detailed experiments on weakly deformed cavitation...

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Experiments in Fluids (2019) 60:33 https://doi.org/10.1007/s00348-019-2679-4

RESEARCH ARTICLE

Detailed experiments on weakly deformed cavitation bubbles

Outi Supponen1 · Danail Obreschkow3 · Philippe Kobel2 · Nicolas Dorsaz2 · Mohamed Farhat2

Received: 18 September 2018 / Revised: 3 January 2019 / Accepted: 13 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019

AbstractWe present high-precision experiments conducted with the aim to better characterise weak deformations of single cavita-tion bubbles. Using two needle hydrophones and a high-speed photodetector, we record the timings of shock waves and luminescence emitted at the collapse of laser-induced bubbles and are able to thereby obtain a precise measurement of their displacement during their lifetime. The bubbles are primarily deformed by variable gravity reached aboard parabolic flights, but we additionally take into account the effect of the nearest surfaces. A time shift of approximately 60 ns is found between the bubble lifetimes measured by the hydrophones and the photodetector for spherically collapsing bubbles, which we believe to be a result of different initial shock wave propagation speeds at the bubble’s generation and at collapse. The normalised bubble displacement is found to follow a �2∕3 scaling law for 𝜁 > 0.001 , where � is the dimensionless anisotropy parameter quantifying the bubble deformation (analogous to Kelvin impulse). Additionally, we quantify the asymmetry of the shock wave generated at the collapse of bubbles with various levels of deformations by comparing the hydrophone signals at two different locations, and find significant variations between the shock peak pressures and energies at 𝜁 > 0.001 . These results consolidate the suggestion to consider � ∼ 0.001 as a practical limit between spherical and deformed bubbles. This limit is probably sensitive to the bubble’s initial sphericity, which is exceptionally high in our mirror-based aberration-free setup.

Graphical abstract

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Time [ms]

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Laser and initial plasma

1 Introduction

Collapsing cavitation bubbles present peculiar and interest-ing fluid dynamics that has astonished researchers for dec-ades. In favourable conditions, they are able to produce shock waves, form high-speed micro-jets, and heat their contents to

* Outi Supponen [email protected]

Extended author information available on the last page of the article

http://orcid.org/0000-0001-6738-0675http://crossmark.crossref.org/dialog/?doi=10.1007/s00348-019-2679-4&domain=pdf

Experiments in Fluids (2019) 60:33

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temperatures of thousands of degrees, emitting light known as luminescence. Traditionally, these bubbles are considered harmful, as they are well-known to cause erosion in hydraulic machinery such as hydraulic turbines, pumps, or ship pro-pellers. However, more recent discoveries have shown that, when harnessed correctly, their power can be beneficial in numerous modern applications. These include biomedical applications, such as lithotripsy using shock waves or high-intensity focused ultrasound (Coleman et al. 1987; Ikeda et al. 2006), or sonoporation assisting drug delivery to targeted cells (Marmottant and Hilgenfeldt 2004; Ohl et al. 2006; Brennen 2015). The promising potential of bubbles in these numer-ous applications has driven efforts in predictive modelling of these violent collapse effects. However, our understanding of the fundamental physics behind cavitation bubbles is far from complete, even more so for bubbles deviating from a spherical shape.

Identifying the fine differences between cavitation bub-bles that collapse in a spherical way and those with even the weakest deformations is a difficult but important challenge that is yet to be addressed. Ohl et al. (1998) commented in their study on luminescence from nonspherically collaps-ing bubbles: “A look into the interior of a collapsing bubble to clarify whether a jet has formed or the sphericity has remained is still beyond the experimental state of the art”, and today, two decades later, this has still not been quite resolved. Many studies assume spherical symmetry for bub-bles located beyond a distance of four or five times their maximum radius from the nearest surface. However, the limitations imposed by spatial and temporal resolution of the measurement instrumentation make it challenging to verify this assumption. Deformed cavitation bubbles have differ-ent damage characteristics than their spherically collapsing counterparts, with the emergence of high-speed jets, multi-ple shock waves, larger rebound bubbles, and displacement (Blake and Gibson 1987; Vogel and Lauterborn 1988; Ohl et al. 1999; Supponen et al. 2018). Most theoretical models for cavitation bubble dynamics assume spherical symmetry, and hence one must understand when exactly such a widely used assumption is justifiable. Therefore, the objective of this study is to quantify the differences between highly spherical bubble collapses and weakly deformed bubbles by experimentally investigating the transition in between them.

It is challenging to experimentally determine the exact limit between spherically collapsing and jetting bubbles, as with such weak deformations, the micro-jet generally forms in the very last stage of the collapse where the bubble interface reaches extremely high speeds and where the spatial scales become too small to be imaged. Here, we present a technique to make high-precision measurements of a cavitation bubble’s displacement even with very weak deformations. The bubble’s displacement is one of the most sensitive measures to quantify a bubble’s deformation (Supponen et al. 2016), and this displacement can

in principle be measured using two hydrophones. We also com-pare the shock wave timings with luminescence emission and aim at determining when the directionality of shock wave emis-sion from non-spherical bubbles becomes important.

