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

Abstract

The dynamics of a laser-induced bubble on top of a solid cylinder is studied both experimentally and numerically. When the bubble is generated close to the flat top along the axis of the cylinder and its maximum radius exceeds the one of the flat top surface, it collapses in the form of a mushroom with a footing on the cylinder, a long stem and a hat-like cap typical for a mushroom head. The head may collapse forming a thin, fast liquid jet into the stem, depending on bubble size and bubble distance to the top of the cylinder. Several experimental and numerical examples are given. The results represent a contribution to understand the behavior of bubbles collapsing close to structured surfaces and in particular, how thin, fast jets are generated.

1. Introduction

Cavitation bubbles of millimeter and sub-millimeter size have been investigated for more than 100 years. The interest arose from their destructive action on solid surfaces, notably propeller blades [1,2]. Since then it is known that cavitation possesses the ability to erode the hardest materials. However, despite considerable efforts to elucidate the erosion process by cavitation bubbles, the precise mechanisms are still under discussion. Hence, there is ongoing interest in cavitation bubble dynamics. When a cavitation bubble is formed, it usually undergoes rapid expansion and—due to ambient pressure or acoustic driving—rapid collapse with subsequent rebound and further collapses and rebounds [3,4]. The dynamics of the bubble is highly influenced by many factors: the properties of the surrounding liquid (density, viscosity), the bubble contents (gas, vapour), the bubble–liquid interface (surface tension, coating), outer factors (pressure, temperature, gravity) and, in particular, the large class of geometrical constraints, i.e., boundaries or objects nearby with different properties from flat to curved or smooth to corrugated and solid to soft. For systematic studies on geometrical constraints, isolated single bubbles are required. The liquid breakdown induced by a focused laser light pulse [5,6] has been used for this purpose. For the beginning of cavitation bubble dynamics studies with laser-seeded bubbles, see [7,8]).

Since the first studies with laser-induced bubbles in constraint geometries [7], vast knowledge has already been accumulated with this method. In particular, the phenomenon of liquid jet formation has been profoundly studied for different geometries, notably flat solid boundaries, but also concave and convex ones [9,10]. When the bubble expands and collapses in the vicinity of constraints, it is pierced by the liquid. The liquid parcel that flows through the bubble then is called a jet. Moreover, besides single spherical bubble expansion and collapse in the vicinity of an extended flat solid boundary (see, e.g., [11,12,13,14,15,16]) with its jet towards the boundary, bubble expansion and collapse in the vicinity of a free boundary (see, e.g., [17,18]) has been studied with its jet away from the boundary. Jet formation has also been studied in the vicinity of an elastic boundary (e.g., [19]) and in a (low) gravity field (see, e.g., [20,21,22]).

First reported experiments on jet formation started in 1961 with the work of Naudé and Ellis [23] and the availability of high-speed photography [7,20,24]. The numerical work started in 1971 with the work of Plesset and Chapman [25] and was essentially promoted by Blake and his group [11,26,27,28,29]. The solid-boundary case with its jets was and still is heavily studied (see, e.g., [13,14,16,18,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]). This jet is called micro-jet and exhibits speeds in the order of 100 m/s. The speed depends on the normalized distance $D^{*}$, that is given by

$$\begin{array}{c}\hfill D^{*}=\frac{D_{\mathrm{init}}}{R_{\max ,\mathrm{unbound}}},\end{array}$$

(1)

where $D_{\mathrm{init}}$ is the initial distance of the (laser-induced) bubble to the solid boundary and $R_{\max ,\mathrm{unbound}}$ is the maximum radius the bubble would attain in an unbounded liquid. More precisely, depending on the normalized distance $D^{*}$, jet velocities between about 35 m/s at $D^{*}$ = 0.3 and about 200 m/s at $D^{*}$ = 3 were measured by [12,58,59] and calculated by [18,19,53,59]. A comparison of velocities of numerical simulations with experiments (with very good agreement) can be found in the work of Koch [58]. A detailed comparison of jet velocities gained by different authors and methods is given in Section 6.

Recently, it was found with the help of laser-induced bubbles that for very low initial distances of the bubble to a solid surface, a different jetting mechanism occurs, producing a fast jet. It was predicted numerically by Lechner et al. [60] and Pishchalnikov et al. [61] and found experimentally already by Benjamin and Ellis [20] with acoustically driven bubbles and recently by Koch et al. [59] with laser-induced bubbles. The fast jets reach speeds in the order of 1000 m/s. They are produced by annular inflow with self-impact and by squeezing the liquid into two opposite directions [62].

After finding the fast jet at an extended flat solid boundary, the question arises as to whether there exist further geometrical configurations of collapsing bubbles at boundaries or objects that provoke a fast jet as well. This is indeed the case. The configuration considered is a bubble expanding and collapsing next to the flat top of a long solid cylinder (a round pillar). Studied is the axially symmetric case, i.e., the bubble center remains on the axis of rotational symmetry of the cylinder. With an extended flat top and a small bubble, this configuration can be considered a slight variant of the extended flat solid boundary. With a shrinking flat top down to the dimensions of a bubble, deviations in the dynamics of the bubble are to be expected. Indeed, a bubble above a solid cylinder was found to have interesting dynamics—with strong association to mushroom shapes. Figure 1 shows the parameters used to characterize the case under study. It contains four independent, geometric parameters: $D_{\mathrm{init}}$—the initial distance of the laser seeded bubble to the flat top of the cylinder; $R_{\max ,\mathrm{unbound}}$—the maximum radius the bubble would attain after expansion in a free, unbounded liquid; $l_{\mathrm{p}}$—the length of the cylinder (height above a planar, solid boundary); $r_{\mathrm{p}}$—the radius of the cylinder. As only the case of a very long cylinder, $l_{\mathrm{p}}\gg r_{\mathrm{p}}$, is considered, the length parameter can be omitted. Thus, there is one more parameter compared to the case of the bubble close to a planar, solid boundary, which is the radius of the cylinder, $r_{\mathrm{p}}$. From these geometric parameters the dimensionless parameter $D^{*}$ has already been defined. A further dimensional parameter for the size of the pillar top surface with respect to the bubble size can be defined as

$$\begin{array}{c}\hfill r_{\mathrm{p}}^{*}=\frac{r_{\mathrm{p}}}{R_{\max ,\mathrm{unbound}}},\end{array}$$

(2)

with $r_{\mathrm{p}}^{*}$ the normalized radius of the flat top with respect to the maximum spherical radius in a free liquid ($R_{\max ,\mathrm{unbound}}$). The reference to the free bubble is chosen—rather than referencing the maximum equivalent radius of the bubble close to the cylinder, $R_{\max ,\mathrm{eq}}$—because the actual bubble expands to an aspherical shape with a volume, whose equivalent radius varies with $D_{\mathrm{init}}$ for the same initial energy (see also [49]). With the reference to $R_{\max ,\mathrm{unbound}}$, the temporal normalization is uniquely defined by the Rayleigh collapse time given by

$$\begin{array}{c}\hfill T_{\mathrm{Rc}}=0.91468·R_{\max ,\mathrm{unbound}}\sqrt{{\rho }_{\infty }/p_{\infty }},\end{array}$$

(3)

where ${\rho }_{\infty }=998.2 \mathrm{kg}/\mathrm{m}{}^3$ and $p_{\infty }=\mathrm{101,315} \mathrm{Pa}$.

$D^{*}=1$ means that the bubble with the reference definition of Equation (2) would just touch the surface, $r_{\mathrm{p}}^{*}=1$ means that the reference bubble in a free liquid has the same diameter as the cylindrical pillar.

