Hard particle erosion and cavitation damage are two main wear problems that can affect the internal components of hydraulic machinery such as hydraulic turbines or pumps. If both problems synergistically act together, the damage can be more severe and result in high maintenance costs. In this work, a study of the interaction of hard particles and cavitation bubbles is developed to understand their interactive behavior. Experimental tests and numerical simulations using computational fluid dynamics were performed. Experimentally, a cavitation bubble was generated with an electric spark near a solid surface, and its interaction with hard particles of different sizes and materials was observed using a high-speed camera. A simplified analytical approach was developed to model the behavior of the particles near the bubble interface during its collapse. Computationally, we simulated an air bubble that grew and collapsed near a solid wall while interacting with one particle near the bubble interface. Several simulations with different conditions were made and validated with the experimental data. The experimental data obtained from particles above the bubble were consistent with the numerical results and the analytical study. The particle size, density, and position of the particle with respect to the bubble interface as well as the bubble position strongly affected the maximum velocity of the particles.

## I. INTRODUCTION

A general problem of hydraulic machinery such as water pumps and turbines is the wear of their internal components. Some of those components that are in contact with a liquid are susceptible to several problems such as solid particle erosion or cavitation damage.^{1–4} For particle erosion, hard particles travel at high speeds in a liquid and hit a solid surface, which causes deformation or in many cases a loss of material. The level of damage depends on various factors such as velocity, attack angle, density, hardness, size, concentration, fracture toughness, and shape factor of the particles in the liquid.^{5,6} The cavitation damage is a product of the collapse of many cavitation bubbles near a solid surface. Cavitation occurs when the liquid pressure due to hydrodynamic effects drops below the vapor pressure of the liquid at a given temperature. Consequently, cavitation bubbles appear and travel with the liquid; when they reach a high-pressure region, they collapse and generate smaller bubbles and pressure waves. These collapses near a solid surface generate micro-jets and high-pressure waves, which directly hit the surface and induce deformation. Such deformation continues to increase due to the successive collapses of bubbles, and the surface material is eventually removed due to fatigue.^{7,8} In some cases, both phenomena act together in a synergistic way, causing more severe damage than that caused by each phenomenon alone.^{9–13} However, in experiments performed using a vibratory apparatus, particles under a critical size inhibit the cavitation damage instead of increasing the damage.^{14} Therefore, the study of the interaction of particles and cavitation bubbles is a key aspect to avoid the damage caused by the synergistic effect of cavitation and particles. Figure 1 shows the damage of some Francis turbine components due to the three aforementioned phenomena.

Several authors showed that the interaction of particles and cavitation bubbles is caused by the bubble dynamics during their growth and collapse. Soh and Willis^{15} developed an experiment to observe the movement of several particles due to the collapse of a bubble. In the experiment, they fixed several particles with cords above a bubble, which was generated by an electric spark near a solid wall, and placed others on the solid wall. They found that the suspended particles were not apparently affected because of the restriction of the cords; however, the particles on the surface were significantly moved from their original position. In 2015, Poulain *et al.*^{16} analyzed the dynamics of spherical particles made of glass, aluminum, brass, and steel with sizes between 1.6 mm and 2.5 mm that were affected by a cavitation bubble of a similar size, which was generated by an electric spark far from any wall. From their observations they defined three phases: first, the bubble grows and pushes the particle away; then, the bubble collapses, and the particle is attracted to it; finally, after a significant amount of time after the collapse, the particle continues moving towards the center of the bubble due to the rebounding of the cavitation bubble at an earlier time. Using an analytic model based on the asymmetric dynamics of the bubble, the authors found that the normalized velocity of the particle showed an inverse-fourth-power-law relationship with the normalized distance between the bubble and the particle.

