4.1. Acoustic cavitation nucleation

The specific shape of the cavitation bubble clusters produced by the passage of the rarefaction wave is highly dependent on ${\Upsilon\hskip -1,05em -\,}$. This is because the negative pressure focuses differently when the original shock wave is emitted from a different location. As the acoustic nucleation only occurs below a certain pressure threshold, the resulting bubble clouds can assume complex three-dimensional structures. Figure 4 presents experimental results showing the temporal evolution of bubble clouds generated for different values of ${\Upsilon\hskip -1,05em -\,}$. In this study the bubble ‘seeding’ position was varied by changing the delay between the drop release and the laser shot, thus shifting the laser focus position along the vertical symmetry axis of the drop.

Figure 4. Acoustic cavitation inside a water droplet. The distribution of bubbles in the liquid changes significantly with the position of the laser-induced bubble. The frame width is 3.15 mm. The time between consecutive frames is 600 ns. Results are shown for (a) ${\Upsilon\hskip -1,05em -\,}=0.65$, (b) ${\Upsilon\hskip -1,05em -\,}=1.1$, (c) ${\Upsilon\hskip -1,05em -\,}=1.7$, (d) ${\Upsilon\hskip -1,05em -\,}=2.5$, (e) ${\Upsilon\hskip -1,05em -\,}=7.5$, (f) ${\Upsilon\hskip -1,05em -\,}=68$, (g) ${\Upsilon\hskip -1,05em -\,}=13$, (h) ${\Upsilon\hskip -1,05em -\,}=5.4$, (i) ${\Upsilon\hskip -1,05em -\,}=1.8$, (j) ${\Upsilon\hskip -1,05em -\,}=0.9$. Full videos of (b,d,f) are available in the online supplementary movies 1–3 at https://doi.org/10.1017/jfm.2023.542.

As aforementioned, the shock waves emitted from the laser focal spot will reflect from the free boundary of the drop as a rarefaction wave. Due to the nearly spherical shape of the drop, the reflected acoustic waves will focus in a region located at a similar distance from its centre $r$ (where the laser bubble was created) but on the opposite side of the drop. In the case where the shock wave originates near the surface (i.e. ${\Upsilon\hskip -1,05em -\,} \lesssim 1$), the resulting pressure distribution is characterised by a negative pressure zone moving close to the liquid surface that produces a spherical shell of tiny cavitation bubbles, as displayed in the panels (a–c) (and also i–j) of figure 4. This phenomenon occurs when the sound reflects multiple times on the drop walls and travels circumferentially near the liquid surface without a significant loss of intensity, which is usually referred to as the ‘whispering gallery effect’ (Raman & Sutherland Reference Raman and Sutherland1922). As the rarefaction waves focus at a similar depth where the shock wave was emitted, it produces explosive cavitation events close to the free boundary and on the drop's vertical axis. The rapid expansion of those larger cavitation bubbles gives rise to the liquid jets shown in the first row of figure 3. A more detailed explanation of the formation and dynamics of this particular type of jet will be published elsewhere.

As the laser focusing depth $d$ is increased, the negative pressure is distributed in larger regions, but still, the nucleation of bubbles predominantly occurs on the side opposite to the laser focus. Additionally, the bubble clusters turn from having the structure of a shell (see panels (g,h,i) of figure 4) into a volumetric cavitation cloud when the laser bubble is generated near the drop centre, as shown in the panels (e,f) of figure 4. This transition can be explained by analysing the pressure distribution dynamics with the numerical simulations (Ando, Liu & Ohl Reference Ando, Liu and Ohl2012; Quinto-Su & Ando Reference Quinto-Su and Ando2013; Gonzalez-Avila & Ohl Reference Gonzalez-Avila and Ohl2016). Figure 5 demonstrates the clear correlation between the evolution of the acoustic pressure profile and the nucleation of secondary cavitation bubbles. Furthermore, this correlation can be used to determine the cavitation pressure threshold of the liquid by comparing the shape and the location of the negative pressure front with the shape of the bubble cloud within the drop. Such a comparison was only possible after applying a numerical algorithm to the simulated results to compensate for the image distortions induced by the drop curvature. The last frames in panels (a,b) of figure 5 display an overlap of both the experimental video frames and the simulated pressure profiles. From the measurements, we found a consistent cavitation threshold of approximately 4.5 MPa. Considering that we did not filter the water sample we assume that the cavitation is most likely heterogeneous.

Figure 5. Acoustic cavitation bubble clouds for laser-induced bubbles at different relative positions in the drop. The frames compare the advance of the shock/tension waves within the drop with the observed nucleation sites. The average drop diameter is ($2.84\pm 0.05$) mm in all cases. The last frame of each series presents an overlay of the frames and the cumulative minimum pressure after the first reflection of the shock wave at the free boundary. The red line indicates the isobar of $-4.5$ MPa, i.e. the approximate nucleation threshold pressure. (a) Here, the bubble is slightly off-centre (i.e. $d\simeq R_{d}$). (b) Results for ${\Upsilon\hskip -1,05em -\,} =5.4$. (c) Change in the cluster dimensions with increasing laser pulse energy (indicated in mJ). (d,e) Present evidence of the formation of complex hollow three-dimensional bubble structures. Here, ${\Upsilon\hskip -1,05em -\,}$ is 3.5 and 1.45, respectively.