2 Methods

2.1 Experimental setup

An illustrative schematic of the experimental setup is shown in Fig. 1. A cavitation bubble is generated with a fre-quency-doubled Q-switched Nd:YAG-laser pulse (Quantel CFR 400, 8 ns, 532 nm, maximum 230 mJ) focused by an immersed aluminium-coated parabolic mirror in the middle of a cubic test chamber ( 18 cm × 18 cm × 18 cm ), which is filled with demineralised and partially degassed water. The centre of the bubble-generating mirror is located on the same horizontal plane as the initial bubble. The vapour bubble forms from the recombination of a plasma produced as a result of the optical breakdown by the laser pulse (i.e., avalanche ionisation by heating of impurities, see study by Vogel et al. 1996b). The shock waves produced by the bubble are recorded by two preamplified piezoelectric nee-dle hydrophones (Precision Acoustics, sensor size 75 μm ), henceforth denoted as hydrophone 1 and hydrophone 2.

Beam expander 10xLaser beam

High-speed camera Light Silver

mirror

White LED light

Vacuum pump

Parabolic mirror

Hydrophone 1

Cavity51

Luminescencecennn

Hydrophone 2

Photodetectorto oscilloscope

45°

30°

Vacuum pumpTransparent

vacuum vessel

Hydrophone 1 Cavity

OscilloscopeAmplifier

AmplifierLaser beam

Hydrophone 2

d2

d1 Parabolicmirror

Fig. 1 Schematic of the experimental setup with the top view (top) and the side view (bottom). d1 and d2 are 34.3 mm and 35.7 mm, respectively


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Hydrophone 1 and 2 are, respectively, placed at a distance of 34.3 ± 0.05mm and 35.7 ± 0.05mm away from bubble centre, and respectively, inclined at 30◦ below and above the horizontal plane. We assume that the hydrophones are sufficiently thin (needle diameter is 0.3mm ) and far away for their presence to have a negligible effect on the bubble dynamics. The hydrophones are connected to an amplifier and an oscilloscope, and their signals are sampled at 100 MHz.

A high-speed photodetector (Thorlabs DET10A/M Si detector, 1 ns rise time) detects the initial plasma upon bub-ble generation and the light emitted during luminescence for a sufficiently spherical bubble collapse. The photodetector’s sensor is placed at the focal point of an aluminium-coated parabolic mirror located outside the test chamber, which reflects the light collected by a second immersed parabolic mirror inside the test chamber. This way assures sufficient signal from a single luminescence event (Supponen et al. 2017a). It should be noted that the luminescence signal amplitude is not corrected for the displacement of the bub-ble away from the focal point of the mirror. However, the only information needed for the purpose of this study is the timing of a luminescence event, and not its amplitude. The photodetector is connected to the oscilloscope and its signal sampled at 1 GHz.

The last collapse stage of the bubble is visualised using shadowgraphy at 10 million frames/s with an ultra-high-speed camera (Shimadzu HPV-X2, exposure time 50 ns), which is post-triggered by the oscilloscope as it detects the collapse shock signal. The illumination is provided by a backlight LED.

The bubbles are primarily deformed by the uniform hydrostatic pressure gradient induced by gravity. This defor-mation is controlled by varying the gravity level during the 67th European Space Agency parabolic flights (gravita-tional acceleration g = 0–2.1ms−2 ), in addition to varying the static pressure inside the test chamber ( p0 = 10–86 kPa ) using a vacuum pump, and the bubble size (maximum bub-ble radius R0 = 1.2–6.2mm ) by adjusting the laser energy. Additional details on the experimental set-up and the para-bolic flights may be found in the study by Obreschkow et al. (2013).

2.2 Vector parameter for bubble deformation

When induced by a gravitational field, the bubble deforma-tion is quantified by the dimensionless anisotropy param-eter � = −�gR0∕�p (where � is the liquid density, g is the gravitational acceleration with a downward direction, R0 is the maximum bubble radius, and �p = p0 − pv is the driving pressure, i.e., the difference between the static pressure out-side the bubble p0 and the vapour pressure inside the bubble pv ). Here, � is a dimensionless counterpart of the translational

momentum accumulated up to the last collapse stage, depicted by the Kelvin impulse (Blake 1988; Blake et al. 2015). This vector quantity provides the direction and the “strength” of the emerging micro-jets (e.g., jet volume, jet speed, bubble migration) (Obreschkow et al. 2011; Supponen et al. 2016), and allows for the comparison between deformations induced by different sources, such as gravity or nearby surfaces. The bubble migration also occurs in the direction of � . For near boundaries, � is defined as a function of the stand-off distance � = h∕R0 (where h is the distance between the bubble cen-tre and the nearest surface) as � = −0.195�−2� , where � is the normal unit vector on the boundary pointing towards the bubble. This relation relies on finding an equivalent uniform pressure gradient that produces the same Kelvin impulse as the boundary (see Supponen et al. 2016, for derivation). Here we account for the combined effects of gravity and the nearest boundaries, i.e., the two immersed parabolic mirrors, when determining � . Therefore, � is the vector sum of the anisotropy parameters defined for these different sources, � = �� + �� . This way of combining different sources of deformation has been justified numerically in the case of simultaneous effects of ultrasound and a neighbouring boundary, where the direc-tion of the resulting Kelvin impulse matches the one of the micro-jet and migration of the bubble (Blake et al. 2015). For �� , we account for the two parabolic mirrors that are located at the same distance away but at different angles from the bub-ble centre. By treating these mirrors as flat boundaries, �� = �� + �� . The free surface is located sufficiently far from the bubble (90 mm) to have a negligible contribution to � . The magnitude and direction of � are then computed as follows:

where x and y are perpendicular axes on the horizontal plane at the height of the initial bubble and z is the vertical axis,

as shown in Fig. 2. Here, |� | =√

||�x||2 + |||�y|||2

+ ||�z||2 is the magnitude of � , and its direction is denoted with � and � ,

(1)

