The dynamics of a single cavitation bubble exposed to a transient acceleration is studied experimentally. A single cavitation bubble is seeded with a pulsed laser in a free-falling and impacting water-filled test tube. After impact, a pressure wave containing compression and rarefaction phases is generated and interacts with the bubble. The bubble dynamics is studied with high-speed imaging and compared to numerical simulations using the Keller–Miksis model. The timing of bubble seeding with respect to the pressure wave is varied, and a regime of enhanced collapse strength is found.
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1 Introduction
The term cavitation describes the formation and successive dynamics of mostly empty bubbles in liquids (Brennen 2013). One interesting aspect of cavitation is the flow following the initial expansion of the bubble, i.e., the pressure-driven rapid shrinkage of the bubble, which is commonly termed collapse. In general, the larger the nearly empty bubble, the more violent the collapse events are. Here, the term violent subsumes the emission of shock waves (Lauterborn and Vogel 2013), the erosion of surfaces (Kim et al. 2014), and even the generation of visible light; all these are caused by the enormous concentration of kinetic energy of the fluid toward the center of the bubble. In cavitation research, one can distinguish between a controlled generation of single and few cavitation bubbles and the seemingly random formation of many voids. Single bubbles are conveniently formed by the deposition of energy that leads to a rapid phase transition from liquid to vapor. This can be achieved with an electric spark, explosive charge, or through heating or ionization of the liquid with a laser. In contrast, random cavitation events occur in fast flows and during impact events. Both are connected to small and large pressure drops below the vapor pressure that expands already existing gas bubbles into larger cavities. Cavitation along a hydrofoil or shock waves that are reflected off an acoustically soft boundary as rarefaction waves are examples of situations where random cavitation occurs. The tube arrest technique is a common method to test how prone liquids are to cavitation nucleation. There random cavitation bubbles are nucleated as a result of the rapid deceleration of the vessel containing the liquid through an impact.
Rapid deceleration can be obtained through various techniques: One common tube arrest method rapidly stops a liquid-filled tube from moving upwards by hitting a target. Here the liquid continues to move as a solid body while the container is suddenly arrested, forming strong tension in the liquid (Chesterman 1952; Williams and Williams 1996; Dular and Coutier-Delgosha 2013). Similarly, when a liquid in a container is at rest and struck by an impulsive force, e.g., through the impact of a mallet on the container holding the liquid, the inertia resisting the movement of the container causes tension in the liquid (Pan et al. 2017; Xu et al. 2021; Daou et al. 2017). In contrast, when a liquid-filled container is free falling and impacting onto a rigid boundary, the liquid is compressed during impact. Antkowiak et al. (2007) demonstrated that the pressure impulse generated may accelerate the meniscus of the free interface into a liquid flow jetting upwards into the surrounding air. Yukisada et al. (2018) found the surface bubbles on the tube can enhance the impact generated jet. Krishnan et al. (2022) studied the second jet phenomena of the water–air interface in such an impact tube. Following the impact, the pressure wave is then reflected from the liquid–air interface as a rarefaction wave which then nucleates cavitation bubbles in the tube (Kiyama et al. 2015). In general, cavitation occurs at seemingly random locations in the liquid and on the submerged surface of the tube. A more controlled nucleation of single bubbles nucleated in the tube was achieved by Dular and Coutier-Delgosha (2013) by placing a small gas bubble attached to an inserted rod. This modification allowed them to study the thermodynamic boundary layer of the expanded bubble as a result of condensation and evaporation. They took advantage that the tube arrest method results in particularly large bubbles, which have also been utilized to study intense collapses. These larger bubbles then ease the investigations on energy focusing during the bubble collapse (Su et al. 2003). A particular clever design was presented by Leighton et al. (2000) using a stationary U-tube where the bubble is placed at one conically shaped end. The pressure difference between the two sides of the U-Tube accelerates the liquid and compresses the bubble. Chen et al. (2004) improved the setup by using low-pressure argon gas and driving the bubble collapse with a liquid of low vapor pressure. This allowed them to enhance the light emission and record the spectrum from a single collapse event.
Fig. 1
Schematic of the experimental setup. a is the side view of the setup, and b is the top view. Laser, high-speed camera, and sensors are connected and controlled by an oscilloscope and a delay generator (BNC525). The tube drops in the guide frame and blocks the red light and activates a light gate with the photodiode. The BNC525 generates delayed trigger signals to the laser and camera. The bubble is generated by an optic breakdown. Bubble seeding time is obtained by another photodiode. The tube impacts on the platform and generates an impulsive pressure wave. The camera records the bubble dynamics with a frame rate of 160 kfps and an exposure time of 1/900000 s
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In the present work, we explore yet another configuration to study cavitation bubble dynamics in a tube undergoing impulsive acceleration. The bubble is generated through optic cavitation in a free-falling liquid container. The precise timing of bubble generation with respect to the impact of the container opens the analysis of the bubble dynamics exposed to a positive pressure wave upon impact or a rarefaction wave once the wave is reflected. Additionally, we can compare the dynamics to that of a cavitation bubble in free fall with a container at rest.