When $r_{\mathrm{p}}^{*}\to \infty$, the limiting case of a bubble in front of a solid plane is attained. When $r_{\mathrm{p}}^{*}\to 0$, the limiting case of a free bubble is approached. Thus the study contributes to the problem of the transition in the dynamics of a single bubble from a free bubble to a bubble restricted in its dynamics by an infinitely extended solid plane boundary. When the length of the cylinder l_p r_p, $l_{\mathrm{p}}$ cannot be neglected. Then it holds for $l_{\mathrm{p}}^{*}=l_{\mathrm{p}}/R_{\max ,\mathrm{unbound}} \to 0$ 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.

In the past, investigation of bubble dynamics near structured objects got less attention than studies on flat or smooth surfaces. However, non-flat and structured surfaces are quite relevant for various applications. Important examples include ultrasonic cleaning [63,64,65,66] and the destruction of urinary stones [61].

Oscillating bubbles in liquids are attracted to solid surfaces (see, e.g., [20], ([1], Ch. 5.9), ([30], Figure 14) and recent work [67]). Together with the fluid flow, jets and shock waves they induce upon bubble collapse, they are ideal objects to loosen and lift off contaminations from those surfaces. These properties of bubbles are widely used for cleaning purposes (see, e.g., [68,69,70]). Furthermore, it is known that cavitation bubbles can reach, clean and also damage crevices, holes, trenches and other complicated surface features (see, e.g., [71,72]). However, how bubbles clean or erode structured surfaces remains fundamentally unknown. Bubble dynamics studies, as prerequisite, are therefore highly welcome. This is done in the present work for the case of a solid cylinder.

Up to now, owing to the large variety of structures and constraints, only a few cases were already investigated for single bubble dynamics. Among the solid boundaries and objects, there are a small hole [73], blind holes/crevices [74], rectangular channels [75], convex surfaces [10], a thin gap (parallel plates) [76,77], rigid spheres [78,79,80,81], a pencil-like electrode [82], ridges and grooves [83], a micro structured riblet [84], edges [85,86] and corners [87].

From the works in the above mentioned list, the work of Tomita et al. [10] and Palanker et al. [82] show mushroom shapes similar to those investigated in the present work, but the fast jets escaped their notice. None of the works on bubbles in constraint geometries of the above list report the fast jet. The maximum axial jet velocity reported in the aforementioned works amounts to 200 m/s in the case of a bubble in a thin gap, whereas in the present work velocities in the order of 1000 m/s will be presented for bubbles on top of pillars.

This work is organized as follows: The experimental methods—seeding bubbles with focused laser light and taking pictures with high-speed photography—are presented in Section 2, and the numerical methods—solving the Navier-Stokes equations with OpenFOAM—in Section 3. The experimental results are given in Section 4: Bubbles on top of the cylinder were produced and photographed according to the arrangement of Figure 2. The numerical results are presented in Section 5. The calculations were done in axial symmetry (2D) with an example in full 3D to confirm the validity of the axial-symmetry approach.

2. Experimental Methods

The experimental setup is depicted in Figure 2. The design is versatile and could be used for the study of bubble dynamics at other objects as well. Here, a single bubble is generated in water by optical breakdown of a nanosecond laser pulse that is focused along the symmetry axis of the cylindrical object.

The laser pulse of wavelength 532 nm is produced by a Q-switched Nd:YAG laser (Litron Nano PIV). The laser pulse duration, $t_p=10 \mathrm{ns} \mathrm{FWHM}$, was measured with a high-speed photodiode (Thorlabs SV2-FC). The impulse response of the photodiode amounts to 0.8 ns. The diode signal was recorded with a 40 Gigasamples per second ($\equiv 40 \mathrm{S}/\mathrm{ns}$), 2.5 GHz ($0.4 \mathrm{ns}$ time resolution) bandwidth oscilloscope (Tektronix DPO 7254). The laser beam is expanded to approximately 2.5 cm in diameter and is then focused into a rectangular glass cuvette (Hellma) filled with de-ionized water at room temperature by means of a lens with a focal length of 35 mm. The laser focus-spot was placed onto the flat top of the cylinder object. The cylinder actually was a sewing pin needle, which was ground to flat-top by diamond grinding. The needle top has a diameter of $(545.6\pm 5.9) \mu \mathrm{m}$.

The complete bubble life-time was captured with a Photron APX-RS camera (model 250K) at a minimum shutter time of 1 µs and frame rates of 21 kiloframes per second (kfps) at 384 × 272 pixels or 100 kfps at 244 × 222 pixels. Simultaneously, pictures of the bubble evolution within a reduced time window of around 1 µs were taken with an Imacon 468 camera. This camera can record up to 100 megaframes per second (Mfps) with a sensor field of 385 × 575 pixels. However, only eight images in total are captured for any time resolution, but all eight images can be set arbitrarily in start time, exposure time and voltage gain. Both cameras were equipped with a K2 Infinity long-distance microscope with large working distances ranging from 54 mm to 200 mm. In combination with the CF-2 and CF-4 modules the K2-objectives allow for resolutions down to 4 µm/pixel (CF-2) and down to $\approx 2 \mu \mathrm{m}$/pixel (CF-4). The illumination was done from the opposite side of the imaging optics (backlighting) with the xenon flash described below. The cylinder top, i.e., the spot of bubble seeding, was put into the center of the water volume inside the glass cuvette.

To circumvent the relatively high jitter of the bubble collapse time, a trigger signal was derived from a He-Ne laser beam passing through the bubble and hitting the high-speed photodiode described above. This triggering method dates back to Lauterborn [8] and has been applied by others, too (see e.g., [88,89]). The method made it possible to capture the bubble dynamics in a repeatable manner at a desired phase, e.g., the collapse phase. The path of the He-Ne laser beam first was directed through the bubble seeding site, and then through various mirrors and pinholes onto the photodiode that was shaded against any stray light from the illumination for the recordings. Depending on the size of the bubble, the light is more or less deflected yielding a photodiode signal, which is highest when the bubble is absent. The low-pass filtered signal was fed into the oscilloscope that could generate a hardware trigger TTL output. When the photodiode voltage crossed a certain level with positive slope, this output then triggered the Imacon camera for megaframes per second (Mfps) recording. This way, the trigger accuracy could be increased to approximately 1 µs precision at the time of minimum bubble volume. This is sufficient to obtain up to 10 Mfps resolution with one single measurement to capture the bubble collapse.

To illuminate the bubbles for megaframes-per-second recordings, the optimum was reached by xenon flash tubes of minimum 30 J light emission in combination with a reflector and a Fresnel lens. The light emission duration is in the range 500 µs to 8 ms, depending on the output energy. The xenon flash used was a photo flash (Mecablitz 36CT2) with a straight flash tube of 35 mm length and about 35 J output, directed onto a Fresnel lens. The amount of light reaching each of the camera objectives had to be adapted according to their specific shutter speeds. This was done by reflecting only part of the light from the flash beam towards the objective of the Photron camera.

3. Numerical Methods

An extended description of the numerical method is given in Koch et al. [42], Lechner et al. [53] and Koch [58]. For the convenience of the reader, the essential details are repeated here with respect to the special situation of the present study, for instance the grid settings.

3.1. Bubble Model

A bubble model for a cold liquid (a liquid far from its boiling point, (cf. [1])) with the following properties is used. The bubble contains a small amount of non-condensable gas to comply with experiments [42,90,91,92]. The vapor pressure is small compared to the ambient pressure and is neglected. The phenomenon of laser-induced breakdown to produce a bubble is not modelled. Instead, an initial bubble of high internal pressure is taken that expands to the experimental maximum radius. The condensation of the vapour part in the bubble is modelled by a reduction of the radius at rest of the bubble, $R_n$. More details are given below and in Appendix A. The validity of the approach has been checked by comparison with experiments in a similar approach [42] and is further demonstrated in Figure A1 in Appendix A. The liquid is taken as compressible for inclusion of pressure waves up to weak shock waves [42]. Thermodynamic effects and mass exchange through the bubble wall are neglected. Gravity can be omitted due to the small size of the bubble. Surface tension does not play a significant role for the phenomena described here and, therefore, is not included, see the discussion in Ref. [53].

