3.1. The influence of bubble–bubble relative distance on the shape of the shock wave

Figure 2 shows the high-speed photographic images on the influence of the air bubble on the shock wave from the first collapse of the cavitation bubble when the bubble–bubble relative distance φ changes.

Figure 2.

Influence of bubble–bubble relative distance on the shape of the shock wave (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 39.94 mm): (a) R_max = 9.83 mm; (b) R_max = 9.49 mm, ε = 0.28, φ = 2.23; (c) R_max = 9.86 mm, ε = 0.25, φ = 1.59; (d) R_max = 9.80 mm, ε = 0.23, φ = 1.43; (e) R_max = 9.74 mm, ε = 0.26, φ = 1.33; (f) R_max = 10.42 mm, ε = 0.22, φ = 1.06; (g) R_max = 10.03 mm, ε = 0.24, φ = 0.54.

In figure 2(a), R_max is 9.83 mm. When t = 2.344 ms, the cavitation bubble contracted to its minimum volume. Because the cavitation bubble was located in a free field, the direction of collapse was not obvious, as shown in figure 2(a ₄). The cavitation bubble emitted a strong shock wave. When there was no external interference, the shock wave in a free field would have a regular spherical shape. This regular spherically symmetric shock wave phenomenon also appeared in the case of laser-induced cavitation collapse (Ohl et al. Reference Ohl, Kurz, Geisler, Lindau and Lauterborn1999). The shock wave shape in figure 2(a ₄) is consistent with the shape of the shock wave reported in Philipp & Lauterborn (Reference Philipp and Lauterborn1998), which is less affected by the wall (γ = 1.9). As for the cavitation bubble in a free field, because the radial velocity of the cavitation surface in all directions was consistent, when the cavitation bubble contracted to its minimum volume, the shock wave induced under the implosion effect would have a regular spherical shape (Beig Reference Beig2018).

In figures 2(b) and 2(c), the air bubble was quite far from the cavitation bubble and the values of φ in two experiments were 2.23 and 1.59, respectively. The cavitation bubbles in two experiments completed their respective first collapses at around 2.216 and 2.305 ms, and during the collapse, such cavitation bubbles emitted a shock wave. From the high-speed photographic image, the shape of the shock wave was quite regular, similar to the shape of the shock wave in figure 2(a). In figure 2(c), when the shock wave reached the air bubble, a darker wave moving in the opposite direction was emitted from the surface of the air bubble, as shown in figure 2(c ₆). When the shock wave passed the air bubble, the colour of the shock wave in the area on the left side of the air bubble was lighter than that of the shock wave in other areas.

In figures 2(d)–2(f), the values of φ were 1.43, 1.33 and 1.06, respectively, and gradually reduced. In the three experiments, all the cavitation bubbles emitted a shock wave. Because the distance between the cavitation bubble and the air bubble was relatively short, in those experiments, the shapes of the shock wave were completely different from that of the shock wave in figure 2(a). Firstly, these shock waves had an obviously stratified structure; and when those stratified waves propagated outwards, the distance between waves within a stratification structure increased as φ was reduced, as shown in figures 2(d), 2(e) and 2(f). Secondly, when the shock wave reached the surface of the air bubble, an obvious wave moving in the opposite direction appeared at the surface of the air bubble, as shown in figures 2(d ₆) and 2(e ₆). Thirdly, when the shock wave passed the air bubble and continued to propagate outwards, in the area behind the air bubble, the shock wave was obviously deformed (as shown in figure 2d ₆), and the degree of deformation decreased as φ was reduced (as shown in figure 2e) and finally became non-obvious (as shown in figure 2f).

As φ was further reduced, during the evolution of the cavitation bubble, the cavitation bubble absorbed the air bubble and a gas-bearing cavitation bubble was formed, as shown in figure 2(g ₂). When the cavitation bubble contracted to its minimum volume, the cavitation bubble and the air bubble completely merged, as shown in figure 2(g ₄). Moreover, compared with other experiments in figure 2, when the cavitation bubble contracted to its minimum volume, the volume of the cavitation bubble was obviously bigger than that in other experiments. Another important phenomenon is that, in this experiment, during the collapse of the cavitation bubble, the shock wave was not visible on the high-speed photographic image.

3.2. The influence of bubble–bubble relative size on the shape of the shock wave

Figure 3 shows high-speed photographic images of the influence of the air bubble on the shock wave from the collapse of the cavitation bubble when ε changes. The radius of the air bubble r changes within the scope of 1.30–2.71 mm.

Figure 3.

Influence of bubble–bubble relative size on the shape of the shock wave (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 39.94 mm): (a) R_max = 9.32 mm, ε = 0.14, φ = 1.27; (b) R_max = 9.52 mm, ε = 0.19, φ = 1.28; (c) R_max = 9.49 mm, ε = 0.21, φ = 1.28; (d) R_max = 9.59 mm, ε = 0.25, φ = 1.28; (e) R_max = 9.69 mm, ε = 0.28, φ = 1.32.