The acoustic cavitation thresholds reported for water in the literature vary strongly, depending on the measurement method, water purity, gas saturation and water temperature. Atchley et al. (Reference Atchley, Frizzell, Apfel, Holland, Madanshetty and Roy1988) used distilled, deionised and filtered ($0.2\ \mathrm {\mu }{\rm m}$) tap water irradiated by pulsed ultrasound and found thresholds between 0.5 and 2.0 MPa, depending on the pulse duration and frequency. Sembian et al. (Reference Sembian, Liverts, Tillmark and Apazidis2016) subjected a water column to a single shock wave and found a cavitation threshold between 0.42 and 2.33 MPa. Biasiori-Poulanges & Schmidmayer (Reference Biasiori-Poulanges and Schmidmayer2023) compared numerical simulations and experiments of a liquid drop subjected to a planar shock wave and found a threshold between 0.37 and 2.4 MPa. A similar shock front can be found when a droplet impacts on a solid surface at a high speed (e.g. higher than $100\ {\rm m}\ {\rm s}^{-1}$) as studied by Kondo & Ando (Reference Kondo and Ando2016) and Wu et al. (Reference Wu, Xiang and Wang2018b, Reference Wu, Liu and Wang2021). Assuming homogeneous nucleation, Ando et al. (Reference Ando, Liu and Ohl2012) and later Quinto-Su & Ando (Reference Quinto-Su and Ando2013) found a cavitation threshold of 60 MPa and 20 MPa, respectively, comparing experiments and simulations of a reflected shock wave at a free boundary. Therefore, the threshold value obtained in this work falls around the middle of the spectrum of values measured by other authors. Figure 5(c) evidences a growth in the secondary bubble cluster with increasing energy of the laser pulse, demonstrating the resulting shift in the location of the cavitation threshold isobar for higher amplitudes of the initial shock wave. It is relevant to point out that VoF simulations are notorious for numerical diffusion that causes the shock wave to smear out over time. Because of this, the simulations may underestimate the pressures reached in the experiments. Please note that the VoF model does not account for phase transitions and the subsequent interaction of nucleated cavitation bubbles with the finite amplitude waves. A model for of high-frequency waves interacting with small cavitation clouds that may be applicable was recently developed by Maeda & Colonius (Reference Maeda and Colonius2019). Finally, panels (d,e) of figure 5 exemplify some of the hollow three-dimensional bubble structures observed in the experiments.

4.2. Bubble jetting

In the second stage presented in figure 3 the laser-induced bubble reaches its maximum radius and then collapses. At this point, it becomes clear that a non-uniform distance between the bubble and the free surface produces an asymmetric collapse, which culminates in a liquid jet. In this section we explore the effect of varying the parameter ${\Upsilon\hskip -1,05em -\,}$ (as performed in § 4.1), but this time we lay focus on the development of the jets, as shown in figure 6.

Figure 6. Bubble jetting is produced by a laser-induced bubble generated at different relative positions inside the drop. The numbers indicate the time in $\mathrm {\mu }{\rm s}$. The length of the scale bars is 1 mm. (a) Spherical oscillation case, ${\Upsilon\hskip -1,05em -\,}=203$. (b) Weak jet case, ${\Upsilon\hskip -1,05em -\,}=3.9$. (c) Standard jet case, ${\Upsilon\hskip -1,05em -\,}=1.5$. (d) Case ${\Upsilon\hskip -1,05em -\,}=0.44$. (e) Bullet jet case, ${\Upsilon\hskip -1,05em -\,}=0.22$. Full videos are available in the online supplementary movies 4–8.

The experiments reveal that as the position of the laser focus is varied between the centre and the surface of the drop, the characteristics of the jetting change smoothly: for large values of ${\Upsilon\hskip -1,05em -\,}$, a spherical rebound of the bubble without any jetting is observed. The values of ${\Upsilon\hskip -1,05em -\,}\gtrsim 3.5$ are accompanied by the formation of a very thin liquid jet crossing through the centre of a weakly deformed bubble. In this ‘weak jet’ case the tip of the jet separates from the main cavity when it starts to collapse during its second oscillation cycle (see figure 6b). For $1.2\lesssim {\Upsilon\hskip -1,05em -\,}\lesssim 3.5$, as in panel (c) of figure 6, the ‘whispering gallery’ effect becomes relevant, causing the inception of larger acoustic bubbles on the side opposite to the laser cavity and the ejection of liquid driven by their expansion. The deformation of the bubble in its rebound phase is significantly stronger than in panel (b) of figure 6. As the laser is focused closer to the drop's surface, i.e. $0.3\lesssim {\Upsilon\hskip -1,05em -\,}\lesssim 1.2$, the expansion of the bubble provokes the onset of a RTI. This can be seen in figure 6(d) by the formation of several ‘spikes’ growing from the thin liquid film trapped between the cavity and the surrounding air. At the same time, the bubble collapse (from $t=66\ \mathrm{\mu }{\rm s}$) results in an elongated cavity, similarly as in the ‘bullet jet’ case (Rosselló et al. Reference Rosselló, Reese and Ohl2022). This behaviour is more pronounced for even smaller stand-off distances, as presented in figure 6(e). The dynamics of this particular jet are described in detail in Rosselló et al. (Reference Rosselló, Reese and Ohl2022) and correspond to the case where the laser cavity is generated almost directly on the surface of the drop (i.e. $0.01\lesssim {\Upsilon\hskip -1,05em -\,}\lesssim 0.3$). Here, atmospheric gas is trapped after the closure of a conical ventilated splash and later dragged into the liquid by the liquid jet that grows from a stagnation point located on the top of a ‘water bell’ (at the bottom of the frame at $t=36\ \mathrm {\mu }{\rm s}$). As a result, an elongated gas cavity is shaped and driven across the drop.

The combined effects of the curved shape of the drop in addition to the diffuse illumination lead to images of the interior of the gas cavity with remarkable clarity. A few examples of this are presented in figure 7.

Figure 7. Detailed view of the interior of a jetting bubble. The time between frames is $2\ \mathrm {\mu }{\rm s}$. (a) Jet formation for ${\Upsilon\hskip -1,05em -\,}=1.6$. The frame width is 1.46 mm. (b) Comparison between experimental data and a simulation performed for ${\Upsilon\hskip -1,05em -\,}=2.9$. The frame width is 1.38 mm. (c) Spray produced by air entering the gas cavity (in which the pressure is lower than the atmospheric pressure) while the jet is formed. The frame width is 2.11 mm.

