We investigate dynamics of a single cavitation bubble in the vicinity of a horizontal wall throughout expansion and collapse using a sharp–interface level-set method. The numerical scheme is based on a finite-volume formulation with low-dissipation high-order reconstruction schemes. Viscosity and surface tension are taken into account. The simulations are conducted in three-dimensional axi-symmetric space. A wide range of initial bubble wall standoff distances is covered. We focus, however, on the near-wall region where the distance between the bubble and the wall is small. We reproduce three jetting regimes: needle, mixed, and regular jets. The needle jets impose a significant load on the solid wall, exceeding the force induced by the collapse of the pierced torus bubble. For intermediate standoff distances, the large delay time between jet impact and torus bubble collapse leads to a significant decrease in the imposed maximum wall pressure. A liquid film between bubble and wall is observed whenever the bubble is initially detached from the wall. Its thickness increases linearly for very small standoff distances and growths exponentially for intermediate distances leading to a significant increase in wall-normal bubble expansion and bubble asymmetry. For configurations where the torus bubble after jet impact reaches maximum size, the collapse time of the cavitation bubble also is maximal, leading to a plateau in the overall prolongation of the cycle time of the bubble. Once the initial bubble is attached to the solid wall, a significant drop of all macroscopic time and length scales toward a hemispherical evolution is observed.

## I. INTRODUCTION

Cavitation is an active field of research due to its wide range of applications. To name only a few application examples, it can be used actively in medicine to deliver drugs into the cardiovascular system,^{1–3} enable needle-free injections with small volumes,^{4} or to destroy kidney stones.^{5–8} It may also serve to enhance the production of water in micro-pumps to deliver a high amount of liquid in a short time interval.^{9} In biology, cavitation can help to remove pharmaceuticals from waste water.^{10} Laser-induced cavitation bubbles near a solid wall enable surface cleaning by the formation of near-wall vortices.^{11–16} This may initiate erosion patterns at the wall due to strong collapse pressures.^{17–19} Furthermore, in the process of laser synthesis and processing of colloids,^{20} cavitation bubbles act as a nanoparticle reactor but also pose a challenge when it comes to the upscaling of the nanoparticle production.^{21}

The dynamics of oscillating or collapsing bubbles near a wall have been studied numerically in depth in the past decades. Initially, the boundary-integral method (BIM) was a common choice to assess bubble dynamics.^{22–32} However, when the bubble undergoes topological changes, for example, due to the formation of a reentrant jet in the vicinity of a wall, BIM faces limitations. Finite-volume methods (FVM) with proper interface capturing schemes^{8,33–41} or implicit interface descriptions^{42–45} have become a favorable approach to simulate cavitation bubble dynamics.^{46} Diffuse interface methods have difficulties in properly resolving bubble dynamics due to the presence of diffusive mixtures zones around the surface of the bubble.^{38} Different stabilization/relaxation/steepening procedures have been proposed to counteract strong diffusive effects.^{33,34,38,47} When the interface is kept sharp and both fluids are treated separately, such modifications are not required.^{48} We apply a sharp–interface level-set method to investigate bubble evolution near a solid wall. Furthermore, the inclusion of compressibility inside the bubble and in the surrounding liquid is an important criterion to properly resolve the bubble dynamics.^{49}

In numerical investigations of bubble collapse scenarios, we can distinguish between the Rayleigh case and the so-called energy-deposit case that has been introduced by Lauterborn *et al.*^{50} In the former, the simulations start with a spherical bubble at maximum volume and a collapse is initiated by a high external pressure. Detailed analyses have been made to study the jetting phenomena,^{51} wall-induced peak pressures,^{8,39,41,42,51} secondary collapse pressures including phase-change,^{45,52} or jet-related temperature rises.^{47} In all investigations, the peak pressure and temperature have been found to be maximal the closer the bubble collapses near the wall.^{39,41,42,47,51} The energy-deposit case includes the initial growth of the bubble and may start with an excess pressure inside the bubble. This case has already been studied for jetting/erosion phenomena in the vicinity of solid boundaries,^{15,16,53–55} free surfaces,^{56} or in bubble clusters.^{27,57} A detailed comparison between the two cases has been made by Lauterborn *et al.*^{50} who investigated single bubbles with standoff distances, $\gamma =D/Rmax>1$. They observe much stronger collapses in the Rayleigh scenario and the bubble shapes at the collapse also differ significantly since the expansion leads to flat bubbles in the vicinity of the wall.

When the initial growth of the bubble is included, a liquid film forms between the solid boundary and the bubble.^{30,58,59} This film may have a significant influence on the further evolution of the bubble, since it blocks the expansion of the bubble toward the wall. Furthermore, occurring jetting dynamics may be affected by the presence of the film as it hinders the jet to directly hit the boundary and initiate splashing effects.^{26,60} Liquid films between a cavitation bubble and a solid wall have been found in simulations,^{16,30,58,61} but only for a limited range of intermediate and large standoff distances. Experimentally the film thickness has been observed by Vogel, Lauterborn, and Timm^{62} and Philipp and Lauterborn^{60} for larger standoff distances. Due to the limited imaging techniques and computational power in early investigations, a liquid film has only been observed for large standoff distances, $\gamma >1$,^{60,63,64} and subsequently, the limit has been pushed toward lower values.^{15,29,65,66} Recently, a more detailed analysis has been made by Reuter and Kaiser^{59} for bubbles closer to the wall. They found that the liquid film thickness decreases with a power law for intermediate standoff distances, $0.45<\gamma <1$, and asymptotically approaches a constant film thickness for vanishing standoff distances, $\gamma \u21920$. Unfortunately, no experimental results are present of below a standoff distance of $\gamma <0.45$.

Jetting phenomena that occur due to the presence of a solid or free surface boundary have been investigated in depth by Obreschkow *et al.*^{67} and Supponen *et al.*^{32,68} They derived universal scaling laws by relating the driving force of the collapse scenario to a Kelvin impulse based on the so-called anisotropy parameter *ζ.*^{69–71} For weak and intermediate jets, universal power laws can be found. In the strong jet regime, when the distance of the bubble to the boundary is small, non-linear trends have been observed. In the strong jetting regime, a further analysis has not been made due to complex surface morphologies that are only hardly treatable by the applied BIM. Recently, Lechner *et al.*^{53,72} have investigated numerically with a finite-volume method this strong jetting regime near a solid boundary in more detail. They proposed three jetting regimes: needle, regular, and mixed jets. Regular broad jets can be found for intermediate and large distances to the solid wall. For very low standoff distances, radial inflow toward the axis of symmetry leads to the formation of fast needle jets. The accompanying circumferential pinching at the bubble head has already been observed in the Rayleigh case in the context of non-hemispherical bubble collapse.^{8,39,42} In the transition zone, both jetting phenomena occur concurrently, leading to scrumbled bubble shapes and mixed jets. Similar trends have been observed by Koch *et al.*^{54} for a bubble collapse near a solid cylinder. In their experiments, Reuter and Ohl^{73} confirmed the existence of such ultra-fast jets, but found lower average jet velocities.