Gas bubbles have been used recently to study their behavior inside a fluid when interacting with several geometric forms.^{17,18} Also, the behavior of a cavitation bubble near a solid wall without the interaction of particles has been studied using computational fluid dynamics (CFD) numerical simulations. Osterman *et al.*^{19} studied the collapse of one single bubble in an ultrasonic field near a wall surface. In their analysis, they used a finite-volume 2D axisymmetric model and the Volume-Of-Fluid (VOF) approach to evaluate the effect of the initial bubble distance from the wall considering an initial bubble with the maximum bubble radius and skipping the growing phase. They validated their simulation with the experimental results of Philipp and Lauterborn^{20} by obtaining consistent shapes of the bubble during the collapse. Their study established that the results were highly sensible to the grid density, and the velocities of the micro-jets developed during the collapse were approximately 100 m/s for a maximum bubble radius of 1.45 mm and a separation of 1.74 mm from the wall. Johnsen and Colonius^{21} simulated the collapse of a gas bubble induced by the shock wave in a free field near a solid surface without considering bubble growth. To validate the simulations, they compared their results with the available theory and experiments, which showed consistency with the bubble dynamics data and propagation of the shock emitted upon the collapse. The induced shock collapse generated notably high velocities in the re-entrant jet, which created a water hammer shock that produced notably high pressures on the wall and could represent potential damage to the neighboring surface. Jayaprakash *et al.*^{22} performed experimental tests and numerical simulations of the interaction of a vertical wall and a bubble. In this case, they simulated the growth and collapse of the bubble using a high initial pressure inside a very small bubble. They validated their results with the data obtained from an experimental bubble generated by an electrical spark. Their studies also showed a notable good correlation between numerical simulations and experimental observations. They concluded that the jet characteristics strongly depended on the standoff distance from the wall. For example, a bubble with a non-dimensional standoff distance between 0.75 and 1 developed the maximum jet momentum, which indicates that the maximum damage can be achieved at this standoff range. Similarly, Chahine and Hsiao^{23} simulated the damage caused by a cavitation bubble collapse on a solid surface of several materials. In this case, they used a boundary condition of a variable pressure and found that at a non-dimensional standoff distance close to 0.75 the energy transferred to the wall is maximum. On the other hand, Chahine *et al.*^{24} simulated the effect of a cavitation bubble in the cleaning process of a surface. They found that the formation and development of a bubble re-entrant jet is a strong source of a localized shear force that can lift and repulse a microscopic particle away from the surface. However, the direct interaction of the particle with the re-entrant micro-jet was not analyzed.

If there are particles near a cavitation bubble collapsing near a solid wall, the generated re-entrant jet can trap them in its velocity field; in some conditions, the particles can be accelerated towards the surface and cause damage on it. To evaluate this effect, Li^{10} proposed a microscopic model that assumed a low concentration of particles suspended in the fluid and a small particle size compared with the cavitation bubbles; thus, the solid particles did not affect the bubbles or flow characteristics. According to this author, the particle is trapped by the jet of the collapsing bubble and accelerated to a high velocity towards the solid surface. In a later investigation, Dunstan and Li^{25} numerically studied the dynamics of only one particle near a cavitating bubble during its collapse. As a result, they verified that the damage potential on a surface was increased because the particle acquired high kinetic energy due to its interaction with the collapsing bubble.

In this work, we perform cavitation experiments with small particles in the surroundings. A simplified analytical model and CFD numerical simulations are used to understand the complicated interaction between a cavitation bubble and the nearby particles. This study might lead to a better understanding of how hard particle erosion is affected by the cavitation process.

## II. EXPERIMENTAL TESTS

Figures 2(a) and 2(b) show the experimental setup to study the interaction of particles and a cavitation bubble. A schematic of the generated bubble is shown in Fig. 2(c) with a reference coordinate centered on the solid surface. A cavitation bubble was created using an electric spark, which was generated when an electric current passes through the tips of two electrodes in contact. A tinned cooper wire of an approximate diameter 0.12 mm was used to generate the spark at the desired location. The wire was obtained from a stranded hook-up wire manufactured by Consolidated Electronic Wire & Cable (Part # 815-5). The electric energy was obtained from a direct-current (DC) source, which enabled one to change the voltage. Then, the energy was stored in a device composed of 4 capacitors with an equivalent capacitance of 23.5 mF, which could be charged to 50 V. To produce the spark, the stored energy was released to the electrodes and a short circuit was produced in the contact zone generating a spark that originated the nucleation of a bubble; different voltages resulted in different bubble maximum sizes. A stream of particles was flowing above the nucleation position. The particles were contained in a syringe connected to a tube with a hollow needle with an inner radius of 0.65 mm (particle feeder) at the end; the tip of the needle and the wires were in the same plane which was verified by releasing some particles in advance and observing the particle falling onto the contact point of the two electrodes. The particles were transported to the desired position by gravity; once the particles reach that position, the bubble was generated to enable their interaction. The bubble was generated near a solid surface (sample) to observe the behavior of the particles that interacted with a cavitation bubble that collapsed near a solid surface. This process was recorded using a Photron FASTCAM Mini APX RS high-speed camera at 50 000 frames per second, which enabled us to track the particles to obtain important variables such as the particle velocity and acceleration. All analyzed particles were tracked using the software Tracker.^{26}