In figure 3(a), ε was 0.14. During the evolution of the cavitation bubble, the air bubble cracked from being squashed and stretched. In the process of collapse, the cavitation bubble emitted a shock wave whose shape is similar to that of the cavitation bubble in figure 2(a). However, compared with figure 3(a), there is an apparent difference in figure 3(b), i.e. when the shock wave reached the area behind the air bubble, the shock wave was apparently deformed, as shown in figure 3(b ₆). Nonetheless, in figures 3(a) or 3(b), with the effect of the air bubble, the shock wave did not have an obviously stratified structure. When ε was further increased to 0.21, as shown in figure 3(c), the shock wave was stratified to some extent, as shown in figure 3(c ₄), and in the area behind the air bubble, the shock wave was deformed to some degree, as shown in figure 3(c ₅). Similar to the phenomenon in figure 3(c), in figure 3(d), the shock wave was stratified to a greater degree, as shown in figure 3(d ₅). Finally, ε was increased to 0.28 in figure 3(e). In the experiment, the shock wave was apparently stratified, and when the shock wave reached the surface of the air bubble, a wave moving towards the centre of the cavitation bubble appeared.

In the experiment on the interaction between the cavitation bubble and the air bubble, an air bubble with a relatively large volume that can float up freely cannot be generated in the water. Thus, when we want to learn about whether an air bubble with a larger volume also has a stratification effect on the shock wave, we generated a cavitation bubble below the free surface so as to observe the effect that an infinitely large air bubble has on the shock wave, as shown in figure 4.

Figure 4.

Influence of free surface on the shape of the shock wave (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 44.03 mm): (a) R_max = 8.99 mm, ε = ∞, φ_s = 0.95; (b) R_max = 8.96 mm, ε = ∞, φ_s = 1.70; (c) R_max = 8.99 mm, ε = ∞, φ_s = 2.78.

In figure 4(a), φ_s was 0.95. Under the condition that the free surface was quite close to the cavitation bubble, the shock wave had an obviously stratified structure, as shown in figure 4(a ₅). The stratified shock wave was reflected by the free surface, thereby forming a reflected wave. The reflected wave spread inwards in the water, as shown in figure 4(a ₇). Based on the shape difference between the shock wave induced by the secondary cavitation bubble in Supponen et al. (Reference Supponen, Akimura, Minami, Nakajima, Uehara, Ohtani, Kaneko, Farhat and Sato2018) and the shock wave in this paper, we can infer that the stratification of the shock wave in figure 4(a) was caused by the influence of the free surface rather than the expansion wave generated during secondary cavitation.

When φ_s was increased to 1.70, the evolution process of the shock wave was as shown in figure 4(b). Compared with figure 4(a), in this experiment, the shock wave still has an obviously stratified structure, as shown in figure 4(b ₄). However, the degree of stratification of the shock wave is lower than that in figure 4(a). The stratification phenomenon of the shock wave from the collapse of the laser-induced cavitation bubble near the free surface also exists in Supponen et al. (Reference Supponen, Obreschkow, Kobel, Tinguely, Dorsaz and Farhat2017). Reflected wave was still formed after the stratified shock wave reached the free surface and such a reflected wave also has a stratified structure. When φ_s was further increased to 2.78, the evolution process of the shock wave was as shown in figure 4(c). In this experiment, the shock wave does not have a stratified structure anymore, and the stratified structure also did not appear after the shock wave was reflected by the free surface. This further indicates that the layered shock wave in figure 4(a) was not caused by secondary cavitation.

From the evolution of the shock wave with the effect of air bubbles of different sizes and the infinitely large air bubble, we can see that the size of the air bubble has a certain influence on the stratification of the shock wave. Under the condition that the distance between the cavitation bubble and the air bubble remained the same, when the air bubble was small, the shock wave was not stratified, but when the air bubble was big, stratification occurred.

4.1. The influence of bubble–bubble relative distance on the strength of the shock wave

Figure 5 shows a high-speed photographic image of the influence of the air bubble on the shock wave under the condition that φ near the wall surface changes. Figure 5(a) is about the shock wave from the collapse of a single cavitation bubble near a wall. In figure 5(b–f), the black area in the middle of each image is air bubbles (and after the cavitation bubble was generated, the black area on the left side of the image is cavitation bubbles) and the boundary area on the right side of the image is the rigid wall surface.

Figure 5.

