# Dynamics of a Laser-Induced Bubble above the Flat Top of a Solid Cylinder—Mushroom-Shaped Bubbles and the Fast Jet

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^{*}

## Abstract

**:**

## 1. Introduction

_{p}r

_{p}, ${l}_{\mathrm{p}}$ cannot be neglected. Then it holds for ${l}_{\mathrm{p}}^{*}={l}_{\mathrm{p}}/{R}_{\mathrm{max},\mathrm{unbound}}\to 0$ that the case of a bubble in front of a solid plane is approached (irrespective of ${r}_{\mathrm{p}}^{*}$). For ${l}_{\mathrm{p}}^{*}\to \infty $ the limit case studied here is obtained. The transition to pillars of small height would need substantial further work and is not studied here. The present work investigates the details of the dynamics in the parameter range $0.047<{D}^{*}<2.009$ and $0.251<{r}_{\mathrm{p}}^{*}<0.893$. This parameter region covers very peculiar bubble dynamics, and, in particular, cases with experimentally observable fast-jet formation are included.

## 2. Experimental Methods

## 3. Numerical Methods

#### 3.1. Bubble Model

#### 3.2. Equations of Motion

`compressibleInterFoam`of the open source package OpenFOAM is used for the implementation of the equations. For details, see the description and validation in Refs. [42,53,59]. Discretization of the above partial differential equations is done with the finite volume method. Because of the segregated nature of the solver, the phase with lower density (gas) undergoes a mass conservation error over time, if not accurately compensated. This was noted already in [42,94]. In the present case, the solver was altered in such a way that the continuity Equation (6) for the gas phase is re-evaluated right after the last iteration over the pressure equation. Details about this procedure are given in Appendix B.

#### 3.3. Initial and Boundary Conditions, Meshes, and Time Steps

`snappyHexMesh`is used to remove the cylinder with radius ${r}_{\mathrm{p}}$ from the mesh in the region $\left(\right)$. The 2D mesh consists of 106,827 cells.

`snappyHexMesh`. The base mesh consists of 46 × 46 × 46 cells with uniform grid spacing. Towards the inner center the cells are refined in domains of concentric spheres such that a final cell edge length of 1.7 µm is achieved, see Figure 3 (right) for a sketch. The innermost refinement domain has a radius of 65 µm and is centered around the origin of the initial bubble. The limited cube domain size of 10 mm edge length clearly lowers the maximum extension of the bubble, therefore the results of the 3D calculations shown here are meant to be understood in a qualitative way. They were done to show the validity of the 2D approach.

`wave transmissive`with a maintenance of an average pressure of ${p}_{\infty}=\mathrm{101,315}\mathrm{Pa}$. At the surface of the cylinder a no-slip condition for the velocity and zero-gradient condition for pressure is imposed. In addition, ${\alpha}_{l}=1$ to account for a liquid layer without the need to precisely resolve it.

#### 3.4. Validation

## 4. Experimental Results

#### 4.1. Typical Mushroom Bubble Dynamics

#### 4.2. Typical Mushroom-Bubble Collapse and Rebound

## 5. Numerical Results

#### 5.1. Comparison of an Experimental Mushroom Bubble with Simulations

#### 5.2. Pressure, Velocity and Flow Fields of a Mushroom Bubble

#### 5.3. General Dynamics of a Mushroom Bubble

#### 5.4. Simulation in Full 3D

#### 5.5. Parameter Study

**E**and

**F**the length of the jet is given, measured from the point of formation to the point of impact onto the pillar surface. When the jet length is put into relation to ${R}_{\mathrm{max},\mathrm{unbound}}$, it is seen that it approaches or exceeds the bubble maximum radius only for higher values of ${D}^{*}$. When the jet length is compared to the cylinder radius (Figure 20F), the jet becomes longer for smaller ${r}_{\mathrm{p}}^{*}$ as well (for a fixed ${r}_{p}$). The jet length varied by less than 10% with grid alterations. It could be an interesting quantity from an experimental point of view, when photographing the jet is planned.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mass Reduction of the Bubble, Validation of the Bubble Model

**Figure A1.**First collapse, rebound and second collapse of a bubble next to a flat solid boundary with ${D}^{*}=1.6$. Comparison of experiments ([12], Figure 2d) and numerical simulations. Experiment: High-speed pictures in gray scale. Time between frames $\mathsf{\Delta}{t}_{\mathrm{exp}}=17.7$ μs, ${R}_{\mathrm{max},\mathrm{exp}}=1450$ μm, frame width = 3.9 mm. Numerical simulation: The color represents the pressure in bar. The white line indicates the interface (liquid volume fraction ${\alpha}_{l}=0.5$). Time between frames 6.1 μs, ${R}_{\mathrm{max},\mathrm{sim}}=500$ μm, frame size 1.345 mm × 1.42 mm (width × height). The solid boundary is located at the lower border of each frame. Simulation with surface tension set to zero, $\sigma =0$.