As listed in Table I, four parameters were evaluated in our experiment: distance from the wall, particle size, particle material, and particle-bubble separation. Each configuration was repeated 3 times, and some were also used to validate the CFD simulations. Table I shows the low and high levels of the evaluated factors in a full factorial experiment. The distance from the wall (*S* in Fig. 2) is the distance between the solid wall and the nucleation center. We tested two particle sizes and two types of particle materials. Here, the two materials have different densities: 2130 ± 210 kg/m^{3} for sand and 3910 ± 350 kg/m^{3} for alumina.^{6} The particle separation parameter (*S*_{p}) is the distance between a particle above the bubble and the bubble nucleation center, which was measured at its maximum radius. During this experiment, the voltage was maintained constant at 38.2 V, which created a maximum radius of 2.5 ± 0.06 mm.

. | Factor level . | |
---|---|---|

Factors . | Low . | High . |

Distance from the wall (S) (mm) | 1.5 | 2.5 |

Particle size (μm) | 53-63 | 75-106 |

Particle material | Sand | Alumina |

Particle-bubble separation (S_{p}) (mm) | 2.52-2.58 | 2.7-2.8 |

. | Factor level . | |
---|---|---|

Factors . | Low . | High . |

Distance from the wall (S) (mm) | 1.5 | 2.5 |

Particle size (μm) | 53-63 | 75-106 |

Particle material | Sand | Alumina |

Particle-bubble separation (S_{p}) (mm) | 2.52-2.58 | 2.7-2.8 |

Other experiments were performed to analyze the effect of the maximum bubble size on the behavior of the particles. Four bubble sizes of 1.5 ± 0.04 mm, 2.5 ± 0.06 mm, 3 ± 0.08 mm, and 4 ± 0.05 mm were generated using charge voltages of 31.5 V, 38.2 V, 43 V, and 47.5 V, respectively. In these cases, the distance from the wall (*S*) was fixed at 1.5 mm, the particle separation from the interface was 0.2-0.3 mm, where the *S*_{p} varied between 1.7 and 4.3 depending on the maximum bubble size, the particle size was 75-106 *μ*m, and the particle material was sand.

## III. NUMERICAL MODEL

We used the geometry of an axisymmetric fluid domain to perform a simplified numerical simulation of the interaction of a solid particle and a bubble subjected to pressure changes near a solid wall. Here, the air is modeled as an ideal gas, which is different from the real case, where the gas is water vapor, and there is mass transfer in the evolution of the bubble. In the simplified case, the evolution of the air bubble is due to pressure changes.

Several experimental investigations generated their bubbles using a spark,^{15,16} a focused laser beam^{20,27} or an ultrasonic field.^{28,29} However, the described simplified simulations have been validated with this type of experimental test.^{19,21} In this simulation, the phenomenon is assumed to be axisymmetric because experiments with a single cavitation bubble^{20,22,30} show that during a collapse near a surface, a micro-jet is developed through the center of the bubble towards the surface and forms an axis of symmetry. Thus, a two-dimensional simulation that spends less computational resource can be used.

The commercial software ANSYS Fluent solver was used to solve Navier-Stokes equations in a transient simulation. This software enables us to use the Volume-Of-Fluid (VOF) model to capture the behavior of the interface between air and liquid water. It also has a variety of turbulence models to choose from; however, a turbulence model was not considered here because the characteristic dimension of the particle is quite small. Therefore, flow around the particle stays in a low Reynolds number regime. Moreover, it is possible to create user-defined functions to define variable boundary conditions or add compressibility effects to the liquid and gas phases.

The governing equations in this simulation are the mass, momentum, and energy conservation equations.