��z�� = ��g + �m2,z�� =

�gR0

�p+

0.195�h∕R0

�2 sin �

��x�� = ��m1 + �m2,x�� = 0.195�h∕R0

�2 (1 + cos � cos �)

��y�� =

��m2,y�� =

0.195�h∕R0

�2 cos � sin �

� = arctan

⎛⎜⎜⎝

��y��

��x��⎞⎟⎟⎠

� = arctan

⎛⎜⎜⎜⎜⎝

��x��2 + ��y

��2

��z��

⎞⎟⎟⎟⎟⎠,


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which are the azimuthal and polar angles of � , respectively. Here, � is the angle between the second parabolic mirror and the xy-plane, � is the angle between the second parabolic mirror and the xz-plane, and h is the distance from the initial bubble centre to both the first and the second parabolic mir-ror ( h = 50.8mm ). As � = 35◦ is constant throughout the parameter space, implying that the relative effect of the two mirrors stays the same, the inclination of � only varies with respect to the horizontal plane. Figure 3 shows the parameter space indicating the regimes where gravity and the mirrors dominate the bubble deformation at normal gravity, deter-mined by comparing their respective components in Eq. (1) (note that in micro-gravity, the mirrors always dominate the deformation).

2.3 Example signal

Typical signals recorded by the two hydrophones and the photodetector during a bubble’s lifetime are displayed in Fig. 4. The first voltage peak, recorded by the photode-tector, corresponds to the laser pulse that generates the initial bubble and saturates the detector. It also includes an initial plasma that can last a few hundred nanoseconds but which, however, cannot be distinguished from the signal caused by the laser. Approximately 24 μs later, hydrophone 1 detects the shock wave that has been formed during the

explosive vapour bubble formation. In the high-speed images, this shock wave consistently appears to have per-fect spherical symmetry. However, it may include some micro-directionality due to the laser focusing, as reported by Tagawa et al. (2016), which could explain the smaller second peak measured by hydrophone 1. The delay cor-responds to the time it takes for the shock wave to propa-gate at speed of sound ( 1472ms−1 at 17◦C ) the distance of 34.3 mm to reach the hydrophone. This is closely fol-lowed by the voltage peak detected by hydrophone 2, cor-responding to the same shock wave having travelled a 1.5 mm longer distance. The subsequent peak in the signal recorded by hydrophone 1 corresponds to the shock wave reflecting back from the bottom of the container. No effect of the shock wave reflections on the bubble dynamics was observed: bubbles far from boundaries and in micro-grav-ity generally collapse almost perfectly spherically without being deformed by the reflected shock waves. The collapse of the cavitation bubble results in a luminescence peak in the photodetector signal, here at t = 1.2ms , followed by the collapse shock wave peaks from the two hydro-phones. The main differences in the waveforms of the col-lapse shock waves compared to the generation shocks are the gradual pressure rise preceding the peak, as well as the tensile part of the shock wave tail. One may observe from these signals that there is a difference between the waveforms of the collapse shock waves recorded by the two hydrophones, implying that this shock wave has some directionality instead of perfect spherical symmetry. This will be investigated in detail in Sect. 3.3.

z

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Fig. 2 Three-dimensional schematic illustrating the displacement �s measured by the two hydrophones. Blue circles illustrate spherical shock waves emitted at the bubble generation, and red circles show shock waves emitted at the bubble collapse, as they reach the two hydrophones at different times. The schematic is not in scale. The angles are � = 38◦ , � = 81◦ , � = 30◦ , and � = 45◦

Gravity-dominated

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Fig. 3 Parameter space of maximum bubble radius R0 , driving pres-sure �p and resulting anisotropy parameter � with regimes where the bubble deformation is dominated either by gravity or nearest neigh-bouring boundary (mirrors), at normal gravity. Conditions with equal influences of gravity and the mirrors are found along the line �p = 85.93∕R0 kPa mm


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2.4 Measurement of bubble displacement

We define the bubble displacement �s as the distance trav-elled by the bubble centroid between the instants of bubble generation and final collapse point (see Fig. 2). For deformed bubbles that split into multiple parts, the centroid position at the collapse is defined as the position of the jet tip at its impact onto the opposite bubble wall. There are several ways that the displacement of the bubble during its collapse can be measured using our experimental setup.

First, the displacement can be directly measured from the images recorded by the high-speed camera. However, for weakly deformed bubble collapses, such measurements may not have sufficient resolution (here, 64μm/pixel) to cap-ture the short displacement. More importantly, if the bubble migrates in a direction away from the focal plane, the image will not be able to accurately quantify such displacement.