Panels (a,b) of figure 7 reveal the temporal evolution of the liquid indentation into the bubble, as well as the toroidal shape acquired by the gas upon its collapse. Moreover, figure 7(b) demonstrates the accuracy of the numerical simulations to reproduce the jetting process. In panel (c) we see how a perforation of the thin liquid sheet between the cavity and the atmosphere resulted in a spray of aerosol droplets ejected into the cavity during jetting. This event can be explained by the lower pressure inside the bubble compared with the atmospheric pressure and the disruption of the liquid on the upper side of the drop caused by the RTI. The spray front spreads into the cavity and collides with the lower wall of the bubble, disrupting the smoothness of the interface.

The bullet jet case of figure 6(e) distinguishes itself from the other cases by its unique features, i.e. its enhanced shape stability during its formation from an open splash, but also by the near robustness against the surrounding fluid and geometry. Bullet jets have been observed in shallow waters (Rosselló & Ohl Reference Rosselló and Ohl2022) and near flexible or rigid materials, without these conditions affecting their dynamics. Furthermore, in a previous work (Rosselló et al. Reference Rosselló, Reese and Ohl2022), we demonstrated that the bullet jet is scalable and independent of the orientation of the surface with respect to gravity. In figure 8 we expand the list of remarkable robustness by showing it to exist of various sizes even within a highly curved and finite volume. Here, the bullet jet size was characterised by the ratio between the radius of the water bell at its base ($R_{wb}$) measured right after its formation and the initial drop radius $R_d$ (i.e. $R_{wb}/R_{d}$).

Figure 8. Scalability of the bullet jet in a millimetric droplet. The measurements, organised in columns, show bullet jets formed from different splash sizes. In each column, the upper frame shows the time at which the water bell closes. In the lower frame, composed of two vertical stripes, the time at which the bullet jet is fully developed is shown on the left, and a frame illustrating the position of the jet tip at an advanced time indicated in $\mathrm {\mu }{\rm s}$ is shown on the right. Results are shown for (a) $R_{wb}/R_{d}=0.16$, (b) $R_{wb}/R_{d}=0.27$, (c) $R_{wb}/R_{d}=0.37$, (d) $R_{wb}/R_{d}=0.56$, (e) $R_{wb}/R_{d}=0.74$.

The images depict that the penetration depth of both the gas and the liquid conforming to the bullet jet is proportional to the initial splash size. For instance, in figure 8(a) the jet loses its momentum and stops around the middle of the drop, but it crosses the drop for the larger splashes shown in panels (c–e). Remarkably, in the latter case the bullet jet occupies almost the entire drop while still preserving its characteristic features.

The physics behind the evolution of the bubble jetting cases classified in figure 6 can be further explained with the aid of numerical simulations, as presented in figure 9.

Figure 9. Numerical simulations of the temporal evolution of jets produced inside the drop for different ${\Upsilon\hskip -1,05em -\,}$. The simulated drop has a height of 2.88 mm and a width of 2.8 mm as measured in the experiments. The plot shows the gas and liquid phases along with the velocity field. The time between frames is $26\ \mathrm {\mu }{\rm s}$ for (a–c) and $30\ \mathrm {\mu }{\rm s}$ for (d) starting at $t=1\ \mathrm {\mu }{\rm s}$ in the first frame. (a) Spherical oscillation case, ${\Upsilon\hskip -1,05em -\,}\rightarrow \infty$. (b) Weak jet case, ${\Upsilon\hskip -1,05em -\,}=3.896$. (c) Standard jet case, ${\Upsilon\hskip -1,05em -\,}=1.518$. (d) Bullet jet case, ${\Upsilon\hskip -1,05em -\,}=0.028$.

Figure 9(a) depicts a purely radial oscillation of both the gas and liquid, found when the bubble is placed in the centre of the drop (i.e. ${\Upsilon\hskip -1,05em -\,}\rightarrow \infty$). The simulations shown in panels (b,c) of figure 9 were computed using the same ${\Upsilon\hskip -1,05em -\,}$ measured from the experimental cases displayed in the corresponding panels of figure 6. In general, the agreement between the simulations and the experiments is excellent, even though small variations in the size of the experimental and simulated bubbles show some differences in the specific timing of their oscillation cycle. The resemblance can be seen in some of the morphological features that characterise the dynamics of each type of jet at different stages, like the width of the indentation formed during bubble piercing, the shape of the cavity after the first rebound, and the way in which the second collapse evolves in each case. More details on noteworthy features are provided below in figure 10.

Figure 10. Detailed collapse dynamics of the gas cavity immediately after the jetting of the laser bubble. Experimental (a) and simulated (b) view of the ‘weak’ jet obtained when ${\Upsilon\hskip -1,05em -\,}=3.9$. The images were taken at 200 kfps. (c) Ring formation after the necking of the cavity typically observed in cases with ${\Upsilon\hskip -1,05em -\,}\approx 1.9$. The images were taken at 500 kfps. (d) Direct comparison between experiment and simulation, revealing the precise flow pattern leading to the ring detachment (indicated by the white arrows). The time between frames is $2.5\ \mathrm {\mu }{\rm s}$. The colour scale in the simulations corresponds to that in figure 9.

In panels (a,c) of figure 9 the bubble is initiated with a much larger pressure than the atmospheric gas outside the drop. This pressure difference, which is constant in all directions, accelerates the liquid between the two gas domains. Since this force is proportional to the pressure gradient, the liquid gets accelerated more strongly between the bubble and the nearest part of the drop surface (where the liquid is thinner), causing the drop to bulge out in that location. Within the first few microseconds of the explosive bubble expansion, the pressure within the bubble decreases rapidly and reaches values much smaller than the atmospheric pressure. Thus, the pressure gradient changes its direction and now accelerates the liquid towards the bubble, which first slows down the cavity's expansion and afterward causes its collapse. In the same way as in the expansion phase, the thinnest part of the liquid experiences the strongest acceleration, which ultimately leads to a liquid jet indenting the bubble from the nearest part of the drop surface.