2 Experimental setup and method
Fig. 2
Schematic of the shock wave generation and propagation in the falling tube for the time series \(T_1\)–\(T_4\). The water-filled tube falls freely until it impacts the platform (\(T_2\)). This generates an impulsive pressure wave that travels upwards and passes the location (\(T_3\)) where a bubble is produced with the laser (\(T_1\)). After reflection at the water–air interface, a tension wave is generated (\(T_4\)). The time \(t = 0\) denotes the time when the impulsive wavefront passes the laser breakdown spot for the first time. The tiny image beside the tube shows the traveling wave at the bubble position. The pulsed laser is focused at a height of \(25\,\text {mm}\). \(\Delta T\) is the difference between bubble seeding time and time of impulsive pressure passing the bubble position, i.e., \(\Delta T = T_1 -T_3\). The sketch shows an example of the bubble being generated before the pressure wave passes the bubble position, i.e., a negative \(\Delta T\) case
Fig. 3
Sample signal of laser breakdown pressure measured with the hydrophone \(2\,\text {mm}\) above the breakdown spot in the static case, i.e., without impulsive pressure generation
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The experimental arrangement for studying the dynamics of a laser-induced bubble in almost zero gravity and during a transient pressure wave is shown in Fig. 1. The bubble is created in a cylindrical test tube with an inner diameter of \(13.5\,\)mm that is filled with degassed water. For observation and better focusing of the laser pulse, optical index matching is achieved by placing the test tube into a cubic housing with glass walls that is filled with water, too. The test tube is placed within a guide frame such that the test tube impacts on the same position on an aluminum platform. A red laser diode beam is used to trigger the fast photodiode (DET10A2, Thorlabs) and an oscilloscope (MSO1104Z, Rigol) in a light gate configuration such that the tube \(3\,\)ms before the impact blocks the laser beam which generates a signal via the photodiode. This signal is used to trigger a delay generator (BNC525, Berkeley Nucleonics Corporation) that starts the further trigger cascade. The cavitation bubble is generated with a pulsed laser (Q2, Quantum Light Instruments, wavelength 532 nm, 6 ns duration, energy 5.0 mJ per pulse with standard deviation 160 \({\upmu}\)J) that is focused on the tube’s axis. The bubble dynamics is captured with a high-speed camera (Mini AX 200, Photron). Another photodiode is used to record the plasma flash. The field of view is illuminated with a LED light (SugarCUBE, Edmund Optics). From the high-speed imaging at each instance of time, a volume equivalent bubble radius R(t) can be estimated. We calculate the cross-sectional area of the gas phase and derive the equivalent radius of a circle of the same area, which then is taken as the bubble radius R(t). To measure the cross-sectional area, the high-speed images are first thresholded, then eroded, and finally dilated by a 2-pixel-square to remove noise. The area of the bubble is obtained by the area of the simply connected region at the bubble position. The test tube was released in all impact experiments from a fixed height of 50 mm.
In essence, and as Fig. 2 shows, after the tube impacts onto the platform, a pressure pulse propagates through the water upwards in the tube. It is reflected at the acoustically soft liquid–air interface as a rarefaction wave and travels back downwards to the bottom of the tube. Both the impulsive pressure and the rarefaction pass through the locus of bubble generation.
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The time difference \(\Delta T\) between the impulsive pressure passing the bubble position and laser-induced bubble generation, \(\Delta T= T_1-T_3\) in Fig. 2, can be varied in a wide range. For precise determination, we measure the time of impact with a piezoceramic sensor (SMD05T04R411, STEMINC) that is glued to the bottom metal plate and close to the point of impact. The sensor records the elastic waves excited at the bottom plate by the impact. The impact-induced pressure transients traveling within the test tube are measured with an axially aligned needle hydrophone (10 MHz bandwidth, Müller-Platte), fixed to the test tube. This hydrophone has a rather high bandwidth of 0.3–11 MHz. Cross-checks with a low-frequency hydrophone (Reson TC4013, bandwidth 1 Hz–170 kHz) showed almost the same pressure signal. As the latter one geometrically blocks the tube, we chose to employ only the needle hydrophone in the further measurements. The relative duration between the impact elastic wave exciting the piezo sensor and the pressure transient passing the locus of the bubble obtained by the hydrophone is used to get the time of the pressure transient passing the bubble locus in the subsequent experiment using the elastic wave.
Fig. 4
Bubble dynamics for the static, and free fall cases. The time with respect to bubble generation is marked on each frame. The common scale is marked on the last frame of the static case. Laser focusing is from the right side of each frame, and the impact pressure propagates from the bottom of the image vertically upwards
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We start with a recording of the impact pressure in absence of the laser breakdown. Therefore, the hydrophone was positioned directly at the location of the laser focus, which is \(25\,\)mm away from the bottom of the tube. The travel time of the pressure transient from the tube’s bottom to the hydrophone position is obtained from the difference between the impact sensor and the first signal on the hydrophone. Similar waves on the aluminum platform are recorded with and without the hydrophone mounted. A typical recording of the pressure pulse at the bubble location is shown in Fig. 6. We see that upon impact the pressure rises to \(8\,\)bar, and after about 150 μs it drops, due to the reflection at the free air interface. From about t=263.8 μs, the pressure has dropped below the static pressure. The rarefaction lasts until \(t\;=\;932.5\,{\upmu}\)s.
Besides the impact, the generation of the laser bubble creates a transient pressure pulse. To distinguish and account for both sources, we separately record the impact pressure in absence of the laser breakdown and the laser breakdown pressure in absence of the impact.
