# High-speed x-ray phase-contrast imaging of single cavitation bubbles near a solid boundary

G. T. Bokman, L. Biasiori-Poulanges, B. Lukić, C. Bourquard, D. W. Meyer,

A. Rack, O. Supponen

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G. T. Bokman, L. Biasiori-Poulanges, B. Lukić, C. Bourquard, D. W. Meyer, et al., High-speed x-ray phase-contrast imaging of single cavitation bubbles near a solid boundary. Physics of Fluids, 2023, 35 (1), pp.013322. 10.1063/5.0132104. hal-05077994

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# High-speed x-ray phase-contrast imaging of single cavitation bubbles near a solid boundary EP

Special Collection: Cavitation

G. T. Bokman 🐶 ; L. Biasiori-Poulanges 🐶 ; B. Lukić 🐶 ; C. Bourquard 🐶 ; D. W. Meyer 🐶 ; A. Rack 🐶 ;
O. Supponen 🐶

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Physics of Fluids 35, 013322 (2023)

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# High-speed x-ray phase-contrast imaging of single cavitation bubbles near a solid boundary EP

Cite as: Phys. Fluids **35**, 013322 (2023); doi: 10.1063/5.0132104

Submitted: 26 October 2022 · Accepted: 23 December 2022 ·

Published Online: 11 January 2023

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G. T. Bokman,<fnref n="1" />,ᵃ ID L. Biasiori-Poulanges,<fnref n="1" /> ID B. Lukić,<fnref n="2" /> ID C. Bourquard,<fnref n="1" />,<fnref n="3" /> ID D. W. Meyer,<fnref n="1" /> ID A. Rack,<fnref n="2" /> ID and O. Supponen<fnref n="1" />,ᵇ ID

## AFFILIATIONS

Note: This paper is part of the special topic, Cavitation.

a) Author to whom correspondence should be addressed: bokmang@ethz.ch

b) Electronic mail: outis@ethz.ch

## ABSTRACT

Laser-induced cavitation bubble dynamics at different distances from a rigid boundary is investigated using high-speed synchrotron x-ray phase-contrast imaging. This is achieved through the design of a tailored experimental chamber specifically designed to reduce the x-ray absorption along the path length in water while mitigating boundary effects. The highly resolved undistorted radiographs are able to visualize a sharp bubble interface even upon complex shapes, which can serve as high-quality benchmarks for numerical simulations. Here, the measured bubble shapes are compared to simulations using the incompressible boundary integral method. The direct optical access to the high-speed liquid jet provides accurate measurements of the evolution of the jet speed, which is contrasted to the simulated results. After the jet has impacted the opposite side of the cavitation bubble, the cavity assumes a toroidal shape, the volume of which can be accurately measured from the radiographs and its temporal evolution compared to the bubble-ring model. Thanks to the clear optical access to the cavity lobes throughout the collapse, non-axisymmetric splashing within the bubble resulting from the jet impact, also known as Blake's splashing, is observed and characterized for stand-off parameters of $\gamma < 1$. Measurements extracted from the highly resolved visualizations provided herein have been validated against scaling laws for droplet impact on a thin liquid film, which contribute to confirm and elucidate the splashing phenomenon.

© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0132104

## I. INTRODUCTION

Cavitation bubble dynamics has been first investigated to understand the damage it causes in industrial domains such as marine transport or hydraulic energy production, where most of the harm has been shown to be originating from the shock waves and high-speed liquid jets resulting from the collapse of cavities near the propeller or turbine blades.<fnref n="1" /> However, time and ever-improved understanding of the physics behind cavitation have extended the possible applications harnessing the damaging dynamics of cavitation for beneficial purposes. Examples of such applications, where expanding and collapsing cavities can interact with different substrates in a favorable way, are sonophoresis, where ultrasound-induced cavitation can help adhesives or drugs penetrate the outermost layer of the skin;<fnref n="2" />,<fnref n="3" /> microfluidic pumping⁴ and

needle-free injections,⁵,⁶ which leverage the rapid flows and production of high-speed jets from laser-induced cavities at microscales; cleaning applications,⁷⁻⁹ where dirt particles and biofilms may be removed from a surface without contact; and particle size reduction,¹⁰ where collapsing cavities can interact strongly with individual particles and cause their fragmentation. The cavitation-induced effects in play in these applications, desirable or not, are largely attributed to the pressures and wall shear stresses induced on neighboring boundaries or objects due to the jetting behavior of the bubble,¹¹ as well as the generation of shock waves.¹²,¹³ Detailed characterization of such effects is still needed today. This drives extensive investigations on single cavitation bubble dynamics near rigid walls, which builds up on the knowledge provided by the pioneering works undertaken more than half a century ago.¹⁴,¹⁵

<footnote n="1">¹Institute of Fluid Dynamics, Department of Mechanical and Process Engineering, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland</footnote>

<footnote n="2">²ESRF—The European Synchrotron, Grenoble F-38043, France</footnote>