A second and more precise method to measure this dis-placement is to use two hydrophones placed at different loca-tions, here above and below the bubble at an inclination. The hydrophones record the timings of the shock waves produced at the generation and at the collapse of the bubble. The shock wave’s arrival time recorded by each hydrophone depends on the distance travelled by the shock wave from the emission centre to the hydrophone sensor (see Fig. 2). At the genera-tion, the distance to both hydrophones is fixed, as the bub-ble is always generated at the same location. The precision of the bubble’s initial location ( ±15 μm ) is verified by the time delay between the bubble generation, as measured by the pho-todetector, and the arrival time of the first shock wave to both hydrophone sensors being close to constant (their difference varies by less than 25 ns) throughout the measurements. At

the collapse, the shock emission centre is different from that of the generation if the bubble collapses in a deformed way, as it migrates away from its initial location in the direction of � . Figure 2 illustrates the hydrophone configuration with the various shock wave measurement timings by each hydrophone.

The bubble’s displacement �s is calculated comparing the two hydrophone signals using the following relation:

where c is the speed of sound in water, which equals c ≈ 1472ms−1 at the water temperature used in the experi-ments (16–18 ◦C ), and

The subscripts 1 and 2, respectively, denote hydrophones 1 and 2, d1 and d2 are the distances between the initial bub-ble centre and the hydrophone sensors, and dc,1 and dc,2 are the distances between the bubble collapse location and the hydrophone sensors. Here, � = 45◦ is the angle between hydrophone 2 and xz-plane, � = 30◦ is the angle

(2)(dc,1 − dc,2) = d1 − d2 + c�T ,

(3)

dc,1 =[(�s cos� sin � + d1 cos �

)2+ (�s sin� sin �)2

+(�s sin � + d1 sin �

)2]1∕2,

dc,2 =[(�s cos� sin � − d2 cos � cos �

)2

+(�s sin� sin � + d2 cos � sin �

)2

+(�s cos � − d2 sin �

)2]1∕2, and

�T =(tc,1 − tg,1

)−(tc,2 − tg,2

).

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Fig. 4 Typical signals from the photodetector (black) and the hydro-phone 1 (blue) and hydrophone 2 (red), and as recorded by the oscil-loscope. The insets show zoomed up signal peaks corresponding

to shocks at the generation and the collapse of the cavitation bub-ble. The bubble parameters here are: R0 = 4.1 mm, �p = 39 kPa, � = 0.014


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of hydrophones 1 and 2 from the xy-plane, �t is the dif-ference between the bubble lifetimes recorded by the two hydrophones, and tc and tg , respectively, denote the recorded time of the collapse and generation shock wave pressure peaks (the first instant at which half of the peak ampli-tude is reached). See Fig. 2 for an illustration of the 3D configuration.

The displacement may also be measured by a third method, which compares the shock wave timings recorded by the hydrophone with the timings of the laser (with the initial plasma) and luminescence, recorded by the photode-tector from sufficiently spherically collapsing bubbles. Such displacement is computed as:

where the subscript H (1 or 2) denotes the hydro-phone, dc,H is the distance travelled by the collapse shock wave to the hydrophone as given in Eq. (3), �T =

(tc,H − tg,H

)−(tc,P − tg,P

) , where tg,P is defined as

the instant of laser detection by the photodetector, and tc,P denotes the instance of the maximum signal amplitude cor-responding to luminescence, and �T0 ≈ 60 ns comprises the unknown time difference between the start of the laser pulse and the formation of the shock wave at bubble generation, as well as that between luminescence maximum amplitude and formation of the shock wave at bubble collapse (discussed in Sect. 3.1).

3 Results

3.1 Event timings

The direct comparison of the bubble’s lifetime ( �t = tcollapse − tgeneration ) as recorded by the two hydro-phones and the photodetector are shown in Fig. 5a. In the ms scale, all these measured times appear identical. The ns scale differences, however, are what can be used to calculate the bubble displacements. Figure 5b shows these differences as a function of the anisotropy parameter � . While the bub-ble lifetime recorded by the photodetector should represent the exact light emission timings, the hydrophone timings depend on the propagation speed and distance travelled by the shock waves from the bubble to the hydrophone sensor. This shock propagation distance, in turn, changes with the bubble displacement, which increases with the bubble’s level of deformation. As expected, the differences in the bubble lifetime measured by the three different instruments increase with increasing bubble deformation. The scatter in Fig. 5b is mainly due to the large variation in bubble sizes and driving pressures included in the data, the 14 ns measurement uncer-tainty (estimated through error propagation in the calcula-tion of �thydro − �tphoto ), and the timing of the luminescence

(4)dc,H = dH + c(�T + �T0)

peak amplitude shifting by approximately 10 ns with respect to the time of the shock wave detection around 24 μs later (Supponen et al. 2017a).

However, there appears to be an intriguing ∼ 60 ns time shift between the photodetector measurement and hydro-phone measurement of the bubble’s lifetime in Fig. 5b even for spherical bubble collapses ( � → 0 ). If the shock forma-tion at the bubble generation temporally coincided with the initial laser pulse, and the shock formation at collapse coincided with the luminescence, the measured bubble life-times should be equal. This would be the case also if the time delay between the initial or luminescence plasma and

0 1 2 3tphoto [ms]

0

0.5

1

1.5

2

2.5

3

t hyd

ro [m

s]

Hydro 1Hydro 2

100 ns(a)

0 1 2 310-3

-200

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50

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ro-

t pho

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(b)