The analysis of Lechner *et al.*^{53} and Reuter and Ohl^{73} provided an initial idea on the formation of these strong jetting phenomena. An analysis on how these jetting phenomena affect the macroscopic time and length scales of the bubble evolution is still missing. In experiments, the prolongation of a full cycle has been shown to increase for smaller standoff distances,^{74–78} but only limited studies are available for very small standoff distances. Recently, Reuter, Zeng, and Ohl^{55} have provided detailed insight into the evolution of the prolongation factor for various small standoff distances. They confirmed their experimental trends with numerical simulations, but did not investigate the collapse and expansion times as these are difficult to assess experimentally. For larger separations, an analytical model has been derived for the collapse time of a bubble near a wall already decades ago by Rattray^{79} and Chahine and Bovis.^{80} The model accurately predicts the collapse time for standoff distances $\gamma >1$. Recently, Lechner *et al.*^{53} observed an oscillating behavior of the collapse time for standoff distances below $\gamma <1$. A clear understanding of the effects that translate into such an oscillating behavior has not been given yet.

In the literature, the collapse of a single bubble in the vicinity of a solid wall has been studied extensively. Individual phenomena, such as the Blake splash, slow and fast axial jets, or the erosion potential due to the water-hammer and collapse pressure, have been investigated. Only recently, more attention has been devoted to standoff distances very close to the solid wall as the complex occurring collapse phenomena are difficult to capture numerically and experimentally due to their inherent small time and length scales. In this work, we show the different phenomena and give a full picture of cavitation bubble dynamics in the vicinity of a horizontal wall. We investigate bubble evolution numerically using a sharp–interface level-set method, including the bubble growth phase. We cover a wide range of standoff distances including small and large bubble separations from the wall. The main focus lies in small standoff distances, where fast jetting phenomena and strong shape transformations occur. Furthermore, our results are brought into the context of existing experimental and numerical analysis to assess their results, enlarge the investigated range, or discuss possible differences.

The paper is structured as follows: after presenting the governing equations and methodology of the sharp–interface method in Sec. II, we give a short summary of the numerical setup in Sec. III. Afterward, the wall-bounded Rayleigh collapse, Sec. IV, and energy-deposit case, Sec. V, are taken to investigate the bubble evolution in the vicinity of a horizontal wall. Finally, in Sec. VI we summarize our findings.

## II. METHODOLOGY

We simulate the bubble dynamics with the open-source code framework ALPACA^{81} that uses a Godunov-type flux-based high-resolution finite-volume formulation. The interface is captured with a sharp–interface level-set method. Details on the implementation, performance, and the validation for various multi-component flows, for example, aero-breakup or Richtmyer–Meshkov instabilities, can be found in Hoppe, Adami, and Adams^{82} and Hoppe *et al.*^{83}

### A. Governing equations

We solve the compressible Navier–Stokes equations including viscous and capillary effects. The general three-dimensional differential form reads as

with the vector of conserved quantities, $U=[\rho ,\rho u,E]T$, the convective fluxes, $F(U)c$, and the viscous fluxes, $F(U)\mu $,

Here, *ρ* is the fluid density, $u=(u,v,w)$ the velocity vector with its three Cartesian components, *p* the thermodynamic pressure, and ** I** the identity matrix. The total energy, $E=\rho e+1/2\rho |u|2$, consists of the specific internal energy,

*e*, and the kinetic part. The viscous stress tensor of a Newtonian fluid,

**, is defined as**

*T*with a constant shear viscosity, *μ*. A bulk or volume viscosity is neglected in all cases. Note that we solve the governing equations separately for each fluid. Forces at the interface, such as capillary forces, do not appear directly in the governing equations. These are included as additional volumetric source terms by the sharp–interface interaction method.

During the expansion and collapse of a bubble, several physical processes may influence the dynamics of the evolution. These occur on a wide range of temporal scales. Viscosity and surface tension effects are included since they strongly affect the bubble evolution in close vicinity to a solid wall.^{53} Compressibility is also included in both the surrounding liquid and the bubble to correctly capture the damping of the bubble evolution and the emission of strong shock waves during the collapse.^{84} Other much slower (e.g., mass diffusion) and faster processes (e.g., evaporation or condensation) are neglected.^{39,84} Similarly, the vapor content inside the bubble and buoyancy effects are neglected.^{53}

### B. Equation of state

To close the system of equations thermodynamically, an equation of state (EoS) is required that relates the internal energy and the density to the thermodynamic pressure, $p=p(\rho ,e)$. We employ the stiffened gas equation of state (SG) for both the liquid and gas phases

where *γ* is the ratio of specific heats, and Π a constant background pressure.^{85} For liquid water, we use $\Pi l=3046\u2009bar$, and $\gamma l=7.15$ to recover the parameters of Tait EoS^{86} that has also been used in earlier studies.^{33,53} Inside the bubble an ideal gas is assumed with $\gamma g=1.4$ and $\Pi g=0\u2009bar$.

### C. Numerical implementation

The governing equations are solved by a flux-based finite-volume approach, where Eq. (1) is integrated in time and over finite volumes. Time integration is done with an explicit third-order Runge–Kutta TVD scheme.^{87} For the flux evaluation at each cell face, a fifth-order WENO reconstruction^{88} together with a positivity-preserving HLLC Riemann solver^{89,90} is used. Signal speeds for the Riemann problem are estimated following Einfeldt.^{91} The viscous stress tensor is evaluated at each cell face with a fourth-order central differencing scheme.^{92} To improve computational efficiency, a block-based multiresolution scheme (MR) is employed.^{93} Hence, regions with strong local variations in the fluid field, such as density discontinuities and interfaces, are resolved with higher resolution. The smoothness of a field is estimated by a wavelet analysis.^{94}

In ALPACA, we capture the interface for multi-component simulations by a signed-distance level-set function. The interface position is given by the zero isoline of the level-set field. Interface velocity and interaction terms are computed by a two-material Riemann problem,^{95,96} and the advection of the level-set field is done with a third-order WENO stencil.^{97} The scalar advection of the level-set is only valid at the interface and might violate the signed-distance property elsewhere. Therefore, an iterative reinitialization procedure is employed.^{98} To be consistent with the level-set advection, we compute the interface normal, $n\Phi $, and interface curvature, *κ,*

with a third-order WENO scheme. In cut-cells, that is, cells cut by zero-level of the level-set function, volume fractions and cell-face apertures are evaluated by linear interpolation of the level-set field.^{39}

For multi-component simulations, the sharp–interface introduces an additional volumetric source on the right-hand side of Eq. (1) owing to the interaction of the two fluids at the contact discontinuity. We employ the sharp–interface interaction method of Hu *et al.*^{99} with an extension to viscous and capillary effects^{96} to balance the stresses in interface-normal direction. Therefore, capillary forces are included with a constant surface tension coefficient, *σ*, and the interface curvature, *κ*. To evaluate spatial gradients across the interface, we employ the ghost-fluid method.^{100} Therefore, we extrapolate all states, ** U**, to the opposite side of the interface with an iterative one-way extrapolation.

^{101}Similarly, interface quantities, such as interface pressure or interface velocity, are extrapolated. To improve efficiency, all level-set operations are computed in a four-cell narrowband around the interface only.

Time integration is done explicitly and requires a stability criterion. After each Runge–Kutta cycle, the time step is computed with an adaptive local time-stepping scheme^{102} that fulfills the Courant–Friedrichs–Lewy (CFL) condition.^{83} To avoid spurious oscillations in small cut-cells, a mixing procedure is employed.^{39,99} If not stated otherwise, a $CFL$ number of 0.6 is used for all simulations carried out in this paper.