<footnote n="3">³Silicon Austria Labs GmbH, Villach A-9524, Austria</footnote>
Numerical simulations of collapsing cavities have significantly improved the community's understanding of the underlying physics by facilitating targeted parametric studies and by enabling unlimited access, e.g., to flow fields or interfaces between different phases. Since the relevant Reynolds numbers are typically high and Mach numbers are sufficiently low, potential-flow-based boundary element or integral methods (BIM) have been widely applied to model the bubble dynamics.¹⁶,¹⁷ Specialized extensions of the BIM method that remain applicable at elevated Mach numbers¹⁸,¹⁹ and beyond jet impact²⁰,²¹ have been devised as well. Furthermore, compressible simulations using finite volume methods and different interface tracking or capturing techniques have been able to access detailed features of the bubble-wall interactions, including shock wave emissions,²²⁻²⁵ quantifying temperatures at the walls,²⁶ characterizing bubble dynamics near boundaries having complex geometries, such as gaps²⁷ or crevices,²⁸ and computing material pitting²⁹ as well as wall shear stresses for bubbles collapsing in water and in viscous fluids.³⁰,³¹ On the other hand, the rapid progress in improving high-speed imaging techniques by pushing the limits of temporal and spatial resolutions has allowed to advance the experimental characterization of cavitation to an impressive extent. There have been recent advances characterizing cavity collapse dynamics near complex geometries, such as thin gaps,³²,³³ corners,³⁴ and slots.³⁵ Ultra-high-speed imaging with MHz acquisition rates has allowed to visualize the complex shock wave emissions and intricate needlelike jet shapes involved in non-spherical bubble collapses,³⁶,³⁷ as well as investigating their relative contributions to the resulting damage to neighboring boundaries.¹² Advanced experimental techniques have enabled temporally resolved measurements of wall shear stresses caused by single laser-induced bubble collapses³⁸,³⁹ and quantification of the flow fields and vortex dynamics⁴⁰ near the collapsing bubbles as well as the film thickness between the bubble and the wall.⁴¹ To overcome challenges imposed by optical distortions due to refractions and reflections in conventional optical imaging, a theory-assisted optical ray tracing technique has also been proposed to access more details on deformed interfaces.⁴² Recently, the use of high-speed x-ray contrast imaging has also made its appearance in imaging fast physical problems, for which optical access is not trivial, including cavitation. It has, for instance, enabled to quantify temporally varying vapor void fractions in cavitating flows through x-ray densitometry,⁴³ to characterize cavitating diesel fuel flows using x-ray micro computed tomography¹⁴,⁴⁵ and to visualize the interaction of a shock wave produced by a gas-gun with a spherical cavity in a solid,⁴⁶ which has revealed similarities to the classical hydrodynamic shock-bubble interaction. Recently, measurements of the temporal evolution of an expanding laser-induced bubble as well as the pressure distribution around it have been achieved via x-ray holography using free-electron laser pulses.⁴⁷ However, to the best of the authors' knowledge, single bubbles collapsing near boundaries are yet to be visualized using x-ray imaging, which could provide improved benchmarking for numerical simulations of this well-studied configuration and reveal some new details of the bubble dynamics.

Here, the classic problem of a laser-induced cavitation bubble collapsing near a solid boundary is investigated while leveraging the time-resolved phase-contrast capabilities offered by the powerful synchrotron x-ray source to capture detailed hidden features of the bubble interface.

## II. METHODS

### A. Cavitation experiments

The experimental setup used to generate cavitation bubbles near a solid boundary is illustrated in Fig. 1. Here, cavities are generated by a high-power single light pulse from a frequency-doubled infrared Q-switched Nd:YAG laser (Lumibird Quantel Q-smart, 532 nm), with a pulse width and energy of 6 ns and 220 mJ, respectively. The laser beam is first expanded with a 10-x beam expander and then focused in de-ionized water using a 90° parabolic mirror to minimize spherical aberrations while maximizing the distance of the bubble from boundaries. This results in a tight focusing of the laser, which is able to ionize the liquid through optical breakdown and to generate a highly point-like plasma containing an extremely high pressure. The plasma, in turn, rapidly expands and forms a spherically growing vapor bubble. Despite the thermal origin and the non-condensable gases generated in the laser-induced bubbles,⁴⁸ they are widely accepted to well represent a cavitation bubble, that is, a vapor bubble that grows due to low liquid pressure rather than high gas pressure, in most practical scenarios. The test chamber containing the water where the cavitation bubble is generated is a 100 × 100 × 350 mm³ high-quality Plexiglas polycrystalline cell made of transparent walls that permits optical access. Two opposing walls are equipped with two telescopic windows consisting of tubes at the end of which circular windows are fastened and sealed with high-vacuum grease. The contact surface between tubes and the container is made water-tight with O-rings. The easily replaceable 0.5-mm-thick high-quality Plexiglas windows are designed to give access to the x-ray beam through the object of interest while reducing the beam path length in water, and thereby the beam absorption (see Sec. II B). Indeed, the windows are made replaceable because as their integral exposure to x-ray light increases, they deteriorate. With time they become tinted in a brown color, resulting in an increased brittleness that can lead to potential cracking. The distance $d_w$ between the windows is adjustable in the 0–100 mm range. The two remaining walls, in the orthogonal direction of the telescopic windows, allow the laser pulse to access the water tank. A metallic plate (anodized aluminum) is positioned at a distance $\delta$ below the focus of the laser to examine the interaction of the cavitation bubble with a solid boundary. The

<figure><img src="image_10.png" /><figcaption>**FIG. 1.** Schematics of the experimental setup (side view). The dashed lines represent the laser path. The x-ray beam follows the y-direction.</figcaption></figure>
stand-off distance between the cavitation bubble center and the metal-
lic plate is varied from $\delta = 0.80-2.59$ mm, and the laser power is kept
constant, which, in the absence of boundaries, results in the gen-
eration of a spherical cavitation bubble of maximum radius
$r_{max} = 2.43$ mm and translates to a stand-off parameter,
$\gamma = \delta/r_{max}$, in the range $0.33-1.07$. Throughout this work, the
value of $r_{max}$ is considered constant despite uncertainties related
to bubble deformations, proximity to the solid boundary and fluc-
tuations in laser energy (measured to be of 1%). However, it
should be noted that when computing an equivalent maximum
radius from a circular area having the size of the observable bub-
ble surface area, the values of the maximum radius decrease in 4%
and 9% for $\gamma = 0.53$ and $\gamma = 0.33$, respectively. Overall, the ambi-
ent pressure in the liquid is taken as $p_{\infty} = 10^5$ Pa, and the temper-
ature is measured as $23^{\circ}$C, which corresponds to a vapor
pressure of $p_v = 2810$ Pa.

B. Imaging techniques

In situ radiographs of the cavitation bubble near the solid bound-
ary are acquired using high-speed synchrotron-based x-ray phase-
contrast imaging capabilities at the 150 m-long ID19 beamline of the
European Synchrotron Radiation Facility (ESRF). Figure 2 displays a
schematic of the x-ray imaging system arrangement. The 16-bunch
filling mode (75 mA ring current) has been exploited, which provides
a continuous train of ~100 ps (FWHM) X-ray pulses with a 176 ns
time interval. Two axially aligned long-period undulators (U32), set to
a minimum 11.50-mm gap, are used to generate the polychromatic
hard x-ray beam. The resulting spectrum is filtered with a set of man-
datory optical elements: a 1.8 mm diamond window and successive
thin carbon and beryllium windows, mounted along the vacuum flight
tube, which suppress the soft x-ray spectral region (<10 keV) and,
therefore, reduce the heat load delivered to the sample. Upstream of
the specimen, two sets of in-vacuum slits are used to trim the x-ray
beam onto a $12 \times 8\text{mm}^2$ field of view.⁴⁹ The characteristic mean
energy of the beam is 30 keV with corresponding flux of $4.6 \times 10^{11}$
photons/s/0.1%BW/mm². X-rays transmitted through the sample are
indirectly imaged using a 500-µm-thick LYSO:Ce scintillator (Ce-
doped Lu₁.₈Y₀.₂SiO₅, Hilger Crystals, UK), which absorbs x-ray pho-
tons and re-emits visible photons with peak emission at 413 nm. The
visible light, emitted both down- and up-range of the scintillator, is
folded away from the x-ray beam path by pellicle beamsplitters
[45:55(R:T) coated for 400-700 nm wavelengths], before being