The case presented in figure 9(d) differs greatly from the previous cases by the fact that now the bubble is close enough to the drop surface to generate an open cavity, allowing the ejection of the initially pressurised gas inside it into the atmosphere, and later the flow of gas into the expanded cavity before the splash closes again. Once the cavity is closed, it remains with an approximate atmospheric pressure, which prevents it from undergoing a strong collapse as it occurs in the previously discussed cases (a–c). The radial sealing of the splash forms an axial jet directed toward the centre of the drop, which pierces the bubble and drags its content through the drop. More details on the mechanisms behind the bullet jet formation can be found in Rosselló et al. (Reference Rosselló, Reese and Ohl2022).

As a consequence of the conservation of momentum, the collapse of the gas cavity gives origin to a stagnation point, from which the liquid flows both inside the pierced bubble and away from it in opposite directions. In particular, the stagnation point is not stationary but moves along the axis of symmetry, following a different trajectory in each case. In the case of figure 9(b) the stagnation point shifts towards the surface as the bubble moves deeper into the drop. For the case in figure 9(c), the stagnation point does not reach the surface and its movement is less pronounced. In the bullet jet case, shown in figure 9(d), the stagnation point forms on the apex of the water bell (i.e. the splash after its closure). It then trails the bell's collapse and remains very close to the drop surface afterward, moving slightly towards the drop centre while the bullet jet moves across the drop.

4.2.2. Influence of $R_d$ on the jet dynamics: behavioural similarity vs structural similarity

The bubble dynamics observed in the falling drop case have many similarities with what is typically seen in bubbles collapsing near a planar rigid surface (Lauterborn et al. Reference Lauterborn, Lechner, Koch and Mettin2018) or a planar free surface (Supponen et al. Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016; Rosselló et al. Reference Rosselló, Reese and Ohl2022). Moreover, the analysis of the values of the stand-off parameter $D^*$ reveals that each type of jet (qualitatively classified according to figure 7 in Rosselló et al. (Reference Rosselló, Reese and Ohl2022)) occurs in a comparable range of values of $D^*$. One example of the latter can be found in figure 11.

Figure 11. Comparison of cases with similar bubble dynamics and a different curvature of the free surface $R_d$. Panels (a,c) show cases where the cavity is produced near a flat free surface. The cases in (b,d) show similar bubbles generated inside a drop with a mean radius of 1.42 mm. Here, the numbers represent the time normalised with the time of collapse of the cavities from each case. Results are shown for (a) $D^*=0.85$, (b) $D^*=0.88$, (c) $D^*=1.6$, (d) $D^*=1.37$.

The parallel found between cases with dissimilar curvature of the liquid surface suggests that, contrarily to the reported observations for bubbles collapsing near concave solid surfaces (Aganin et al. Reference Aganin, Kosolapova and Malakhov2022), $R_d$ does not have a dominant role in the particular jetting regime adopted by the cavities when the bubbles are located near the free boundary. This statement was confirmed by the numerical simulations depicted in figure 12. There, the dynamics of identical bubbles expanding and collapsing near the surface of the drop, or the flat free surface of an ideally infinite pool, are compared for three stand-off distances $D^*$.

Figure 12. Jetting dynamics of identical bubbles produced near a flat surface or the curved surface of a droplet ($R_d = 1.42\ \mathrm {\mu }{\rm m}$). The non-dimensional time, indicated by the numbers, was normalised with the collapse time of each bubble. Results are shown for (a) $D^*=0.61$, (b) $D^*=1.02$, (c) $D^*=1.23$.

The simulations show that the correspondence between the flat and curved surface cases is gradually lost when the bubble is placed further away from the drop surface. The deviation between the two cases is already visible in figures 11(c) and 11(d). There, the jet dynamics is matched only when $D^*$ takes a higher value for the flat free surface measurement. The simulations indicate that this discrepancy starts at around $D^*=1.2$ (shown in figure 12c) and keeps growing for higher values. We can portray these changes as being enclosed between two extreme scenarios: (1) the bubble is produced right on the liquid surface, generating a bullet jet, which is not affected by the characteristics of the boundaries and, thus, is independent of $R_d$. As the cavity is placed closer to the drop centre, the surface curvature becomes increasingly relevant to the jet dynamics. This is consistent with our definition of ${\Upsilon\hskip -1,05em -\,}$, since $D^*$ and ${\Upsilon\hskip -1,05em -\,}$ take similar values for lower values of $d$, and grow apart as the cavity is placed deeper in the drop. (2) When the bubble is almost at the drop centre (i.e. ${\Upsilon\hskip -1,05em -\,}\,\xrightarrow \,\infty$), there is no jetting for the curved case. However, in the flat surface case the jetting still occurs for comparable values of $D^*$ (e.g. $D^*\sim 2$), demonstrating how the curvature weighting factor $\chi$ becomes increasingly relevant.

It is important to stress that if the bubble is placed near the drop boundary, for instance, at $D^*\lesssim 1.4$, the discrepancies found in the jetting dynamics of a bubble in the ‘semi-infinite’ liquid pool when compared with the droplet case are mainly provoked by the surface curvature, and not by the dissimilar extension of the liquid below the gas cavity. This particular point is corroborated in the Appendix by means of complementary measurements and numerical simulations of jetting bubbles in the proximity of a hemispherical tip of a cylindrical water column.