Influence of bubble–bubble relative distance near the wall on the shape of the shock wave (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 42.68 mm; the black area on the right of each image is rigid wall surface): (a) R_max = 10.23 mm, γ = 2.850; (b) R_max = 10.19 mm, φ = 0.546, γ = 2.767, ε = 0.234, θ = 8.956 °; (c) R_max = 9.73 mm, φ = 1.056, γ = 2.867, ε = 0.298, θ = 1.991 °; (d) R_max = 10.49 mm, φ = 1.305, γ = 2.800, ε = 0.215, θ = 29.934 °; (e) R_max = 10.27 mm, φ = 1.502, γ = 2.873, ε = 0.183, θ = 46.087 °; (f) R_max = 10.08 mm, φ = 1.934, γ = 2.836, ε = 0.238, θ = 12.314 °.

In figure 5(a), γ is 2.850. When collapsing, the shock wave is a regular spherical shape. When the shock wave reached the wall surface, it impacted on the wall surface. Because γ > 1 in figure 5(a), during the first collapse of the cavitation bubble, the re-entrant jet would not impact on the wall surface (Philipp & Lauterborn Reference Philipp and Lauterborn1998).

In figure 5(b), φ is 0.546. The cavitation bubble merged with the air bubble and formed a gas-bearing cavitation bubble. After the gas-bearing cavitation bubble contracted to its minimum volume, its volume was apparently bigger than the minimum volume of the cavitation bubble in the experiment of figure 5(a), and there were no signs of the shock wave from the images, as shown in figures 5(b ₄)–5(b ₆).

In figure 5(c), φ is 1.056. The air bubble was apparently deformed, the air bubble had a cone-shaped surface (as shown in figure 5c ₃). At that moment, the air bubble and the cavitation bubble were not merged. Shock waves had an obviously stratified structure, as shown in figure 5(c ₅). The shock waves from secondary cavitation appear at multiple points at the same time (Supponen et al. Reference Supponen, Akimura, Minami, Nakajima, Uehara, Ohtani, Kaneko, Farhat and Sato2018), while in figure 5(c ₅), the third layer shock wave only has one layer and such a shock wave has a relatively regular shape. Thus, we can infer that the third layer of the shock wave came from the point where the cavitation bubble collapsed. The evolution of the cavitation bubbles in figure 5(d,e) is similar to that in figure 5(c), these shock waves had a stratification structure. The difference among the three experiments above is that, as φ increased, the degree of shock wave stratification weakened. In figure 5(f), the value of φ was 1.934. In the area between the air bubble and the wall surface, the shock wave did not have any apparent deformation. Instead, it kept a completely spherical shape, similar to the situation in figure 5(a).

From the above-mentioned influence of the air bubble on the shock wave under the condition that φ changes, the air bubble has a certain stratification effect on the shock wave. As for the strength of the stratified shock wave, we need to carry out an analysis based on the impact pressure. The process of a shock wave impacting on the wall surface in figure 5 is shown in figure 6.

Figure 6.

Process of the shock wave impacting on the wall surface under different bubble–bubble distance conditions. The images above concern the process of the shock wave impacting on the wall surface in the experiments of figure 5. The red dotted line on the right of the images represents the location of the sensor, the shock waves in all the impact processes have been marked with a white arrow.

In figure 6(a), it was obvious that the impact of the spherical shock wave on the wall surface had one peak value that reached 2.390 MPa. Vogel et al. (Reference Vogel, Busch and Parlitz1996) studied the relationship between the velocity of the shock wave from the laser-induced cavitation bubble ${u_s}$ and its pressure ${p_s}$, and found they satisfy the following equation:

(4.1)\begin{equation}{p_s} = {c_1}{\rho _0}{u_s}({10^{({u_s} - {c_0})/{c_2}}} - 1) + {p_\infty },\end{equation}

where ${\rho _0}$ denotes the density of the water that has not been compressed by the shock wave, ${c_0}$ denotes the acoustic speed in water (22 °C, ${c_0} \approx 1482.9\;\textrm{m}\;{\textrm{s}^{ - 1}}$), ${c_1}$ and ${c_2}$ are 5190 m s⁻¹ and 25 306 m s⁻¹, respectively, and ${p_\infty }$ denotes the hydrostatic pressure. From figure 5(a ₅,a ₆), we can get that the velocity of the shock wave is approximately 1486.726 m s⁻¹. Thus, we estimated that the impact pressure of the shock wave is approximately 2.808 MPa, which is close to the data given in this paper by means of measurement. Beig et al. (Reference Beig, Aboulhasanzadeh and Johnsen2018) conducted precise numerical simulations on micrometre-scale cavitation bubbles when the conditions of γ are different and analysed the pressure profile. The pressure profile is similar to that measured in the study of this paper. The value of γ in this paper is much larger than that in Beig et al. (Reference Beig, Aboulhasanzadeh and Johnsen2018). Therefore, it is unfortunate that a precise comparison of peak pressure cannot be made.