3.2. Equations of Motion

The equations of motion of the two-phase flow are formulated in the “one-fluid” approach, i.e., with one density field $\rho (x,t)$, one velocity field $\mathit{U}(x,t)$, and one pressure field $p(x,t)$, satisfying the Navier-Stokes Equation (4) and the continuity Equation (5):

$$\begin{array}{ccc}\hfill \frac{\partial \left(\rho \mathit{U}\right)}{\partial t}+\nabla ·(\rho \mathit{U}\otimes \mathit{U})& =& -\nabla p+\nabla ·\left[\mu \left(\nabla \mathit{U}+{(\nabla \mathit{U})}^T-\frac{2}{3}(\nabla ·\mathit{U})\mathbb{I}\right)\right],\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \frac{\partial \rho }{\partial t}+\nabla ·\left(\rho \mathit{U}\right)& =& 0.\hfill \end{array}$$

(5)

∇ denotes the gradient, $\nabla ·$ is the divergence, and ⊗ the tensorial product. $\mu (x,t)$ denotes the viscosity field and $\mathbb{I}$ the unit tensor. Viscosity is included as it was found essential for the dynamics of a bubble close to a solid boundary [53,60].

In order to distinguish between liquid (l) and gas (g), volume fraction fields ${\alpha }_l(x,t)$ and ${\alpha }_g(x,t)$ are introduced with ${\alpha }_l=1$ in the liquid phase, ${\alpha }_l=0$ in the gas phase, and ${\alpha }_g(x,t)=1-{\alpha }_l(x,t)$. The position of the interface is then given implicitly by the transition of ${\alpha }_l$ from 1 to 0. The viscosity field $\mu (x,t)$ can be written as $\mu (x,t)={\alpha }_l(x,t){\mu }_l+{\alpha }_g(x,t){\mu }_g$ (see, e.g., [93]). The dynamic viscosities ${\mu }_l$ of the liquid and ${\mu }_g$ of the gas are taken to be constant (${\mu }_l=1.002\times {10}^{-3} \mathrm{kg}{\mathrm{s}}^{-1}{\mathrm{m}}^{-1}$, ${\mu }_g=1.86\times {10}^{-5}\mathrm{kg}{\mathrm{s}}^{-1}{\mathrm{m}}^{-1}$). The density field $\rho (x,t)$ is given by $\rho (x,t)={\alpha }_l(x,t){\rho }_l(x,t)+{\alpha }_g(x,t){\rho }_g(x,t)$ with ${\rho }_l$ and ${\rho }_g$ the densities of the liquid and gas, respectively. As there is no mass transfer between bubble interior (gas) and exterior (liquid), the respective phase-fraction density fields ${\alpha }_l{\rho }_l$ and ${\alpha }_g{\rho }_g$ separately obey the continuity equation

$$\frac{\partial \left({\alpha }_i{\rho }_i\right)}{\partial t}+\nabla ·\left({\alpha }_i{\rho }_i\mathit{U}\right)=0,i=l,g.$$

(6)

The equations of motion are closed by the equations of state for the gas and the liquid. For the gas in the bubble, the change of state is assumed to be adiabatic,

$$\begin{array}{c}\hfill {\rho }_g\left(p\right)={\rho }_{gn}{\left({\displaystyle p/p_n}\right)}^{1/{\gamma }_g},\end{array}$$

(7)

with $p_n$ and ${\rho }_{gn}$ the pressure and the density of the gas in the bubble of equilibrium radius $R_n$ at normal conditions, respectively, and ${\gamma }_g=1.4$ the ratio of the specific heats of the gas (air). For the liquid, the Tait equation of state for water is used (see, e.g., [90]):

$$\begin{array}{c}\hfill {\rho }_l\left(p\right)={\rho }_{\infty }{\left((p+B)/(p_{\infty }+B)\right)}^{1/n_T}\end{array}$$

(8)

with $p_{\infty }$ the atmospheric pressure, ${\rho }_{\infty }$ the equilibrium density, the Tait exponent $n_{\mathrm{T}}=7.15$ and the Tait pressure $B=305\mathrm{MPa}$.

The segregated, pressure-based two-phase solver 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

Initially, a spherical bubble with high internal pressure is placed at a distance $D_{\mathrm{init}}$ from the top of the cylinder in a still liquid under ambient pressure $p_{\infty }$. The initial bubble radius is set to $R_{\mathrm{init}}=20\mu \mathrm{m}$ for all simulations. The bubble gas content is defined by the equilibrium radius $R_n$, when mass exchange across the interface is neglected, as in the present model. In reality, rapid evaporation, condensation and gas diffusion phenomena may play a role at the bubble interface. A simple approximation was chosen to account for it. In [95], a good agreement between simulations with the Gilmore model of a spherical bubble and experiments could be achieved by reducing the equilibrium bubble radius $R_n$ by 60 % during each local maximum of the bubble radius. In order to model both the strong first expansion and strong collapse of the bubble, a very high initial gas content, hence high $R_{n,1}$, is chosen that is linearly reduced in a specified time interval to approximately one third, hence $R_{n,2}$. Details on this procedure are given in Appendix A. The initial pressure in the bubble is obtained by compressing the gas content in the bubble adiabatically from the equilibrium radius $R_{n,1}$ to $R_{\mathrm{init}}$, i.e., $p_{g,\mathrm{init}}=p_n{\left({\displaystyle R_{n,1}^3/R_{\mathrm{init}}^3}\right)}^{{\displaystyle {\gamma }_g}}$, resulting in values of a few GPa for the cases studied here. Details on how the initial data are set are given in Appendix C. For a fixed $R_{\mathrm{init}}=20\mu \mathrm{m}$ the following equation describes the relation between the initial data ($R_{n,1}$) and the resulting $R_{\max ,\mathrm{unbound}}$:

$$\begin{array}{c}\hfill R_{\max ,\mathrm{unbound}}[\mu \mathrm{m}]=3.1662·R_{n,1}[\mu \mathrm{m}]-83.1[\mu \mathrm{m}].\end{array}$$

(9)

The Equation (9) is found for $R_n\in [97 \mu \mathrm{m},227.25 \mu \mathrm{m}]$. When including larger values of $R_n$ the relation becomes slightly non-linear. Simulations were mainly carried out in axial symmetry (“2D”). Distinct simulations were performed in full 3D. A sketch of the numerical set-up together with the grid parameters of the meshes is given in Figure 3. The mesh consists of a symmetry axis on the y-axis, as well as a central region, $x,\left|y\right|<\mathrm{Xi}=80\mu \mathrm{m}$, with Cartesian orientation of cells and uniform grid spacing $\mathsf{\Delta }x=1 \mu \mathrm{m}$ and $\mathsf{\Delta }y=1\mu \mathrm{m}$. Via a transition region between Xi and Xii, the Cartesian grid is matched to a polar grid. The polar grid starts from $r=\sqrt{x^2+y^2}=\mathrm{Xii}\approx 136 \mu \mathrm{m}$. Cells in the polar grid have an aspect ratio of unity up to a distance of $r=\mathrm{X}\approx 1.04 \mathrm{mm}$. Further out, a stronger radial edge grading $g_f\approx 5$ of the cells is applied, so that the cells become longer in the radial direction. This numerically dampens outgoing shock waves, while keeping the overall cell amount low. The outer boundary of the computational domain is located at a distance of 52 mm from the origin. The utility snappyHexMesh is used to remove the cylinder with radius $r_{\mathrm{p}}$ from the mesh in the region $\left\{x\in \{0,r_{\mathrm{p}}\},y\in \{-52 \mathrm{mm},-D_{\mathrm{init}}\}\right\}$. The 2D mesh consists of 106,827 cells.