So far, we have classified and compared the characteristics of the jetting regimes produced at different $D^*$ qualitatively, i.e. based on their general morphological features as presented in previous works (Supponen et al. Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016; Rosselló et al. Reference Rosselló, Reese and Ohl2022). In the following, we will refer to this as behavioural similarity. An alternative and more precise way of analysing the spatial correlation between the dynamics of two different jets can be achieved by contrasting the pixel distribution on the video frames to find common features between images. This quantitative comparison method is usually referred as structural similarity analysis and can be implemented using different image scanning algorithms (Sampat et al. Reference Sampat, Wang, Gupta, Bovik and Markey2009). Here, we use the complex wavelet structural similarity index (CW-SSIM) (Zhou & Simoncelli Reference Zhou and Simoncelli2005) to evaluate the correlation between the temporal evolution of two different jetting cavities. The CW-SSIM approach has some advantages over direct pixel to pixel comparison methods (e.g. intensity based) or the simpler versions of the structural similarity index (e.g. SSIM). For instance, it accounts (up to some point) for both intensity variations and non-structural geometric distortions like object translation, scaling and rotation (Sampat et al. Reference Sampat, Wang, Gupta, Bovik and Markey2009). The CW-SSIM index can take values ranging from zero (if there is no correlation at all) to one (when the images are identical).

In figure 13 we contrast the dynamics of two bubbles initially located at a distance $D^*$ from a flat or a curved surface as already done in figure 12, but this time using the CW-SSIM index to evaluate their similarity. The non-dimensional time ($t^*$) was computed using the collapse time of the bubbles for each case. Figure 13 presents plots of the temporal evolution of the similarity index next to a series of selected frames at $t^*=0.3, 0.6, 0.9, 1.05, 1.2, 1.35, 1.5, 1.7$ and 1.9, which illustrate and compare the shape of the cavities in the flat boundary case (grey background on the left side) and the drop case.

Figure 13. Structural similarity of the dynamics of jetting bubbles produced near a flat surface or the drop boundary at an identical stand-off distance. Here, the temporal evolution of the CW-SSIM index is presented for three examples corresponding to cases with $D^*=0.47$, $D^*=0.88$ and $D^*=1.85$. The insets show a comparison of the images of both simulated bubbles. The frames are centred at specific non-dimensional times (blue vertical lines), displaying a half-frame corresponding to the flat surface case (grey background on the left) and a half-frame taken from the drop case with the same $D^*$.

The results expose the differences between the behavioural similarity and the structural similarity approaches, i.e. two bubbles can have the same jetting regime but still have dissimilar structures. This is observed in bubbles at lower stand-off distances like the case with $D^*=0.47$ in figure 13. The discrepancy can be explained by the higher degree of fragmentation of the bubble after jetting, acquiring an elongated shape in regimes with a ventilated cavity, or where the liquid layer between the gas in the bubble and the atmosphere is affected by the RTI. In particular, the cases producing an open cavity (i.e. $D^*\lesssim 0.35$) were not suitable for the structural similarity analysis. Here, the fluctuations of the splashing dynamics observed in the numerical simulations and the impossibility to define a collapse time due to the non-collapsing nature of those cavities (see figure 6d,e) prevented us performing a reliable assessment of the CW-SSIM index.

As the laser bubble is produced deeper into the liquid, both structural and behavioural approaches lead to the same conclusions (previously discussed in figure 12). The similarity found in the development of bubbles near surfaces with or without curvature is excellent for some stand-off distances, e.g. around the middle point located between the liquid surface and the drop centre. One example of this is shown in the central panel of figure 13 corresponding to $D^*=0.88$. As we already mentioned, near the centre of the drop (i.e. $D^*\simeq 2$) the difference in the anisotropy in both cases produces dissimilar bubble oscillations (see the lower panel of figure 13).

A more general overview of those three scenarios is presented in figure 14, where the mean value of the CW-SSIM index is plotted along with $D^*$ and ${\Upsilon\hskip -1,05em -\,}$. Considering that all the bubbles have a very similar initial expansion phase, only the times corresponding to the first collapse and the complete second oscillation cycle were computed in the mean value of CW-SSIM. After the second collapse, the bubble is heavily fragmented and there is no longer a recognisable structure. Figure 14 confirms that the structural similarity is rather poor for bubbles near the surface (i.e. $0\lesssim D^*\lesssim 0.7$). Around $D^*=0.9$, the similarity index reaches a peak where the match is excellent. A good agreement is sustained over a range of $D^*$ values between approximately 0.7 and 1.7, meaning that even when the features of the cavities are not identical they have a similar distribution of the gas phase (and the same jetting regime). For $D^*\gtrsim 1.7$, the similarity index suffers an abrupt fall and the value of ${\Upsilon\hskip -1,05em -\,}$ diverges as the bubble seeding position gets closer to the drop centre. This is consistent with the definition of ${\Upsilon\hskip -1,05em -\,}$, which relates its magnitude to the influence of the drop curvature in the bubble dynamics.

Figure 14. Structural similarity between cases with bubbles seeded at different stand-off distances from a flat free surface or inside the drop. The curve shows the mean value of the ‘total’ CW-SSIM index (green) computed as the average of the mean indices observes in the first bubble collapse (red) and in its second oscillation cycle (blue). The sudden increase in ${\Upsilon\hskip -1,05em -\,}$ (black markers) as we seed the bubble close to the drop centre is linked to a decay in the similarity between the cavities.

The experimental results displayed in figure 11 suggest that there is a correspondence between the dynamics of a bubble seeded with a given $D^*_{flat}$ in the flat surface case and the temporal evolution of a bubble with $D^*_{drop}$ inside the droplet. We explore this apparent ‘equivalence’ between values of $D^*_{flat}$ and $D^*_{drop}$ in figures 15 and 16. Since the result of the CW-SSIM analysis is affected by a significant translation of the objects being compared, the initial positions of the bubbles were matched by performing a vertical shift on the drop case simulations. Figure 15 shows evidence of the mentioned correspondence by presenting two examples, one with $D^*_{flat}=1.02$ and $D^*_{drop}=0.95$, and a second one where the bubble is closer to the drop centre, i.e. $D^*_{flat}=2.40$ and $D^*_{drop}=1.68$.

