the particle; therefore the jet length and the displacement of the bubble center gradually decrease. When γ > 1.4, the jet length increases with γ due to the fact that the jet impact occurs in the rebound phase.
In addition, Table 5 compares the results of this paper with the literature. Table 5. Comparison of the results of this paper with the literature.

Conclusions
In this research, the jet phenomenon of a cavitation bubble near a single particle is numerically investigated based on the finite volume method and the volume of fluid method. The numerical model takes into account thermodynamic effects, phase transitions, and the surface tension of the liquid. And the numerical results are in good agreement with the experimental results. Based on the detailed analysis of the numerical results, the following concluding remarks could be given: (1) The jet phenomena near the particle are categorized into three cases. For Case 1, no jets are generated in the first period, but a jet is generated in the rebound stage. For Case 2, a jet is directed towards the particle. For Case 3, two jets are facing each other. (2) The ranges for dimensionless distances between the bubble and the particle are given for three cases, which are 0.2 ≤ γ ≤ 0.45, 0.5 ≤ γ ≤ 1.0, and 1.05 ≤ γ ≤ 1.6, respectively. (3) As γ increases, the jet impact occurs in the rebound stage for Case 1, in the first period for Case 2, and from the first period transition to the rebound stage for Case 3. Heat capacity of liquid (J/(kg K)) C p,v Heat capacity of vapor (J/(kg K)) K Kinematic energy (J) λ l Thermal conductivity of liquid (W/(m·K)) λ v Thermal conductivity of vapor (W/(m·K)) R l Liquid constant (m) R v Vapor constant (m) R max The maximum radius of the cavitation bubble (mm) R p Particle radius (mm) R d Distance between the right end point of the cavitation bubble and the origin (mm) l The center distance between the particle and the bubble at the initial moment of the simulation or experiment (mm)