The 3D mesh (see Figure 3, right) consists of 5.2 M cells. The domain is a simple cubical box of 10 mm edge length discretized with Cartesian grid cells. The cylinder with its flat top in the center of the cube is cut out of the mesh again with 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.

The boundary condition for the outer boundary is set to be 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.

The Courant number of the flow, Co, is calculated to determine the time step size $\mathsf{\Delta }t$ for adaptive time stepping. Moreover, the Courant number of the interface, ${\mathrm{Co}}_{\mathrm{i}}$, and the acoustical Courant number, ${\mathrm{Co}}_{\mathrm{a}}$, are introduced in a similar manner:

$$\begin{array}{c}\hfill \mathrm{Co}\left(\mathbf{x}\right)=\frac{\mathsf{\Delta }t}{\mathsf{\Delta }x\left(\mathbf{x}\right)}·\left|\mathbf{U}\right|\left(\mathbf{x}\right),{\mathrm{Co}}_{\mathrm{i}}\left(\mathbf{x}\right)=\frac{\mathsf{\Delta }t}{\mathsf{\Delta }x\left(\mathbf{x}\right)}·\left|\mathbf{n}\mathbf{U}\right|\left(\mathbf{x}\right),{\mathrm{Co}}_{\mathrm{a}}\left(\mathbf{x}\right)=\frac{\mathsf{\Delta }t}{\mathsf{\Delta }x\left(\mathbf{x}\right)}·(c+\left|\mathbf{U}\right|)\left(\mathbf{x}\right),\end{array}$$

(10)

with $\mathbf{n}$ the normal vector on the interface and $c\left(\mathbf{x}\right)$ the local speed of sound. The time-step criterion is set such that the maximum flow Courant number, Co, and interface Courant number, ${\mathrm{Co}}_{\mathrm{i}}$, should stay below 0.2 and the maximum acoustic Courant number, ${\mathrm{Co}}_{\mathrm{a}}$, should be below 1 during periods of bubble evolution, where compressibility effects are important [42].

3.4. Validation

The code has been validated extensively by comparison with ordinary differential equation models for bubble collapse in spherical symmetry (the Gilmore model) in [42] and by comparison with experimental data also in axial symmetry and full 3D for the asymmetric collapse of a bubble close to a flat solid boundary in [42,53,58,59]. All these comparisons demonstrate excellent agreement for the evolution of the bubble shape between the experiments and the numerical simulation. A further example of a visual comparison for a bubble next to a flat solid boundary is given in Appendix A. The jet velocities versus $D^{*}$ found by different authors via simulation and measurements are given in Section 6. The jet velocities gained by the present code are compared to measurements and to values of the simulations of totally different codes with excellent agreement. A comparison of the bubble shape between experiment and numerical simulation for a bubble close to the top of a cylinder, as investigated here, is given in Section 5.1.

4. Experimental Results

The typical mushroom-bubble dynamics with its special collapse and rebound on top of a long pillar is presented in the following. Usually, two to five ultra-high-speed Imacon measurements of the bubble collapse phase were taken for concatenating the short sequences. They were complemented with the respective photodiode signals (see Figure 2) and lower-speed Photron recordings for an overview of the bubble dynamics from bubble generation to rebound. The time from bubble generation to collapse is about 160 µs for the bubble studied.

4.1. Typical Mushroom Bubble Dynamics

As an introduction to the dynamics of a bubble on top of a rigid cylinder, the image sequence in Figure 4 is given. It depicts a stacking/interleaving of four measurements from the Photron APX-RS camera view, each recorded at 21,000 fps with a resolution of 384 × 272 pixels. By interleaving repeated recordings of bubbles of the same size, an approximate time resolution of 84 kfps is obtained.

The main aspects of the mushroom-bubble dynamics are visible in Figure 4. At first, the bubble is overlapping and embracing the cylinder edge during expansion. During collapse, a neck is formed, together with the mushroom’s head. The final bubble collapse and the rebound turn the mushroom shape into a blurred one, but the neck seems to persist for a longer time during rebound. In the rebound phase, the collapsed former mushroom head expands and gets quite large, but stays smaller than the expanded bubble before. Finally, it detaches from the cylinder like a projectile. When viewing the raw video of one of the experiments, a shaking of the needle in axial direction due to the momentum transfer can be observed after the bubble is gone.

4.2. Typical Mushroom-Bubble Collapse and Rebound

The ultra-high-speed Imacon camera recording shown in Figure 5 (corresponding to the measurement presented in Figure 4) allows for a more detailed insight into the dynamics of the peculiar mushroom collapse and rebound phase.

These images were taken at frame rates of 317 kfps and 870 kfps. The exposure time was 150 ns and the sequence starts from shortly after maximum bubble volume. The observer may expect a violent neck closure in the middle of the neck in frame 7, where it is narrowest. However, this neck closure will not happen. Even more astonishing is the apparent, perfectly straight cylindrical neck in frame 8 and the mushroom cap, thin and transparent like a jellyfish. In the early rebound phase in frame 10, secondary cavitation leads to tiny bubbles around the neck, reminding of the mushroom annulus in real mushrooms. Afterwards, the mushroom’s head expands further and turns into a cloudy projectile, not reported on the time scale of Figure 5, but seen in the last frames of Figure 4.

In Figure 6, five frames of five prominent measurements are put together to show the moment of collapse. In view of the repeatability of the measurements, it seems that the bulk gas “under the cap” of the mushroom cap shrinks to the top below the cap while the stem homogeneously becomes more tapered on the full length. The terminology convention for the different parts of the bubble is given in Figure 7.

To obtain more insight into the very moment before and after each frame in Figure 6, a closer look into the stem dynamics at higher time resolution is needed. In Figure 8, the sequences of further experiments are stacked to elucidate the dynamics of neck formation and its subsequent evolution. The exposure time has been reduced to 30 ns and a recording speed of 10 Mfps was applied. It is observed that between frame 4 and 5 of Figure 8 the neck closure is expected to happen but it just stays stable for several frames and develops into a long gas cylinder reminding of a mushroom stem. Actually, in the further frames, the lower part of the stem recedes to form a “bottom stand foot”. Its peak seems to slide down the stem. The bottom stand foot further decreases its height, while the mushroom stem collapses first in the upper half, subsequently in the lower half. The bottom stand foot has not reached zero height yet, when the mushroom stem is already expanding again as an irregular line of small bubbles in their rebound phases.

5. Numerical Results

The CFD simulations were performed in axial symmetry (89 cases studied) and, for a distinct case, in full 3D to show the validity of the axial-symmetry approach. For all cases, the surface tension was neglected. The radii $r_{\mathrm{p}}$ of the cylinder were set to 200 µm and 160 µm. In the following sections the simulations are related to the experimental results, afterwards the general dynamics of the mushroom-shape formation are presented and lastly the set of 89 cases is evaluated.

5.1. Comparison of an Experimental Mushroom Bubble with Simulations

The code was tested for its validity in several studies on bubble dynamics by comparison with experiments [42,53,59]. Here, a further comparison, now with an experimental mushroom-shaped bubble, is given for the convenience of the reader to trust the results.