Figure 3 depicts the pressure transient created by the breakdown only. Here the hydrophone was located about \(2\,\)mm above the breakdown spot, and the tube was placed on the metal platform. The signal is caused by the shock wave emitted upon laser breakdown, the bubble collapses, and their reflections (Lauterborn et al. 2018). The bubble lifetime \(T_\textrm{L}\) can be read from the time span between plasma shock wave and collapse shock wave and is about 97 μs. During both events, a shock wave is emitted seen as distinct peaks in the curve. The impact pressure and the reflected pressures from bubble generation are later combined to reconstruct the pressure at the bubble position. Unfortunately, a direct measurement and placement of the hydrophone at the location of bubble generation would physically interfere with the bubble. Yet, to compare the bubble dynamics with the Keller–Miksis model, we need a good estimate of the pressure at the location of the bubble, which is discussed in Appendix A.
In the later experiments, the hydrophone was located at a larger distance of approximately \(10\,\text {mm}\) above the laser breakdown position. This allowed monitoring the pressure transients thus checking for constant conditions during the course of experiments. The motion of the bubble centroid from seeding to collapse was below \(0.3\,\)mm in all cases, i.e., the distance between the bubble and hydrophone can be considered constant here.
3 Results and discussion
Fig. 5
a Radius–time curves of selected experiments. The experimental data are plotted as symbols and the numerical simulations using the Keller–Miksis model as solid lines. The radius in the experiments is the equivalent bubble radius R(t). The six cases plotted are shown in Figs. 4, and 7. The error bars represent the uncertainty from image processing. b Same experiments as in a but now plotted over a longer time span to cover also the later cavitation cloud dynamics
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3.1 Static and free fall
Figure 4 (top row) shows the laser-generated bubble dynamics in the test tube at rest (static case) and serves as a reference for the dynamics recorded in free fall and with impact. For all experiments, the time \(t=0\) is the time when the laser pulse is fired, thus when the bubble is formed. After nucleation, the bubble expands here to a maximum radius of \(R_\text {max}=0.69\,\)mm and collapses at \(t=96.1\,\upmu \textrm{s}\); it reaches its maximum volume at \(t \approx 39.9\,{\upmu}\)s. Thus, the bubble expansion and collapse phase are not symmetric and thus are different from what is typically found for laser-induced bubbles far from boundaries. We explain this behavior with the interaction of the expanding bubble with the shock waves the bubble emitted during bubble generation and which then reflected back from the glass walls. While reflections of shock waves in larger containers also occur, they have typically planar surfaces. Thus, the shock waves do not become refocused. Their effect on the later bubble dynamics can thus in general be ignored. Here, however, the bubble is generated in the center of the test tube with a circular cross section. The emitted shock wave is partially refocused and interacts with the expanding bubble. The approximate round-trip time for this shock wave is about 10 μs; thus, it interacts with the bubble during the expansion phase. This has a significant effect on the expansion dynamics and is discussed in more detail in Appendix A, where the magnitude of the reflected pressure is also estimated.
Coming back to the bubble dynamics, after \(t = 96.1\;{\upmu} \textrm{s}\) the bubble rebounds and reaches a smaller second maximum radius of \(R_\text {max2}=0.43\,\textrm{mm}\) at \(t=139.9\,{\upmu} \textrm{s}\) and collapses a second time at \(t=164.9\,{\upmu} \textrm{s}\). Thereafter, several smaller oscillations can be observed until about \(t=400\,{\upmu} \textrm{s}\) during which the bubble moves slowly upwards due to buoyancy. The measured volume equivalent radius and the simulated radius–time curve are shown in Fig. 5a as black symbols and as a line, respectively.
Figure 4 (bottom row) presents the free fall case during the first bubble oscillation. The bubble dynamics of the static and the free-fall case are rather similar. This is also evident from the corresponding radius time evolution in Fig. 5a where the red dots from the free-fall nearly overlap with the black symbols from the static case. The slight differences between both curves lie within the repeatability of the experiment. This behavior is expected as the influence of gravity on the bubble dynamics for the present geometry is small. We can estimate the gravity-induced jetting using the anisotropy parameter \({\varvec{\zeta }}=-\rho g R_{\text{ max }} \Delta p^{-1}\) as defined by Supponen et al. (2016). Setting \(R_\text {max}=0.7\,\)mm and the ambient pressure \(p_a=1\,\)bar, we obtain \(\zeta = 7\times 10^{-5}\), which is far less than the value of \(\zeta = 10^{-3}\) that is considered as the threshold for weak jetting (Supponen et al. 2016), see also Ref. (Obreschkow et al. 2006).
3.2 Transient pressure driven
Fig. 6
a Impulsive pressure, \(p_\text {i}\), measured at the breakdown region but without laser bubble seeding (average over ten measurements). Compression phases are indicated in red, and tension phases (from reflections) in blue. b Timeline of the frames shown in Fig. 7. Each frame time is marked with \(\diamond\), the green solid diamonds marked with the letter ”L” indicate the respective instances of laser shooting in
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Figure 6a depicts the impulsive pressure measured at the location of bubble generation in absence of a cavitation bubble. The high-pressure region lasts for \(263\,\upmu\)s followed by a rarefaction period till \(t=932\,\upmu\)s and some oscillations. The pressures are up to \(8\,\)bar positive and \(-5.5\,bar\) negative. We will now discuss the bubble dynamics for four phases \(\Delta T\) with respect to the time of bubble generation. The laser-induced bubble generation is indicated with the letter ’L’ in Fig. 6b. Please note the first oscillation in a static liquid lasts for about \(96\,\upmu\)s, thus fitting within the first positive cycle of the impulsive pressure. The four cases are (a) bubble generated prior to the positive pressure, (b) bubble generated during maximum positive pressure, (c) bubble generated just prior to the start of the negative pressure, and (d) bubble generated during the start of the negative pressure. The selected cases are stated on the left in Fig. 6b, and the instants of time of the selected frames are indicated with open diamond symbols.