The presented simulations are conducted in three-dimensional axi-symmetric domains. This requires a transformation from the Cartesian space (*x*, *y*, *z*) to the reduced cylindrical space (*r*, *z*). With this transformation, singular terms of type $1/r$ are introduced by the divergence operator. To avoid the flux evaluation at the symmetry axis, all terms that have a dependency on $1/r$ are treated as geometric source terms on the right-hand side of Eq. (1).^{35} This allows to account for cylindrical geometry effects with an unaltered left-hand side of the governing equations. Details of this approach can be found in Meng^{103} and in the Appendix.

## III. NUMERICAL SETUP

### A. Domain description

In Fig. 1, a sketch of the full three-dimensional setup is given together with the main geometric parameters and boundary conditions. We only simulate the reduced axi-symmetric space (*r*, *z*) (light gray plane in Fig. 1) to minimize the computational effort. The *z* axis is the rotation axis.

In all cases, an initial spherical bubble with radius, *R*_{0}, is placed in a cubic domain with edge length, *L*. Initially, the bubble is placed at a distance, *D*_{0}, normal to the wall. The size of the initial radius depends on the scenario and will be described in more detail below. On all far-field boundaries, zero-gradient boundary conditions are imposed to isolate the bubble dynamics from possible reflections of outgoing shock waves. The boundary on the plane at *z* = 0 a wall condition is employed.

### B. Mesh

Two different scenarios are used to simulate wall-bounded cavitation bubble dynamics. All simulations employ an adaptive grid refinement provided by the multiresolution algorithm. The entire domain consists of quadratic cells, which allows adapting the mesh efficiently to the interface evolution and to strong variations in the fluid field. For completeness, we give a short summary of the relevant parameters of the multiresolution algorithm. Further details can be found in Hoppe, Adami, and Adams.^{82}

In ALPACA, the domain is decomposed into a number of blocks per direction, $Nb$, where all blocks are of equal size, $Lb$, and contain a fixed amount of internal cells, $IC$. Adaptation of the grid is achieved by refining the grid successively on different levels, *l*. Higher levels correspond to finer resolutions and vice versa. Between two refinement levels the cell size, $\Delta x$, differs by a factor of two. The cell size on a single level, $\Delta xl$, can be computed from

The finest level is defined by the maximum number of refinement levels, $lmax$.

We set the number of internal cells per block to $IC=16$ and the number of blocks on level zero to $Nb=4$. The maximum level is $lmax=9$ to give proper resolutions, when the bubble reaches its minimal size. In Fig. 2, an example of a grid for the computational domain is given. Additionally, three stages during bubble evolution are represented to show grid adaptation to the bubble interface (indicated by the red curve). To increase numerical dissipation in the far field of the domain in order to reduce numerical shock reflections at the outer boundaries, a high compression ratio is desired. Therefore, we gradually coarsen the grid outside a cubic box of $R>2Rmax$, see Fig. 2.

### C. Initial conditions

Two different scenarios are investigated. First, the Rayleigh collapse case is used for validating the numerical scheme. Second, the preceding expansion of the bubble is included to simulate the full growth and collapse of the cavitation bubble. This scenario will be called “energy-deposit scenario” to be consistent with Lauterborn *et al.*^{50} In all scenarios, the surface length of the domain is $L=80Rmax$ to match Lauterborn *et al.*^{50} and Lechner *et al.*^{53} In the near-wall region, viscous and capillary effects may play an important role. Therefore, we employ realistic dynamic viscosities in the liquid, $\mu l=1\u2009mPa$, and in the gas phase, $\mu g=0.017\u2009mPa$. The surface tension coefficient is chosen to be $\sigma =0.072\u2009N/m$.

#### 1. Rayleigh collapse scenario

In the Rayleigh collapse case, the bubble evolution starts with a bubble at maximum size, $R0=Rmax$, and internal pressure, *p*_{0}. Due to the high external pressure, $p\u221e\u226bp0$, the bubble collapses to reach a minimum size and exhibits rebound cycles of expansion and collapse until it reaches an equilibrium state.

We apply the same initial conditions that have been used by Lauterborn *et al.*^{50} The bubble initially has a maximum radius of $R0=Rmax=500\u2009\mu m$ with an initial pressure of $p0=1521\u2009Pa$. At equilibrium, the density and pressure of the bubble are $\rho eq=1.204\u2009kg/m3$ and $peq=1\u2009atm$, respectively. The external pressure is $p\u221e=1\u2009atm$, and the density of the liquid water is set to $\rho \u221e=998\u2009kg/m3$ yielding an initial moderate pressure ratio of $p0/p\u221e\u22480.015$. In all simulations, the cell size on the finest level is $\Delta xmin=1\u2009\mu m$.

#### 2. Energy-deposit scenario

Compared to the Rayleigh-collapse case, in the energy-deposit scenario the bubble is initiated at minimal radius, $R0=Rmin$, and expands to its maximal size.^{50} After that, a collapse is initiated again and subsequent cycles follow. To initiate the cycle, the small bubble contains a high internal pressure $p0\u226bp\u221e$.

Here, we adapt the initial conditions of Lechner *et al.*^{53} ($R0=20\u2009\mu m,\u2009p0=11\u2009000\u2009bar$). Compared to the Rayleigh collapse, a slightly smaller cell size is used to ensure a minimum number of cells around $Nmin\u224825\u221230$ at the initial radius to properly resolve cavitation dynamics including the expansion phase.^{42} Compared to Lechner *et al.*,^{53} we do not enforce a liquid film directly at the wall, but use a symmetry boundary condition of the level-set field, which mimics a constant contact angle of $90\xb0$. A contact between the bubble and the wall is only possible if the bubble approaches the wall within the first cell layer.

To account for potential condensation/evaporation between the bubble and the liquid, Lechner *et al.*^{53} apply a numerical mass correction step that reduces the mass inside the bubble in time to decrease the equilibrium radius. In our simulations, we do not use such modifications. By suppressing evaporation/condensation, the focus is on the evolution during the first expansion cycle. Lauer *et al.*^{39} have shown that such effects are mainly relevant for the subsequent expansion cycle.

### D. Dimensionless standoff distance

In both scenarios, the bubble is initially placed at a distance, *D*_{0}, normal to the wall. To obtain a dimensionless standoff distance, one needs to define a characteristic radius, $Rref$. In the Rayleigh case the simple choice, $Rref=R0=Rmax$ has often been used.^{50} More difficulties arise in the energy-deposit case, where the real maximum equivalent radius is not known *a priori*. One possible definition follows from the Rayleigh scenario, where the maximal equivalent radius of a bubble that expands in an unbounded liquid, $Rref=Rmax,ub=3/(4\pi )Vmax,ub1/3$, is chosen.^{50,53,54} This leads to the dimensionless standoff distance *γ,*

In the energy-deposit case, another important definition relates the initial standoff distance to the initial radius of the bubble, *R*_{0,}

Since the initial radius is lower than the maximal bubble size, the condition, $\gamma 0<\gamma $, is always fulfilled. In most scenarios, even $\gamma 0\u226a\gamma $ is a valid approximation.