captured by two ultra-high-speed Shimadzu Hyper Vision HPV-X2
cameras.⁴⁶ The cameras are each equipped with 50 mm tandem lenses
in 1 : 1 magnification ratio (Nikon Nikkor AF 50 mm f/1.4), which
results in a 32 µm/pixel spatial resolution. A combination of two syn-
chronized cameras is used in series to capture radiographs of the phe-
nomena at an exposure time of 200 ns while recording 256 frames at
the speed of 200 kHz. Here, the cameras do not allow for synchroniza-
tion to an external source, and double exposure may occur during the
exposure time of the high-speed photography (176 < 200 ns).
Consequently, the flux is exploited for exposure instead of the bunch
structure, resulting into moderate intensity variations between frames.
The imaging system is located 7.5 m downstream of the sample, so
that radiographs are recorded in the near-field Fresnel diffraction
regime. This ensures the propagation-based x-ray phase-contrast
imaging, such that the interference between transmitted x-rays
results in an increased edge contrast due to the (partial) spatial
coherent illumination, while preserving the geometrical represen-
tation of imaged phenomena.⁵⁰ The image analysis of time and
space resolved radiographs of a liquid-gas interface problem
requires a relatively strong signal to accurately detect discontinu-
ities related to phase changes. This implies that the x-ray beam
path length through the sample must be minimized to reduce the
absorption. The physical problem herein addressed involves
shock, compression, and expansion waves, which, when interact-
ing with the wall and the windows of the test chamber, may mod-
ify the driving mechanisms at play. While the telescopic windows
have been designed to reduce the x-ray path length, they locally
confine the bubble-boundary configuration and, thus, may
induce unwanted boundary effects. For this reason, the present
experiments require to compromise between absorption and
wall effects. Considering the sound speed in water at
23 °C, $c_w = 1489$ m/s, and the timescale of the experiment
~500 µs, wave reflections upon the walls have time to reach the
bubble for $d_w \le 745$ mm.

However, noting that boundary effects in cavitation bubble
dynamics are mainly dependent on the distance between the bubble
center and the windows normalized to the maximum radius of the
cavitation bubble $\xi$, the inter-window distance is constrained by this
parameter. In fact, boundary effects have been demonstrated to have
little effect on bubble dynamics⁵¹,⁵² for $\xi > 3$, which translates into
a minimum inter-window distance of $d_w = 2\xi > 2 \times 3r_{max}$
$\approx 14.58$ mm. However, in the case of highly symmetric experimental
conditions (relying on mirror-focused laser as herein and/or micro-
gravity conditions³³) boundary-induced deformations have been noted
for $\xi > 5$ or, in this case, $d_w > 24.30$ mm. Figure 3 plots the normal-
ized signal intensity measured on radiographs as a function of the
inter-window distance $d_w$. It shows complete absorption of the flux for
$d_w \gtrsim 100$ mm, and that a workable signal is achieved for $d_w \le 20$ mm.
A one-term exponential model is fitted to measurements, yielding an
absorption coefficient, $\alpha$, of $-0.037 \text{ mm}^{-1}$. The x-ray path length in
water is, therefore, the limiting parameter in the determination of the
optimal inter-window distance. In this work, the inter-window dis-
tance is set to 20 mm, which corresponds to the minimal absorption
within the $\xi > 3$ domain displayed as the shaded area of Fig. 3.
Considering the proximity of the flat wall for $\gamma \le 1$, it is assumed that
it will be dominating the deformations, and other boundary effects are
assumed negligible.

<figure><img src="image_11.png" /><figcaption>FIG. 2. Schematics of the experimental setup (top view), showing the x-ray imaging system arrangement. X-ray light is absorbed in-between the telescopic windows over the distance $d_w$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>FIG. 3. Functional dependency of the image intensity over the inter-window distance $d_w$ measured from the radiographs (circle markers). The solid line shows the one-term exponential interpolation of the experimental data to determine the absorption coefficient (Beer–Lambert law). The shaded region indicates the range of inter-window distances, for which boundary effects felt by the cavitation bubble are considered negligible.</figcaption></figure>

### C. Boundary integral simulations

Experimental results are complemented with simulations from the inviscid and axisymmetric boundary integral method (BIM), thanks to the Reynolds numbers being of $O(10^3)$ for the flows under investigation. In the BIM method, the liquid flow is derived from the potential-flow theory or more precisely Green's representation theorem,⁵⁴ that is

$$ \phi(\vec{x}) = \int_S \left[ -\phi(\vec{y}) \frac{\partial}{\partial \vec{n}} g(\vec{x}, \vec{y}) + \frac{\partial \phi(\vec{y})}{\partial \vec{n}} g(\vec{x}, \vec{y}) \right] dS(\vec{y}), \quad (1) $$

with Green's function

$$ g(\vec{x}, \vec{y}) = \frac{1}{2\pi|\vec{x} - \vec{y}|}, \quad (2) $$

which is solved at every time step for the potential gradient normal to the liquid/vapor interface $\partial\phi/\partial\vec{n}$ (see Fig. 4). In order to do so, the axisymmetric interface S is discretized into disjoint ring-like sections $\Sigma_i$ as depicted in Fig. 4, and the potential and potential normal gradients are assumed to remain constant within each section. Consequently, the potential on the left side of Eq. (1) takes a finite number of values on the surface sections $\phi_i$ with $i = 1, 2, \dots, N$, and the surface integral on the right side can be decomposed into N section integrals, which is commonly referred to as Nyström discretization.⁵⁵ After simplifying the section integrals (for more detail, see Taib et al.¹⁶] the linear equation system reads

$$ \phi_i + \sum_{j=1}^{N} \phi_j \underbrace{\int_{\Sigma_j} h(x, z, x_i, z_i, \vec{n}_j) dl}_{h_{ij}} = \sum_{j=1}^{N} \underbrace{\frac{\partial \phi}{\partial \vec{n}} \int_{\Sigma_j} g(x, z, x_i, z_i) dl}_{g_{ij}}, \quad (3) $$

where h is the gradient of Green's function normal to the interface.