## Appendix B. Mass Error Compensation

## Appendix C. Initial Conditions

## Appendix D. Calculation of the Advected Color Layer Map

## Appendix E. Further Parameter-Study Plots

**Figure A2.**Mushroom bubble at ${r}_{\mathrm{p}}^{*}=0.314$ for four different normalized distances ${D}^{*}$ with the same ${R}_{\mathrm{max},\mathrm{unbound}}$. Cylinder radius is 200 µm. Large overlap at maximum bubble volume over the pillar rim. At large overlap and small ${D}^{*}$, the mushroom stem gets long and slim (see the experiments). Note the stack of (torus) bubbles that is shot upwards away from the pillar (last column).

**Figure A3.**Mushroom bubble at ${r}_{\mathrm{p}}^{*}=0.382$ for four different normalized distances ${D}^{*}$ with the same ${R}_{\mathrm{max},\mathrm{unbound}}$. Cylinder radius is 200 µm. Less overlap than in Figure A2.

**Figure A4.**Mushroom bubble at ${r}_{\mathrm{p}}^{*}=0.893$ for two different normalized distances ${D}^{*}$ with the same ${R}_{\mathrm{max},\mathrm{unbound}}$. Very small overlap at maximum volume of the bubble over the pillar rim, no overlap for ${D}^{*}$= 1.116. The mushroom shape is gradually abandoned with increasing distance of the bubble from the pillar top. At least the jet in the last row is a standard jet with involution of the distal part of the bubble and not by annual inflow with self-impact. Cylinder radius is 200 µm.

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**Figure 1.**Sketch of the parameters for classification of bubbles close to a solid cylinder. The bubble is initiated by a hot plasma spot generated by a focused laser pulse. The radius ${R}_{\mathrm{max},\mathrm{unbound}}$ is understood as the maximum radius of a spherical bubble with the same initial energy in an unbounded liquid. The radius of the spherical bubble with the same volume the bubble attains at maximum volume with the cylinder present (blue area) is the equivalent maximum radius ${R}_{\mathrm{max},\mathrm{eq}}$. Note that ${R}_{\mathrm{max},\mathrm{unbound}}$ and ${R}_{\mathrm{max},\mathrm{eq}}$ generally have different values.

**Figure 2.**Sketch of the experimental arrangement. The He-Ne laser allows for triggering onto bubble dynamics phases with approximately 1 µs precision. A typical bubble “shadow” curve with trigger level is shown in the lower right inset. The xenon flash tube refers to the Mecablitz device.

**Figure 3.**Sketch of the numerical set-up (not to scale) and grid parameters of the 2D mesh (

**left**) and sketch of the computational domain and refinement regions for the Cartesian 3D mesh (

**right**). The 2D mesh consists of 106 827 cells. The Cartesian 3D mesh consists of 5.2 million cells.

**Figure 4.**Image sequence of a typical mushroom bubble generated in the center of the top surface of a metal cylinder at ${D}^{*}\approx 0$. Radius of the cylinder top is ${r}_{\mathrm{p}}=272.8\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$. Exposure time is 1 µs. Stacked and interleaved sequences of four recordings with the Photron camera at 21 kfps each. The times for one of the sequences are given. The frames without a time tag are taken from different measurements as an interpolation. The bubble has a maximum horizontal width of $1689\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}\pm 18\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$ at 95.2 µs.

**Figure 5.**Experimental images of a mushroom bubble at ${D}^{*}\approx 0$ similar to Figure 4, but recorded with the substantially higher frame rate of the IMACON camera. Exposure time is 150 ns. Sequences of three recordings are stacked together. The respective recording numbers are denoted in parentheses. Frame rates for the sequences are: (1) $317\phantom{\rule{0.166667em}{0ex}}\mathrm{kfps}$ and for (2) and (3) $870\phantom{\rule{0.166667em}{0ex}}\mathrm{kfps}$. Background of the images was subtracted; then the space occupied by the needle is shown in white pixels. Frame width is $633.5\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}\pm 6.9\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$.

**Figure 6.**Repeatability of the experiments. Five experimental images from five different recordings of mushroom bubbles at ${D}^{*}\approx 0$ similar to Figure 5, extracted at time instants, where the mushroom shape is thinnest. Background of the images was subtracted. Frame width is $633.5\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}\pm 6.9\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$.