The presence of a wall has a significant influence on the evolution of the bubble and has a retarding effect on the collapse time. Rattray^{79} has analyzed such scenarios analytically and found a relation for the wall-bounded prolongation factor, $kc,Rat$. This factor relates the collapse time to the Rayleigh collapse time, $TcRay=0.915R0\rho l/p\u221e$, as a function of the dimensionless standoff distance

Since the Rayleigh collapse time has been deduced for purely inertial collapses of bubbles containing only vapor, it is not a proper estimate when effects of viscosity, surface tension, and non-condensable gas are included as they delay the collapse. Therefore, the Rayleigh time in Eq. (8) is replaced by the collapse time of the simulation in an unbounded liquid, $Tc,ub$. This implicitly adapts the correlation such that is valid also for scenarios that do not comply with Rayleigh jetting. A similar correlation has been found by Chahine and Bovis^{80} by matched asymptotic expansions. They have shown that these first-order correlations are only valid for large standoff distances $\gamma \u22651$.

## IV. WALL-BOUNDED RAYLEIGH COLLAPSE

For the Rayleigh collapse, we vary the standoff distance in the range $\gamma \u2208[0,3]$ to cover a sufficient range of short- and far-wall distances. In Fig. 3, the wall-bounded prolongation factor, $kc=TVmin/Tc,ub$, is plotted over the standoff distance. The reference time has been deduced from a Rayleigh collapse in an unbounded liquid, $Tc,ub=46.11\u2009\mu s$.

We observe that for detached bubbles, $\gamma >1$, the collapse time is well predicted by the analytical correlation of Rattray,^{79} Eq. (8). Only small deviations at lower collapse times between correlation and simulation are observed. This is in good agreement with Lauterborn *et al.*,^{50} Vogel and Lauterborn,^{104} and Johnson and Colonius,^{51} who have found similar trends. For attached bubbles, two different regimes can be distinguished. For larger standoff distances, $0.75\u2264\gamma \u22641$, the correlation is still in good agreement, but the simulations overpredict the collapse times by about 2%. Compared to the analytical model, in the simulations compressibility and viscosity are included which is known to retard the collapse.^{49,105} When the bubble moves even closer to the wall, $0\u2264\gamma <0.75$, the theory of Rattray^{79} fails since it is not valid for fully attached bubbles.^{80} In the limiting case, *γ* = 0, the simulation predicts a collapse time that resembles that of an unbounded liquid.^{42} However, the analytical solution, Eq. (8), diverges. A more suitable correlation for attached bubbles can be deduced considering the energy content of the bubble. For attached bubbles, the wall cuts a spherical cap, $Vcap$, which reduces the bubble volume, $V<V0=4/3\pi \u2009R03$, and hence, its energy content, $E=p0V$. The modified bubble volume can be computed by

In an unbounded liquid, the collapse time scales with the initial radius of the bubble, $Tc,ub\u223cR0$. Relating the collapse time to the limiting case of an attached bubble, *γ* = 0, yields^{53}

Introducing Eq. (9) into Eq. (10) finally gives a dimensionless correlation for the collapse time of a bubble that is cut by a wall as a function of the non-dimensional standoff distance

The simulation closely follows this correlation in the range $0\u2264\gamma <0.75$; that is, the collapse time is mainly controlled by the energy content of the bubble, whereas wall effects only play a minor role. In the transition zone, $0.75\u2264\gamma \u22641$, the wall starts to affect the collapse behavior to shift the collapse times toward the analytical solution of Rattray.^{79} Hence, the prolongation of the collapse has an upper limit of $kwallmax(\gamma =1)=21/3$.

## V. WALL-BOUNDED BUBBLE EXPANSION AND COLLAPSE

In the energy-deposit case, the initial expansion of the bubble is considered during the simulations to represent the cavitation growth and collapse after optical breakdown of laser-induced cavitation bubbles more realistically.^{50} In the following, we give detailed insight into the different aspects of bubble motion near a solid wall including shape transformations, jetting phenomena, and the formation of a liquid film between bubble and wall. Therefore, we investigate the expansion and collapse of the bubble for standoff distances in the range of $\gamma \u2208[0,4]$ with finer incremental steps in the near-wall region.

### A. Time evolutions

In Fig. 4, the bubble evolution is given for five different standoff distances, $\gamma ={0,0.2,0.76,1,3}$, at five different time instances, $\tau =t/Tcyc,ub$, with $Tcyc,ub=91.61\u2009\mu s$ as the unbounded cycle time. The evolution is visualized by contours of the bubble after the bubble has fully expanded upon collapse and jet formation (outer to inner contours).

At very low standoff distances, $\gamma \u21920$, Fig. 4(a), the bubble expands approximately hemispherically. When the initial standoff distance is below a certain threshold, contact of bubble and wall occurs immediately or at early stages of the expansion, see outermost contour in Fig. 4(a). A contact angle of $90\xb0$ is imposed by the applied boundary condition of the level-set field. Due to the velocity difference in the boundary layer, the outer part of the bubble expands faster than the near-wall fluid. Thus, a small liquid layer is pulled below the outer rim of the bubble during the expansion. When the bubble reaches its maximum volume, the bubble starts to contract radially. Compared to the outer surface of the bubble, the liquid layer near the wall still continues to expand. This leads to the formation of an acute base of the bubble. In the late stages of the collapse, a cusp forms near the bubble head due to the fast radial inflow.^{53} The cusp retracts downward and generates a fast needle jet at the axis of symmetry accompanied by the detachment of small toroidal structures, see the innermost contour in Fig. 4(a).

At higher standoff distances, $\gamma ={0.76,1,3}$, the needle jet turns into a broader and more regular jet from the head of the bubble toward the wall. In the transition zone, $\gamma =0.2$, both the flow from the top and the side reach the symmetry axis at a similar time. Thus, the bubble is almost flat during collapse.^{53} During the expansion process, flattening of the bubble by the wall reduces with increasing standoff distance. Hence, the fully expanded bubble has an almost spherical shape already at *γ* = 1 [outermost contour in Fig. 4(d)]. Except for very low standoff distances, in all cases a thin liquid film forms between the bubble and the wall, which delays the wall-normal contraction of the bubble near the wall. Therefore, the base of the bubble remains flat and close to the wall over the entire cycle. Only for larger standoff distances, *γ* = 3, the geometric center of the bubble has a stable position at the initial standoff distance and the contraction of the bubble is initiated over the entire bubble surface. The combination of the reduced radial contraction from the lower side, but larger bubble sizes at the maximum, lead to an increase in liquid jet size with increasing standoff distance in the range $0.2<\gamma <1$. Once the bubble reaches approximately spherical shape in its maximum at $\gamma >1$, the further increased contraction from the lower side reduces the jet size again.

