A review of bubble collapse near particles

Bubble–particle interactions are among the most important features of cavitation and bubble dynamics in silt-laden flows (Gao , 2022; Zhang , 2016; Duan and Karelin, 2003; Zhou , 2009; Dular and Petkovšek, 2015). For example, the jets and shock waves generated by bubble collapse near particles can cause damage to components in various types of pumps and turbines, such as pumps for lifting water and hydroturbines for electricity generation (Lai , 2022; Geng , 2021; Luo , 2018; Bai , 2008; Yang , 2021). In this context, a review of current progress in research on the interaction between bubbles and particles will be of great use.

The mechanisms of interaction between particles and bubbles formed during cavitation have been investigated through theoretical analyses, numerical simulations, and experimental studies. Most theoretical approaches assume spherical cavitation bubbles and are based on the classic Rayleigh–Plesset equation for bubble dynamics or developments of this equation (Besant, 1859; Rayleigh, 1917; Plesset and Prosperetti, 1977; Plesset, 1949; Kirkwood and Bethe, 1942; Keller and Miksis, 1980; Herring, 1941; Gilmore, 1952). Numerical simulations commonly use the volume of fluid and finite volume methods through the OpenFOAM solver. In experimental studies, cavitation bubbles are usually generated by electric spark excitation (Lauterborn, 1972; Kling and Hammitt, 1972; Lauterborn and Bolle, 1975; Vogel , 1996) or laser excitation (Liu , 2002; Akhatov , 2001; Lim , 2010). For example, ionization of water molecules using a pulsed laser produces a high-temperature, high-pressure plasma, with subsequent generation of cavitation bubbles (Sinibaldi , 2019; Wen , 2023).

The present paper provides an overview of the physical mechanisms of bubble collapse near particles, with an emphasis on jet and shock wave phenomena. It is organized as follows. For the reader’s convenience, bubble collapse near a flat wall is first discussed as an example that provides a general picture of cavitation behavior near a boundary. The reason for the choice here of collapse near a flat wall is that there is an extensive literature on this topic, which has been investigated in great detail over the years (Ma , 2017; Wu , 2018; Wang , 2018; Aganin , 2016; Sagar and el Moctar, 2020; Lindau and Lauterborn, 2003). Three aspects of bubble collapse near a particle are then reviewed: (a) the bubble collapse process, involving the inception, growth, collapse, rebound (first, second, etc.), and finally disappearance of the bubble (Wang , 2022; Zheng , 2023a); (b) jets, including the inception jet, counter jet, and double jets; (c) shock waves, including incident and reflected shock waves (e.g., that reflected by the particle), jet-induced shock waves, and jet-split shock waves. For each of these, a number of examples are given, based on results obtained by our group. A summary of research to date on the collapse of cavitation bubbles near walls or particles is presented in Table I. More detailed reviews can be found in Yu  (2023b, 2023c).

Figure 1 shows the collapse processes of a typical cavitation bubble near a flat wall at different distances. A high-speed camera was used in order to observe the details of the collapse and of jet formation. It can be seen that at the start of the collapse, a torus bubble ring is formed. This continues to develop as the bubble interface shrinks. Finally, the collapse of the bubble finishes with jet formation. Because of the proximity of the bubble to the flat wall, the jet is directed toward the wall (Benjamin and Ellis, 1966; Vogel , 1989). The maximum radius of the bubble determines the width of the jet. The jet speed can be very high, even exceeding 100 m/s, and it is therefore not surprising that damage can be caused quite rapidly to the wall, even if it is protected by an anti-cavitation film. This is one of the many industrial problems caused by cavitation, which has long proved to be a hazard—from the first trial of the turbine-powered steamship Turbinia to the giant hydroturbines in the Three Gorges Dam.

The last frame in Fig. 1 shows another kind of jet, which is directed away from the wall and hence is usually termed a “counter jet.” Although this jet is away from the wall, it can have a significant impact on the cavitation process. The generation of the counter jet depends strongly on the bubble–wall distance. For a bubble far from the wall, the counter jet will be weak or even nonexistent.

When a bubble collapses in a liquid of infinite extent, the compression ratio (the ratio between the maximum and minimum bubble volumes) can be extremely high, as much as 200 000. In the special case of oscillating gas bubbles (usually containing argon) driven by acoustic waves, the compression ratio can be even higher, and luminescence may occur. From one viewpoint, such a bubble can be treated as an effective energy converter, transforming input mechanical energy into the motion of the flow surrounding the bubble. However, for a bubble collapsing near a wall, the compression ratio is dramatically reduced to as low as 600, depending on the stand-off distance (see below). This provides a clear illustration of the significant influence of the wall on bubble behavior.

In fact, the presence of the wall will delay the bubble’s collapse. As an example, for γ = 1, $T1stcollapse*$ is 1.2, whereas for γ = 2, it is 1.11 (Lauterborn and Kurz, 2010). Thus, the closer the bubble is to the wall, the more prolonged will the collapse process be.

The extreme conditions inside the bubble during collapse can lead to luminescence. This phenomenon has been observed experimentally for decades. In the case of a single well-controlled air bubble driven by acoustic waves, the luminescence can persist for several days in what is usually termed “sonoluminescence.” The presence of a wall can have an effect on the emitted luminescent energy. In general, it is found that the normalized luminescent energy (defined as the light energy emitted by the collapsing bubble in the presence of a wall divided by that emitted by a spherical bubble collapsing in a liquid of infinite extent) increases with increasing stand-off distance.