**Figure 8.**Neck and stem dynamics of a mushroom-shaped bubble for ${D}^{*}\approx 0,\phantom{\rule{4pt}{0ex}}{r}_{p}=272.8\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$. Experimental images of a bubble similar to Figure 4 and Figure 5, but with ultra-high-speed frame rate (10,000,000 fps and 5,555,556 fps). Exposure time is 30 ns. Five recordings are stacked together (recording number in parentheses). Frame width is $633.5\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}\pm 6.9\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$.

**Figure 9.**Imacon camera frame sequence of a mushroom bubble at a time during ring jet impact at the top of the mushroom cap. Times denote the delay to the camera trigger. Overlay with the volume fraction field ${\alpha}_{l}$ from the axisymmetric, numerical simulation of a bubble at ${D}^{*}=0.057$ and ${r}_{\mathrm{p}}^{*}=0.306$ (mind that the numerical simulation is represented by a cut through the bubble). Background of the experimental images was subtracted. Frame width is $766\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}\pm 10\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$. The cylinder radius ${r}_{\mathrm{p}}$ in the experiment is 272.8 µm, in the simulation ${r}_{\mathrm{p}}$ is 200 µm.

**Figure 10.**Simulation results for the pressure p [bar] of a bubble with ${R}_{\mathrm{max},\mathrm{unbound}}=472.57\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$, ${D}_{\mathrm{init}}=30\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$ and a cylinder radius of ${r}_{\mathrm{p}}=200\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$, resulting into ${D}^{*}$ = 0.06 and ${r}_{\mathrm{p}}^{*}$ = 0.423. White areas belong to the bubble. Cross sections along the center of axial symmetry. In the first frame (upper left frame), the shock wave from the bubble initiation is visible as is the low-pressure wave from diffraction at the rim of the cylinder. The last two frames contain a zoom inset into the region of maximum pressure at the flat top surface of the cylinder. Rayleigh collapse time ${T}_{\mathrm{Rc}}$ is 42.90 µs.

**Figure 11.**Simulation results for the velocity $\mathbf{U}$ of the same bubble and cylinder as in Figure 10. In the first frame (upper left frame), it is seen that the shock wave from bubble initiation induces an outward flow ahead of the expanding bubble surface. The last two frames contain a zoom inset into the region of maximum velocity (the fast jet). In the insets the velocity arrows are omitted for the sake of visibility. Cylinder radius is 200 µm.

**Figure 12.**Flow field simulations presented with the help of a warped color layer tracer field of the same bubble and cylinder as in Figure 10. White areas belong to the bubble. Cross sections along the center of axial symmetry. Note the palmette-like flow structure in the last frame. Cylinder radius is 200 µm.

**Figure 13.**Enlargement of the bubble on top of the pillar at time 85.906 µs with the warped color layer tracer field of Figure 12. The white area is the bubble foot stand. It still collapses further. A stack of vortices develops from the upwards flow.

**Figure 14.**General dynamics of a mushroom bubble, part 1 shown using the same bubble as in Figure 10, Figure 11, Figure 12 and Figure 13 (${R}_{\mathrm{max},\mathrm{unbound}}$ = 472.57 µm, ${D}^{*}$ = 0.063, ${r}_{\mathrm{p}}^{*}$ = 0.423, ${T}_{\mathrm{Rc}}=42.90\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{s}$): expansion and begin of collapse, plotted with the modulus of the liquid velocity field. The red arrows indicate the essential directions of the flow field. The dashed red circles in the upper right frame indicate (torus) droplets that impinge onto the inner bubble wall. The dashed red circles in the lower left frame point to the area of the shedding of sub-resolution (torus) bubbles from the rim of the collapsing mushroom head. They finally form the umbrella left after collapse of the mushroom head. (see also Figure 15). Cylinder radius is 200 µm.

**Figure 15.**General dynamics of the mushroom bubble, part 2 (${R}_{\mathrm{max},\mathrm{unbound}}$ = 472.57 µm, ${D}^{*}$ = 0.063, ${r}_{\mathrm{p}}^{*}$ = 0.423, ${T}_{\mathrm{Rc}}=42.90\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{s}$): fast jet formation, bubble collapse and rebound, plotted with the modulus of the liquid velocity field. The red arrows indicate the essential directions of the flow field. The fast jet in this case reaches more than 700 m/s (see dashed rectangle on top of the velocity scale in frame 2). The dashed red circles in the lower left frame denote the region, from where shock waves are emitted. Cylinder radius is 200 µm.

**Figure 16.**Full 3D simulation for ${D}^{*}=0.057,\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{p}}^{*}=0.382$ (large overlap of the bubble over the pillar rim), showing the “mushroom dynamics” including the projectile. Cylinder radius is 200 µm. The quarter volume of the bubble pointing to the front has been cut out for visualizing the bubble shape by a black contour line with the fast jet inside, first to be seen at 92.124 µs. The liquid is the white background for best visualization of the bubble and its contour.