In figure 5(b), the cavitation bubble became a gas-bearing cavitation bubble, the minimum volume to which such a cavitation bubble contracted was relatively large and, during the collapse, the shock wave was not visible on the image. However, from the measurement data, we can see two peak values, and these two peak values are similar, i.e. 0.366 and 0.372 MPa.

In figure 6(c), there are three peak values, i.e. 0.309, 0.390 and 0.119 MPa. The three peak values can be located on the experimental images. In the pressure profile, because the peak value is tiny, the shock wave was stratified, and the stratified shock wave was further shielded by the air bubble. Thus, the maximum pressure peak value in figure 6(c) was close to the pressure peak value under the condition of merger in figure 5(b). As for figure 6(d), we can see two layers of shock waves, their impact pressures were 0.589 and 0.548 MPa respectively. As for figure 6(e), there was only one obvious main peak value that is 1.541 MPa. Due to the tiny interval between the two peaks on the image, we can infer that the second shock wave is approximately 0.257 MPa. As for figure 6(f), the shape of the shock wave has no obvious stratification, and the peak value of the shock wave is approximately 1.581 MPa. Compared with figure 5(a), there was no obvious stratification in figure 5(f), but the peak value of the shock wave is apparently lower than 2.390 MPa. By analysing figure 5(f ₆), we found that, in the area between air bubble and wall surface, the colour of the shock wave became lighter in the foregoing area after the shock wave was hindered by the air bubble. It can be seen from the above data that when φ is larger, the air bubble can effectively attenuate the intensity of the shock wave.

4.2. The influence of bubble–bubble relative size on the strength of the shock wave

In this part, we will further analyse the shock wave under the condition that ε changes. Figure 7 is about the evolution process of the shock wave when ε gradually increases.

Figure 7.

Influence of bubble–bubble relative size near the wall on the shape of the shock wave (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 42.68 mm): (a) R_max = 10.02 mm, φ = 1.305, γ = 2.715, ε = 0.102, θ = 30.920 °; (b) R_max = 9.90 mm, φ = 1.357, γ = 2.906, ε = 0.158, θ = 13.804 °; (c) R_max = 10.03 mm, φ = 1.264, γ = 2.657, ε = 0.233, θ = 7.771 °; (d) R_max = 10.06 mm, φ = 1.305, γ = 2.715, ε = 0.269, θ = 17.102 °.

In figure 7(a), the radius of the air bubble is approximately 0.1 times the maximum radius of the cavitation bubble. From figure 7(a ₄), we can see that the shock wave had two layers, but the distance between the two layers was tiny. In figure 7(b ₅), the shock wave had a stratified structure. Moreover, when the shock wave impacted on the surface of the air bubble, a reflected wave appeared, as shown in figure 7(b ₆); and, in the area behind the air bubble, the shock wave had a shape that is slightly oblate. The value of ε was gradually increased to 0.233, as shown in figure 7(c), and the shock wave still had a stratified structure. However, from the high-speed photographic image, the trace of the shock wave is not as obvious compared with figures 7(a) and 7(b). In figure 7(d), ε was increased to 0.269. When the cavitation bubble collapsed, the shock wave was still stratified, but the brightness of the shock wave was further reduced. Similarly, in the area between the air bubble and the wall surface, it is hard to observe the shock wave, as shown in figure 7(d ₆).

Figure 8 is about the process of the shock wave impacting on the wall corresponding to figure 7. In figure 7(a ₄), we can see two layers of shock wave, but it is difficult to identify the shock wave when it impacted on the wall. From the evolution of the shock wave in figure 8(a), we can see that the shock wave had one peak value that is 2.480 MPa. In figure 7(b ₅), we can identify that the shock wave was comprised of two layers, and their peak values of the impact pressures were 1.420 and 0.308 MPa, respectively. In figure 8(c), the impact had two obvious peak values, which were 0.737 and 0.656 MPa. In figure 8(d), the peak value of the impact pressure was 0.702 MPa. It is worth noting that the pressure peak value in figure 7(b) is approximately twice the pressure peak value in figure 8(c), while the γ value in figure 7(b) is slightly bigger than that in figure 8(c). From this, we can see that the air bubble has an obvious attenuation effect on the pressure peak value of the shock wave.

Figure 8.

Process of the shock wave impacting on the wall surface under different conditions of air bubble size. The images above concern the process of the shock wave impacting on the wall surface in the experiments of figure 7. The red dotted line on the right of the images represents the location of the sensor. In the images, the shock waves in all the impact processes have been marked with a white arrow.

4.3. The critical condition of the stratification effect

From § 3 and the §§ 4.1–4.2, we can see that when the relative distance φ and the relative size ε meet certain conditions, the air bubble will have a certain stratification effect on the shock wave. We analysed the relationship between φ and ε under the two circumstances, i.e. in a free field and near a wall surface, as shown in figure 9. The value of ε is varied in the range 0.09–0.56, and φ is varied in the range 0.42–5.14.