The mass conservation equation can be written as

where *ρ* is the fluid density and $v\u2192$ is the velocity vector. With the axisymmetric simplification, the mass conservation equation is

where *y* is the axial coordinate, *r* is the radial coordinate, $v$_{y} is the axial velocity, and $v$_{r} is the radial velocity. The momentum conservation equation is

where *p* is the static pressure; $\rho g\u2192$ and $F\u2192$ are the gravitational and external body forces, respectively. The shear stress tensor $\tau \u0332\u0332$ is defined by

where *μ* is the dynamic viscosity, *I* is the unit tensor, and the second term of the right-hand side is the effect of volume dilation.

In the axisymmetric case, the axial and radial momentum conservation equations are defined as

and

where

The energy conservation equation in its general form can be written as

where

The first three terms on the right-hand side of Eq. (8) represent the energy transfer by conduction, species diffusion, and viscous dissipation, where *k*, *T*, *h*, and *J* are the thermal conductivity, temperature, sensible enthalpy, and diffusion flux of species, respectively. The last term (*S*_{h}) is a volumetric heat source. In the VOF model, there is no species diffusion, no condensation or evaporation, and the heat source is zero.

The VOF two-phase model describes the behavior of a primary phase in a secondary phase assuming that the phases do not mix with each other. In this study, this model is used to observe the behavior of an air bubble (primary phase) in liquid water (secondary phase).^{19} The tracking of the interface surface between the two phases is obtained by solving the continuity equation for the volume fraction (*α*) of the secondary phase

Because of the limitations of the VOF model, the right-hand side of Eq. (10) is zero since there is no mass transfer of the source terms of evaporation or condensation. The volume fraction of the primary phase is calculated considering that the sum of the two volume fractions is one. Given the known value of the volume fraction of one phase in a computational cell, the fields for all variables and properties are shared by the two phases and represent a volume-averaged value at each location.

The two phases in the VOF model are considered compressible. The primary phase is air, and its properties are defined using the ideal gas law, whereas the density of secondary phase (water) depends on the pressure according to the following expression:

where *K* is the water bulk modulus (2.2 GPa), *ρ*_{0} is the reference density (1000 kg/m^{3}), and Δ*p* is the pressure difference regarding the reference pressure (1 atm).^{19}

The simulated domain was a rectangle of 200 mm × 200 mm, which represents a 400-mm-diameter cylinder in the axisymmetric case. The selected size helps avoiding boundary effects in the small region where the phenomenon occurs (*S*/*R*_{b} ratio of 0.37-1);^{21} additionally, in test simulations of domain sizes 50 mm × 50 mm, 200 mm × 200 mm, and 500 mm × 500 mm, the results obtained with the last two sizes had a difference in maximum velocity of the micro-jet less than 3%. Therefore, to reduce the computational cost, the size of 200 mm × 200 mm was selected. The domain was discretized using a hexahedral structured mesh, which was refined in the region where the bubble grew and collapsed. A small air bubble of radius 60 *μ*m was placed on the symmetry axis at the beginning of the simulation near a solid wall at several positions according to the numerical experiment described in Table II. In a similar way as done by Chahine and Hsiao,^{23} the size of the bubble nucleation was chosen arbitrarily. However, we needed to take into account that it should cover more volume than a spherical particle of diameter 60 m, since one of the simulations starts with a particle at the center of the bubble. During the growth of the bubble, a maximum size is achieved, which depends on the initial pressure in the fluid domain to be simulated.

Particle size (μm)
. | Particle material . | S_{p} (mm)
. | S (mm)
. | R_{b} (mm)
. |
---|---|---|---|---|

60 | Sand | 2.52-2.58 | 1.5 | 2.5 |

60 | Sand | 2.52-2.58 | 2.5 | 2.5 |

90 | Sand | 2.52-2.58 | 2.5 | 2.5 |

90 | Alumina | 2.52-2.58 | 2.5 | 2.5 |

60 | Sand | 1.5 | 1.5 | 2.5 |

60 | Sand | 1 | 1.5 | 2.5 |

60 | Sand | 2.5 | 2.39 | 2.5 |

Particle size (μm)
. | Particle material . | S_{p} (mm)
. | S (mm)
. | R_{b} (mm)
. |
---|---|---|---|---|