Fig. 5 a Hydrophone versus photodetector measurement of the bub-ble’s lifetime. b Difference between bubble lifetime recorded by the hydrophones and the photodetector versus the anisotropy parameter � . The varied parameters here are R0 = 1.2–6.2 mm , �p = 7.2–83.2 kPa , and g = 0–2.1 ms−2


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the corresponding shock formation was constant. However, the difference of 60 ns implies that the bubble’s lifetime is intrinsically different when measured with the photodetector or the hydrophone. The exact instant of bubble generation with respect to the laser pulse cannot be captured by the photodetector, although this is expected to happen within a few nanoseconds only, as observed by Vogel et al. (1996a). It has been suggested that the shock formation results from a focusing of a density wave inside the bubble, which, after reaching the its centre, reflects and propagates outwards in the liquid (Magaletti et al. 2015). Schanz et al. (2012) reported through molecular dynamics simulations that the light emission occurs within 100 ps before the instant at which the bubble reaches its minimum size, and that the reflected density wave reaches the bubble wall at this very instant. However, the ensuing shock wave, both at the bub-ble generation and at the collapse, is expected to initially propagate with supersonic speeds (Schoeffmann et al. 1988). We have previously estimated that the average shock speed from high-speed videos during the exposure time of the first frame after the collapse of a spherical bubble to be approxi-mately 3000ms−1 over a distance of 300 μm , making the shock wave reach the hydrophone 100 ns earlier than if it was propagating the entire distance at sound speed (Sup-ponen et al. 2017b).

While the shocks produced both at the bubble generation and at collapse are expected to reach the hydrophone ear-lier, the exact intial speed of the collapse shock wave could be faster compared to the former shock wave, since there is an important pressure rise in the liquid near the bubble during the last stages of its collapse, which is not present during the bubble generation (Lauterborn and Vogel 2013; Supponen et al. 2017b). This high pressure causes light to deflect near the bubble wall at its last collapse phase, as can be seen as the shaded region in the second frame of the high-speed images of the bubbles collapsing in micro- and normal gravity in Fig. 6. This high pressure can also be seen in Fig. 4 as a gradual pressure rise preceding the shock front in the hydrophone signals of the collapse shock wave (this being more pronounced for hydrophone 2). Speed of sound in water has been reported to be higher at high pressures; for example, Hidalgo-Baltasar et al. (2012) have measured a speed of sound in water as high as 2400ms−1 at 700 MPa at room temperature. Rayleigh’s incompressible theory predicts pressures of hundreds of MPa near the bubble wall prior to the final collapse point for spherically collapsing bub-bles (Rayleigh 1917). Figure 7 shows the distribution of the speed of sound in water as a spherical bubble collapsing at 100 kPa reaches the size R∕R0 = 0.03 , computed by combin-ing Rayleigh’s theoretical model and the model provided by the International Association for the Properties of Water and

Fig. 6 Shadowgraph images of the final collapse stage and the emit-ted shock wave in micro-gravity at � = 0.0013 (top), in normal grav-ity at � = 0.004 (middle), and in hyper-gravity at � = 0.009 (bottom).

R0 ≈ 5mm and �p ≈ 10 kPa . The exposure and interframe times are 50 ns and 200 ns, respectively. The black bar in the first frame shows the 1 mm scale


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33 Page 8 of 13

Steam (IAPWS) (see Appendix for details Wagner and Pruß 2002). This bubble size corresponds to one that we clearly observe in the final collapse stage for highly spherical bub-bles (the actual minimum size of the spherical bubble is smaller but not observable), while the characteristic size of the bubble at the final collapse stage for the most deformed bubbles in Fig. 5b at � ∼ 0.003 is R∕R0 = 0.01 (Supponen et al. 2016). The speed of sound is found to be a lot higher than 1472ms−1 (speed of sound at initial static pressure) over a relatively large distance from the bubble at its collapse in Fig. 7; at a radial distance as far as 0.5R0 from the bubble centre, the sound speed equals 1596ms−1 for this relative bubble size. The collapse shock wave, which travels through this high pressure region, could thereby maintain a higher propagation speed than the sound speed far from the bubble over a longer distance compared to the shock wave produced at the generation, which travels through a medium of the ini-tial static pressure.1 This could explain why the hydrophones consistently measure a shorter bubble lifetime compared to the photodetector measurements.

3.2 Bubble’s centroid displacement

Figure 8 compares the different ways to measure the bub-ble displacement, i.e., from the bubble lifetime difference between two hydrophone signals, from the the bubble life-time difference between a hydrophone and a photodetector signal, or as measured directly from the recorded high-speed images. Within the significant error bars, the measured dis-placements are in good agreement. While the photodetector signal has a higher temporal resolution than the hydrophone signals, the scatter in the computed displacements is sig-nificant. As mentioned earlier, this can be explained by the luminescence pulse peak intensity varying by 10 ns with respect to the shock wave detection, and by the event con-taining multiple peaks (Ohl 2002; Supponen et al. 2017a), making the exact collapse time troublesome to determine. This time shift is incorporated in the vertical error bars, encompassing thus the scatter [note that the time shift of 60 ns observed in Sect. 3.1 is accounted for in the computation of the displacement, see Eq. (4)]. Furthermore, displace-ments could only be measured using the photodetector for relatively spherical bubbles at 𝜁 < 0.003 , as luminescence could not be detected for more deformed bubbles. Displace-ments from the high-speed images, on the other hand, could only be extracted at higher levels of deformation due to lim-ited resolution, but within the significant uncertainty they