Figure 9.

Critical conditions for stratification of shock waves by air bubbles in a free field and near the wall.

From figure 9, we can see that the stratification effect of the air bubble on the shock wave is consistent under the two conditions. When ε remained the same and φ gradually increased, the following phenomena appeared in turn, i.e. the shock wave was stratified under the condition that the air bubble and the cavitation bubble were merged. The shock wave was stratified under the condition that the bubble–bubble distance was quite small, and no stratified structure occurred under the condition that the bubble–bubble distance was large. Subject to the experimental condition, no more experiments could be conducted under the condition that the size of the air bubble was larger.

In addition, in figure 9, when φ < 1, which is equivalent to the situation where the air bubble has occupied part of the space for the development of the cavitation bubble before the generation of the cavitation bubble, it is easy for the air bubble and the cavitation bubble to merge. Under such circumstance, the shock wave would be easily stratified, and the shock wave even could not be visible on the high-speed photographic images in some experiments. When 1 < φ < 1.5ε + 1.25, the two bubbles did not merge in the process of evolution, but the shock wave was stratified. When φ > 1.5ε + 1.25, the air bubble had little influence on the evolution form of the shock wave and the shock wave was not stratified.

From the discussion above we can see that the relationship between the φ and (1.5ε + 1.25) becomes the basis for determining the critical condition for the stratification effect that the air bubble has on the shock wave from the first collapse of the cavitation bubble. From the physical properties of the surrounding medium distribution of the cavitation bubble, we can see that, because the air bubble and water are different in compressibility, the air bubble within the range of effect of the cavitation bubble changes the consistency of the surrounding medium of the cavitation bubble. Compared with the collapse of the cavitation bubble in a free field, the inconsistency of the surrounding medium of the cavitation bubble ultimately leads to an asymmetry of the shape of the cavitation bubble when it collapses. From the mechanism of generating the implosion shock wave and the water hammer shock wave, we can see that when φ > 1.5ε + 1.25, the distance between two bubbles is relatively large and the influence that the air bubble has on the shape of the cavitation bubble when the cavitation bubble collapses is small. Beig (Reference Beig2018) carried out a fine numerical simulation on the collapse of the cavitation bubble near the wall surface. From the research findings thereof, we can see that when the relative distance between the cavitation bubble and the wall surface γ > 4, the water hammer shock wave and implosion shock wave are nearly indistinguishable. When 1 < φ < 1.5ε + 1.25, the existence of the air bubble will affect the evolution of the cavitation bubble and the shape thereof when the cavitation bubble collapses. When the cavitation bubble collapses, because the re-entrant jet impacts on the surface of the opposite side of the cavitation bubble, a water hammer shock wave, i.e. the outer shock wave in the experimental images, is formed. As the cavitation bubble further collapses and rebounds, an implosion shock wave appears. Such a stratification phenomenon of the shock wave also exists when the cavitation bubble near the wall surface collapses. The shock wave from the collapse of the laser-induced cavitation bubble near the wall surface in Lindau & Lauterborn (Reference Lindau and Lauterborn2003) and the result of the high-precision numerical simulation on the collapse of the cavitation bubble near the wall surface in Beig (Reference Beig2018) verify the formation mechanism of the stratification of the shock wave from the collapse of the cavitation bubble. When φ < 1, the cavitation bubble absorbs the air bubble during its evolution, which leads to the existence of a huge amount of non-condensable gas inside the cavitation bubble that increases the minimum volume of the cavitation bubble to which the cavitation bubble contracts during its collapse, and the surface of the cavitation bubble is very irregular after it merges with the air bubble. The joint effect of such factors effectively attenuates the shock wave induced by the implosion effect of the cavitation bubble, leading to the stratification of the shock wave under the condition that the two bubbles merge, which even cannot be seen in the high-speed photographic images.

6.1. The influence of the air bubble on the shape of the shock wave from the rebounded cavitation bubble

In numerous experiments, we found that, when the air bubble existed, a shock wave was also observed during the second collapse of the cavitation bubble, and that, under some conditions, such a shock wave was also stratified. The multiple peak values in the experiment conducted by Supponen et al. (Reference Supponen, Obreschkow, Kobel, Dorsaz and Farhat2019) perfectly reflect the characteristic of the gradual reduction in the strength of the collapse of the cavitation bubble. Thus, this part mainly analyses the shock wave from the second collapse of the cavitation bubble.