By combining the resulting potential gradients normal to the interface $\partial\vec{n}\phi|_i$ with the gradients tangential to the interface and com- putable from $\phi_i$, the advection speed $\vec{u} = \nabla\phi$ of the interface nodes can be numerically estimated and the bubble interface advected in time. The corresponding temporal evolution of the potential is derived from the Bernoulli equation¹⁶ as

<figure><img src="image_13.png" /><figcaption>FIG. 4. Schematic of the numerical representation of the liquid/vapor interface used in the boundary integral method (BIM). The potential and potential gradient normal to the interface surface S are $\phi_i$ and $\partial\vec{n}\phi|_i$, respectively. The surface is discretized into edges, $\Sigma_i$ with $i = 1, 2, \dots, N$. The white markers show the edge centers and the dark ones show the nodes.</figcaption></figure>

$$ \frac{D\phi}{Dt} = \frac{p_{\infty} - p_v}{\rho} + \frac{\vec{u}^2}{2}, \quad (4) $$

where the density of water is $\rho$. Additionally, effects associated with surface tension, $\sigma$, are considered negligible when comparing the driving pressure gradient $\Delta p = p_{\infty} - p_v$ to $p_{\sigma} = \sigma/r_0$ the pressure induced by surface tension on a bubble of initial radius, $r_0$. The difference between both is of three orders of magnitude. To account for compressibility effects, the BIM can be extended to a compressible BIM method (CBIM)¹⁸ based on low-order perturbation expansions developed in terms of the maximum Mach number,

$$ M_a = \sqrt{\frac{\Delta p}{\rho c_w^2}}. \quad (5) $$

However, the bubbles considered in this work are characterized by a very low $M_a$ on the order of $O(10^{-3})$, which makes differences between the BIM and CBIM negligible.

As a slight extension from classical BIM implementations for cavitation bubble applications,¹⁶,¹⁷ a node rearrangement scheme has been implemented to keep numerical errors equal along the cavity interface. Here, nodes are equidistantly re-distributed once deviations from a uniform node pattern have become too large. To avoid triggering numerical instabilities from spurious oscillations along the interface, a linear interpolation scheme is applied during the rearrangement step for node positions and potential levels. As an additional stabilizing measure, smoothing or more specifically local averaging is applied to node velocities and potential levels.²⁴,⁵⁶ To resolve the cavity geometry, 36–48 grid points are used. For the time integration of the liquid/vapor interface nodes and the node potentials $\phi_i$ [see Eq. (4)], a four-stage Runge–Kutta integration scheme is applied in combination with equally-sized time steps of 2.5 µs that resolve the rapid initial expansion. The time integration starts from spherical initial bubbles having radii of 0.43, 0.45, 0.48, and 0.44 mm, which is extracted from first frames for stand-off distances $\gamma = 0.33, 0.53, 0.79$, and 1.07, respectively, and with
the initial potential defined by Eq. (37) in the work of Taib et al.<sup>16</sup> The maximum radius used for all cases is $r_{max} = 2.43$ mm. Finally, to approach the bubble collapse, where, in some cases, the jet tip hits the opposite bubble interface, the time step is sequentially reduced at the end of each simulation. In this work, no effort has been made to adapt the numerics to the rotational flow following the collapse.<sup>21</sup>

### III. RESULTS

#### A. Bubble dynamics and the boundary integral method

In this work, the ability of high-speed x-ray imaging to unravel the hidden aspects of complex fluid dynamics problems involving multiple phases is demonstrated on the classic, well-studied case of a cavitation bubble near a rigid boundary. The radiographs allow us to accurately visualize the sharp liquid/vapor interface of the cavity even when it assumes more complex shapes, and they can be directly compared to corresponding bubble shapes simulated using the boundary integral method, as displayed in Fig. 5 (Multimedia view). Here, four cases of interest are displayed, where the maximum bubble volume is assumed constant corresponding to an equivalent maximum radius of $r_{max} = 2.43$ mm, for stand-off parameters of $\gamma = 0.33, 0.53, 0.79$, and 1.07, despite deformation-related uncertainties for the two smallest stand-off distances. The problem is assumed axisymmetric, and the image sequences display on the left of the vertical axis a snapshot taken from the BIM simulations, and on the right, the corresponding radiograph. For each case, the collapse dynamics is shown up to the last simulated time step before the jet impact onto the opposite bubble wall, and the solid boundary is located at the bottom. Qualitatively, the measurements show interesting features, which are labeled on the images and briefly discussed here. The presence of smaller, secondary cavitation bubbles induced by the laser can be consistently seen for every case above and below the principal cavitation bubble [see Fig. 5, indications (1)]. They are generated in the path of the laser, which is directed from top to bottom as indicated in Fig. 1 and can be seen migrating with the cavitation bubble-induced flow. When the main cavitation bubble is close to the solid boundary, these smaller secondary bubbles are pressed against the wall and are more visible. These are difficult to distinguish but present in past experiments involving laser-induced cavitation bubbles near solid walls.<sup>5,15,21</sup> This is either due to their increased volume caused by a drop in pressure near the wall originating from the liquid rapidly escaping from the space between the bubble and the wall,<sup>57</sup> or simply by a higher concentration of nuclei at this location. In some cases, these small bubbles merge with the main bubble. This coalescence process results in the ejection of a thin liquid film from the hard wall toward the top of the bubble [see Fig. 5, indications (2)]. This phenomenon is presumably common in laser-induced cavitation bubble experiments but difficult to image due to it occurring in close proximity to the solid wall and, at later stages, inside the cavitation bubble. The thin liquid film eventually reaches the gas/liquid interface, where the many smaller fragmented microdroplets are shot against the inner bubble surface resulting in perturbations rather uniformly across the entire interface [see Fig. 5, indications (3)]. A close up sequence of images showing the thin liquid film breakup into microdroplets is displayed in Fig. 6 for $\gamma = 0.33$, which can be better appreciated in the corresponding online multimedia. This phenomenon explains bubble interface perturbations that may occur before the jet has even formed and obstruct optical access

to the bubble interior when using conventional imaging techniques. The probability of this event occurring seems to increase as the stand-off parameter decreases, starting from $\gamma = 0.53$. For the two cases above this threshold, the cavitation bubble does not press the secondary bubbles against the wall, and, instead, the smaller gas bodies are transported by the flow along the wall [see Fig. 5, indication (1) for $\gamma = 0.79$].