A more detailed analysis on the characteristic time and length scales of the bubble evolution as well as the geometric shape transformations is given below. To validate our numerical results, we compare the evolutions with experimental data of Reuter and Kaiser^{59} and with our own experiments similar to Kalus, Barcikowski, and Gökce.^{106} In Fig. 5, the bubble evolution is shown for three standoff distances *γ* = 0, $\gamma =0.56$, and $\gamma =1.05$ at six different time instances. The experimental images (left half in each frame) are overlaid by the central cut of the simulations (red solid line) and compared to the side view of the simulations (right half). The solid boundary is located at the lower edge of each image, and each time instance is displayed as non-dimensional time, $\tau =t/Tc,ub$, in the upper left corner of each frame. Since the maximum radius in the experiments, $Rmax,\u2009exp$, differs from that of the simulations, $Rmax,sim$, we have scaled the radius and time by $Rmax,\u2009exp\u2009/Rmax,sim$ to match both evolutions.^{53}

We observe excellent agreement between simulation and experiment until collapse for the three standoff distances. When the bubble is attached to the wall, *γ* = 0, Fig. 5(a), the bubble expands hemispherically. At approximately $\tau =0.5$, the expansion is finished, and the collapse is initiated. During this phase, only the upper half contracts, which leads to the generation of radial inflow and the formation of a cusp near the bubble head ($\tau =0.94$). This cusp than turns into the needle jet and a very flat and small toroidal bubble at the collapse ($\tau =0.99$). We also observe good agreement for the near-wall evolution of the bubble. During expansion, liquid is pulled below the bubble leading to the formation of a constriction. In the collapse phase, the acute base is visible in both the simulations and experiments.

For lower standoff distances, $\gamma =0.56$, Fig. 5(b), the bubble remains flat near wall over the entire evolution. Again, the upper half of the bubble expands nearly as a hemispherical cap and the upper half contracts during the collapse, which leads to the formation of a kink on the outer side of the bubble. During the late collapse stages, $\tau =0.96$, a jet penetrates the bubble in the center, forming a flat toroidal bubble. When the outward induced flow by the jet meets the inward motion due to the collapse, a splashing effect (Blake splash) can be observed.^{25,26} During the rebound phase, splashing remains visible for a long time and splits the torus in two parts. This cannot be seen from the side view projections of the bubble.

For larger standoff distances, $\gamma =1.05$, the evolution is different, see Fig. 5(c). During the expansion, the flattening effect due to the presence of the wall is not as pronounced as for lower standoff distances. In collapse phase, $\tau >0.5$, the wall prevents spherical symmetry, $\tau =0.75$, and the liquid jet forms. Since the liquid film is thicker, the reentrant jet has a more destructive character leading to a strong disintegration of the torus bubble in the rebound phase, $\tau =1.11$. Experiments do not provide sufficient resolution, which makes a comparison with the simulations difficult. Bubble shapes in wall-parallel and wall-perpendicular directions agree well, however.

### B. Liquid layer between bubble and wall

At small standoff distances, the bubble is stretched along the wall and a thin liquid film forms between the bubble and the wall. In Fig. 6, the temporal evolution of the normalized film thickness, $h\u2217=h/Rmax$, is given for three standoff distances, $\gamma ={0.56,0.76,0.99}$, and the numerical results are compared with data of Zeng, An, and Ohl,^{16} and Reuter and Kaiser.^{59} Note that at jet impact, $t/Tjet=1$, the film ruptures.

In all cases, the film thickness decreases monotonically for all standoff distances. For the lowest given standoff distance, $\gamma =0.56$, a stable film quickly establishes once the bubble has reached its maximum size at $t/Tjet\u22480.4\u22120.5$. With increasing distance toward the wall, the film continues to decrease until jet impact, even beyond maximal bubble expansion indicating a down-shift of the bubble centroid toward the wall. Comparing the numerical results with the experimental data of Reuter and Kaiser,^{59} we observe good agreement for the lowest considered standoff distance. For larger *γ,* we believe that the experiments overpredict the film thickness due to limitations in measurement capabilities for the curved bubble shapes until jet impact.^{59} Similar trends have been observed by Zeng, An, and Ohl.^{16} For the two smallest standoff distances, we are in good agreement with their results. At $\gamma =0.99$, larger discrepancies are observable, where our results are closer to the experimental measurements and lie within the experimental tolerance for thicker liquid films.^{59}

In Fig. 7, the evolution of the film thickness at jet impact, $hjet\u2217=hjet/Rmax$, is shown as function of the standoff distances in the near wall region, $\gamma <1.1$. Comparisons are made with the results of Lechner *et al.*^{53} and experimental data of Reuter and Kaiser.^{59}

With decreasing standoff distances, also the film thickness decreases. In the range $0.6\u2264\gamma \u22641$ the decrease in the film thickness follows the analytical correlation, $hjet\u2217(\gamma )=0.071\gamma 4.86+0.011$, that has been deduced by Reuter and Kaiser^{59} based on their experimental data. A similar trend has been obtained by Lechner *et al.*^{53}

Below $\gamma \u22480.6$ and above $\gamma \u22481$, the results of the simulations and the experimental correlation differ significantly, which is not surprising since only a few experimental data are present in this regime. For $\gamma \u22640.6$, the correlation overpredicts the film thickness and levels at a constant value, $hjet\u2217\u22480.011$, in the limit of vanishing standoff distances. In our simulations, we observe an approximately linear decrease in the film thickness, which tends to zero for very small *γ*. The reduction can mainly be attributed to the emission of the initial shock wave due to the high-energy content of the bubble. After emission, the shock impacts on the wall, is reflected, and leads to subsequent reflections between bubble and wall. At every impact on the bubble surface, the expansion of the bubble toward the wall is decelerated. The strength of the deceleration depends on the attenuation of the shock wave over its traveled distance. Most of the shock energy dissipates into the liquid at very short distances with decays above geometric attenuation.^{107–109} Hence, for larger distances, the influence is less significant and increases for smaller standoff distances $\gamma <0.6$. Overall, this leads to a decrease in the film thickness with decreasing standoff distance and prevents a rupture of the film.

When the initial bubble is cut by the wall, $\gamma <\gamma 0$, both are in contact with each other and no liquid film forms, which gives a lower limit of the film thickness, $hjet\u2217(\gamma <\gamma 0)=0$. The finite size of the initial bubble may be an important characteristic in the generation of cavitation bubbles. For example, in laser-induced cavitation the creation of the cavitation bubble follows after an initial plasma has formed.^{107} Hence, further investigations are required to clarify whether there is a real lower limit of the standoff distance, where the formation of a liquid film is prevented or the film ruptures during the very early stages of the expansion as in the context of bubble pairs^{27} or near a free surface.^{110}

Above *γ* = 1, the experimentally deduced correlation underpredicts the film thickness significantly. For even larger values, the film thickness further increases and asymptotically approaches linear behavior $hjet\u2217=\gamma $, see Fig. 8.