In Figure 9, eight frames from an Imacon recording of are compared to a simulation in axial symmetry with matching set of $r_{\mathrm{p}}^{*}$ and $D^{*}$. Shown is the moment of self-impact of the annular inflow at the top of the mushroom cap. The experiments show a pronounced brim of the hat-like gaseous structure, the "umbrella". The simulations confirm the brim by a remainder bubble at the end of the chain of isolated, sub-resolution bubbles that form the umbrella. Note that the bubbles of the umbrella in the simulation images are ring shaped as well due to axial symmetry. Sub-resolution here means that the cells in question are filled only to a slight extent with gas, resulting in ${\alpha }_l\to 1$ but not ${\alpha }_l=1$. This is also seen in the experimental case as a transparent, grey cap, presumably consisting of many tiny bubbles. A very good agreement of the simulation with the experiment can be stated.

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

The metamorphosis of an expanding bubble on top of a pillar into a mushroom-shaped bubble upon collapse is given in Figure 10, Figure 11 and Figure 12 together with the pressure, velocity and flow fields in the liquid (water), respectively. The bubble starts with high internal pressure that drives the expansion of the bubble and gives rise to a radiated shock wave (frame 1 of Figure 10). At the circular edge of the pillar the shock wave is reflected with the emission of a low pressure acoustic torus wave. At about maximum expansion (frame 2 of Figure 10) the bubble is surrounded by a low pressure field.

The collapse dynamics are best followed in the velocity-field sequence (Figure 11) and the warping of a color layer field that is advected with the flow (Figure 12). From the velocity distribution around the bubble in the second frame of Figure 11 it is seen that the collapse of the bubble starts from the overlap part of the bubble upwards. The flow is directed and guided upwards along the cylinder and at the rim of the cylinder inwards to form a gaseous stem and neck with a head. That way, the mushroom shape of the bubble is formed. At the sharp curvatures of the mushroom head high pressures are generated by liquid inflow through flow focusing (frames 5 and 6 in Figure 10). The ring-shaped pressure squeezes the top of the bubble head inwards towards the axis of symmetry. The self-impact of the flow leads to a fast jet downwards into the stem and a flow upwards, to be seen only later in time (Figure 12, last frame). Simultaneously, a small bubble may be left moving upwards. The collapse of the gaseous stem looks like receding downwards with leaving a foot stand (last frames in Figure 10, Figure 11 and Figure 12). The foot stand collapses further and the upwards flow become visible together with a palmette-like flow structure with a stack of (torus) bubbles, as seen in Figure 12 (last frame).

Figure 12 shows a sequence of the warping of a color layer tracer field to make the liquid flow and layer mixing better visible in its entirety. It is observed that liquid is sucked upwards from below the cylinder sides already in the early collapse phase (frame 3 to 6 in Figure 12). The liquid for the fast jet derives from a liquid sheet just above the flat top of the cylinder (frame 7 and subsequent frames), whereas the liquid surrounding the receding neck derives from the liquid aside and below the cylinder (frame 4 and subsequent frames). The self-impact of the ring jet (between frame 6 and 7) not only leads to the fast jet downwards but also to a liquid jet upwards, as seen best in the last frame. Moreover, a peculiar palmette flow structure develops reminding of ornaments on antique ceramic vases.

The argument that there is indeed a fast jet in the experiments as well, is put in the following way: by comparing the simulations to the photographs of the experiment (e.g., in Figure 5, Figure 8 or Figure 9), it becomes evident, why there is no neck closure observed in the experiment. At first, the observer of the bubble in Figure 8 sees a tapering of the neck by about 1.2 pixels per 0.1 µs, so that a zero width is expected latest at 0.9 µs. Instead, a stable stem develops with a finite width. In the numerical simulations in 3D, shown later in Section 5.4, this stem can best be seen for comparison. When looking closely at a magnification of an intermediate frame of Figure 12 at $t=85.906\mu \mathrm{s}$, shown in Figure 13, it is observable that the sideways inflow, denoted by the purple liquid layer, is hindered to reach the axis of symmetry by the fast-jet inflow denoted by the blue and almost black liquid layers.

The conclusion is therefore that the momentum of the fast jet seems to dominate the flow direction at the axis of symmetry due to the about twenty-fold velocity, even though the mass transported by the jet is supposedly much less than the one of the sideways inflow. Thus, the stable stem in the experiments can be explained by the phenomenon of the fast jet. In this respect, the fast jet is also seen in the experimental photographs, even though it is not evident to the eye without the knowledge of the simulations.

5.3. General Dynamics of a Mushroom Bubble

The general dynamics process of the formation of a mushroom-shaped bubble is explained in Figure 14 and Figure 15 with a higher temporal resolution. It was found to be grossly similar for all $D^{*}$-cases investigated, as will be seen in Section 5.5.

In Figure 14, the typical expansion and begin of collapse of a bubble on top of a cylinder with $D^{*}\approx 0$ and $r_{\mathrm{p}}^{*}<1$ (the radius of the cylinder is smaller than the maximum radius of a free bubble, $R_{\max ,\mathrm{unbound}}$) is shown. During expansion, when the bubble interface crosses the cylinder rim, it swirls around it, ejecting liquid droplets (in axial symmetry torus ring drops) into the bubble body. Once passed, these droplets hit the outer bubble wall, inducing surface waves there (dashed circles in frame 3). Due to the boundary layer around the cylinder, the bubble never touches the solid, but “swims” on the boundary layer. This is evident from simulations close to any solid boundary, but cannot be easily observed in the current experiment. However, there exists one measurement by Reuter et al. [96,97] for a bubble close to an extended plane solid boundary and a comparison with simulations by Lechner et al. [53] with excellent agreement.

When the bubble starts collapsing, the outer waist is lifted almost parallel to the cylinder. This flow produces the two annular inflows that form a) a neck and b) the extreme curvatures at the mushroom cap rim. As denoted by the red arrows and circles in frame 7, Figure 14, the flow that forms the mushroom neck has also a component upwards that tapers the cap to a thin gas film. The phenomenon of flow focusing comes into play [53,98], flow focusing generating strongest acceleration where curvatures are highest. An annular jet is formed that runs along the top part of the shrinking mushroom cap, leaving trails of dim remnant gas that form a thin umbrella. Numerically, this results into a tearing of the interface, leaving areas where $0.9<{\alpha }_l<1$. In the end, this annular liquid jet impacts in the zenith of the umbrella, producing a fast jet. The fast jet actually is the reason why the neck will not impact onto itself, as stated before concerning Figure 13. It can be seen in Figure 15 that the fast jet here reaches values of more than 700 m/s. In some cases, it is more than 2000 m/s, as will be shown later. The liquid inflow from the top now, starting from frame 1, Figure 15 on, makes all sideways inflows at the neck negligible, changing the subsequent dynamics to a zipper-like collapse. The neck is tapered from inside rather than from outside flows. In the experiment, only the aforementioned “bottom foot stamp” is observed here, because the outside bubble surface has too many wrinkles to see the jet inside. The minimum volume happens from top-down, thus the top gas fragments are already in the rebound phase, when the lower ones collapse and emit shock waves (not seen here – taking place in frame 7, Figure 15, as indicated by the dashed red circle in the frame). Therefore, the upper bubbles are “kicked” and squeezed upwards. Thereby, a layered structure of (torus) bubbles is observed.

5.4. Simulation in Full 3D

One simulation is given in full 3D, i.e., without resorting to half or quarter spaces. Figure 16 shows an example—the dynamics of a mushroom bubble in a series of selected pictures plotted in a 3D view. It reveals that the axisymmetric calculations capture the main dynamics very well, in particular the mushroom shape and also the palmette-like bubble structure forming at and after collapse (the “projectile” experimentally, Figure 7). A small calculation domain size has been chosen—a box of 10 mm edge length with the bubble in the center—to cope with calculation times. The contour surfaces of ${\alpha }_l=0.9$ (light grey, transparent) and ${\alpha }_l=0.5$ (green, transparent) are plotted. The higher value for ${\alpha }_l$ accounts for the dim gas remnants that are sub-resolution. They form the “umbrella” left after the collapse of the mushroom head (see Figure 14 and Figure 15). The intention of the simulation was to show the qualitative agreement with the results in axial symmetry and the experimental observations.