60 | Sand | 2.52-2.58 | 1.5 | 2.5 |

60 | Sand | 2.52-2.58 | 2.5 | 2.5 |

90 | Sand | 2.52-2.58 | 2.5 | 2.5 |

90 | Alumina | 2.52-2.58 | 2.5 | 2.5 |

60 | Sand | 1.5 | 1.5 | 2.5 |

60 | Sand | 1 | 1.5 | 2.5 |

60 | Sand | 2.5 | 2.39 | 2.5 |

The boundary conditions are shown in Fig. 3. The pressure boundary condition at the top edge of the domain is a step function that varies with time. In this boundary, an initial pressure of 110 kPa immediately decreases to 1 kPa and is maintained during 4 × 10^{−4} s; then, the pressure changes to 100 kPa and remains constant until the end of the simulation to enable the bubble collapse. A similar boundary condition was used by Chahine and Hsiao,^{23} which is closer to an actual process inside hydraulic turbines or pumps; a small air bubble enters a region of low pressure thereby growing fast and then collapses when it goes through to a high pressure region (close to the blades).

The time, in which the low pressure was maintained, was chosen considering the time of the bubble’s growth and collapse obtained by Poulain *et al.*^{16} for a bubble of radius 2.5 mm. The particle was modeled as a wall boundary with a circular shape and located in the symmetry axis at different positions from the bubble (see Table II); this particle could move along the symmetry axis. To model the interaction between the particle and the fluid, the Six DOF solver of the ANSYS Fluent software coupled with a dynamic mesh was used, which enabled us to calculate the forces and moments of an object of six degrees of freedom immersed in a fluid.

The pressure and velocity were coupled using a coupled scheme that included the volume fractions of air and water. In the spatial discretization, the pressure was interpolated using a PRESTO! scheme,^{31} whereas the density, momentum, and energy were interpolated using a second-order scheme, and a compressive scheme was used in the case of the volume fraction. Finally, the temporal discretization was set to be first-order implicit using a time step size of 10^{−8} s. Table II shows the performed numerical simulations to validate the results and evaluate several parameters in the interaction of the particle with the cavitation bubble.

Before performing the validation with the experimental data, the mesh size of the proposed simulation in Sec. III was evaluated in a mesh independence study. Therefore, the maximum cell size to obtain reliable results in the zone where the bubble grows and collapses is a square of side 10 *μ*m.

## IV. RESULTS AND DISCUSSION

### A. Experimental test results

Figure 4 shows a sequence of the evolution of the bubble and particles with one of the tracked particles highlighted. Figure 5 shows the evolution of the vertical position of the highlighted particle and bubble interface, and the particle velocity and acceleration during the growth and collapse of the bubble for testing, where the particle size is 75-106 *μ*m, the particle material is sand, the distance from the wall is 1.5 mm, and the maximum size of the bubble is 2.53 mm. The bubble dynamics enables particle movement during the growth (time 0-0.42 ms in Fig. 4) and collapse (time 0.48-0.8 ms in Fig. 4) of the bubble. It was not possible to observe the particle movement during most part of the bubble growth phase due to the bright spark in every test; however, the collapse process, where the tracked particle reached its maximum speed, was well observed. Here, we noted that the particle motion is significantly affected at the beginning of the bubble collapsing phase. However, at a later stage, when a micro-jet is formed, the distance between the particle and the cavitation bubble interface is large enough that its effect could not be that strong, as observed in Fig. 4.

The normalized time *t** in Fig. 5 was calculated using the required time of bubble growth and collapse as the reference, which was defined as the instant immediately before the spark appeared (*t*_{i}) until the moment when the bubble interface reached the initial position of the bubble’s center (*t*_{gc}). The expression is

Figure 6 shows the experimental test results for all evaluated cases. The graphs in panels (a) and (b) show the magnitude of the maximum velocity and maximum acceleration of particles above the bubble in the vertical direction. Neither velocity nor acceleration significantly varied with the distance of the wall in most cases. In fact, an ANOVA study of the performed factorial experiment reveals that the most significant parameters from least to greatest relevance in the particle velocity are the particle size, material, and position with respect to the bubble interface. Table III shows the *F* and *p* values for the main effects and their combined effects on the particle maximum velocity and acceleration; the *p* values below 0.05 are significant. Thus, both particle size and material significantly affect the particle velocity because of their effect on the particle mass since larger and denser alumina particles (whose density is 3910 kg/m^{3}) are more difficult to move than small and lighter sand particles (whose density is 2150 kg/m^{3}). Moreover, particles near the bubble interface are highly affected by the velocity field generated from the bubble growth and collapse, whereas a smaller effect is observed on the particles far from the bubble interface.