Fig. 7
Bubble and cluster dynamics for four different phase positions of the bubble seeding with respect to the impulsive pressure wave. For each row, the phase position is indicated via the time delay \(\Delta T\), i.e., the time between bubble seeding and impulsive pressure wave arrival. The pressure pulse and respective timing are sketched in Fig. 6, and the respective time evolution of the radius is shown in Fig. 5. Blue marked \(t(p-)\) shows the time of the pressure transform to negative in each case. A scale bar is indicated in the last frame of each series. Case a\(\Delta T=-50.2\,\upmu \textrm{s}\) shows the bubble seeded before the impulsive compression but influenced in the first oscillation cycle. Case b\(\Delta T=77.8\,\upmu \textrm{s}\) shows a bubble generated in the high-pressure stage. Case c\(\Delta T=238.6\,\upmu \textrm{s}\) shows the bubble generated around the time when the pressure changes from positive to negative. Case d\(\Delta T=267.3\,\upmu \textrm{s}\) shows a bubble generated during the negative pressure
Fig. 8
Color-coded bubble radius evolution R(t) as a function of bubble seeding delay \(\Delta T\). The time \(t=0\) denotes the time of bubble generation. The vertical axis shows the bubble generation delay \(\Delta T\), where negative values indicate bubble generation prior to the transient pressure pulse. The red–white dashed line indicates the start of impulsive pressure. a Experimental data. b Simulated bubble response
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We now discuss these 4 cases shown with the selected frames in Fig. 7. In Fig. 7a, the impulsive pressure reaches the bubble when it is at maximum expansion. This results in a pressure gradient aligned with the direction of gravity. This higher pressure accelerated the collapse, resulting in a shorter lifetime than in the static or free fall cases, see also the R(t)-curves in Fig. 5). Also, the rebounds of the bubble are resulting in a smaller radius due to the persistent positive pressure. At \(t=89.2\,\upmu\)s jetting in a vertical direction opposite to gravity can be seen, see arrow in Fig. 7a. This jetting is likely induced by the spatial gradient of the impulsive pressure. Let us estimate this effect again using the anisotropy parameter for this case as \({\varvec{\zeta }}=\nabla p R_{\text {max}} \Delta p^{-1}\). From the slope in Fig. 6 and the speed of sound in water \(c=1483\,ms^{-1}\), we can estimate \(\nabla p \approx 436\,\)bar/m. This gives an anisotropy value of \({\varvec{\zeta }} \approx 0.30\) during the first bubble collapse where the jet develops. The gradients are considerably smaller than in shock wave driven cavitation bubble collapse (Sankin et al. 2005); thus the jet becomes only visible during the re-expansion of the bubble at \(t=89.2\,{\upmu}\)s. The bubble collapses a second time around the frame \(t=126.7\,\upmu\)s. Yet once the tension at \(t=314\,\upmu\)s sets in, minute fragments expand into large bubble clusters. This cluster lives till about \(t=1500\,\upmu\)s.
Figure 7b presents a bubble seeded during the positive pressure phase as indicated in Fig. 6b. The bubble reaches a maximum radius around \(t=9.5\,\upmu\)s that is much smaller than in the static case. In addition, the lifetime is further reduced due to the high pressure, which in addition also reduces the amplitude and duration of later rebound oscillations. Here the first collapse results in horizontally aligned bubble fragments, which over time spread out. Likely, a horizontal flow is generated during the collapse that transports the bubble fragments. Eventually, when the tension sets in around \(t=240.7\,\upmu\)s in Fig. 6b, the fragments expand into a large cluster of bubbles. Interestingly, the collapse of this cluster around \(t=1441\,\,\,\,\upmu\)s reverses the orientation and the bubble fragments are now aligned vertically.
In Fig. 7c, the bubble is generated at the end of the high-pressure phase which reduces the maximum expansion as compared to the static case. Yet, during shrinkage, the pressure drops sufficiently to prevent a collapse and re-expand the bubble. During this, re-expansion that starts around \(t=55\,\,\,\,\upmu\)s also nuclei along the laser cone, i.e., left and right of the main bubble, is expanded into a cavitation bubble. These may be the result of a non-optimal laser focus with absorption not only in the main laser plasma (Fu et al. 2018), which means it is caused by the dual effect of the negative pressure and laser heating. Some of the thereby generated bubbles remain below the optical resolution. The tension, however, expands these microscopic bubble fragments. Four bubbles expand and presumably coalesce to a maximum size around \(t=900\,\,\,\,{\upmu}\)s and collapse at \(1300\,\,\,\,{\upmu}\)s, see also Fig. 5.
While for cases (a) to (c) in Fig. 7, the bubble is seeded at ambient or during high pressure, for case d) the bubble is generated during the tension phase. The tension lasts for 668 \({\upmu}\)s during which the bubble is expanded to a much larger size as compared to the static case, i.e., to \(R_\textrm{max}=2.22\,\)mm as compared to \(R_\textrm{max}=0.7\,\)mm. At \(t=932.5\,\,\,\,{\upmu}\)s, the pressure becomes positive again, which is around the time when the bubble shrinkage starts. The bubble collapses as a prolate ellipsoid with the minor axis aligned with the direction of gravity. We explain this particular shape with the effect of boundaries hindering the inflow from the sides due to the presence of test tube walls. Likely the pressure gradient induces an upward jetting flow nicely visible at \(t=1544\,\,\,\,{\upmu}\)s in Fig. 7d.