Figure 15. Similarity study of the jetting dynamics of bubbles near a flat surface (grey background) or a drop (white background) with a different $D^*$. The initial position of the bubbles was matched by performing a vertical shift on the drop case simulations. For each value of $D^*_{flat}$, there is one value of $D^*_{drop}$ with similar dynamics, i.e. producing the maximum CW-SSIM index when a simulation of a given $D^*_{flat}$ is compared against simulations with every possible value of $D^*_{drop}$. The numbers indicate non-dimensional time $t^*$. (a) For $D^*_{flat}=1.02$, the maximum average CW-SSIM index was achieved with $D^*_{drop}=0.95$. (b) The best match for $D^*_{flat}=2.40$ was $D^*_{drop}=1.68$. (c) Temporal evolution of the CW-SSIM index for the cases in (a,b).

Figure 16. Best match between the dynamics of bubbles in the flat boundary and the drop cases for different stand-off distances, i.e. $D^*_{flat}$ and $D^*_{drop}$. For each value of $D^*_{flat}$, the best match was obtained by finding the corresponding value of $D^*_{drop}$ that maximises the mean CW-SSIM index. As indicated by the parameters of the linear fit, the best match is found at similar values of $D^*$ at the lower depths, but they become increasingly different as $d\rightarrow R_d$, where the bubble is seeded at the drop centre (indicated with a vertical dotted line). The dashed grey line was added as a visual reference. The vertical error bars represent the deviation of CW-SSIM from the perfect similarity case (i.e. $\text {CW-SSIM} = 1$).

These two examples in figure 15 prove that there are pairs of $D^*$ values where the similarity between the dynamics of bubbles produced near two surfaces with uneven curvature is remarkable, at least during the whole period comprised in the first two oscillation cycles. This correlation analysis was performed for an extended range of values of $D^*$ to find that, for each value of $D^*_{drop}$, there is one value of $D^*_{flat}$ with similar dynamics, i.e. which maximise the CW-SSIM index when a simulation made with that particular $D^*_{flat}$ is compared against simulations with every possible value of $D^*_{drop}$. As shown in figure 16, the dependence of the equivalent stand-off distance starts as a linear function in the proximity of the surface and grows rapidly as the bubble is placed deeper in the liquid. Interestingly, the linear fit performed on smaller values of $D^*$, which meet the conditions for the CW-SSIM analysis, projects a ratio between equivalent $D^*_{drop}$ and $D^*_{flat}$ near 1 when the seeding position approaches the surface. Now, bearing in mind the definition of ${\Upsilon\hskip -1,05em -\,}=\chi D^*$, the previous observation is consistent with the limiting case at the surface where $\chi \rightarrow 1$, meaning that the curvature does not play a significant role for the bubble jetting dynamics. At the other extreme, i.e. as $d \rightarrow R_d$, ${\Upsilon\hskip -1,05em -\,}$ diverges, indicating a strong influence of the drop geometry on the bubble evolution. At this point, it is important to stress that in this context ‘equivalence’ does not mean that the dynamics is identical, but their structure is ‘as similar as it could be’ for the matching of $D^*_{flat}$ and $D^*_{drop}$. The same clarification applies to bubbles with the same ${\Upsilon\hskip -1,05em -\,}$, which is a multivalued function as explained in § 2.1.

4.3. Radial bubble oscillations

In the previous section we studied features found in the dynamics of an axisymmetric jetting bubble. Let us now have a closer look at the only case with spherical symmetry, i.e. where the laser cavity is placed in the centre of the drop (${\Upsilon\hskip -1,05em -\,}\,\xrightarrow \,\infty$). In this scenario the bubble undergoes several spherical oscillations with a decaying amplitude, as commonly observed in laser bubbles created in unbounded liquids (Liang et al. Reference Liang, Linz, Freidank, Paltauf and Vogel2022). Figure 17(a) presents a comparison between an experiment and simulated data computed using the VoF solver, finding an excellent agreement. As for the previously simulated results, here we applied the correction script that accounts for the distortion induced by the drop curvature.

Figure 17. Direct comparison between the experiment and a numerical simulation for a case where the laser bubble is placed at the centre of the drop. (a) The median diameter of the drop is 1.42 mm. The simulated images showing the velocity field have been remapped to account for the distortion provoked by the curvature of the drop. The numbers indicate time in $\mathrm {\mu }{\rm s}$ and the colour scale is given in ${\rm m}\ {\rm s}^{-1}$. (b) Radial dynamics of the experimental and simulated bubbles. The experimental radius was obtained by fitting a circle on the bubble. The radius in the simulations was estimated using the gas volume (i.e. the spherical equivalent radius). The results were compared with the unbounded case to find that the bubbles inside the drop have a shorter expansion/collapse cycle. (c) Bubble dynamics is obtained with a modified Rayleigh–Plesset model for different drop sizes. The radii of the larger drops remain almost unaltered during the bubble oscillation.

Figure 17(b) depicts the temporal evolution of the bubble radius $R(t)$ for the examples in panel (a). In addition, it presents $R(t)$ calculated for a case of a drop of an ideally infinite size, which corresponds to the case of an unbounded liquid domain. The initial conditions in the VoF model were chosen to match the experimental $R^*_{max}$, and then the other cases were simulated maintaining the same parameters while changing the drop size. Notably, the bubble computed with CFD reaches a slightly larger maximum radius as the liquid layer thickness is increased to infinity (i.e. an unbounded bubble case) and, thus, also has a larger collapse time. This might be due to the effect produced by the consecutive (and alternating) tension and pressure waves interacting with the bubble during its expansion (see figure 17a).