. | Maximum . | Maximum . | ||
---|---|---|---|---|

. | velocity . | acceleration . | ||

Source . | F . | p . | F . | p . |

S | 6.34 | 0.017 | 5.5 | 0.025 |

Particle size | 350.18 | 0 | 51.06 | 0 |

Particle material | 663.12 | 0 | 102.06 | 0 |

S_{p} | 814.01 | 0 | 122.3 | 0 |

S * particle size | 42.35 | 0 | 0.56 | 0.46 |

S * particle material | 0.2 | 0.661 | 10.61 | 0.003 |

S * S_{p} | 1.35 | 0.253 | 0.94 | 0.34 |

Particle size * particle material | 62.18 | 0 | 9.28 | 0.005 |

Particle size * S_{p} | 123.64 | 0 | 31.53 | 0 |

Particle material * S_{p} | 192.02 | 0 | 60.77 | 0 |

S * particle size * particle material | 2.08 | 0.159 | 9.62 | 0.004 |

S * particle size * S_{p} | 6.36 | 0.017 | 0 | 0.995 |

S * particle material * S_{p} | 0.95 | 0.336 | 0.23 | 0.636 |

Particle size * particle material * S_{p} | 72.05 | 0 | 14.97 | 0.001 |

S * particle size * particle material * S_{p} | 6.88 | 0.013 | 4.62 | 0.039 |

. | Maximum . | Maximum . | ||
---|---|---|---|---|

. | velocity . | acceleration . | ||

Source . | F . | p . | F . | p . |

S | 6.34 | 0.017 | 5.5 | 0.025 |

Particle size | 350.18 | 0 | 51.06 | 0 |

Particle material | 663.12 | 0 | 102.06 | 0 |

S_{p} | 814.01 | 0 | 122.3 | 0 |

S * particle size | 42.35 | 0 | 0.56 | 0.46 |

S * particle material | 0.2 | 0.661 | 10.61 | 0.003 |

S * S_{p} | 1.35 | 0.253 | 0.94 | 0.34 |

Particle size * particle material | 62.18 | 0 | 9.28 | 0.005 |

Particle size * S_{p} | 123.64 | 0 | 31.53 | 0 |

Particle material * S_{p} | 192.02 | 0 | 60.77 | 0 |

S * particle size * particle material | 2.08 | 0.159 | 9.62 | 0.004 |

S * particle size * S_{p} | 6.36 | 0.017 | 0 | 0.995 |

S * particle material * S_{p} | 0.95 | 0.336 | 0.23 | 0.636 |

Particle size * particle material * S_{p} | 72.05 | 0 | 14.97 | 0.001 |

S * particle size * particle material * S_{p} | 6.88 | 0.013 | 4.62 | 0.039 |

Figure 7 presents the behavior of particles with the variation in the maximum bubble size for sand particles of size 75-106 *μ*m; the initial position of the bubble from the wall was 1.5 mm, and the separation of the bubble interface was 0.2-0.3 mm. An increase in the bubble radius escalates the maximum velocity of the particles above the bubble, whereas the maximum acceleration does not significantly vary.

### B. Validation of CFD simulations

Similar to the study of Poulain *et al.*,^{16} a simplified analytical approach, which considers the spherical collapse of a bubble and away from any solid surface, was developed to understand the behavior of the particles above the bubble during the collapse phase. However, in contrast to the previous work, here we study the dynamics of particles at the micron scale.

In this analysis, the particles were assumed to have a sphere-shape. Then, we assumed that the flow created by the evolution of the bubble is incompressible.^{32} Using spherical coordinates centered at the nucleation site, radial velocity *u* can be written as

where *R*_{b} is the bubble radius. Considering negligible gravitational force, an analysis of the forces that act on the particle because of the flow leads to the following expression:

where *m*_{p} is the particle mass, *r*_{p} is the radial position of the particle, $r\u0308p$ is the particle acceleration, and *F*_{drag}(*t*) is the drag force, which is defined as

where *R*_{p} is the particle radius, *ρ* is the density of water, and *C*_{D} is the drag coefficient, which is assumed constant at a value of 0.47 according to the experimental data reported by NASA.^{33} The particle velocity can be evaluated by integrating Eq. (14) over the bubble collapse phase. As a simplification, a negligible displacement is assumed in the drag expression. Similarly, Poulain *et al.*^{16} used the same assumption in which the particle movement was limited by only 5% from its initial position. However, in our case, the particle moves up to 44%. This rough assumption is one of the reasons why the maximum velocity in the analytical model does not match with experimental values accurately. Thus, *r*_{p} = *r*_{p,i}, where *r*_{p,i} is the initial position of the particle immediately before the bubble begins to collapse. Therefore, the particle velocity can be written as

where *t*_{i} is the initial time; $R\u0307$_{b} is the velocity of the bubble interface, which is a function of time and can be evaluated using Rayleigh-Plesset equation^{7}