Besides the experimental data, Fig. 5 shows the respective radii from numerical simulations. For the simulations, the same initial conditions were assumed, implying the same initial energy content in all bubbles. This is justified by the small scatter in plasma shock wave peak pressure, which was 4.08 bar with a standard deviation of 2.3% for the six cases in Fig. 5, respectively. In general, the experimental and numerical curves are in fair agreement for the first oscillation. Later differences can be observed, which stem mainly from neglecting the fragmentation of the bubble into a cluster and the approximation of a spherical bubble. The cylindrical confinement of the liquid domain limits the expansion of the bubble in the experiment, which is not considered in this simple model.
3.3 Phase–response
Figure 8a and b displays the response of cavitation bubbles by varying continuously the phase \(\Delta T\) between bubble generation and the impulsive pressure for the experiment and the simulation, respectively. The bubble radius R(t) is color coded. For a phase \(\Delta T=0\), the bubble is created at the moment the impulsive pressure passes the laser focus. Negative \(\Delta T\) refers to bubbles generated prior to the wave passage. The dashed line indicates the time the impulsive pressure reaches the bubble. Consequently, right from this line, after some time, a blueish region is found indicating the respective bubble compression.
For phases between −250 and −200μs, however, the first and second expansion occurs before the pressure pulse has passed. As a result, the bubble dynamics is not a function of \(\Delta T\) and bubbles in this phase range show essentially the same dynamics, e.g., they all collapse around \(100\,\,\,\,{\upmu}\)s. Yet, the rebound of bubbles that are created 100μs before the pressure wave passes is affected. Bubbles created closer to the wave passage, e.g., \(\Delta T>-50\,\,\,\,{\upmu}\)s, show a strong reduction in the first bubble expansion and only minute rebounds. For \(\Delta T>100\,\)μs (lower right with \(t>140\,\,\,\,{\upmu}\)s in Fig. 8a), the bubble expands again due to the passage of the rarefaction wave.
Let us now compare the measured bubble response with the simulations shown in Fig. 8b. For all simulations, the initial conditions for the Keller–Miksis model are \(R(t=0)=78\,\,\,\,{\upmu}\)m and \(\dot{R}(t=0)=540\,\)m/s and the equilibrium radius, i.e., the amount of non-condensable gas, is \(R_\textrm{n}=90\,\,\,\,{\upmu}\)m. For details, we refer to Appendix A. Overall, we find good qualitative and quantitative agreement between the experiment and the spherical bubble model until the first collapse. In particular, the shortening of the bubble lifetime with increasing \(\Delta T\) is well reproduced. Also, the onset of re-expansion of the bubble for large \(\Delta T\) and large t in the lower right region in Fig. 8b is confirmed in the simulations. The good agreement also demonstrates that the experiment of bubble nucleation and impulsive pressure generation is highly repeatable.
There are also some quantitative discrepancies between simulations and experiments concerning the rebounds, in particular, the rebound periods last about 40% longer. The main reasons lay in the lack of spherical symmetry of the rebounded bubble in the experiments (see Fig. 7) while the Keller–Miksis model assumes one spherical bubble and the refocusing of the collapse shock wave after reflection at the tube walls in the cylindrical symmetry (as described in Appendix Fig. 11), which we do not account for. Furthermore, due to the lack of initial bubble symmetry, the experimental collapse may be ’weaker,’ i.e., with less energy dissipation through acoustic emission. Then, more energy would be available for the re-expansion and the bubble expands to a larger radius. Furthermore, the test tube geometry prevents spherically converging flows and likely results in smaller liquid velocities as compared to a larger liquid domain.
3.4 Analysis of bubble clusters dynamics
In the experiments, we have seen that besides the shortening of the bubble collapse, large bubbles and bubble clusters can also be induced when the rarefaction part of the impulsive pressure interacts with the laser seeded bubble.
Here, we analyze in more detail the dynamics of the large bubbles and bubble clusters as a function of the phase \(\Delta T\) covering now a larger range of \(\Delta t\) and time t in Fig. 9 as compared to Fig. 8. Figure 9a shows the experimental radius time data and Fig. 9b the bubble dynamics from the simulations. Please note the different ranges for the color scale in the two figures. Figure 9a in the upper left, i.e., small \(\Delta T\) and small t, is reproducing Fig. 8a. The start of the positive pressure is indicated with a dashed red line labeled \(p+\). Additionally, the start of the rarefaction wave is plotted with a dashed blue line and labeled with \(p-\). The most prominent feature of Fig. 9 is the large red blob that forms just past the \(p-\)-line. Its maximum is reached about \(500\,\,\,\,{\upmu}\)s after the beginning of the tension phase. The blueish region between the \(p+\) and \(p-\)-line indicates that only very small bubbles or no bubbles at all are visible. Still, nuclei are present that are expanded into bubbles once the rarefaction waves pass. This region is termed “bubble cluster” in Fig. 8a because here multiple bubbles expand and merge. The equivalent radius plotted here must be read as an approximate one. It is obtained by integrating in the high-speed frames the pixel areas covered with bubbles. This area is converted into a radius assuming a single projected bubble.