We can identify three different scaling regions of the film thickness at jet impact as function of the standoff distance. In the inner region, $\gamma <0.6$, the normalized film thickness decreases linearly and tends toward zero for attached bubbles. In the outer region, $\gamma >1.1$, the thickening of the film asymptotically tends toward the linear relation $hjet\u2217=\gamma $.^{53} The transition zone, $0.6\u2264\gamma \u22641.1$, shows an exponential growth of the film thickness and follows the analytical correlation of Reuter and Kaiser.^{59}

A liquid layer with linearly decreasing thickness has been found already in earlier experimental studies. Due to imaging resolutions, it has been deduced that contact between the bubble wall may be initiated at standoff distances $\gamma \u22480.9\u22121$.^{60,63,64,76} Nevertheless, some experiments and numerical investigations hinted that a liquid layer might exist even for lower standoff distances.^{65,66} Boyd and Becker^{58} have concluded that an initially detached bubble should always remain separated from the wall. The existence of the liquid layer has been proven experimentally by Reuter and Kaiser^{59} until $\gamma \u22480.5$. From our numerical results, we can confirm these claims and further state that the liquid layer exists even for very small standoff distances, corroborating the suggestions of Boyd and Becker.^{58} Hence, a liquid film is suppressed only if the initial bubble is directly attached to the solid wall or if the distance is below a limit that initiates rupture of the liquid film.^{27,111}

The thickness of the film may play an important role in splashing after jet penetration. In the outer region, the film is too thick to initiate splashing.^{26} When the bubble moves closer to the wall, the liquid film reduces significantly and an inward flow of the collapse and outward motion due to the jet impact initiate a splash at the bubble base.^{26} For small standoff distances, $0.3<\gamma <0.8$, we observe a significant influence of the splash on the subsequent collapse, where the torus bubble is completely penetrated by the jet. Only for very low standoff distances $\gamma <0.2$, the thin liquid film prevents splashing due to a stabilizing radial outward motion.

### C. Characteristic bubble-shape evolution

During its evolution, the bubble undergoes several shape transformations, where its surface is characterized by different curvatures. Especially, at jet impact or shortly before, different bubbles shapes can be observed, see Fig. 4. In Fig. 9, the bubble shape close to jet penetration is shown together with the contour at maximum expansion for different standoff distances in the near wall regime, $0\u2264\gamma \u22640.9$. Additionally, characteristic regimes of the jet and types of the bubble head and base are given. The formulations and limits for the different jetting regimes have been taken from Lechner *et al.*^{53} and are analyzed in further detail below. In Fig. 10, a detailed view is given on some bubbles in the characteristic regimes to highlight the different occurring geometric shapes.

For very low standoff distances, the formation of needle jets has been observed by Lechner *et al.*^{53} In such cases, the collapse of the bubble is characterized by the formation of a cusp at the bubble head due to the fast radial inflow. This leads to the creation of a reflex angle, $\alpha >\pi $, near the bubble head, see (A) in Fig. 10. These radial jetting phenomena have already been observed in the context of non-hemispherical bubble collapse.^{8,39,42,112} There, the initially oblate bubble initiates circumferential pinching at the wall as the collapse velocity reaches a maximum. A similar reflex angle can be found at the base of the bubble, where the velocity difference across the wall boundary layer and the inward motion of the bubble during the collapse are superimposed.

When the standoff distance increases, $0.2\u2264\gamma \u22640.24$, the circumferential flow reduces and both, the radial and axial inflow, meet near the symmetry axis at similar times leading to the formation of scrumbled bubble shapes.^{53} At the bubble head, the formation of a small kink can be observed, which is a result of the convergent cylindrical flow.^{73} Due to the thin liquid film between bubble and wall, the outer rim of the bubble base shows similar shapes as for lower standoff distances.

With further increasing standoff distances, $0.3\u2264\gamma \u22640.4$, the jet becomes broader and reduces the cylindrical inflow to ease the formation of a step near the bubble head, see (B) in Fig. 10. For even larger standoff distances, this kink is smoothed out by surface tension. Since the film thickness increases, more fluid is pulled below the bubble shaping the base into a single acute ring, $\alpha <\pi /2$. When more and more fluid fills the gap between wall and bubble, a secondary kink is created above leading to a straight, cylindrical base, see (C) in Fig. 10. These sharp corners are counteracted by surface tension above $\gamma \u22650.9$. In the transition zone, $0.6\u2264\gamma \u22640.9$, a single obtuse kink is visible above the wall.

We observe that the initially formed cusp/kink at lower standoff distances reduces once the broader regular jet suppresses the radial inflow near the bubble head. Its position actually moves away from the wall, since the size of the bubble at jet penetration increases for larger *γ*. At the base, initially a reflex angle, $\alpha >\pi $, is formed due to the superposition of the velocity field near the wall and the flattening of the bubble. When the bubble base moves away from the wall, an acute rim forms, $\alpha <\pi /2$. For even larger *γ,* this rim turns into a straight base followed by an obtuse kink further away from the wall. Once the expansion toward the wall increases, capillary forces smoothen the bubble surface completely. The presence of the kink at the bubble base is also supported by experiments of Lindau and Lauterborn^{64} who observe concentric rings at the outer rim of the bubble for $\gamma <0.7$. Similar trends have also been found experimentally by Reuter and Ohl^{73} and Kröninger^{77} who see the formation a kink for standoff distances below $\gamma <0.75$. The presence of kinks at the bubble surface for even higher standoff distances is mainly due to high interface resolution by the level-set approach.

### D. Jet characteristics

Jetting phenomena that occur during bubble collapse near a rigid wall have been analyzed in depth by Lechner *et al.*^{53} They have identified three different jetting regimes: needle ($\gamma <0.2$), regular ($\gamma >0.24$), and mixed jets ($0.2\u2264\gamma \u22640.24$), Fig. 9. In the regular jet regime, the interface at the symmetry axis is accelerated continuously during the collapse phase forming a broad jet from the bubble head toward the wall. After the jet has penetrated the bubble, it impacts on the wall and is deflected parallel to it. Since the collapse of the bubble continues, that is, the outer surface of the bubble still contracts, a splashing effect (Blake splash) might occur when inward and outward motion meet at the wall.^{25,26}

In the mixed and needle jet regime, the jetting phenomena are more complex. Therefore, in Fig. 11, these regimes are visualized by three characteristic standoff distances $\gamma =0.1$ (needle jet), $\gamma =0.2$ (mixed jet), and $\gamma =0.225$ (mixed jet).

The source of the needle jets is completely different, shown in Fig. 11(a). During the collapse, the outer fluid produces a radial inflow near the bubble head that leads to the formation of a cusp.^{53} When this inflow impacts on the axis of symmetry, the cusp is separated from the main bubble and a shock wave is emitted. This shock hits both interfaces and leads to a fast acceleration of the bubble surface near the interface and generates the needle jet. Compared to the regular jet, the needle jet decelerates from the time the shock wave impacts on the bubble surface toward the impact of the jet on the wall. In addition to the penetration of the main bubble, also the head of the cusp is pierced by a fast shock-accelerated jet forming a toroidal bubble.^{53} Similarly to the regular jet regime, the collapse of the bubble continues after impact. In the experimental images of Reuter, Deiter, and Ohl^{17} [Fig. 4(b) in Ref. 17] the needle jet is indicated inside the bubble accompanied by a cluttered/rough surface. From numerical results [Fig. 11(a), $\tau =1.229\u22121.232$], we see that the shock-wave from the radial inflow is transmitted into the bubble and undergoes several interactions/reflections with the bubble interface. This leads to complex wave patterns and high velocities inside the bubble, which may be visible in the experiments.

In the transition zone, Fig. 11(b), both jet characteristics have a similar strength. Hence, the radial inflow and the axial jet meet at the axis of symmetry at similar times and close to the wall. This leads to a circumferential pinching by a wall-parallel jet similarly to the formation process for negative standoff distances in the Rayleigh collapse scenario.^{8,39,42} When the wall-parallel flow approaches the symmetry axis, a strong upward accelerated jet is initiated due to the deflection of the flow. Finally, this jet penetrates the almost flat cusp and jetting phenomena toward the wall are mostly suppressed. Compared to the two other regimes, the collapse of the bubble is finished before the upward jet is generated.