-0.3 -0.2 -0.1 0 0.1 0.2 0.3x/R0

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0.2

0.3

y/R

0

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400c [m/s]

Bubble

Fig. 7 Speed of sound in the liquid surrounding the collapsing bubble at R∕R0 = 0.03 according to Rayleigh’s incompressible theory (Ray-leigh 1917) and the IAPWS-95 model for speed of sound in water at 17◦C versus pressure (Wagner and Pruß 2002)

0 0.2 0.4 0.6 0.8 1sH1+H2 [mm]

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m]

Hydro 1 + photoHydro 2 + photoFilm

Fig. 8 Comparison of the bubble displacement �s measured from the bubble lifetime difference between two hydrophones (H1 + H2) with that measured from the bubble lifetime difference between a hydrophone and the photodetector, and directly from the high-speed recording. Some representative measurement uncertainties are shown by error bars. The varied parameters here are R0 = 1.2–6.2 mm , �p = 7.2–83.2 kPa , and g = 0–2.1 ms−2

1 As a rough estimation of time scales, we can compare a shock wave travelling at 2000ms−1 (speed of sound at 370 MPa, which is compa-rable to the mean pressure of the pressure field predicted by Rayleigh corresponding to Fig. 7) and another travelling at 1470ms−1 (speed of sound at initial static pressure) over a distance of 400 μm (0.1R

0 for

a 4 mm radius bubble, comparable to the length of the high pressure region due to buildup prior to collapse) to find a ∼ 70 ns time differ-ence.


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Page 9 of 13 33

mostly agree with the displacements measured by the two hydrophones. Therefore, as comparing the two hydrophone signals allows for a larger range of bubble deformations and a better resolution of the bubble displacements (smaller hori-zontal error bars compared to vertical error bars in Fig. 8), we choose this method to further investigate the bubble dis-placement as a function of the bubble deformation.

The normalised displacement of the bubble’s centroid during its lifetime as a function of � is displayed in Fig. 9, as measured by comparing the shock wave timings recorded by the two hydrophones. As expected and in agreement with previous observations of many different types of cavitation bubbles (for example, Tomita et al. 2002; Gregorčič et al. 2007; Hung and Hwangfu 2010; Zhang et al. 2015), the dis-placement of the bubble increases with its level of deforma-tion. At 𝜁 > 0.001 , this increase follows remarkably well a �s∕R0 ∝ �

2∕3 scaling law. This can be explained by the dis-placement (a length) being proportional to the characteristic radius of the bubble at the final collapse point, r = R(t)∕R0 , which has been shown to scale as r ∝ �2∕3 in a scale-free framework (Supponen et al. 2016). The scale-free approxi-mation is justified by most of the bubble’s motion occurring right before and after the bubble has reached its minimal size at weak deformations, and indeed, potential flow theory also predicts such scaling for displacements (assuming a jetting bubble) as � → 0 . However, we have previously found dis-placements of deformed bubbles to scale with �3∕5 accord-ing to measurements made through high-speed imaging

(Supponen et al. 2016). This applies to more deformed bub-bles up to � ∼ 0.1 , as a considerable fraction of the bubble’s motion then occurs earlier in the bubble’s lifetime at larger bubble radii, causing a decrease in the power law index com-pared to weakly deformed collapses.

At lower levels of deformation ( 𝜁 < 0.001 ), however, the trend for the displacement in Fig. 9 becomes less clear and the data points more scattered. This is partly due to the increased relative measurement error (a 10 ns error in time corresponds to a 15 μm error in distance), combined with the error in the initial bubble location ( ±15 μm ). It may, how-ever, also be a result of the delicate weak jet regime, where it is unclear from the high-speed images whether the bubble has deformed or kept a spherical shape through the collapse, and where the eventual small displacement is highly sen-sitive to uncontrollable micro-variations in the initial con-ditions. As � → 0 , the bubble compression ratio R0∕Rmin (where Rmin is the minimum radius of the bubble) increases to the point where the gas compression inside the bubble overcomes the bubble’s shrinking motion before a jet can form. The observation that at 𝜁 < 0.001 the measurements consistently show lower displacements than the theoretical scaling supports the idea that the jets of the bubbles in this regime are hindered, therefore transitioning to spherical col-lapses. The highest level of bubble deformation at which we compute a null displacement within the measurement resolution is at � = 0.0014.

3.3 Directionality of shock waves

To assess the directionality of the shock waves, we com-pare the peak pressure and energy measured by the two hydrophones, one of which generally captures shock waves emitted from the jetting side of the bubble, and the other one detects those departed from the opposite side. From the hydrophone voltage signal U(t)(V), we define the maximum peak pressures measured from the bubble generation and the collapse as the corresponding signal peak amplitudes, pg = Ug,max and pc = Uc,max . The shock wave energy is computed in arbitrary units from each shock wave signal as E = ∫ U(t)2dt (Vogel and Lauterborn 1988; Supponen et al. 2017b), where the integration bounds are selected to comprise the full waveform of the shock wave. It should be noted that these values have not been corrected for the non-linear dissipation, as the shock waves propagate in water and hence may contain some error. However, for the purpose of this study, the uncorrected relative values are sufficient for determining when directionality becomes important.