Fig. 9
Comparison of experimental and numerical results as in Fig. 8, but this time for longer time series including the cluster dynamics. a Experimental R(t) curves from 254 measurements. The cases of Fig. 7 are indicated by the arrows on the y-axis. bR(t) curves from the Keller–Miksis model. As the model overestimates the radius, the color scale is capped in this image to allow for better comparability. The red–white dashed lines indicate the zero time when the impulsive pressure reaches the bubble location. The blue-white dashed lines indicate the instance when the pressure transits from positive to negative. The black–white dashed curve indicates the approximate instance of bubble collapse, while the white dashed line separates the bubble cluster and single bubble regions
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The origin of the multiple nuclei forming the bubble cluster is the fragmentation of the laser-induced cavitation bubble during its first and later collapses. As a result, microscopic gas fragments have formed that dissolve due to diffusion. This dissolution time is much longer than the time span between \(p+\) and \(p-\), i.e., thus the fragments act as cavitation nuclei once the rarefaction wave passes.
The phases \(\Delta T\) of the four cases presented in Fig. 7 are marked on the vertical axis. While case (a) and case (b) result in an earlier collapse and a re-expansion of bubble fragments into a cluster, case (c) and case (d) lead to the expansion of a single bubble. Case (c) is particularly interesting as here the short time the positive pressure acts on the bubbles is not sufficient to induce a full collapse; the bubble after a period of shrinkage re-expands under the action of the negative pressure as a single bubble. Interestingly, this single bubble has a clearly longer lifetime than the bubble clusters created for slightly shorter phases \(\Delta T\).
When the bubble is seeded completely in the tension phase (see \(\Delta T\ge 263.8\,\,\,\,{\upmu}\)s), the bubble expands to an even larger single bubble with a longer first oscillation period, which is limited here by the duration of the tension phase. This is confirmed by the decreasing first oscillation duration for increasing phase \(\Delta T\). The transition from bubble cluster to large single bubble happens around case (c), see horizontal dashed line in Fig. 9a.
Experimental and numerical data show a similar qualitative behavior even though the numerical model only considers single, spherical bubbles. The collapse times of the bubble cluster regime are overestimated in the experiments. We attribute this to the effect of the boundary of the test tube, limiting the expansion of the bubbles in the experiments. As a result, the bubble does not expand as much as in the simulations where they also collapse later. Yet, one could argue that the bubbles in simulations expand approximately two to three times more, which would correspond to at least twice the collapse time. In the experiments, the collapse time is only at most 20% shorter. This may be explained that in constrained geometries where the bubble is of similar size as the container, the bubble lifetime is prolonged, too (Quinto-Su et al. 2009).
The shortest lifetime of the bubble is obtained for \(80\,\,\,\,{\upmu}\)s\(\,\le \Delta T \le 160\,\,\,\,{\upmu}\)s where it is reduced to less than \(20\,\,\,\,{\upmu}\)s. This finding is reproduced in the simulations. Yet, for the collapse of the bubble cluster, we see lesser of agreement between experiments and simulations. In the simulations, Fig. 9b, following the first large expansion, all bubbles re-expand in the same way, just shifted by the phase \(\Delta T\). In the experiments, there is only a considerable re-expansion observed for \(\Delta T\) between \(150\,\,\,\,{\upmu}\)s and \(300\,\,\,\,{\upmu}\)s. This clearly shows that the cluster dynamics can only be modeled to some extent with this simple model. Also, the re-expansion is determined to a large extent by the bubble content. The model does not account for mass transport, condensation, and evaporation. This is particularly important for bubbles containing predominantly vapor as is the case here (Akhatov et al. 2001).
3.5 Collapse enhancement
Fig. 10
Peak pressure of the first collapse shock wave (blue) and bubble lifetime (red) as a function of \(\Delta T\)
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The forced collapses may lead to an enhanced compression of the bubble as compared to a collapse solely driven by constant ambient pressure.
To measure the collapse intensity, we study the (first) collapse peak pressures as a function of \(\Delta T\) in Fig. 10. To account for potentially varying seeding energies, each collapse peak pressure is normalized on its respective plasma shock wave peak pressure. To read the peak pressures from the data, we remove the impulsive pressure from the signal first, using a 40 kHz cut-off high pass. The bubble lifetime is shown in red.
The maximum of the blue curve around \(\Delta T\approx -30\,\,\,\,{\upmu}\)s shows that the maximum collapse enhancement is obtained when the bubble can mostly expand at atmospheric pressure and the transient compressive pressure starts acting on the bubble as it starts shrinking (see the waveform in Fig. 6). In comparison, the collapse peak pressure for the static bubble without transient pressure is only \(\approx 0.2\), see times \(\Delta T<-100\,\,\,\,{\upmu}\)s, i.e., where the first collapse occurs already before the impulsive pressure acts on the bubble. Thus, the collapse enhancement can be considered about 1.6/0.2=eight-fold in pressure amplitude and 64-fold in terms of energy. Here, a differently shaped waveform, i.e., with a steeper front, could likely further improve the collapse enhancement, and we expect that in this case, the collapse peak maximum would shift to \(T_\textrm{L}/2\).
When \(\Delta T\) is increased from this optimum enhancement, the bubble lifetime decreases as the bubble is expanding already in the compressive pressure resulting in smaller collapse peak pressures. After \(\Delta T\approx 230\,{\upmu}\)s, \(T_\textrm{L}\) increases sharply to more than \(1\,\)ms, because the collapse stage is prolonged by the rarefaction wave. A greater scatter of collapse shock wave pressure for \(\Delta T \approx 230\,{\upmu}\)s, i.e., for the bubbles that are expanded to large sizes, can be observed. We identify two scenarios that lead to this effect. First, it was shown for single bubbles with controlled asymmetry introduced by a solid boundary that larger and more deformed bubbles tend to split during the final stage of their collapse, i.e., in the last tens to hundreds of nanoseconds. In that case, instead of one collapse, a “burst” of several smaller, subsequent collapses can be observed, temporally sufficiently separated such that their individual shock waves do not interfere, i.e., exhibiting significantly lower peak pressures (Reuter et al. 2022). In addition, along the laser beam cone before and after the focus, cavitation nuclei are generated which are also expanded by the impulsive pressure and may merge with the main bubble. Both processes include statistic components and can vary from run to run.