Recently, it has been shown that the use of the compressible boundary integral method outperforms the classical incompressible version in simulating the bubble growth and collapse near a boundary.<sup>19</sup> This has been attributed to the compressible effects accounting for the energy dissipation during the growth phase of the bubble. Therefore, for similar initial conditions, i.e., initial bubble radius and interior over-pressure, the incompressible BIM may grow to a larger maximum radius, which, in turn, will delay the overall dynamics of the bubble. However, due to the very low Mach number of $O(10^{-3})$ associated with the current configuration, compressible effects are herein considered negligible.<sup>19</sup> Interestingly, for $\gamma < 1$, the numerically simulated dynamics is ahead of the experiments, which is clearly attributed to the bubble deforming less in the vertical axis for the numerical cases. As a result, the simulated bubbles end up being flatter with a wider base at the end of their collapse. Another consequence of the higher vertical deformity of the experimental bubbles is that more momentum is present in this direction, and the jet moves faster than predicted by the BIM as will be seen quantitatively in Sec. III B. However, for $\gamma > 1$, the numerical results appear to be slightly lagging in the two final frames when compared to the experiments, which is rather surprising. On the fourth frame of the $\gamma = 0.33$ case, one notices a kink on the surface, which is emphasized by numerical artifacts but comes from an actual sharp angle on the bubble surface, as can be seen on the radiograph. Finally, one can appreciate the results obtained from x-ray images for the validation of numerical methods, thanks to the sharp interfaces, the absence of distortion, and the enhanced optical access to the interior of the cavities.

#### B. Liquid jet velocity

The undistorted optical access to the liquid jet allows for quantification of the speed of its tip with respect to time. This useful quantity of the jet evolution can clearly be extracted with uncertainties being only dependent on the image resolution, thanks to the use of phase-contrast x-ray imaging. To do so, the location on the bubble interface where the jet tip forms is measured on each frame, and the variation in position between two frames divided by the interframe time is defined as the jet speed. This is plotted in Fig. 7 for the four stand-off distances presented in Sec. III A, from a dimensionless time of 1–2.5, although, in the early stages, the jet velocities are negligibly small. The error bars are computed from the spatial and temporal uncertainties using a 95% confidence Monte Carlo error propagation. The dark measurement point of each case marks the instant at which the jet is visibly formed. This instant is determined by finding the first frame, on which a darker section of the radiograph indicates that the bubble has become concave, and that the x-ray light is traveling through water inside the cavity. The recorded speed is directly compared to results taken from the numerical simulations and shows some discrepancies at the final stages. This is to be expected from the mismatch in bubble shape as witnessed in Fig. 5. The maximum recorded jet speeds are in accordance with previously reported values.<sup>13,30,52,58</sup> Note that for
<figure><img src="image_14.png" /><figcaption>FIG. 5. X-ray image sequence of the growth and collapse of a cavitation bubble of $r_{max} = 2.43$ mm, for different stand-off parameters $\gamma$ (right). These snapshots are directly compared to simulations obtained from the boundary integral method at the corresponding times (left). The dimensionless reference times are $t/(r_{max}\sqrt{\rho/\Delta\rho}) = 0.02, 0.10, 0.41, 0.81, 1.22, 1.62$, and $1.83$. The last frame of each image sequence is taken from left to right, at 2.03, 2.05, 2.05, and 2.01. The labels indicate: (1) smaller, secondary cavitation bubbles induced by the laser, (2) the ejection of a thin liquid film during the coalescence process between the main cavitation bubble and the secondary cavitation bubbles, and (3) the impact of microdroplets, originating from the breakup of the thin liquid film, against the cavity surface. Multimedia views: https://doi.org/10.1063/5.0132104.1; https://doi.org/10.1063/5.0132104.2; https://doi.org/10.1063/5.0132104.3; https://doi.org/10.1063/5.0132104.4</figcaption></figure>

$\gamma = 0.53$ and 0.79, the last measurement has a lower speed than all the previous points. This is imputed to the fact that the jet has likely already impacted the solid wall on the last frame considered, and using a higher frame rate would help to mitigate these uncertainties.

Here, the jet impact time, which is defined as the moment the jet reaches the opposite side of the cavity, is found to be almost constant for $\gamma \le 0.79$, i.e., as the bubble deformation becomes more pronounced, which is again in accordance with previous work.$^{22,53}$ The local minima
<figure><img src="image_15.png" /><figcaption>FIG. 6. Image sequence showing the breakup of a thin liquid film, resulting from the coalescence of the main cavitation bubble with smaller ones, into microdroplets which end their course against the inner bubble surface. The images are given for $\gamma = 0.33$ at $t/(r_{max}\sqrt{\rho/\Delta\rho}) = 0.57, 0.85, 1.14$, and $1.42$.</figcaption></figure>

appearing in the BIM results (see Fig. 7) approximately indicate the moment at which the bubbles become concave and form a jet.

### C. Toroidal bubble dynamics

X-ray images of the final stages of a cavitation bubble collapsing at different stand-off distances $\gamma = 0.33, 0.53, 0.79$, and $1.07$ from the solid boundary are displayed in Fig. 8. On the first frame of each image sequence, the jet reaches the opposite side of the cavity, and the bubble becomes toroidal. The volume of the toroidal bubble can be computed from the x-ray images (assuming axial symmetry), thanks to the sharp bubble interface as well as that of the jet inside the bubble, at least up to the final collapse stage. During the rebound, the bubble interface is too perturbed to get a reasonable measurement from the image. The cross section $S$ of the toroidal bubble, clearly visible as two distinct lobes on the x-ray image, is extracted on each frame, and the volume is deduced by assuming

$$V = \pi A_0 S, \qquad (6)$$

where $A_0$ is the distance from the lobe center to the torus center and is measured to stay approximately constant through the collapse phase of the vortex ring. The temporal evolution of the extracted toroidal bubble volume is quantified up to the final collapse and displayed in Fig. 9 for the four stand-off distances. The maximum initial torus volume is found for a stand-off parameter of $0.33 < \gamma < 0.79$, which has previously been suggested by numerical simulations,22,53 where the volume of a cavitation bubble near a flat rigid boundary at jet impact is maximized for $\gamma \approx 0.6$. This is further supported by the Rayleigh prolongation factor,59 which is a value quantifying the prolongation of the collapse time of a bubble located in the vicinity of a solid boundary and is maximum for $\gamma \approx 0.69$. When normalizing the volume by the toroidal bubble’s initial volume, $V_0$, recorded on the frame of jet impact, and the time by the vortex ring’s collapse time $\tau_c$ (time from the jet impact on the