Figure 10 shows the shock peak pressure ratio and the shock wave energy ratio from the bubble collapse as a function of � , measured at different gravity levels. As the two hydrophones are located at different distances from the bubble, we divide these ratios with the peak pressures

10-3 10-210-3

10-2

10-1

100s/

R0

2.5 3/5

3.7 2/3

Fig. 9 Normalised bubble displacement �s∕R0 , measured from the bubble lifetime difference between two hydrophones, as a function of � . The solid lines represent the previously established model for normalised bubble displacement (green) and the scaling for a charac-teristic length as � → 0 (pink) (Supponen et al. 2016). The error bars show the measurement uncertainty. The varied parameters here are R0 = 1.2–6.2 mm , �p = 7.2–83.2 kPa , and g = 0–2.1 ms−2


1 3

33 Page 10 of 13

and energies measured at the bubble generation, which we assume to produce spherically symmetric shock waves and to be proportional to the bubble energy, similarly to the collapse shock waves for spherical bubbles (Vogel and Lauterborn 1988). Note that � is not zero at micro-gravity, as the nearest boundaries, the two parabolic mirrors, are also accounted for as a source of bubble deformation. There is a clear asym-metry in the emitted shocks from aspherical bubble col-lapses. The portion of the shock wave (or waves) captured by hydrophone 2 is weaker compared to that measured by hydrophone 1 as the bubble deformation increases beyond � ∼ 0.001 , with its peak pressure reducing to barely a half of that measured by hydrophone 1. However, beyond � ∼ 0.005 , this ratio increases again, reaching unity at � ∼ 0.01 . Inter-estingly, 𝜁 > 0.005 corresponds to a regime where gravity is the dominant source of deformation for most of the data, and beyond � ∼ 0.0059 , all data are gravity-dominated according to Fig. 3. The data obtained in normal and hyper-gravity col-lapse in an orderly fashion to a single curve, while in micro-gravity there is a strong scatter of data, in particular between � ∼ 0.001 and 0.003. This can be explained by the direction of the jet being different with respect to the hydrophones when dominated by the nearest surface, which is directed sidewards, rather than the downward-pointing pressure gradi-ent producing an upward-pointing jet. Therefore, the hydro-phones are able to detect different ‘sides’ of the spherically asymmetric shock wave. The dominant source of deformation for bubbles in micro-gravity is always the nearest bound-ary, this being shown in Fig. 6 that displays an example of a bubble in micro-gravity producing a weak jet towards the left side, which corresponds to the direction of the parabolic mirrors in the image plane. Indeed, the range of angles of the jet from the vertical axis, as computed through Eq. (1) is large, � = 20◦–80◦ . Here, we see that, in micro-gravity, hydrophone 2 may detect shocks with up to 30% higher peak pressures compared to hydrophone 1, while at higher gravity

levels at that same � , hydrophone 1 detects about 30% higher peak pressures compared to hydrophone 2. We may observe an equivalent behaviour with the shock wave energy ratios, the highest differences observed in micro-gravity being in the order of 50%. At � ∼ 0.003–0.005, the hydrophone 2 consist-ently measures shock wave energies as low as 35% of those measured by hydrophone 1.

4 Discussion

The transition between spherical and deformed bubbles is a complicated regime to characterise. However, our results imply that (i) below � ≈ 0.001 , the measured displacements become significantly smaller and deviate from the scaling law based on scale-free jetting theory, and hence are less predictable (Fig. 9); and (ii) below � ≈ 0.001 , the shock waves are spherically symmetrical but above it, they show important directionality (Fig. 10). While we were unable to prove that no jets were formed at 𝜁 < 0.001 , our observations consolidates further our previous suggestion (Supponen et al. 2016) that this regime acts as a practical transition limit between spherical and deformed bubbles. There are, of course, many factors that can affect this limit, including experimental conditions such as initial sphericity of the bub-ble. Large laser-induced bubbles typically have an important growth in shape instabilities in the final stage of the collapse, likely emerging from anisotropy and perturbations in the initial plasma [for instance, due to the laser focal region being cone-shaped as observed by Vogel et al. (1994) and more recently by Tagawa et al. (2016)], which can yield bub-ble splitting (Baghdassarian et al. 1999, 2001). The bubble surface during its growth phase is generally rather stable, allowing a bubble that is initially not spherical to assume a highly spherical shape at its maximum expansion. During the collapse, however, the surface modes become amplified,

Fig. 10 a Shock peak pres-sure ratios (pc2∕pc1)∕(pg2∕pg1) and b shock energy ratios (Ec2∕Ec1)∕(Eg2∕Eg1) as a func-tion of � , measured at different gravity levels. The dashed line shows the limit ( � = 0.0059 ) beyond which all of the measured bubble deforma-tions are gravity-dominated. Here, 0 g, 1 g and 1.8 g include data measured in the ranges 0–0.5 ms−2 , 0.5–1.2 ms−2 , and 1.2–2.1 ms−2 , respectively. The other varied parameters here are R0 = 1.2–6.2 mm and �p = 7.2–83.2 kPa

10-4 10-3 10-2 10-10

0.5

1

1.5

(pc2

/pc1

)/(p g

2/p g

1)

(a)

0 g1 g1.8 g

10-4 10-3 10-2 10-10

0.5

1

1.5

(Ec2

/Ec1

)/(E g

2/E g

1)

(b)