It can be seen that the aforementioned torus-droplet splashing (Section 5.3) also takes place in 3D (frame 2 to 4, Figure 16), but the torus shape is broken to droplet fragments, as expected. This seems to be the main reason for asymmetric distortion of the bubble interface and the aforementioned wrinkles (Section 5.3).

5.5. Parameter Study

The parameter study is done with the intention to find the range in the two parameters $D^{*}$ and $r_{\mathrm{p}}^{*}$, where mushroom bubbles and their fast jets appear. A set of initial distances $D_{\mathrm{init}}$ = 30, 90, 150, 250, 350, 400 and 450 µm was chosen, as well as a set of bubble sizes determined, $R_{\max ,\mathrm{unbound}}$ = 224.02, 249.98, 285.70, 333.32, 400.00, 472.57, 523.58 and 636.42 µm, generated by the respective initial radii of $R_{n,1}$ = 97, 105.20, 116.48, 131.52, 152.58, 175.5, 191.61 and 227.25 µm. From these sets, a subset of all combinations was chosen. The cylinder radius $r_{\mathrm{p}}$ was varied from 200 µm to 160 µm for the whole subset, as well. This results in a total of 89 cases studied. The specific values for $R_{n,1}$ and $R_{\max ,\mathrm{unbound}}$ result from the conversion of the wanted values for $r_{\mathrm{p}}^{*}$ = 0.5, 0.6, 0.7 and 0.8 for $r_{\mathrm{p}}=200\mu \mathrm{m}$. The other values for $r_{\mathrm{p}}^{*}<0.5$ grew historically in the course of the study.

From this data set, Figure 17 and Figure 18 show four examples of bubbles with mushroom dynamics for $r_{\mathrm{p}}^{*}$ = 0.423 and $D^{*}$ = 0.063, 0.190, 0.317 and 0.529. The first row in Figure 17 shows the same bubble as in Figure 14 and Figure 15, just for comparison. The bubbles are plotted with their liquid velocity field for four prominent time instances each in a row: 1—lifting the hips above the cylinder edge (after bubble maximum volume), 2—at about annular jet impact, 3—fast jet impact on the top of the cylinder and 4—at about bubble minimum volume. When looking at the equivalent radius at maximum volume of the mushroom bubbles, $R_{\max ,\mathrm{eq}}$, it is seen that $R_{\max ,\mathrm{eq}}$ tends to be smaller than $R_{\max ,\mathrm{unbound}}$. As expected, some simulations showed that the bubbles lose their mushroom shape and approach a spherical shape at large $D^{*}$ (refer to e.g., Figure A4). A complete description and scan of the transition region is left for future work.

The Figure 17 and Figure 18 reveal, in their first column, that for bubbles close to the top of the cylinder the flow upon collapse is first directed upwards along the solid cylinder side to form a gaseous cylinder above it with a spherical cap larger than the gaseous cylinder diameter—the mushroom shape. With increasing length of the gaseous cylinder (the mushroom stem), the liquid flow along it is redirected inwards towards the axis of symmetry (as stated before concerning Figure 14). This redirection stems from the radial liquid inflow due to the bubble collapsing. In 68 of the 89 cases studied (five of them shown in Figure 17, Figure 18 and Figure 19, as well as some more in Appendix E), the bubbles attain a mushroom shape on their way from their maximum volume to final collapse.

In the second column of the figures, the thick mushroom stem has developed into a neck, and the mushroom head has collapsed or almost collapsed by annular inflow of liquid. A characteristic feature of the collapse of the head is the formation of an “umbrella” of tiny bubbles. The umbrella persists some time with collapse of their bubbles and their rebound, as seen in the third and fourth column of the figure.

In the third column, a thin, fast jet has developed and reaches the cylinder surface. The maximum jet velocity surmounts 2100 m/s in some range of distances, for the current grid setup. For complete curves or manifolds of the jet velocity in dependence on the distance from the surface and on bubble and cylinder size, huge extra work would be necessary.

The jet velocity was found to depend on many factors for the mushroom case, e.g., the torus droplets that remain from the swirling of the bubble around the cylinder rim (see Figure 14, frame 2 and 3) and hinder the annular inflow. Furthermore, it was observed that remainders of these droplets can gather at the axis of symmetry and hinder the fast jet in its progression. The precise occurrence of these droplets is of rather chaotic nature and therefore extremely grid dependent. The velocities of the fast jets of the bubbles investigated range from 189 m/s to 2164 m/s and fluctuate by several 100 m/s per case, depending on the grid. It was already shown in [60] that the precise velocity of the fast jet for $r_{\mathrm{p}}^{*}\to \infty$ cannot be given yet, since it does not converge with increasing resolution in space and time.

In the fourth column of Figure 17 and Figure 18, the mushroom stem has collapsed with small remains of the foot stand. The small bubbles of the umbrella are still present. The annular self-impact not only gives rise to the fast jet towards the pillar top, but also initiates a fluid flow in the opposite direction (see Figure 12 and Figure 13). A stacked sequence of (torus) vortices is formed by the confined, jet-like upwards flow (Figure 13 and the lower row of Figure 15). Thereby, a vertically stacked sequence of bubbles develops.

The examples show mushroom bubble dynamics with (fast) jet formation by annular inflow with self-impact. Overall, it can be stated that the special type of mushroom-bubble formation, collapse and fast jet formation occurs in a large range of parameters, at least for bubbles on a long pillar. Over the parameter range studied, the only significant differences for the mushroom bubbles are the curvature of the outer interface of the mushroom stem at self-impact of the annular inflow, as well as the height of the stem at the time of fast jet formation.

In the present work it becomes evident that fast jet formation is not restricted to the case of bubbles very close to a surface of an extended solid as first found numerically in [53,60,61] and confirmed experimentally in [58,59]. What can be noted here, too, is that the parameter range, where a fast jet occurs, can be extended to at least $D^{*}=0.875$ in the case of the mushroom bubbles, compared to $D^{*}=0.2$ in the case of a bubble in the vicinity of a flat solid boundary ($r_{\mathrm{p}}^{*}\to \infty$).

Extending the parameter range to higher values of $r_{\mathrm{p}}^{*}$, that is to bubbles with small or no overlap over the pillar rim at maximum volume, a minor change in bubble dynamics occurs. An example is given in Figure 19 for $r_{\mathrm{p}}^{*}$ = 0.893 and $D^{*}$ = 0.134 for a bubble with small overlap. The bubble hardly develops a mushroom shape (first frame). Only a broad short stem develops (second frame). But a fast jet with a velocity of 301.5 m/s develops with a receded stand foot (third frame). There are only few remains of a sort of mushroom head (fourth frame). Considering the results of Ref. [53] for a flat solid boundary (corresponding to $r_{\mathrm{p}}^{*}\to \infty$), it can be stated that for $D^{*}=0.134$ and $r_{\mathrm{p}}^{*}>0.893$ a fast jet will most probably occur, albeit not all of them being of the mushroom type. It is conjectured that there will be a gradual transition from a mushroom-type collapse to a “$r_{\mathrm{p}}^{*}\to \infty$”-type collapse [53] with both types exhibiting a fast jet. The results already show that annular-inflow-induced fast jets in bubble dynamics are a robust phenomenon.