In figure 12(a), the relative distance between the cavitation bubble and the air bubble φ < 1. According to the critical condition of the air bubble stratifying the shock wave from the collapse of the cavitation bubble in figure 9, when the cavitation bubble contracted to its minimum volume, the air bubble had completely merged into the cavitation bubble. The minimum volume of the cavitation bubble is slightly bigger than that when there is no air bubble (comparing figures 5a ₃ and 12a ₃). After the cavitation bubble entered the second oscillation cycle, during its evolution, the cavitation bubble started to have an irregular shape (Lindau & Lauterborn Reference Lindau and Lauterborn2003), and during its second collapse, the shock wave is invisible on the images.

Figure 12.

High-speed photographic image of the shock wave from the collapse of the rebounded cavitation bubble (frame rate: 180 000 f.p.s., exposure time: 0.25 μs, frame width: 42.68 mm): (a) R_max = 10.18 mm, φ = 0.727, γ = 2.489, ε = 0.209; (b) R_max = 10.28 mm, φ = 0.949, γ = 2.562, ε = 0.108; (c) R_max = 10.55 mm, φ = 1.264, γ = 2.575, ε = 0.222; (d) R_max = 10.33 mm, φ = 1.460, γ = 2.784, ε = 0.101; (e) R_max = 10.06 mm, φ = 2.036, γ = 2.867, ε = 0.169; (f) R_max = 10.47 mm, φ = 2.250, γ = 2.667, ε = 0.209.

In figure 12(b), the bubble–bubble relative size ε is 0.108. In the first and second collapses of the cavitation bubble, the shock wave had the characteristics of a stratified structure, as shown in figures 12(b ₃) and 12(b ₅). In figure 12(c,d), when the cavitation bubble contracted to its minimum volume during the second cycle, it is obvious that the volume in figure 12(c) is bigger than that in figure 12(d), as shown in figures 12(c ₄) and 12(d ₄). The most important point is that, in figure 12(d), during its second collapse, the shock wave showed characteristics of a stratified structure, while in figure 12(c), we did not find any trace of a shock wave. In figure 12(e,f), during the first two oscillation cycles, the cavitation bubble did not merge with the air bubble. During the second collapse, the shock waves in the two experiments were different. In figure 12(e), the shock wave from the second collapse was comprised of two layers, as shown in figure 12(e ₅), while in figure 12(f), the shock wave from the second collapse had only one layer, as shown in figures 12(f ₅) and 12(f ₆).

Figure 13 concerns the process of the shock wave from the second collapse impacting on the wall surface, corresponding to that in figure 12. As for figure 13(a), during the second collapse of the cavitation bubble, we did not find any shock wave in the high-speed photographic image; accordingly, the pressure did not have any obvious peak value (the peak value was only 0.158 MPa) and the time taken for the whole process of the impact is relatively long. In figure 12(b), during the second collapse, the cavitation bubble still emitted a shock wave that had a stratified structure, and that the major peak value was approximately 1.290 MPa.

Figure 13.

Process of the shock wave from the collapse of the rebounded cavitation bubble impacting on the wall surface. The images above concern the process of the shock wave from the collapse of the rebounded cavitation bubble impacting on the wall surface in figure 12. The red dotted line on the right of the images represents the location of the sensor. In the images, the shock waves in all the impact processes have been marked with a white arrow.

By comparing figure 12(c,d), we found that the size of air bubble in figure 12(c) was smaller than that in figure 12(d). During the second oscillation cycle, the shape and the size of the cavitation bubble were apparently changed after the cavitation bubble absorbed the air bubble. Due to these changes in the shape and size of the cavitation bubble, shock waves were not found in figure 12(c) during the second collapse. From the corresponding process of pressure, there were two obvious peak values during the process of the impact in figure 12(d), i.e. 0.770 and 0.838 MPa.

As for figure 12(e,f), during the first two cycles, the cavitation bubble did not merge with the air bubble. Accordingly, during the second collapse of cavitation bubble, in figure 13(e), the peak value of the impact on the wall surface was 0.270 MPa (because the distance between the two peak values is tiny, it is hard to identify the second peak value). Moreover, the time taken by the shock wave to impact on the wall surface in figure 13(e) was slightly longer than that in figure 13(f). The peak value of the impact pressure that the shock wave with a single layer in figure 13(f) had on the wall surface was 0.339 MPa.

Based on the characteristics of the shock wave from the second collapse with the effect of the air bubble, it can be found that the characteristics of the shock wave from the collapse of the rebounded cavitation bubble is closely related to φ and ε. We systematically collected experimental data on the influence of the air bubble on the collapse shock wave of the rebound cavitation bubble, as shown in figure 14.

Figure 14.

Relations between the bubble–bubble dimensionless parameters and the stratified shock wave from the collapse of the rebounded cavitation bubble.