0 g1 g1.8 g


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Page 11 of 13 33

as their growth rates are sensitive on the initial perturbations (see stability analysis by Plesset and Mitchell 1956). The effect of the initial shape of laser-induced bubbles on their collapse dynamics has in fact been previously harnessed for the purpose of engineering micro-fluidic flows on demand by Lim et al. (2010). However, using a high-convergence parabolic mirror and thereby producing a highly point-like initial plasma, we delay the emergence of the shape insta-bilities during the bubble collapse. This could explain why we are able to observe jets in bubbles for at a much lower level of deformation (for bubbles near rigid surfaces, beyond � ∼ 10 ) compared to other experiments (typically � ∼ 5 ). Indeed, the data obtained in micro-gravity in Fig. 10, where the bubble deformation is dominated by the parabolic mir-rors with the smallest stand-off parameter being � ∼ 8 for the largest bubbles, already show important shock wave direc-tionality at 𝜁 > 0.001.

5 Conclusion

Interesting details in the collapse of weakly deformed cavita-tion bubbles are disclosed owing to the new high-resolution measurements. We have obtained a precise measure of the displacement of weakly deformed bubbles from their gen-eration to collapse location using two hydrophones, which detect the passages of the respective shock waves. The defor-mation of bubbles is controllably adjusted by varying the gravity level aboard parabolic flights. We carefully charac-terise the 3-D configuration of the experiment to compute a precise measure of the vector parameter � , the dimensionless quantity providing the “strength” of the bubble deformation and the direction of its displacement. The normalised dis-placements for weak deformations are found highly repro-ducible for 𝜁 > 0.001 and to scale remarkably well with �2∕3 despite the large variations in bubble radii, gravity levels and driving pressures. With lower levels of deformations, we observe a strong scatter in the displacements, which may be due to higher measurement uncertainty but also to the higher sensitivity of the bubble to micro-disturbances. The displacement measurements obtained with the hydrophone signals are more accurate compared to the photodetector signal, which measures the bubble’s liftetime as defined by the initial laser pulse and the luminescence event at col-lapse. This is due to the shock wave emission at collapse being more reproducible in time compared to luminescence, which may exhibit multiple peaks. The bubble’s lifetime as measured by the photodetector is also consistently found to be 60 ns longer than that measured by both hydrophones, likely due to the differences in the shock wave propagation speeds at the generation and at the collapse.

In addition, the two hydrophones are used to quantify the asymmetry of the shock waves generated at the bubble

collapse. A ratio as low as 50% is observed between the peak pressures measured in a single shock wave event from two opposite sides as measured by the two hydrophones. We have therefore concluded that the directionality of the shock waves is non-negligible as the bubble deforms and produces jets, i.e., collapses at 𝜁 > 0.001 . Efforts will be made to cover a controllable range of angles to map the pressure distribution along the shock waves from aspherical bubble collapses in the future.

Acknowledgements The authors gratefully acknowledge the support of the Swiss National Science Foundation (Grants no. 200020-144137 and P2ELP2-178206), the University of Western Australia Research Collaboration Award (PG12105206) obtained by D.O. and M.F., and the European Space Agency. They also thank Henry Hollenweger for his valuable assistance with the experiment and the parabolic flights.

Appendix: Speed of sound in water at elevated pressure

The equations of the IAPWS-95 model (Wagner and Pruß 2002) are available in the R-package “IAPWS95”. The speed of sound can be computed as a function of temperature and pressure using the function wTp. The model fits the experi-mental data from Hidalgo-Baltasar et al. (2012) with high accuracy ( < 0.3% ), as can be seen in Fig. 11.

Rayleigh’s incompressible model for the pressure dis-tribution in the liquid surrounding a spherically collapsing bubble is given as follows (Rayleigh 1917):

where p0 is the pressure at infinity, R is the bubble radius, R0 is the maximum bubble radius, and r is the radial distance from the bubble centre.

(5)p

p0= 1 +

R

3r

(R30

R3− 4

)−

R4

3r4

(R30

R3− 1

)

0 100 200 300 400 500 600 700p [MPa]

1500

2000

2500

c [m

/s]

Fig. 11 Speed of sound in water c at room temperature versus pres-sure p. Solid line corresponds to the IAPWS-95 model (Wagner and Pruß 2002) and the symbols are experimental measurements from Hidalgo-Baltasar et al. (2012)


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33 Page 12 of 13

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1 3

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Affiliations

Outi Supponen1 · Danail Obreschkow3 · Philippe Kobel2 · Nicolas Dorsaz2 · Mohamed Farhat2

1 Department of Mechanical Engineering, University of Colorado, 1111 Engineering Drive, Boulder 80309, USA

2 Laboratory for Hydraulic Machines, École Polytechnique Fédérale de Lausanne, Avenue de Cour 33bis, 1007 Lausanne, Switzerland

3 International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia

http://orcid.org/0000-0001-6738-0675

Detailed experiments on weakly deformed cavitation bubblesAbstractGraphical abstract1 Introduction2 Methods2.1 Experimental setup2.2 Vector parameter for bubble deformation2.3 Example signal2.4 Measurement of bubble displacement

3 Results3.1 Event timings3.2 Bubble’s centroid displacement3.3 Directionality of shock waves

4 Discussion5 ConclusionAcknowledgements References

detailed experiments on weakly deformed cavitation...

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