When the radial inflow hits the wall further away from the symmetry axis, Fig. 11(c), the deflection fully hinders primary jetting phenomena at the axis. Instead, the bubble splits into two parts, the outer toroidal bubble and a small bubble in the center. Whereas the toroidal bubble expands as in the regular jet regime, the center bubble develops with a mushroom-like shape due to the pushing effect of the outer torus. This leads to a secondary circumferential pinching at the bottom of the inner bubble. As a result, a secondary needle jet is created close to the wall. Again, an upward and downward motion is initiated, but the strength of these jets is lower.

For even larger standoff distances, $\gamma \u22650.25$, the increase in regular jet velocity hinders the impact of the radial inflow on the wall. Hence, secondary jetting phenomena are avoided at the symmetry axis. This corresponds well to the definition of the transition zone between needle and regular jets, $0.2\u2264\gamma \u22640.24$, of Lechner *et al.*,^{53} and Reuter and Ohl.^{73} Since we do not reduce the equilibrium radius during the evolution of the bubble or include phase-change across the interface, the effects in the mixture zone might shift toward lower/higher standoff distances. Furthermore, the driving pressure may influence the extent of the transition zone.

To quantify the characteristics of the different jet phenomena, we analyze the jet based on its velocity, $ujet$, the bubble volume at jet impact, $Vb,jet$, and the delay time between jet impact and bubble collapse, $\Delta Tjet=Tjet\u2212Tc$. For these quantities, Supponen *et al.*^{32} have provided scaling laws in the weak jet regime based on experimental and numerical data obtained from a boundary-integral method. To allow comparisons between different scenarios, they introduced an anisotropy parameter, *ζ*, that relates the driving pressure of the collapse to a Kelvin impulse.^{69} For a rigid wall, this anisotropy factor is given as function of the non-dimensional standoff distance, *γ*, and reads

In Fig. 12, we compare the different normalized jet characteristics in the strong jet regime, $\zeta >0.1$, with the numerical results of Lechner *et al.*,^{53} and Supponen *et al.*^{32} All data are obtained at the instant the jet impacts the wall.

Good agreement is found for the volume of the bubble at jet impact. The peak value, $Vb,jet/Vmax=0.1$, lies around $\gamma \u22480.6$ and agrees well to the results of Lechner *et al.*^{53} and Supponen *et al.*^{32} Toward the transition zone between needle and regular jets, a sharp drop is visible indicating the influence of the simultaneous radial and axial inflow. In the needle jet regime, the collapse continues after jet impact. With decreasing standoff distance, the needle jet becomes sharper and the bubble volume is less affected. Hence, the bubble volume at jet impact has a maximum when the standoff distance tends to zero. For bubbles at large standoff, the collapse and jet impact occur simultaneously, and the bubble volume reduces. The minimal bubble volume is reached at the lower limit of the transition zone between needle and regular jets, $\gamma \u22480.2$. The radial inflow has its largest influence on the remaining inner bubble and outer torus bubble, see Figs. 11(b) and 11(c).

When the jet forms and penetrates the bubble, a toroidal shape remains and the collapse process continues. For weaker jets, $\zeta \u22480.1$, the jet and collapse occur almost at the same time, $\Delta Tjet/Tc,ub\u22480.01$.^{32,68} With decreasing standoff, the delay time first increases and reaches a flat plateau, $0.34\u2264\zeta \u22640.54,\u20090.6\u2264\gamma \u22640.7$. For even higher anisotropy factors, the delay time decreases again due to the increase in radial inflow and the transition to a different jet regime. In the needle jet regime, the delay times level out at approximately 1%. These findings are in good agreement with Lechner *et al.*,^{53} Tomita and Shima,^{113} and Tong *et al.*^{114} Compared with Supponen *et al.*,^{32} we observe a shift of the delay time toward higher standoff distances, but a quantitative accordance. This may be related to the different numerical schemes. In their boundary-integral approach, compressibility, viscosity, and surface tension effects are not considered, which affect the collapse time and also the jetting time. In the transition zone between regular and needle jets, $0.2\u2264\gamma \u22640.25$, negative delay times may arise due to the impact of the radial flow on the wall before the regular jet pierces the bubble, see Fig. 11(c). This agrees with recent observations of Tong *et al.*^{114} who found negative delay times around $0.1<\gamma <0.3$.

Considering the jet speed, we observe a decrease for smaller standoff distances, that is, larger anisotropy factors. For anisotropy values of $\zeta \u22480.1$, jet velocities around $ujet\u2248100\u2009m/s$ are reached and decrease to one third at $\zeta \u22483,\u2009\gamma \u22480.25$. Similar velocities have been obtained by Lechner *et al.*^{53} near a rigid wall and by Koch *et al.*^{54} for a bubble collapse near a cylinder. Furthermore, they are in good agreement with experimental data of Philipp and Lauterborn^{60} and Vogel, Lauterborn, and Timm.^{62} The velocities of Supponen *et al.*^{32} are only marginally higher in the regular jet regime, but the deviation tends to increase when the bubble moves closer to the wall as complex bubble shapes cannot be resolved properly by their applied numerical method.^{77} In the fast jet regime, $\zeta >5,\u2009\gamma \u22640.2$, jet velocities around $ujet\u22481000\u2009m/s$ can arise. Compared with Lechner *et al.*,^{53} the observed velocities are around two times lower. Recently, the existence of the needle jets has been confirmed experimentally by Reuter and Ohl^{73} who found average jet velocities around $850\u2009m/s$, which agree fairly well with our numerical results. The high jet velocities in Lechner *et al.*^{53} can be attributed to the evolution of the jet velocity over time, see Fig. 13.

In the regular jet regime, $\gamma \u2208{0.5,1,1.5}$, the upper pole of the bubble is continuously accelerated during the bubble collapse. The downward acceleration reaches its maximum shortly before the jet is initiated. Afterward, the acceleration reduces and the jet velocity reaches a plateau until impact on the wall. Hence, the determination of the jet velocity is possible over a larger time interval reducing the measurement errors. In the needle jet regime, however, the continuous acceleration before jet initiation is followed by an impulsive acceleration once the radial inflow impacts on the symmetry axis. At creation, the jet has its maximum velocity, which, then, decreases until jet impact. Therefore, a quantitative analysis of the jet velocity mainly depends on the temporal resolution and chosen time instance. We observe peak values at $\gamma \u22480.1$ up to $1500\u2009m/s$, which corresponds to the velocities obtained by Lechner *et al.*^{53} Nevertheless, our observed average values are lower and lie in the range $750\u22121050\u2009m/s$, which better agrees with the experiments.^{73}

Another difference between the needle and regular jet regime can be elucidated by the geometry of the jet. Therefore, in Fig. 14 the jet width, $Wjet$, height, $Hjet$, and volume, $Vjet$, is shown as a function of the standoff distance at the moment the jet penetrates the bubble. The geometric extent of the different length scales is indicated by the image in the lower right corner of the plot and follows the definitions of Jayaprakash, Hsiao, and Chahine.^{115}

In the regular jet regime, the continuous acceleration of the jet during the collapse leads to an increased jet width for decreasing standoff distances. The maximum is reached shortly before the transition regime toward the needle jet at $\gamma \u22480.4$. The width of the jet always remains below 40% of the maximum bubble size and reaches approximately 25% at $\gamma \u22481$. This corresponds well to recent experimental results of Reuter and Ohl^{73} who found a jet width of $Wjet/Rmax\u22480.22$ at $\gamma \u22480.98$. Similar trends have also been observed by Jayaprakash, Hsiao, and Chahine^{115} who found a maximum jet width of 35% at around $\gamma \u22480.3$.