opposite cavity side until the total cavitation bubble collapse), the measurement approximately collapses to a single line following:

$$\frac{V}{V_0} = 1 - \frac{t}{\tau_c}. \qquad (7)$$

This almost linear behavior is induced by the toroidal bubbles having a non-zero radial velocity at the time of formation.57 The measurements are also compared to the inviscid bubble-ring equation,60

$$(r\ddot{r} + \dot{r}^2)\log\left(\frac{r}{8A_0}\right) + \frac{1}{2}\dot{r}^2 = \frac{p_\infty - p_B}{\rho}, \qquad (8)$$

where $r$, $\dot{r}$, and $\ddot{r}$ are the torus minor radius, speed, and acceleration, respectively, and the pressure of the bubble at the interface is

$$p_B = p_v + p_{g,0} \left(\frac{r_0}{r}\right)^{2\kappa} + \frac{\rho}{2} \left(\frac{\Gamma}{2\pi r}\right)^2 - \frac{\sigma}{r}. \qquad (9)$$

This last term includes an adiabatic formulation of the toroidal compression of the non-condensable bubble contents, where $p_{g,0}$ is the initial partial pressure, $r_0$ is the initial torus tube radius, $\kappa$ is the polytropic exponent, and $\Gamma$ is the circulation. The bubble-ring equation is solved numerically using a Runge-Kutta fourth order method and displayed as the dark lines in the top of Fig. 9. The initial conditions are computed from the initial radiographs shown in Fig. 8, which yields an initial radius of the ring measured as

$$r_0 = \sqrt{\frac{S}{2\pi}}, \qquad (10)$$

and an initial radial speed

$$\dot{r}_0 = \frac{\Delta r_0}{\Delta t}, \qquad (11)$$

where $\Delta r_0$ is the change in the torus tube radius between the frame where the torus forms and the previous one and $\Delta t$ is the interframe time of $5 \mu s$. The circulation, $\Gamma$, however, is an unknown parameter in the experiments, which has to be fitted using Eq. (8). These experimental values are compared from the ones extracted at the end of the BIM simulations when the jet impacts the opposite side of the bubble. There, the shape produced by the nodes yields the initial minor and major radius of the torus similarly as in experiments. The speed is computed by averaging the velocity vectors of each node and the circulation by computing the jump in potential across the contact point at the tip of the jet when it reaches the bottom interface and defined as Eq. (10) in the work of Wang et al.20 Table I summarizes all above-mentioned quantities. As can be expected from the last frames of the image sequences displayed in Fig. 5, the differences in bubble shapes when comparing the experiments to the BIM result in differences in $r_0$ and $A_0$, which are, however, considered relatively mild. The disparities in the torus radial speed are also quite important, which is a direct consequence of the mismatch in bubble shape. Interestingly, the circulations found by best fit are, on the other hand, fairly close to the ones obtained by the BIM. They are, however, three orders of magnitude higher than the ones previously measured in a PIV experiment,40 which can be explained by the much larger bubble-rings and earlier stages of the dynamics investigated herein. Circulation acts in a similar way to viscosity, damping the collapse in its latest stage, which explains
<figure><img src="image_16.png" /><figcaption>**FIG. 7.** Temporal evolution of the jet speed taken for various stand-off parameters, $\gamma$. The speed and time are normalized to the bubble characteristic speed, $\sqrt{\Delta p/\rho}$, and time, $r_{max}\sqrt{\rho/\Delta p}$, respectively. The dark curves correspond to jet speeds numerically computed using the boundary integral method. The dark measurement point indicates the first instance at which the bubble adopts a concave shape, and a jet is visibly formed. Different marker shapes are associated with different stand-off distances.</figcaption></figure>

the slight deviations of the torus volume from the $f(x) = 1 - x$ in the normalized representation of Fig. 9.

Qualitatively, a few aspects are worth mentioning. The jets can be seen propagating at a slight angle from the vertical axis, which may result from slight effects from the test chamber boundaries. On the frames subsequent to the collapse, showing the rebound phase, a non-coherent mass of vapor can be seen spreading along the rigid wall. Furthermore, for $\gamma = 1.07$, the bubble separates in two distinct cavities upon collapse [see Fig. 8, indication (1)], which has previously been reported<sup>58</sup> for a similar stand-off parameter of $\gamma = 1.1$. The appearance of a protuberance [see Fig. 8, indications (2)] on the lower periphery of the flattened cavitation bubble is visible for $\gamma \le 0.53$, which has been suggested to result from the hindered motion of the local flow due to its closer proximity to the wall.<sup>61</sup>

## D. Blake's splashing

As the jet impacts the opposite side of the cavity, and thus, the aluminum plate, Blake splashing<sup>54,61,62</sup> [see Fig. 8, indications (3)] occurs and can clearly be distinguished as a ring-like splash structure in the cavity lobes. This last dynamical aspect is herein further investigated, as its morphology has yet to be fully understood, with consequences both on the suitability of numerical models relying on