From figure 14, we can see that when ε and φ are quite small, the cavitation bubble merges with the air bubble during the first oscillation cycle and emits a shock wave with a stratified structure during the collapse of the rebounded cavitation bubble; when φ is quite small, if the size of the air bubble is large, the cavitation bubble will also merge with the air bubble in the first cycle and may not emit a shock wave when the rebounded cavitation bubble collapses. As φ increases, under the condition that the air bubble is small, the shock wave emitted when the rebounded cavitation bubble collapses will basically have a stratified structure, and if the air bubble is large, the rebounded cavitation bubble will not emit a shock wave. As φ further increases, the rebounded cavitation bubble will emit a shock wave. However, under such circumstances, sometimes the shock wave is not stratified and sometimes the shock wave is stratified.

Comparing figures 14 and 9, it can be seen that the critical condition for stratification of the shock wave from the collapse of the rebounded cavitation bubble is different from that of the shock wave from the first collapse of the cavitation bubble. One reason is that, after the first collapse, the form of evolution of the cavitation bubble is not symmetric in shape anymore. The other reason is that the influence that the air bubble has on the cavitation bubble is further accumulated, which leads to a certain change in the shape and volume of the cavitation bubble during its second collapse. Thus, the critical condition for stratification of the shock wave is not obvious when the two bubbles do not merge. However, when ε is relatively small, no matter whether the merger takes place in the first period or the second period, the shock wave from the collapse of the rebounded cavitation bubble will be stratified. When ε is relatively large, it is difficult for us to recognize the shock wave in the images.

As for the characteristics, i.e. whether the rebounded cavitation bubble emits a shock wave when it collapses and whether the shock wave will be stratified, by analysing the size of cavitation bubble when it collapses and whether coalescence occurs, we find that, during the evolution of the cavitation bubble, if the cavitation bubble absorbs non-condensable gas at an earlier stage, the amount of non-condensable gas in the cavitation bubble will increase when the cavitation bubble contracts to its minimum volume and the minimum volume of the cavitation bubble will also increase, which means that the degree of contraction of the cavitation bubble will decrease compared with the case when there is no air bubble. Moreover, the bigger the volume of the air bubble is, the more obvious the reduction in the degree of contraction will be. Besides, after the air bubble is absorbed by the cavitation bubble, the shape of the cavitation bubble will be very complex when the cavitation bubble contracts to its minimum volume. The smaller the air bubble is, the smaller the influence that it has on the shape of cavitation bubble when the cavitation bubble contracts to its minimum volume will be; and the bigger the air bubble is, the bigger the influence that it has on the shape of cavitation bubble when the cavitation bubble contracts to its minimum volume will be.

6.2. The influence of air bubble on the strength of the shock wave from the rebounded cavitation bubble

Section 6.1 observed and measured the influence that the air bubble has on the shock wave from the collapse of the rebounded cavitation bubble and found that the stratification phenomenon of the shock wave from the rebounded cavitation bubble is also affected by the factors such as the bubble–bubble relative distance φ and the bubble–bubble relative size ε. Thus, this section will analyse the strength of the shock wave from the collapse of the rebounded cavitation bubble.

According to the classification of the shock wave in figure 14, we analysed the relationship between the relative pressure peak of the shock wave from the collapse of the rebounded cavitation bubble under different modes, as shown in figure 15. From the distribution of ${P_{max}}_{,2}/{P^{\prime}_{max,2}}$ in figure 15, we can see that the ${P_{max}}_{,2}/{P^{\prime}_{max,2}}$ is closely related to whether the two bubbles merge or not. After the cavitation bubble and the air bubble merge in the first period, the pressure peak of the shock wave from the collapse of the rebounded cavitation bubble has decreased to 20 % of the corresponding value under the circumstance where there is no air bubble. When the cavitation bubble and the air bubble merge in the second period, the pressure peak of the shock wave from the collapse of the rebounded cavitation bubble has decreased to less than 40 % of the corresponding value under the circumstance where there is no air bubble. When the cavitation bubble and the air bubble do not merge in the first two periods, the pressure peak of the shock wave from the collapse of the rebounded cavitation bubble decreases to different extents compared with that under the circumstance where there is no air bubble. If the shock wave is stratified, the degree of decrease will be larger.

Figure 15.

The relationship between $\alpha \varphi /\varepsilon \gamma $ and ${P_{max}}_{,2}/{P^{\prime}_{max,2}}$.