To shed some light on this matter, we use a spherical bubble model based on a modified Rayleigh–Plesset model (RP) (Obreschkow et al. Reference Obreschkow, Kobel, Dorsaz, de Bosset, Nicollier and Farhat2006; Zeng et al. Reference Zeng, Gonzalez-Avila, Ten Voorde and Ohl2018) that accounts for the finite droplet size, viscosity of the liquid and interfacial tension. It is worth noting that in these previous works the millimetre-sized droplet was sitting on the top of a blunt needle or deformed into an ellipsoidal shape by a strong levitating acoustic field. This leads to non-spherical boundary conditions that affect the bubble dynamics. In the present analysis the droplet is nearly perfectly spherical, thus matching with the purely spherical RP model within a droplet. The results, presented in figure 17(c), show that the bubble grows up to almost the same size independently of the drop size. For the same initial conditions set in the VoF model ($p_g(t=0)=1.69$ GPa and $R_b(t=0)=17.3\ \mathrm {\mu }{\rm m}$), the work is almost completely done against the surrounding pressure ($p_\infty =1$ bar) while surface energy and viscous dissipation is negligible. Yet, for smaller droplet volumes, the inertia is reduced and, therefore, the expansion time to maximum bubble radius and the almost symmetrical collapse reduce, too.

For the particular initial conditions, both models agree on the elongation of the oscillation cycle; however, they predict dissimilar results on the maximum radius reached by the bubbles. The simple RP is used to give a comparison to the VoF simulations and to evaluate the impact of the shock wave (and its reflections) on the bubble dynamics. From figure 17(c) we can infer that the maximum expansion of the bubble is nearly independent of the droplet size, while in the VoF simulations it is not. The VoF model accounts for the reflected wave, thus, the discrepancy suggests that upon the acoustic soft reflection of the shock wave momentum is imparted on the droplet interface. The importance of reflected waves on cavitation nucleation in confined liquid samples was recently also found for an acoustic hard reflection where the bubble expansion was lowered (Bao et al. Reference Bao, Reuter, Zhang, Lu and Ohl2023). A more comprehensive formulation for the bubble dynamics than the RP model that incorporates both the bubble–shock wave interaction and compressibility effects can be found in Zhang et al. (Reference Zhang, Li, Cui, Li and Liu2023).

4.4. Drop surface instabilities

In the previous sections the formation of radial liquid jets growing from the drop surface in the shape of ‘spikes’ was mentioned. As explained above, this phenomenon stems from an initial perturbation of the liquid interface and the posterior ejection of liquid produced by the RTI. This kind of instability occurs when the rapid expansion or the collapse of the bubble wall accelerates a thin liquid layer trapped between the cavity and the atmospheric gas, producing a pattern of ripples on the drop surface that grow further in the consecutive bubble oscillations. A clear example of the events leading to the onset of this kind of instability in this particular experiment is shown in figure 18(a). There, the acoustic emissions from the laser dielectric breakdown nucleate a cloud of bubbles within the drop. As the cavity expands, all these smaller bubbles are incorporated (by coalescence) into the main bubble, producing a series of dimples on the bubble surface, visible at $t=50\ \mathrm {\mu }{\rm s}$ of figure 18(a). These dimples may contribute to the later destabilisation of the drop surface, which is highly dependent on the ratio $R^*_{max}/R_d$. Additionally, $R^*_{max}/R_d$ determines the liquid layer thickness and its acceleration by the bubble/drop dynamics. The ripples in the drop surface become noticeable just after the first bubble collapse (i.e. $t=130\ \mathrm {\mu }{\rm s}$) and grow significantly during the bubble re-expansion, as shown at $t=190\ \mathrm {\mu }{\rm s}$. However, the most dramatic events take place after the second bubble collapse (i.e. at $t=230\ \mathrm {\mu }{\rm s}$). There, the ripples grow into liquid ‘spikes’ that lead to the detachment of small droplets due to the action of the Rayleigh–Plateau instability, as shown in figure 18(b). At the same time, the second bubble collapse releases a strong shock wave in the radial direction. This shock wave interacts with the array of meniscus-shaped pits on the liquid surface to produce fast radial jets (see the frame at $t=420\ \mathrm {\mu }{\rm s}$). The later sequence is clearly captured in the frames of figure 18(c). It is important to note that this complex phenomenon not only depends on the shock wave strength but also requires certain conditions to be met (Tagawa et al. Reference Tagawa, Oudalov, Visser, Peters, van der Meer, Sun, Prosperetti and Lohse2012; Peters et al. Reference Peters, Tagawa, Oudalov, Sun, Prosperetti, Lohse and van der Meer2013), like a minimum depth and curvature of the pits, which may explain the absence of ‘spikes’ during the first bubble collapse.

Figure 18. Drop surface destabilisation mechanisms. The mean drop radius is 1.42 mm and the numbers represent time in $\mathrm {\mu }{\rm s}$. (a) As the main bubble expands, the secondary, acoustic cavitation bubbles produce small dimples on the gas cavity surface (e.g. at $50\ \mathrm {\mu }{\rm s}$). These may promote the formation of a series of ripples during the bubble collapse (at $130\ \mathrm{\mu }{\rm s}$). As the bubble re-expands, the RTI causes the growth of liquid ‘spikes’ that later lead to the detachment of small drops due to the Rayleigh–Plateau instability, as indicated with a green arrow in (b). There, the frame width is $570\ \mathrm {\mu }{\rm m}$. At the same time, the second collapse of the bubble enhances the surface irregularities and pits that appear in the areas between the ripples. The shock wave emitted during the second collapse gives origin to fast liquid jets ejected from the centre of the pits, as highlighted with a blue arrow in (c). The frame width in this sequence is $490\ \mathrm {\mu }{\rm m}$. The full video is available in the online supplementarymovie 9.

To further analyse the onset of these instabilities, we varied the energy of the laser pulse, hence producing bubbles with various sizes and, thus, with distinct ratios $R^*_{max}/R_d$. The results are presented in figure 19. Even when the extreme image distortion produced near the drop interface prevent us obtaining an accurate value of the bubble radius, these measurements make evident that the amplitude of the ripples increases with increasing $R^*_{max}$ and with each consecutive bubble oscillation.