axisymmetric boundary conditions to predict cavitation events near a solid surface and on the erosion efficiency of the jetting bubble. In fact, the splash can considerably affect the pressure exerted on the wall.<sup>63</sup> By exhibiting a non-axisymmetric fingerlike structure, the damage probability is increased as each finger reaching the bubble interface generates a water-hammer shock wave. In addition, the impact of the splash structure on the solid boundary eventually results in multiple splashes, with potentially high destructive effects. This mechanism, first reported by Tong *et al.*,<sup>52</sup> occurs when the liquid jet, piercing a bubble at relatively low stand-off distances, impacts the thin liquid layer separating the vapor bubble from the rigid boundary. Upon impact, the outward radial flow propagating from the jet axis meets the inflow induced by the bubble shrinking, which eventually generates the splash. Numerical evidence has strongly supported this phenomenology.<sup>19,62</sup> However, clear observations are experimentally challenging as the structure develops inside the bubble, and conventional imaging of the splash is often performed at an angle or aligned in the direction of motion of the jet.<sup>12,61</sup> In this work, radiographs of Blake's splashing, coupled to the jet dynamics, provide a unique opportunity to experimentally quantify the phenomenon. It has been reported for a range<sup>12,61,64</sup> of $0.12 \le \gamma \le 1.02$ and is visualized herein for all stand-off distances in this domain, of which only the largest is quantitatively studied due to its increased resolution. As seen in Fig. 8
<figure><img src="image_17.png" /><figcaption>FIG. 8. X-ray image sequence of the collapse of toroidal bubbles resulting from the impact of a liquid jet with the opposite side of a cavitation bubble of $r_{max}$ = 2.43 mm, for different stand-off parameters $\gamma$. The first frame of each sequences shows the moment the jet reaches the opposite side of the cavity [at $t/(r_{max}\sqrt{\rho/\Delta\rho})$ = 2.03, 2.09, 2.07, and 2.07] after bubble generation, from left to right] and the dimensionless interframe time is 0.02. The labels indicate (1) the bubble separation in two distinct cavities upon collapse, (2) the appearance of a protuberance, and (3) Blake's splashing.</figcaption></figure>

for $\gamma$ = 0.79, the splash is, in the more advanced stages, not axisymmetric and displays a three-dimensional “fingerlike” morphology. Observations show similar traits to the splashing of a droplet on a thin liquid film (see Fig. 10), upon which an analogy is drawn. From low to

moderate Weber and Reynolds numbers, droplets impacting a thin liquid film may experience splashing.⁶⁵ Upon impact, the splashing regime, also usually referred to as the crown or corona splashing, is initiated with the formation of a cusp at a radial distance $r_c$ from the
<figure><img src="image_18.png" /><figcaption>**FIG. 9.** (Top) evolution of the volume, V, of a toroidal bubble with respect to time, t, for different stand-off distances, γ. The solid lines show the numerically computed time evolution of the bubble volume obtained from the bubble-ring equation using the initial minor radius of the torus, r₀, major radius of the torus, A₀, initial radial ring speed, ˙r₀, extracted experimentally and a fitted value for the initial circulation, Γ₀. (Bottom) same but the volume and time are, respectively, normalized to the initial volume and collapse time of the torus, V₀ and τc. The dotted line displays f(x) = 1 - x.</figcaption></figure>

droplet center. The cusp lies at the intersection between the droplet and the upper boundary of the impacted liquid film. The outward motion of the cusp develops into a splash-induced sheet-jet, whose tip is located at a radial distance r<sub>j</sub> and grows to form a corolla propagating radially from the droplet center. The edge of the corolla eventually becomes unstable, which develops multiple fingerlike jets that are ejected outward, resulting in a crown pattern. Capillary-driven mechanisms finally breakup the fingers, generating secondary droplets. The spreading radius of the cusp, r<sub>c</sub>, formed during the impact of a droplet onto a thin liquid film has been shown<sup>65</sup> to scale with r<sub>c</sub> ~ t<sup>1/2</sup> and is well approximated by the time-dependent jet radius,

<table><caption>**TABLE I.** Initial conditions of the bubble-ring equation for different stand-off parameters, γ. The four values obtained from the BIM are compared to the ones extracted experimentally, where the circulation is a fitted parameter using Eq. (8). The bubble ring's initial minor radii, r₀, and major radii A₀, are given in mm. The initial radial velocity, ˙r₀, is given in m·s⁻¹ and the initial circulation Γ₀ in m²·s⁻¹.</caption><thead><tr><th>γ</th><th>1.07</th><th>0.79</th><th>0.53</th><th>0.33</th></tr></thead><tbody><tr><td>r<sub>0,BIM</sub></td><td>0.29</td><td>0.54</td><td>0.59</td><td>0.60</td></tr><tr><td>A<sub>0,BIM</sub></td><td>0.64</td><td>0.86</td><td>1.18</td><td>1.31</td></tr><tr><td>˙r<sub>0,BIM</sub></td><td>54.62</td><td>27.94</td><td>17.37</td><td>11.78</td></tr><tr><td>Γ<sub>0,BIM</sub></td><td>0.10</td><td>0.08</td><td>0.05</td><td>0.04</td></tr><tr><td>r<sub>0,Exp</sub></td><td>0.40</td><td>0.68</td><td>0.69</td><td>0.51</td></tr><tr><td>A<sub>0,Exp</sub></td><td>0.56</td><td>0.83</td><td>1.00</td><td>0.87</td></tr><tr><td>˙r<sub>0,Exp</sub></td><td>26.56</td><td>12.31</td><td>13.01</td><td>14.11</td></tr><tr><td>Γ<sub>0,Fit</sub></td><td>0.10</td><td>0.07</td><td>0.09</td><td>0.05</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>**FIG. 10.** Analogy between (left) the splashing phenomenon during the impact of a droplet onto a thin liquid film and (right) the collapse of a cavitation bubble near a rigid boundary at relatively low stand-off distances (here, γ = 0.79). The left-hand side shows extracted contours from the numerical simulation reproduced with the permission from C. Josserand and S. Zaleski, “Droplet splashing on a thin liquid film,” Phys. Fluids **15**, 1650–1657 (2003). Copyright 2003 AIP Publishing. In both cases, the impact of a liquid body upon a thin liquid film results in splashing, which adopts the form of a jet evolving from an initial cusp located at r<sub>c</sub>.</figcaption></figure>

$$r_c \approx r_j(t) = \sqrt{d_0 U t}, \qquad (12)$$

where d₀ is the drop diameter and U is the impact velocity.

By analogy, here, the bubble jet plays the role of the droplet and the trapped water between the bubble and the rigid boundary corresponds to the thin liquid film, and the spreading of Blake's splashing is determined by computing the radius r<sub>s</sub> of the splash over time for γ = 0.79 (see Fig. 11, top). The comparison with the corresponding prediction reports a very good agreement and suggests that Blake's splashing follows the same scaling law reported for droplet splashing. The bubble jet velocity and radius used for the theoretical prediction are the jet impact speed and the jet radius measured on the first frame showing the splash. The computed theoretical splash-induced jet velocity v<sub>j</sub> agrees unsurprisingly well with the experimental measurements. Here, the vertical error bars come from the image resolution,

<figure><img src="image_20.png" /><figcaption>**FIG. 11.** (Top) spreading of Blake's splashing given by the time-dependent splash radius, r<sub>s</sub>, normalized to r<sub>j,j</sub>, the bubble main jet radius at the time of impact. (Bottom) splash velocity vs. normalized to the bubble main jet speed, u<sub>j</sub>. The dimensional time is non-dimensionalized with respect to τ = 2r<sub>j,j</sub>/u<sub>j</sub>.</figcaption></figure>
while the horizontal bars indicate the time interval within which the velocity has been computed (circle markers being the mean time coordinate). When contrasting the radial position and speed of the splash with the scaling laws derived for droplet jetting upon impact on a thin liquid film, it appears that cavitation bubbles collapsing near a rigid boundary for relatively low stand-off distances are phenomenologically similar.