In addition, if the cavitation bubble and the air bubble do not merge in the first two periods, the relationship between ${P_{max}}_{,2}/{P^{\prime}_{max,2}}$ and $\alpha \varphi /\varepsilon \gamma $ is similar to the change in ${P_{max}}_{,1}/{P^{\prime}_{max,1}}$ and $\alpha \varphi /\varepsilon \gamma $. The empirical relationship between ${P_{max}}_{,2}/{P^{\prime}_{max,2}}$ and $\alpha \varphi /\varepsilon \gamma $ may be obtained by a power function with the Levenberg–Marquardt method (Madsen et al. Reference Madsen, Nielsen and Tingleff2004)

(6.1)\begin{equation}\frac{{{P_{max,2}}}}{{{{P^{\prime}}_{max,2}}}} = a{\left( {\frac{1}{\gamma }} \right)^b}{\left( {\frac{{\alpha \varphi }}{\varepsilon }} \right)^b}.\end{equation}

When the cavitation bubble and the air bubble did not merge in the first two periods and the shock wave from the collapse of the rebounded cavitation bubble was not stratified, the adjusted R ² value in (6.1) is 0.816 and the values of a and b are 0.985 and 0.141, respectively. When the cavitation bubble and the air bubble did not merge in the first two periods and the shock wave from the collapse of the rebounded cavitation bubble was stratified, the adjusted R ² value in (6.1) is 0.882 and the values of a and b are 0.773 and 0.218, respectively.

According to (5.2), we evaluate the energy of the shock wave from the collapse of the rebounded cavitation bubble under the effect of the air bubble and make a classification based on the modes of influence that the air bubble has on the shock wave from the collapse of the rebounded cavitation bubble so that the relationship between ${E_2}/{E^{\prime}_2}$ and $\alpha \varphi /\varepsilon \gamma $ is obtained, as shown in figure 16.

Figure 16.

The relationship between $\alpha \varphi /\varepsilon \gamma $ and ${E_2}/{E^{\prime}_2}$.

From figure 16, it can be seen that, if the ways that the air bubble affects the shock wave from the collapse of the rebounded cavitation bubble are different, the degrees of attenuation of the energy of the shock wave from the collapse of the rebounded cavitation bubble will also be different. No matter whether that cavitation bubble merges with the air bubble in the first period or in the second period, the energy of the shock wave from the collapse of the rebounded cavitation bubble will obviously decrease. When the cavitation bubble and the air bubble do not merge in the first two periods, the energy of the stratified shock wave will be slightly less than that of the unstratified shock wave.

In figure 16, the empirical relationship between ${E_2}/{E^{\prime}_2}$ and $\alpha \varphi /\varepsilon \gamma $ may be obtained by a power function with the Levenberg–Marquardt method (Madsen et al. Reference Madsen, Nielsen and Tingleff2004)

(6.2)\begin{equation}\frac{{{E_2}}}{{{{E^{\prime}}_2}}} = a{\left( {\frac{1}{\gamma }} \right)^b}{\left( {\frac{{\alpha \varphi }}{\varepsilon }} \right)^b}.\end{equation}

When the cavitation bubble and the air bubble did not merge in the first two periods and the shock wave from the rebounded cavitation bubble was not stratified, the adjusted R ² value in (6.2) is 0.825 and the values of a and b are 0.673 and 0.275, respectively. When the cavitation bubble and the air bubble did not merge in the first two periods but the shock wave from the rebounded cavitation bubble was stratified, the adjusted R ² value in (6.2) is 0.865 and the values of a and b are 0.522 and 0.345, respectively. It should be noted that when the cavitation bubble and the air bubble merged in the first period or the second period, the energy of the shock wave from the collapse of the rebounded cavitation bubble decreased to less than 40 % of that under the circumstance where there is no air bubble, or even lower, and the number of experiments where the shock wave could be recognized in the photographic images were fewer. Thus, in figure 16 no fitting analysis was conducted in such situations. However, from figure 16, we can see that, under such situations, the energy of the shock wave is approximately 20 % of that under the circumstance where there is no air bubble.

Comparing figure 11 with figure 16, it can be found that, under the effect of the air bubble, the energy of the shock wave from the collapse of the cavitation bubble gradually decreases from the first collapse to the second collapse. One reason is that, as the cavitation bubble collapses, its energy is continuously released. The other reason is that the decrease in the relative energy is also connected with the way that the air bubble affects the cavitation bubble.

In short, when the air bubble – cavitation bubble – measuring point system parameters are different, the ways that the air bubble affects the shock wave from the collapse of the cavitation bubble will be different, which leads to an obvious change in the strength of the shock wave. The primary cause is that the air bubble and the water are different in compressibility, which leads to asymmetry of the surrounding medium of the cavitation bubble. When the cavitation bubble is quite close to the air bubble, the existence of the air bubble will lead to a temporal and spatial delay in the water hammer shock wave induced by the re-entrant jet and the shock wave induced by the implosion when the cavitation bubble contracts to its minimum volume, which will finally result in the stratification phenomenon of the shock wave that is photographed in the experiment. When the cavitation bubble is very close to the air bubble, the two bubbles will merge, and the non-condensable gas absorbed will increase the minimum volume to which the cavitation bubble contracts and change the original shape of the cavitation bubble when it collapses, which will finally result in the stratification phenomenon of the shock wave.