Figure 19. Onset of the drop surface instabilities for bubbles produced with different laser pulse energies. The mean drop radius is 1.42 mm and the numbers represent time in $\mathrm {\mu }{\rm s}$. The energy of the laser pulse is (a) $L=1.9\ {\rm mJ}$, (b) $L=3.1\ {\rm mJ}$ and (c) $L=3.9\ {\rm mJ}$. For this energy, the RTI affects the drop surface enough to produce liquid ejection after the second bubble collapse. For the remaining panels, the energy of the laser pulse is (d) $L=4.6\ {\rm mJ}$, (e) $L=5.2\ {\rm mJ}$ and (f) $L=6.4\ {\rm mJ}$. Note that (d–f) are shown in wider frames than (a–c) to show the larger ‘spikes’. A full video of (f) is available in the online supplementary movie 10.

In figure 19(a) the expansion of the bubble is not sufficient to visibly disturb the drop's spherical surface. In the case shown in figure 19(b) the bubble's first collapse does not break up the drop surface, however, a mild wave pattern is observed on the surface after the bubble re-expansion (at $t=420\ \mathrm {\mu }{\rm s}$). In spite of the presence of these low amplitude ripples, no radial jets are ejected from the drop upon the second bubble collapse. When the ratio $R^*_{max}/R_d$ is further increased, as shown in figure 19(c), we find very similar dynamics of the bubble/drop system, but now the valleys between the ripples (and the acoustic pressure wave) are deep enough to trigger the radial jetting. This confirms the existence of threshold conditions for the ‘spikes’ to be formed. In the remaining cases presented in panels (d–f) of figure 19, the general dynamics of the bubble/drop system are very similar to the previous cases, although as the laser pulse energy is increased the instabilities become perceivable at an earlier time. For example, in figure 19(f) liquid ‘spikes’ are already formed after the first bubble collapse (Zeng et al. Reference Zeng, Gonzalez-Avila, Ten Voorde and Ohl2018).

Figure 20 shows VoF simulations of the RTI found on the drop surface. From figure 20(a) it is clear that the instability is grown by the volumetric oscillation of the bubble, while the shock waves emitted from the bubble upon its creation (and later at its collapse) accelerates the ripples on the drop surface and form the thin ‘spikes’. This kind of simulation was previously performed by Zeng et al. (Reference Zeng, Gonzalez-Avila, Ten Voorde and Ohl2018) for an ellipsoidal droplet with the RTI manifesting only in a reduced region of the surface located on the drop poles. In the present work we study a nearly spherically symmetric case where the spikes have no preferred origin, i.e. they escape the droplet isotropically.

Figure 20. Numerical simulations of the instabilities development at the surface of the drop. (a) Selected frames for a case with $R_0=35\ \mathrm {\mu }{\rm m}$ (left) and $R_0=45\ \mathrm {\mu }{\rm m}$ (right). The non-dimensional time $t^*$ is shown on the top left of each frame. (b) Temporal evolution of the RTI spike height for various $R_0$. An ad hoc threshold for the instability onset is indicated by a dashed line at 1 % of $R_d$. (c) Selected frames of a zoomed view of the drop surface for $R_0=45\ \mathrm {\mu }{\rm m}$ (frame window indicated by a red square in b), showing the RTI and the Rayleigh–Plateau instability.

The instability was quantified by defining the spike height as half of the difference between the maximum and minimum radial deviations from the initial drop shape, which was then normalised with the average drop radius, $R_d=1420\ \mathrm {\mu }{\rm m}$. In figure 20(b) it is evident that the instability is formed immediately after bubble creation, as it grows during the bubble's initial expansion. It starts shrinking at $R_0\leq 40\ \mathrm {\mu }{\rm m}$ during its first collapse and grows again upon its rebound.

For $R_0=25.7\ \mathrm {\mu }{\rm m}$, the normalised spike height stays below 0.2 %, meaning that the instability does not further develop in the first bubble oscillation cycles. As the initial radius $R_0$ is increased in steps of $5\ \mathrm {\mu }{\rm m}$, the spike height approximately doubles. Thus, the spike height is exponentially related to the bubble size, i.e. $\text {spike height}/R_d\sim e^{R_0 \times \text {const.}}$. Considering that the instability increases continuously with increasing laser energy, a threshold for the onset of the instability can only be chosen arbitrarily. Here, we choose an ad hoc threshold value as the normalised spike height of 1 % of $R_d$, around which the spike height does not shrink during most of the bubble's first collapse.

Similarly to that observed in the experiments, the simulations of figure 20(b) show how the spikes are ejected earlier in time as the maximum radius reached by the bubble is increased. For $R_0=35\ \mathrm {\mu }{\rm m}$, the spikes nearly cross the threshold (indicated in the plot with a dashed line) in the second oscillation cycle, while for $R_0=40\ \mathrm {\mu }{\rm m}$, the threshold is crossed shortly after the first collapse, and for $R_0\geq 45\ \mathrm {\mu }{\rm m}$ it is already exceeded during the first oscillation cycle. Figure 20(a) compares the instability for a case below ($R_0=35\ \mathrm {\mu }{\rm m}$) and above ($R_0=45\ \mathrm {\mu }{\rm m}$) the established threshold, showing a strong increase in the spike size as well as the ejection of droplets for a larger bubble. This droplet separation from the spikes, highlighted in the bottom row of figure 20(c), was previously discussed in figure 18(b) as an example of the effect of the Rayleigh–Plateau instability. An upper limit of the spike height is reached at $\approx 1\,R_d$ for $R_0=65\ \mathrm {\mu }{\rm m}$, where the outer spikes reach about twice the drop size, while the inner spikes breach the liquid layer that separates the bubble from the outside air. Because of this, the bubble interior is partially filled with atmospheric gas and the cavity ceases to oscillate. At this point, the drop can no longer be defined as such, as shown in the experiments of figure 19(f) where the liquid mass becomes an intricate collection of spikes and a significant portion of it is ejected away as smaller droplets.