Here, $pv$ is the vapor pressure of water at the operating temperature; *p*_{∞} is the pressure in the bulk of the surrounding liquid, which is a function of time; *p*_{g0} is the initial partial pressure of the gas inside the bubble; *R*_{0} is the initial radius of the bubble; *γ* is the ratio of the heat gas capacities *c*_{pg} and $cvg$; and *S* is the surface tension.

In the case of the bubble collapse, for simplicity, we neglected the effects of non-condensable gas and the viscosity because the inertial force is dominant.^{7} Also, we did not take the surface tension into account because its effect is important only for very tiny bubbles.^{32} Therefore, the last three terms on the right-hand side of Eq. (17) are removed, and it can be integrated to yield

During the collapse of the bubble, $Rb\u0307$ is negative, so

In this analysis, Eq. (19) was solved using numerical integration with the trapezoidal rule to find the collapse time and evaluate the evolution of the bubble radius with time. Then, this solution was used to solve Eq. (20) using numerical integration to evaluate the particle velocity. After several tests of convergence, the integration interval was divided into 2000 sub-intervals to achieve reliable results.

The previous analysis helps us to understand the behavior of the particle because of the bubble dynamics effect during the collapse phase; however, there is no bubble after the collapse, so the particle begins to decelerate because of the drag caused by the fluid, which is assumed to be static, until the particle stops as observed in the experiments. To evaluate the particle velocity after this phase of the movement, the momentum equation of the particle was used to obtain

The integration of Eq. (21) results in

which enables us to evaluate the particle velocity after the bubble collapse to compare with experimental data.

Figure 8 presents an overlay comparing an image sequence of the bubble collapse obtained experimentally and the results obtained through CFD simulation, and a good agreement of the bubble interface and the particle position between CFD and experimental data is observed. Figure 9 shows the magnitude of the particle velocity in several graphs to compare the experimental data with numerical CFD and the analytical solutions obtained with Eqs. (20) and (22) in several conditions during the bubble collapse. In Fig. 9, we used the normalized time of Eq. (12) to better compare the results. It is observed how, due to the bubble dynamics, the velocity of the particle increases until a maximum value and then starts to decrease. Additionally, we note that the CFD simulations predict the behavior of the particle affected by the bubble dynamics reasonably well, particularly after the particle reaches its maximum velocity. The analytical analysis provides a good prediction of the particle’s behavior above the bubble and under the conditions of a spherical collapse despite all assumed simplifications. This analysis helps us to identify two phases for the behavior of a particle immersed in the field near the bubble during its collapse. In the first phase, which begins at the maximum radius of the bubble, the particle velocity increases and reaches its maximum when the bubble radius is near zero. In the second phase, the fluid was assumed to be static as observed in the high-speed images, and the velocity of the particle decreases, which indicates that the fluid decelerates the particle.

### C. Additional CFD results

In Sec. IV B, we have employed experimental data and an analytical approach to validate a CFD simulation of the interaction of a solid particle with a collapsing bubble near a solid wall. In this situation, the bubble collapse is not spherical and a micro-jet is developed towards the wall, as has been observed by other authors.^{19,21,23} The obtained results using the simplified analytical approach showed that particles located just above the bubble have a similar behavior for a spherical collapse, when the bubble is far away from any solid wall, and for non-spherical collapse, when the bubble is near a solid wall and develops a micro-jet. However, if the particle location is changed, the situation changes and the developed analytical approach does not work.