Extending the parameter range to $r_{\mathrm{p}}^{*}\to 0$ is left to a future study. A complicated transition scenario is expected to other forms of bubble dynamics. The reason is that the bubble dynamics in free space is approached with an unknown transition scenario, as finally the mushroom bubbles must disappear. The study of this transition is beyond the scope of the present work.

Figure 20 shows results of the parameter study by means of an interpolated heat map with isocurves of the extracted quantities. The data points are plotted in their respective colors that represent their values. 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. The values of the velocities of the fast jet are given in Figure 20A, but they are to be understood as tentative results, since they are not converged values, as stated before. Variations in the jet velocity of up to several hundred meters per second were found, depending on the torus droplet-formation dynamics, grid structure and grid resolution. For evaluation of the velocity the definition given in [53] was applied: The velocity of the jet is defined by the distance from the location of the jet formation to the location of the jet impact onto the pillar surface divided by the time in between the two events. The corresponding water hammer pressures ($\rho c{v}_{\mathrm{jet}}$) would range from 0.3 GPa to 3 GPa. The time interval from fast-jet formation till the impact is given in Figure 20B. It is linked to the fast-jet velocity and therefore to be regarded as a tentative result. The only interpretation that can be drawn from Figure 20B is that the interval is in the range of 1 percent of the Rayleigh collapse time computed with Equation (3).

Four different quantities that seem more robust are given in Figure 20C–F. In Figure 20C the prolongation factor of the collapse time, the time from maximum bubble volume until the minimum volume, is shown, normalized by the Rayleigh collapse time. The prolongation factor seems to increase with increasing $r_{\mathrm{p}}^{*}$ and depend less on $D^{*}$.

In (Ref. [53], Figure 16) the prolongation factor for the case of a flat solid boundary is given as a function of $D^{*}$ including experimental values. For the case of bubbles very close to the boundary ($D^{*}$→ 0), the prolongation factor reaches 1.27. In the case of mushroom bubbles, it is seen in Figure 20C that already for $r_{\mathrm{p}}^{*}\approx 0.9$ the value of 1.2 is obtained.

The maximum equivalent radius of the bubbles is smaller than $R_{\max ,\mathrm{unbound}}$ for all mushroom bubble cases but three, as shown in Figure 20D. For some cases at $r_{\mathrm{p}}^{*}>0.8$ and $D^{*}>1$ the relation $R_{\max ,\mathrm{eq}}>R_{\max ,\mathrm{unbound}}$ applies, too. However, the respective bubbles are not of a mushroom shape (refer to e.g., Figure A4).

In frame 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 $R_{\max ,\mathrm{unbound}}$, 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

The case of an expanding and collapsing bubble on top of a long cylindrical pillar on a plane solid boundary has been studied for the purpose of investigating whether the cylindrical constraint would provoke a fast jet as well. Experiments were done for bubbles very close to the flat top surface of the pillar ($D^{*}\approx 0$), with bubbles embracing the pillar top upon expansion. The accompanying numerical study is covering the cases of bubbles with diameters about two to three times the diameter of a very long cylinder and bubbles close to the flat top of the cylinder in the range $D^{*}=0.047$ to $D^{*}=2.009$. In 68 of the 89 cases studied, the bubble attains a mushroom shape with a fast jet into the stem and an oppositely directed flow that shoots a bubble cluster away from the pillar, the projectile (see Figure 7).

The previously published simulations where a fast jet occurs, were all done for bubbles close to or at an extended flat, solid boundary (i.e., $r_{\mathrm{p}}^{*}\to \infty$). The simulations in [53,58,59,60] were done with a small initial bubble of high internal pressure to simulate the expansion and collapse of laser-breakdown-induced bubbles (energy-deposit case [49]). The simulations in Pishchalnikov et al. [61], on the other hand, were done by inserting a bubble with prescribed shape and deliberate internal pressure into the liquid (a variant of the pure Rayleigh boundary case where a spherical bubble is inserted [49]). In the following, the results of Lechner et al. [53,60] and Pishchalnikov et al. [61] are compared: The prescribed-shape Rayleigh-boundary bubble in Ref. [61] was given the form of an oblate spheroid with a flat part at the solid boundary. This shape was derived from the photographed shape of a bubble oscillating in a sound field at the surface of a urinary stone. It is very reminiscent of the shape of an energy-deposit bubble at maximum volume after expansion very close to a solid surface [60] that expanded and collapsed at an ambient static pressure of $p_{\infty }$ = 101,315 Pa. The Rayleigh-boundary bubble in Ref. [61] was made collapsing at $p_{\infty }$ = 1.55 MPa, derived from the pressure of the acoustic field applied in the corresponding experiment. Given an internal pressure of $P_{02}$ = 100 kPa (notation in [61]), a fast jet towards the solid boundary with a velocity of $v_{jet}$ = 1164 m/s was found. When the (experimentally unknown) internal pressure is reduced to $P_{01}$ = 10 Pa (almost empty bubble), a velocity of $v_{jet}$ = 4117 m/s is obtained. In Ref. [53], fast-jet velocities of 1327.18 m/s for $D^{*}=0.1$ and 2061.3 m/s for $D^{*}=0.048$ were found. Because of too different conditions between Refs. [53,61], a quantitative comparison could not yet be done. It must be stated, too, that none of the above mentioned fast-jet velocities converged with higher grid resolution owing to the singular nature of the self-impact after annular flow focusing, even when viscosity (energy dissipation) and the compressibility of the liquid (energy storage) are included in the model.

Experimental results for the fast jet can be found in Benjamin Ellis [20], Pishchalnikov et al. [61], Koch [58] and Koch et al. [59]. In the following, the results therein are discussed. In Ref. [20], the bubble was subjected to highly reduced ambient pressure, which made it possible to photograph the slowed-down phenomenon already in 1966. In Ref. [61], an experimental bubble is shown developing the typical constriction upon collapse as also present numerically [60,61]. Direct comparisons of experimental and numerical bubble shapes show very good agreement up to the very collapse [61]. The fast jet was not caught in the experiment there, but when experiments and simulations fit over a large part of the dynamics, it is legitimate to interpolate the dynamics in between with the appropriate equations, here the Navier-Stokes equations (see the same arguments in [53] with examples). Thus it can be predicted from the numerical interpolation of the experimental collapse sequence that a fast jet was present in the experiment in Ref. [61].

Koch et al. [59] compared an experimental bubble with simulations over the whole cycle of expansion and collapse with a real camera and also a virtual camera by inserting the simulated bubble into a digitized version of the experimental setup. They could even catch the fast jet in the real experiment and with better resolution in the simulated experiment. With a spatial resolution of 1.8 µm, the full 3D simulation of the experimental bubble predicts a maximum jet speed of at least 732 m/s. The corresponding measurement of Philipp and Lauterborn [12] amounted to 150 m/s as a lower bound owing to the limitations of the experimental instrumentation. When trying to compare the different jet speeds, the different experimental and numerical conditions must be respected. However, when extending the range of $D^{*}$ towards higher values, the simulations of Supponen et al. [18], Lechner et al. [53] and Koch [58], as well as the experiments of Philipp and Lauterborn [12], Brujan et al. [96], Koch [58] and Koch et al. [59] can be compared as shown in Figure 21.

The diagram of Figure 21 shows the jet speeds of (expanding and) collapsing bubbles obtained on flat solid boundaries given in dependence on the normalized distance, $D^{*}$, of the bubble to the boundary. There is excellent agreement over large part of the diagram ($D^{*}$ up to 2) with deviations only where the limits of instrumentation were approached (about 150 m/s). There is an interesting experimental point at $D^{*}$ = 0.1 with $v_{\mathrm{jet}}$ = 150 m/s. According to the numerical simulation, it well could have been a fast jet, just off the experimental possibilities at that time. The measurements repeatably gave this unexpected off-lying high value at the limit of instrumentation.