A bubble collapsing in a liquid of infinite extent mainly undergoes radial oscillations, while maintaining a spherical shape. However, bubbles collapsing near particles can exhibit quite different shapes. For example, for the close distance shown in Fig. 2, a mushroom shape with a clear neck characteristic is observed. Furthermore, with increasing distance, different shapes are also possible, such as the pear-shaped bubble shown in Fig. 3. At large distances, the bubble collapses with a spherical shape like that observed in a large liquid volume.

As shown in Fig. 4, one of the distinct aspects of bubble collapse near a particle is the bubble splitting during the second collapse. This usually occurs at a certain distance when the bubble is not in contact with the particle. Usually, the bubble splits into two or more parts, which further rebound and collapse again. It is evident from frame 4 in Fig. 4 that owing of the influence of the particle, the left part is bigger than the right part. Both parts continue to expand and subsequently separate from one another. These daughter bubbles can interact with each other, leading to even more complex phenomena.

The presence of a particle can also extend the duration of bubble collapse. Figure 5 shows the variation of the dimensionless second period of the bubble (normalized in terms of the Rayleigh collapse time) with the stand-off distance. The black squares correspond to short and medium distances (cases 1 and 2, respectively). The red squares and circles correspond to the left and right sections of split bubbles (cases 3 and 4, respectively). As can be seen, the general trend is that the bubble period decreases with increasing standoff distance, although the behavior is more complex when bubble splitting occurs (cases 3 and 4), owing to interactions between the daughter bubbles and between these bubbles and the wall.

For a certain range of distances, a jet (called the “forward jet” to distinguish it from the counter jet) emerges from the bubble. Usually, this jet has a high velocity and contains a large amount of energy. When it hits the particle, the particle is accelerated dramatically. One of the physical mechanisms of the abrasion caused by cavitation is the combined effect of the jet and the accelerated particle. Figure 6 shows images from high-speed photography during the second period of bubble collapse that capture the forward jet (Wang , 2023b), with its tip directed toward the particle. In contrast to the jet formed during the first period, the energy of this jet is limited.

Figure 7 shows the counter jet as captured by high-speed photography. In frame 4, the jet can be seen to hit the particle surface, with a small amount of vapor forming at the right side. This counter jet is formed by the extreme shrinkage of the bubble during collapse. As the bubble continues to collapse, the counter jet will develop further.

In the case of multiple particles, the formation of a neck between particles can lead to some interesting jets. Figure 8 shows the temporal development of bubble shape during the first phase simulated using OpenFOAM with the solutions of the equations for bubble dynamics near particles and boundary conditions given in Hu  (2023). In frame (c), the neck forms with a maximum volume. It then starts to collapse, with a final break at the middle in frame (h). As can be seen in frame (i), two clear jets are formed, directed toward the upper and lower particles, respectively. In some cases (see Fig. 9), the neck can also split into three or more parts, with the jets penetrating the middle part of the vapor volume.

Figure 10 depicts the shock wave evolution for a bubble collapsing near a particle as simulated by OpenFOAM. For more detailed investigations of incident shock wave, readers are referred to Yu , 2023a. In frame 1, the bubble is at the inception stage. Because of the high pressure and temperature within the bubble, the surrounding liquid is rapidly pushed away during bubble inception. Hence, a strong shock wave is formed (termed the incident shock wave) and propagates in a spherical manner.

As shown in frame 2 of Fig. 10, the incident shock wave undergoes intense reflection from the particle surface, and the reflected shock interferes with the original wave. In the subsequent collapse process, the jet impacts with the particle, with the subsequent formation of a reflected wave. Impact with the bubble interface can also lead to reflected waves. However, these waves are generally weak and cannot be observed in detail on the scales considered here.

Figure 11 shows the shock wave evolution in the case of bubble neck shrinkage between two equal-sized particles as simulated by OpenFOAM. Many types of shock waves can be observed, including shock waves 1 and 2 located respectively outside and inside the bubble. The neck split point can be identified as the source of shock wave 1. In Fig. 11(c), shock wave 2 is reflected near the tail of the jet.

With regard to perspectives for future research, the following directions are suggested.

1. A theoretical model for jet prediction should be developed. We have already developed a Kelvin impulse model for jet prediction near particles and have validated this model and used it to obtain several successful predictions (Wang , 2022; Zheng , 2023b).

2. Bubble collapse near particle clusters is important in many industrial-scale phenomena. However, research to date has only considered clusters consisting of two or three particles. Study of the effects of clusters with many more particles (of the order of tens or hundreds) is an important albeit challenging task.

3. Shock-capture algorithms are required for the detailed analysis of bubble–particle interactions. Current methods are able to capture some primary shock waves, but to obtain a detailed description of interference between different shocks, more advanced algorithms are necessary.

4. Investigation of the characteristics of particle motion is another important topic (Su , 2021; Wu , 2021; Ren , 2022). Currently, owing to technical difficulties with high-speed observations, as well as for simplicity, particles are generally fixed with respect to a stationary wall (Lv , 2019). In the future, multi-scale photographic techniques need to be developed to capture different aspects of motion in detail.

5. Because the damage caused by cavitation bubbles is usually to the wall surface, particular attention needs to be paid to the bubble–particle–wall interaction. Our previous experimental investigations (Zhang , 2019a) have shown that the presence of a wall can alter many characteristics of collapsing bubbles, such as the interface evolution (Fig. 12) and the direction of the jet (Fig. 13).

This research was financially supported by the National Natural Science Foundation of China (Project No. 51976056).

The authors have no conflicts to disclose.

Jiaxin Yu: Conceptualization (equal); Investigation (equal); Validation (equal); Writing – original draft (equal). Jinxin Luo: Formal analysis (equal); Investigation (equal). Yiming Li: Data curation (equal). Yuning Zhang: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data are available from the corresponding author on reasonable request.