**Figure 17.**Mushroom bubbles with the magnitude of the flow velocity of the liquid. Bubbles at the same ${r}_{\mathrm{p}}^{*}=0.423$ for two different normalized distances ${D}^{*}$ with the same ${R}_{\mathrm{max},\mathrm{unbound}}$. The individual color scales give the maximum velocity in the respective frame. Note the small bubble in the second row, second frame, on top of the bubble stem and the stack of small (torus) bubbles that are shot upwards away from the pillar by the upwards flow after annular-flow impact (last column). They can be identified with the projectile in the experiment. When comparing with the experiment, note that the numeric pictures are presented as a cut through the bubble and should be complemented by rotation of the cut for a full bubble view. Cylinder radius is 200 µm.

**Figure 18.**Mushroom bubbles with the magnitude of the flow velocity of the liquid. Bubbles at the same ${r}_{\mathrm{p}}^{*}=0.423$ for two further normalized distances ${D}^{*}$ with the same ${R}_{\mathrm{max},\mathrm{unbound}}$, in addition to Figure 17. Cylinder radius is 200 µm.

**Figure 19.**Mushroom bubble with the magnitude of the flow velocity of the liquid. Bubble at ${r}_{\mathrm{p}}^{*}=0.893$ and ${D}^{*}$ = 0.134. Very small overlap at maximum volume over the pillar rim. More examples with this ${r}_{\mathrm{p}}^{*}$ in Figure A4 in Appendix E. Cylinder radius is 200 µm.

**Figure 20.**Extracted quantities from the parameter study including data points for both ${r}_{\mathrm{p}}=160\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$ and ${r}_{\mathrm{p}}=200\phantom{\rule{0.166667em}{0ex}}\mu \mathrm{m}$. Plotted are (

**A**) the tentative results of the jet velocity for the given mesh (results may differ substantially for other meshes), (

**B**) the tentative results of the time interval from jet formation till jet impact for the given mesh (results may differ substantially for other meshes), (

**C**) the collapse time normalized by the Rayleigh collapse time, the number being called the prolongation factor, (

**D**) the actual, equivalent maximum bubble radius normalized by ${R}_{\mathrm{max},\mathrm{unbound}}$, (

**E**) the jet length normalized by ${R}_{\mathrm{max},\mathrm{unbound}}$, (

**F**) the jet length normalized by the cylinder radius ${r}_{p}$. White data points denote cases, where either the bubble dynamics was different from the mushroom case or a standard jet by involution of the bubble wall was observed.

**Figure 21.**Jet velocities of a bubble in the vicinity of a flat solid boundary, measured and simulated by different authors: Philipp and Lauterborn [12] (measurements), Brujan et al. [96] (measurements), Supponen et al. [18] (Boundary Element simulation), Lechner et al. [53] and Koch [58] (simulations with the present code), as well as Koch [58] and Koch et al. [59] (measurement of the fast jet). The experimental data in [12,96] were obtained for laser-generated cavitation bubbles in water under normal ambient conditions (${p}_{\infty}=\mathrm{101,325}$ Pa and ${\rho}_{l}=998\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$). In the numerical work in [53,58,59], the same ambient conditions were assumed. The dimensionless values given in [18] have been incorporated into this diagram, using the values $\mathsf{\Delta}p={p}_{\infty}-{p}_{v}$ with ${p}_{\infty}=\mathrm{101,325}$ Pa, ${p}_{v}=2337$ Pa and ${\rho}_{l}=998\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$.

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## Share and Cite

**MDPI and ACS Style**

Koch, M.; Rosselló, J.M.; Lechner, C.; Lauterborn, W.; Mettin, R.
Dynamics of a Laser-Induced Bubble above the Flat Top of a Solid Cylinder—Mushroom-Shaped Bubbles and the Fast Jet. *Fluids* **2022**, *7*, 2.
https://doi.org/10.3390/fluids7010002

**AMA Style**

Koch M, Rosselló JM, Lechner C, Lauterborn W, Mettin R.
Dynamics of a Laser-Induced Bubble above the Flat Top of a Solid Cylinder—Mushroom-Shaped Bubbles and the Fast Jet. *Fluids*. 2022; 7(1):2.
https://doi.org/10.3390/fluids7010002

**Chicago/Turabian Style**

Koch, Max, Juan Manuel Rosselló, Christiane Lechner, Werner Lauterborn, and Robert Mettin.
2022. "Dynamics of a Laser-Induced Bubble above the Flat Top of a Solid Cylinder—Mushroom-Shaped Bubbles and the Fast Jet" *Fluids* 7, no. 1: 2.
https://doi.org/10.3390/fluids7010002