However, the structure of the splash has yet to be elucidated. In line with the droplet impact analogy, one can assume the corolla to destabilize under an acceleration-driven mechanism and to form fingers (i.e., Rayleigh-Taylor instability). In Fig. 8 for $\gamma = 0.79$, two fingers are clearly visible in the fourth frame on either side of the cavitation bubble's main jet, while dark discontinuities are perceptible in-between, suggesting additional fingers at the back of the jet, inside the cavity. After image processing involving contrast enhancement and signal normalization, the signal intensity profile along the splash diameter $d_s$ is extracted to detect discontinuities. The resulting intensity profile is given as a solid line in Fig. 12. This demonstrates the three-dimensional morphology of the splash, which, at this stage, is no longer axisymmetric but consists of periodically distributed fingers. The dimensionless spatial period $\ell/d_s$ of the signal is extracted by fitting a Fourier series, represented as the dashed line in Fig. 12 and found to be $0.2 \pm 0.06$, where $\pm 0.06$ is determined using a peak detection algorithm of the raw signal. However, due to signal integration, these results are insufficient to discuss the possibility of fingers not being side-by-side, but overlapping. Therefore, the spatial period measured is a minimum that could possibly be doubled. Figure 12 qualitatively supports the droplet impact analogy, for which the Rayleigh-Taylor instability is inferred to be the underlying mechanisms for the finger formation, but additional measurements are still necessary to validate it. Nonetheless, the results confirm previous suggestions¹⁹ that axisymmetric models are unable to capture the detailed, multiscale 3D features of the bubble collapse near a boundary.

Finally, at the latest stage, while the cavitation bubble is still collapsing and its wall is contracting, the fingerlike splash structure interacts with the surface of the cavity. The latter, initially smooth, becomes ruffled. Radiographs show evidence of the fingers crossing the bubble interface, as clearly visible in Fig. 13 for $\gamma = 0.79$. Four distinct fingers, denoted as $f_1, f_2, f_3$, and $f_4$, are observed on the left-hand side of the

<figure><img src="image_21.png" /><figcaption>**FIG. 12.** Detection of the splash-induced fingers based on the signal intensity profile along the splash spanwise diameter (see Fig. 8, 4th frame for $\gamma = 0.79$).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**FIG. 13.** The initially smooth bubble surface becomes corrugated when the destabilized splash interacts with it. Four fingers $f_1, f_2, f_3$, and $f_4$ are visible on the left-hand side of the bubble. Note that no fingers are distinguishable on the right-hand side, where the bubble interface still exhibits some surface perturbations. This image corresponds to the sixth frame of Fig. 8 for $\gamma = 0.79$.</figcaption></figure>

bubble. The distance between $f_3$ and $f_4$ is $\sim 1.9\ell$, which confirms the order of magnitude of the inter-finger distance. It also suggests that fingers overlap on the radiograph, which results in a doubled dimensionless spatial period in Fig. 12, but the discrepancy might also originate from curvature effects. Nevertheless, the observation of the latter stage of the fingers complements the first confirmation on the 3D nature of Blake's splash.

## IV. CONCLUSION

In this work, high-speed synchrotron x-ray phase-contrast imaging has been used to investigate the classical problem of laser-induced cavitation bubble dynamics at different distances from a rigid boundary. To do so, a tailored experimental setup has been designed, which is capable of reducing the x-ray absorption distance of water, keeping losses in light intensity below 70%, while mitigating boundary effects on the experiment. Thanks to the x-ray, unprecedented optical access to the liquid/gas interface results in a clear visualization of the liquid jet perforating the cavity as well as the temporal evolution of the cavity lobes after the jet has impacted the opposite bubble wall, which makes the bubble assume a toroidal shape. The unique, highly resolved images allow for a one-to-one comparison between the boundary integral method and the experiments, which displays a reasonable match considering the inviscid and incompressible nature of the code. The radiographs also enable an accurate measurement of the temporal evolution of the liquid jet speed as well as the toroidal bubble volume. The liquid jet speed displays some expected discrepancies with the BIM. The inviscid bubble-ring equation is fitted to the measurements with the circulation as a fitting parameter and based on experimentally extracted initial conditions. The fitted circulation is in reasonable agreement with the one computed from the boundary integral method despite the variations in bubble shape and liquid jet speed.

By sharply resolving the gas-liquid interface, radiographs offer a unique visualization of Blake's splashing by revealing its morphology consisting of periodically distributed fingers. Experimental data have been validated against scaling laws for droplet impact on a thin liquid film, which contribute to confirm and elucidate the splashing phenomenon. The signal intensity fluctuation along the splash diameter, illuminating the fingers, allows for the determination of their spatial
period. This latter quantification is of major importance for future
works aiming to model the finger formation.

## ACKNOWLEDGMENTS

The authors acknowledge the help of Michael Egli in building
the experimental setup, the financial support from ETH Zurich, and
the ETH Zurich Postdoctoral Fellowship program as well as the
European Synchrotron (ESRF) for providing the synchrotron
radiation facilities and the beamline ID19 (Proposal MI-1397, pilot
of a BAG project supported by the European Union's Horizon 2020
research and innovation program under Grant Agreement No.
870313, Streamline).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Guillaume Thomas Bokman:** Data curation (equal); Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal). **Luc Biasiori-Poulanges:** Formal analysis (equal); Methodology (lead); Visualization (equal); Writing – original draft (equal). **Bratislav Lukic:** Investigation (equal); Writing – original draft (equal). **Claire Bourquard:** Investigation (equal). **Daniel Werner Meyer:** Data curation (equal); Formal analysis (equal); Software (lead); Writing – original draft (equal). **Alexander Rack:** Project administration (equal); Resources (equal). **Outi Supponen:** Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available
within the article.

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