Figure 10 shows the simulation results of the collapse of a bubble located at 1.5 mm from the wall with a sand particle of diameter 60 *μ*m located at the center of the bubble at the beginning of the simulation. In Fig. 10, we can see how the particle is trapped by the velocity field of the re-entrant jet during its development, leading to an acceleration of the particle towards the solid wall. Figure 11 shows the CFD results of the evolution of the position, velocity, and acceleration of the particle shown in Fig. 10. It is observed that most of the time the particle maintains its position and begins to move once the bubble reaches its time of collapse (t* = 1). From this time, the particle is strongly accelerated due to the effect of the micro-jet. Then, it begins to decelerate as it approaches the wall. However, its velocity before it touches the surface is 71 m/s, which is a much higher velocity than the maximum velocities developed during the operation of a hydraulic turbine (between 10 and 40 m/s^{34–36}).

The location of the particle at the center of the bubble is an idealized situation. Generally, the bubble is nucleated at impurities, such as small air bubbles or at the crevices located on the surface of the particle,^{37} which causes the particle to be located outside the bubble. Hence, particles located a different positions below the bubble were simulated to verify its behavior. Figure 12 shows the simulation of a bubble located 1.5 mm from the wall with a particle located 1 mm from the wall at the beginning of the simulation, and Fig. 13 shows the CFD results of position, velocity, and acceleration of the particle. It is observed how the bubble interface reaches and accelerates the particle. After some time, the velocity of the interface begins to decrease because it is approaching its maximum radius; however, the particle continues with a higher velocity and is separated from the bubble interface. A similar behavior was found experimentally by Arora *et al.*^{38} The velocity of the particle before it reaches the wall is 1.25 m/s, which is low enough to cause significant damage; instead, the particle will be able to block the jet momentum and can reduce the damage due to cavitation. This can explain the results obtained by Lian *et al.*^{14} who performed experiments of cavitation in a fluid with a certain concentration of particles of different sizes using a vibratory apparatus; in this case, the bubbles grow and collapse relatively close to the surface. They found that particle sizes under an approximate value of 35-48 *μ*m inhibit the damage caused by cavitation when the concentration of sediment increases, however, when the particles are larger than this size, the damage increases with an increase in the concentration.

On the other hand, Fig. 14 shows the simulation of a bubble located 2.5 mm from the wall with a particle of diameter 60 *μ*m located 2.39 mm from the wall, which is directly under the bubble. Figure 15 shows the CFD results of the position, velocity, and acceleration of the particle. It is observed that the particle is absorbed by the growing bubble, and after the collapse, it is reached and accelerated by the jet up to 109 m/s. Then, as the particle approaches the wall, its velocity decreases until 81 m/s, just before it touches the surface. This value is higher than the value obtained with the bubble located at 1.5 mm and is consistent with the result obtained by Jayaprakash *et al.*^{22} who found that at non-dimensional standoff values of the bubble (*γ* = *S*/*R*_{bmax}) between 0.75 and 1 (in our case 1), the maximum damage on the surface can be achieved. Additionally, a clear difference of almost three orders of magnitude is observed in the values of acceleration with respect to the case in which the particle is located 1 mm from the wall. This indicates that the initial position of the particle as well as the position of the bubble have a strong influence on the velocity and acceleration of the particle due to the micro-jet effect. This situation can lead to an inhibition or an acceleration on the damage of a surface exposed to cavitation and sand erosion. Therefore, further work is required to obtain a detailed description of the influence of the standoff distance of the particle and the bubble on the particle velocity and the damage that this can cause on the surface.

## V. CONCLUSION

Experimental and numerical approaches were used to study the interaction of particles with a cavitation bubble collapsing near a solid wall. This analysis helps us to understand the effect of the bubble dynamics on particles of different sizes and densities. When the particles are above the bubble, their density and size have a strong effect, whereas the bubble separation from the surface is not significant for the results of maximum velocity. Additionally, the bubble size has a small effect on the velocity of the particle located above the bubble. The experimental results and a simple analytical study validated the numerical simulations. After the validation, additional numerical simulations of the interaction of a collapsing bubble and a particle were performed to analyze situations that could not be observed in the experimental tests. In this case, the influence of the micro-jet on the particle was observed and it was found that the initial position of the particle and the bubble has a strong influence on the particle velocity, which can lead to the inhibition or acceleration of damage caused by the collapse of a cavitation bubble.

## ACKNOWLEDGMENTS

The authors acknowledge Colciencias, Universidad del Valle, and Virginia Tech for their support during the development of this project. This research was partially supported by the National Science Foundation (Grant No. CBET-1604424).