Thus, a reversed wave trace with leading rarefaction and trailing compression phase could further improve the collapse enhancement. This could be realized for example via an additional reflector exhibiting a sound soft boundary to achieve the necessary phase shift of the transient pressure wave.
4 Conclusion and outlook
A simple setup to study a single laser-induced bubble for zero gravity was presented using a falling tube. The same setup was used to study the interaction of the single bubble with an impulsive pressure wave produced once the free-falling tube hit the platform followed by a tension phase upon reflection at the water surface. The experimental bubble dynamics observed by high-speed imaging were compared to a simplified spherical bubble model where the pressure is taken from the experiment. A parametric study of the bubble dynamics in dependence of the phase between bubble seeding and impulsive pressure wave was performed. As a result, the maximum bubble radius varied between 100 and \(2500\,{\upmu}\)m. When hitting the bubble location after the first collapse, the rarefaction wave expands not a single bubble but a cluster of bubbles. Also, small bubbles that are produced due to the non-ideal laser focus may also be expanded by the tension wave. If the tension wave passes the bubble during its collapse, it may be halted and the bubble re-expands as a single bubble. Moreover, the compression strengthens the bubble collapse, generating higher collapse shock wave pressures. Here, certain pressure profiles are promising with the aim to achieve maximum collapse compression.
Overall, the single bubble dynamics could be well described by a Keller–Miksis model and to some extent also the cluster dynamics, which implies that the cluster behaves similarly to a single bubble. Yet, for this good agreement, it was necessary to consider that the shock wave generated during the dielectric breakdown is reflected on the tube walls and focused back on the bubble. While generally in experiments with laser-induced bubbles, this effect can be ignored, here due to the cylindrical geometry and the rather small radius of the test tube, it is significant.
This simple experimental technique of dropping the container where the laser-induced bubble is generated allows one to expose bubbles to rather large negative pressure amplitudes at low frequencies. This technique may be advantageous for enhancing energy focusing of collapsing laser-induced cavitation bubbles to improve earlier attempts (Ohl 2000; Duplat and Villermaux 2015). This could be checked by registering the light emission from the bubble collapse, particularly in the single bubble regime, i.e., here for \(\Delta T > 250\,{\upmu}\)s.
Declarations
Conflict of interest
We declare that the authors have no conflict of interest as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
Ethical approval
Not applicable.
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We make use of the Keller–Miksis model to describe the bubble dynamics (Keller and Miksis 1980; Prosperetti and Lezzi 1986). It is applicable for radial oscillations of single, spherical bubbles in an infinite large liquid and exposed to a varying pressure p(t). It accounts for an ideal gas content of the bubble, surface tension, viscosity of the liquid, and the compressibility of the liquid, which becomes important for strong collapses (Lauterborn and Vogel 2013):
\(R\) denotes the bubble radius, \(\dot{R}\) the bubble wall velocity, \(\ddot{R}\) is the acceleration of the bubble wall, while \(\rho\) is the density of the liquid, and \(C\) is the local speed of sound, which we treat as constant. \(p_\text {stat}\) is the atmospheric pressure, \(\mu\) is the dynamic viscosity, \(\sigma\) is the surface tension coefficient of water, \(b\) is the van der Waals parameter, \(\kappa\) is the adiabatic exponent, and \(p(t)\) is the impulsive pressure. We use the following values for these parameters: \(\rho\) = 998 \(\textrm{kg} \textrm{m}^{-3}\),\(C\) = 1483 \(\textrm{m} \textrm{s}^{-1}\), \(p_\text {stat}\) = 101325 Pa, \({\upmu}\) = 0.001 Pa s, \(\sigma\) = 0.0725 \(\textrm{Nm}^{-1}\),\(b\) = 0.0016, \(\kappa\) = 1.4. For the calculations, we neglect the confinement of the liquid domain. To reduce computational cost and increase stability, p(t) is digitally filtered using a low pass filter with cutoff frequency \(25\,\)MHz.
The equilibrium radius of \(R_\text {n}=90\,{\upmu}\)m for the laser energy that was kept constant was obtained from the experiment. It is identified with the radius of the remaining bubble after a number of oscillations, i.e., when it starts to rise slowly due to buoyancy. We do not distinguish between vapor and gas here. Next, we chose a proper pair of initial conditions, i.e., the radius \(R(t=0)\) and the wall velocity at \(\dot{R}(t=0)\) to fit the experimental maximum bubble radius in the static case. For all simulations, these initial values are set to \(R(t=0)=54.5\,{\upmu}\)m with a bubble wall velocity of \(\dot{R}(t=0)=1280\,\)m/s. These values fit the static case well and are in agreement with measurements, see (Kröninger et al. 2010; Lauterborn and Vogel 2013; Bao et al. 2021).