When comparing the fast jets on pillars here with the fast jets on a plane solid boundary in [53], it is noticed that the range of fast jet formation with respect to the normalized distance $D^{*}$ is extended to at least $D^{*}=0.875$. For a plane solid boundary, the range is restricted to below $D^{*}\approx 0.2$ [53]. The reason for the extension of the range of fast jets in the case of pillars seems to be the liquid that additionally is sucked up by the collapsing bubble from aside and below the top of the pillar. After having passed the flat top the liquid is redirected towards the axis to eventually form the neck. The liquid flow, via neck formation, produces a sharp curvature by constriction of the at first spherical cap to form the mushroom head. By flow focusing in the course of the proceeding collapse, an accelerating annular inflow is generated. During the proceeding annular inflow, the numerical approach of an interface with finite thickness comes in favour, since it indicates the flat sheets of gas in the mushroom umbrella that are sub-resolution. Upon self-impact of the annular inflow, the bubble is divided into a very small bubble on top and the main bubble below that sits on the top of the pillar. Simultaneously, a thin, fast jet is generated into the gaseous stem (see Figure 10, Figure 11, Figure 12 and Figure 13), while bubble collapse still progresses. Interestingly, the liquid of the fast jet stems from the liquid above the top line of the pillar, whereas the liquid around the gaseous stem originates from liquid aside and below the top of the pillar (Figure 12 and Figure 13).

The upwardly directed momentum of the liquid from neck formation pushes the remains of the bubble head upwards as best seen in Figure 4 for an experimental bubble and in Figure 16 for a numerical 3D-bubble. The final formation mechanism for the fast jet is the same as with a plane solid boundary, but owing to a different history in forming the sharp curvature at the rim of the mushroom head. The sharp rim is only formed in the course of collapse of the bubble, whereas in the case of a plane solid boundary it is formed upon expansion of the bubble by the viscous boundary layer [53].

Due to the singular event of the impact of the annular inflow, the numerical results of the present work are not evaluated for the exact numbers of the fast-jet velocities. The focus of this work is on mushroom-bubble dynamics and the comparison towards experimental photographs, as well as proving the existence of the fast jet in the case of bubbles above pillars. Thereby a second type of fast jet formation by annular inflow with self-impact has been found. It is coupled to the development of the mushroom shape, in particular to the sharp curvature of the rim of the mushroom head.

Mushroom-like dynamics like the ones presented here can be found, for instance, in Tomita et al. ([10], Figure 14b), Palanker et al. [82], Li et al. [78,80] and Zhang et al. [79]. In all those works, single cavitation bubbles are investigated close to curved, rotationally symmetric objects. However, those authors were only interested in the whole bubble cycle rather than the collapse details as in the present work. In Ref. [10], the authors investigate different solid-boundary curvatures and in the case of high curvatures the bubble shape is close to the shapes in the case of a cylinder. The bubble was generated at $D^{*}\approx \gamma =0.33$ on top of a convex object with a curvature radius slightly larger than the bubble maximum radius. In comparison to the present work, it seems that the bottom foot stand (see Figure 7) of the mushroom shape is missing and the stem has a rather cone shape. This shape is reminiscent of the bubble in Figure 19 that for larger $r_{\mathrm{p}}^{*}$ is on the way to shapes at a flat solid boundary. This comparison persists for the other works, too. The authors of Ref. [82] investigated a similar structure as the previous authors for the purpose of investigating the application for eye surgery. The bubble in their work is generated on top of a pencil-like electrode and the subsequent dynamics looks very similar to the ones that is presented here. However, the authors focus on the phenomenon occurring much later than the first bubble collapse, called “jet” in their work and “projectile” in the present work. The authors of the Refs. [78,79,80] investigate bubbles close to rigid spheres of similar size as the bubble. It seems that the bubble shape that is given in the present work, which includes the bottom foot stand and the straight stem, is determined by the sharp edge of the cylinder object. According to our experience (Figure 19) with bubbles of similar sizes of bubble and object, they do not develop a real mushroom shape with a long stem. However, interpreting the dynamics visible in the above works, it could be guessed that a fast jet might occur there as well, because of the formation of a mushroom-like cap and the inevitable, following flow focusing.

A nearby sharp edge can lead to interesting phenomena in the early stages of bubble evolution, as has been investigated experimentally by Senegačnik et al. [86]. A secondary cavity can form directly below the edge caused by the low pressure region of the bubble induced overflow of the liquid over the edge. Albeit our model does not include phase transition, we notice the formation of a low pressure region directly below the edge (not shown). Furthermore, using diffuse illumination, Senegačnik et al. [86] visualize and measure the liquid injection into the bubble that follows the expansion of the bubble over the sharp edge (see Figure 14).

Another example of an annular liquid inflow is given by a bubble trapped in a narrow gap [77]. The jet velocities, however, stay below 200 m/s, probably due to the flow restriction of the liquid domain.

For micro-sized bubbles (radius 1 µm) Zevnik et al. [81] find very different shapes in dependence on the distance from a solid sphere and on the sphere size. It can be deduced from the radius-over-time curves of the bubbles that the collapse is not very strong. This may be the cause for the different dynamics without a fast jet. The higher influence of surface tension for smaller bubbles leads to high internal pressures in the bubble at rest and thus a cushioning of the collapse.

As a result of the study, it can be stated that different preconditions may initiate bubble shapes that provoke a converging annular inflow with self-impact and fast jet formation.

7. Conclusions

A laser-induced bubble on top of a long cylinder with a radius at maximum volume larger than the radius of the cylinder shows a dynamics very different from a bubble on an extended flat surface. After having embraced the cylinder top upon expansion, upon collapse it develops a mushroom shape with a head, a long stem and a footing. The special fluid flow leading to the mushroom shape could be reproduced in numerical studies by solving the Navier-Stokes equations for a Tait-compressible liquid with OpenFOAM. The simulations were carried out in axial symmetry (2D), as shown sufficient by comparison with a simulation in full 3D. The simulation code has been validated again by comparison of an experimental with a numerical mushroom shape (Figure 9). Moreover, the code has been compared with other, independent codes with respect to jet velocities (Figure 21). The bubble model has been validated by comparison with a complete high-speed experimental sequence comprising bubble expansion, the first collapse, the rebound and the second collapse of a laser-induced bubble with $D^{*}$ = 1.6 in the vicinity of a flat solid boundary (Figure A1). A quite large range of parameters has been studied ($0.047\le D^{*}\le 2.009$ and $0.251\le r_{\mathrm{p}}^{*}\le 0.893$). In 68 of 89 cases investigated, the bubble forms a mushroom shape and the mushroom head collapses with forming a fast jet (velocity ranging from 189 m/s to 2164 m/s) into the gaseous stem from the self-impact of an annular inflow by two-dimensional flow focusing. The existence of the fast jet in the experiments has been proven by comparing the experiments with the numerical simulations with respect to stem stability. The mixing of the liquid by the bubble has been shown by warping of a color layer tracer field that is advected with the flow. The range of parameters shows that fast jets of annular-inflow type (studied numerically in [53,60,61] and experimentally in [58,59] for bubbles close to a plane solid boundary) are not a singular phenomenon, but rather a robust one. It is expected to occur in many other configurations as well, and to alter the view on cleaning and erosion of surfaces. It may find applications in laser-induced ablation in liquids with pillars of various shapes as targets for laser synthesis and processing of colloids [99]. The fast jet phenomenon also bears a clear potential for erosion of solids, since the water hammer estimates reach to several GPa.