Here, the time-varying pressure p(t) results from a superposition of the impulsive pressure \(p_\text {i}\) due to the impact and the reflections of the plasma shock wave produced during bubble seeding \(p_\text {s}\), i.e., \(p(t)=p_\text {i}(t)+p_\text {s}(t)\). The impulsive pressure can be measured easily by placing the hydrophone at the bubble position but without bubble generation as described in Figure 3. The latter also has a significant effect on the bubble here due to the cylindrical tube geometry that focuses the shock wave back onto the bubble. This contribution is experimentally more difficult to obtain because the hydrophone cannot be placed at the location of the bubble (and the plasma). To reconstruct \(p_\textrm{s}(t)\), i.e., the pressure from the reflected shock wave at the bubble position, the tube is maintained static and the plasma is generated in close proximity, i.e., \(2\,\)mm from the hydrophone. An example of this recording is shown in Fig. 3a and discussed in more detail in Fig. 11b. It starts with the plasma shock wave (the largest, leftmost peak) and is followed by reflections of this wave on the tube walls. Between \(t=26\,{\upmu}\)s and \(t=48\)s, further diffuse reflections appear, and at \(t=98.5\,{\upmu}\)s the shock wave from the bubble collapse is detected. From the time interval between the plasma shock wave and the collapse shock wave, a bubble lifetime of \(T_\text {L}=97\,{\upmu}\)s is measured. An illustration of the shock wave propagation and reflections in the tube demonstrates the need of accounting for the reflected pressure acting back on the bubble. For this, we simulate a simplified acoustic wave propagation with a finite element method (COMSOL Multiphysics). The linear wave equation is solved in cylindrical symmetry where energy losses and dispersion are neglected. The geometry is shown in Fig. 11b. The glass tube walls are modeled as sound-hard boundaries. The lower horizontal boundary has a symmetry condition to reduce the computational cost. The upper boundary is impedance matched as the water column exceeds the simulation geometry in the vertical direction. Plasma and bubble are shown in the lower left corner, and the shock wave is modeled via a pressure condition using a short positive Gaussian pulse (FWHM=\(0.58\,{\upmu}\)s) followed by a negative pulse emitted from the bubble wall. To account for a sound soft boundary bubble wall, the boundary condition is switched after the emission from a radiating surface to a sound soft boundary. After reflection of the shock wave from the cylindrical tube wall, it is focused back onto the axis of symmetry resulting in rather large pressures on the bubble at \(t_4\). In the experiments, the hydrophone could not be aligned perfectly above the bubble and along the axis of symmetry resulting in an underestimation of the measured pressure and the pressure the bubble experiences. To account for this discrepancy, we multiply the recorded pressure with a geometric focusing factor k to match the measured bubble radius with the observed bubble radius. The best fit up to the first bubble collapse in a least square sense is achieved for \(k=8.3\). This value is used in our simulations for re-scaling the measured pressure recorded at a \(2\,\)mm distance, i.e., \(p_\text {s}(t)=k\, p^{\mathrm {2\,mm}}_{\textrm{hyd}}(t)\)..
Fig. 11
Effect of refocusing of the pressure wave emitted from the laser-induced breakdown back on the bubble dynamics. a Measurement (top) and reconstructed reflected pressure (bottom) factor k for re-scaling. b Simulation of the shock wave reflections shown for subsequent times \(t_{1} \ldots t_{4}\). A strong focusing due to the cylindrical geometry is present. c Measurement of the radial bubble dynamics R(t) (squares) and fits according to the Keller-Miksis model (lines) for three cases: ignoring the reflections, using the measured pressure at the remote location, and using the reconstructed pressure
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The re-scale measured pressure from the first till the fourth reflection, i.e., from \(8.5\,{\upmu}\)s till \(38\,{\upmu}\)s, is accounted for, see the orange and yellow boxes in Fig. 11a. Outside the yellow box, we ignore the reflections as the signal approaches the noise limit of the hydrophone; therefore, \(k=1\) for \(t>38\,{\upmu}\)s.
To illustrate the importance of accounting for the reflected and re-focused pressure on the bubble dynamics, we fit the radial dynamics using different initial conditions to achieve the same bubble lifetime for three cases \(p(t)=k\,p^{\mathrm {2\,mm}}_{\textrm{hyd}}(t)\), \(p(t)=p^{\mathrm {2\,mm}}_{\textrm{hyd}}(t)\), and \(p(t)=0\) in Fig. 11c. The following initial conditions were used: \(R(t=0)=54.5\,{\upmu}\)m and \(\dot{R}(t=0)=1280\,\)m/s, \(R(t=0)=54.5\,{\upmu}\)m, and \(\dot{R}(t=0)=540\,\)m/s, and \(R(t=0)=54.5\,{\upmu}\)m and \(\dot{R}(t=0)=260\,\)m/s, respectively. While all of them result in a similar lifetime of around 94\(\,{\upmu}\)s, only the curve using the factor \(k=8.3\) is able to reproduce the asymmetry of oscillation seen in the experiment.
Fig. 12
Bubble cluster oscillation calculated using the Keller-Miksis model for the measured impulsive pressure for different time delays between bubble seeding and impulsive pressure generation (color coded)
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Figure 12 shows the resulting Keller-Miksis R(t)-curves for different phase positions between bubble seeding and impulsive pressure given by the time delay \(\Delta T\). Without the impulsive pressure, the bubble expands only to about \(700\,{\upmu}\)m around \(100\,{\upmu}\)s (see the curves at \(\Delta T<-100\,{\upmu}\)s). The bubble expands to maximum radii when its seeding temporally coincides with the beginning of the rarefaction; then, it can reach more than \(8\,\)mm at maximum expansion. The numerical curves overestimate the measured bubble sizes by about 4 times, which we attribute to the effect of tube walls and the resulting geometric confinement of the bubble.
Supplementary Information
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