Contrarily to the width, the height of the jet reaches its maximum at around *γ* = 1.^{115} At this point, two effects are superimposed by each other. First, the bubble reaches approximately spherical shape at maximum expansion; that is, it has reached its maximum size. Second, the liquid layer between bubble and wall at maximum expansion is still small enough to prevent radial contraction from the lower side of the bubble. For smaller standoff distances, the bubble height reduces, and for larger distances, the contraction from the lower bubble wall increases. Both effects reduce the height of the jet.

The opposing evolutions of the jet height and width lead to an approximately constant jet volume in the full near-wall regular jet regime, $0.3<\gamma <1$. This has been indicated by Jayaprakash, Hsiao, and Chahine,^{115} but they provide only a very few data points in this regime, which makes a full comparison difficult. Quantitatively, we are in very good agreement with their data. Finally, for even larger standoff distances the jet volume reduces and presumably limits into a linear decrease as has been found by Obreschkow *et al.*^{67} and indicated by the evolution of the film thickness. As the velocity of the jet decreases for smaller standoff distances, the kinetic energy of the liquid jet also decreases for $\gamma <1$ and reaches a local maximum around $\gamma \u22481$.^{76,116}

In the transition toward the needle jet regime, all characteristic scales reduce significantly. First, the height of the jet drops due to the radial inflow at the bubble head preventing a further piercing of the bubble by the jet. When the annular inflow increases for lower standoff distances, also the width of the jet reduces. The minimum jet volume is reached at around $\gamma \u22480.1\u22120.15$ as the size of the cusp is maximal there, and the jet penetrates the bubble with the highest velocity.

When we compare the remaining volume of the bubble at jet penetration, $Vb,jet$, with the jet volume, $Vjet$, we find a minimum at around $\gamma \u22480.6$ as the volume of the torus bubble at jet penetration reaches a maximum and the delay time between jet and collapse is maximal. For smaller and larger standoff distances, the delay reduces leading again to an increased ratio between jet and bubble volume. At around $\gamma \u22481.5$, jet and bubble volume are almost identical. In the transition zone, a drop is again visible limiting in a nearly constant ratio in the needle jet regime.

### E. Characteristic length scales

Collapse and expansion are mainly affected by geometric shapes of the bubble during its evolution. When the bubble reaches its maximum size, the initial bubble energy has been transformed into potential energy. In Fig. 15, the equivalent radius $Rmax=3/(4\pi )Vmax1/3$, the height, $Hmax$, the width, $Wmax$, and the height to width ratio, $Hmax/Wmax$, of the bubble at maximum volume are plotted as function of the standoff distance. All lengths have been normalized by the maximum size of a bubble expanded in an unbounded liquid, $Rmax,ub$.

In the limiting case *γ* = 0, the expansion of the bubble is hemispherical leading to similar extensions in wall-parallel and wall-perpendicular direction, $Wmax\u2248Hmax$. Since the initial volume is a hemisphere, the maximum equivalent radius gives the theoretically predicted value of $Rmax/Rmax,ub=(1/2)1/3$. This resembles the observations from spherical bubble collapse and aspherical Rayleigh collapse at *γ* = 0.^{42} The maximum equivalent radius, $Rmax\u2248Rmax,ub$, is approximately reached, once the initial bubble detaches from the wall, $\gamma \u2248\gamma 0$. Until around $\gamma \u22480.3$ the wall prevents an expansion of the bubble perpendicular to it. Hence, the bubbles are highly stretched along the wall with $Hmax/Wmax\u22480.5$ resembling hemispherical expansion with a larger size compared with *γ* = 0. When the standoff distance is further increased the confinement effect of the wall weakens and the bubble expands both in wall-parallel and wall-perpendicular direction. This leads to a strong increase/decrease in bubble height/width and a reduction of the bubble asymmetry at maximum expansion. A spherical shape is recovered only for large standoff distances, $\gamma \u22652.5$. For all investigated standoff distances, the bubble volume/equivalent radius remains lower than the volume of a bubble that expands in an unbounded liquid.^{58} Only for very large standoff distances, $\gamma \u2192\u221e$, the expansion would be spherical with the unbounded radius. As the wall blockage is most effective until $\gamma \u22480.3$, the minimal bubble volume is reached at this point. The reduced width of a bubble for increasing standoff distances has also been observed by Kröninger^{77} who studied the laser-induced collapse near a solid wall for standoff distances $\gamma \u2208[0.5,0.7]$. In the experiments, the size of the bubble significantly increased around $\gamma \u22480.5$.

The blockage of the wall is also visible in the evolution of the bubble centroid. In Fig. 16, the relative movement of the centroid, $\Delta zce=zce\u2212zce,0$, from the original position $zce,0$ is plotted as function of the standoff distance for several positions during the bubble evolution. Shown are the centroid position at the maximum centroid movement, $\Delta zce,max$, at maximum bubble volume, $\Delta z\u2009exp$, at the moment the jet penetrates the bubble, $\Delta zjet$, and at bubble collapse, $\Delta zc$.

For all standoff distances, the centroid migrates away from the wall during the expansion with decreasing movement for increasing standoff distances.^{24,116} When the distance to the wall is small, $\gamma <0.3$, the maximum displacement of the centroid equals that at maximum expansion, $\Delta z\u2009exp\u2009=\Delta zce,max$. At this point, the blockage effect of the wall reduces and the bubble expansion is predominantly perpendicular to the wall. This leads to a shift between $\Delta z\u2009exp$ and $\Delta zce,max$. For all larger standoff distances, $\gamma >0.3$, the centroid position reaches its maximum before the expansion is completed and already migrates toward the wall in the final stages of the bubble growth. The displacement reduces significantly for larger standoff distances and is below 1% once the expansion of the bubble is almost spherical at $\gamma \u22482.5$. The remaining finite movement of the centroid for larger standoff distances is confirmed by data of Supponen *et al.*^{32} who found that the centroid translates following a power law even in the weak jet regime $\gamma >10$.

During the collapse phase, the bubble migrates toward the wall due to the pressure difference between the low-pressure region at the wall and the higher pressure on the opposite side.^{60} The translation of the bubble is then initiated by secondary Bjerknes forces.^{8} At the instant when the jet penetrates the bubble, the centroid has significantly moved toward the wall.^{116} The absolute displacement, $\Delta ze$, depends mainly on the thickness of the liquid film between the bubble and the wall, and the initial standoff distance. At $\gamma \u22481$, the migration is maximal and reduces for larger standoff distances due to an increase in the liquid film. This agrees well with the results of Sun *et al.*^{78} who found a maximum displacement of $\Delta zc=\u22120.93$ at $\gamma \u22481$. For all standoff distances, the bubble migration continues after jet impact on the wall, $\Delta zc>\Delta zjet$. The maximum difference is reached, when the delay time of the jet $\Delta Tjet$ is maximal at $\gamma \u22480.6$.

### F. Bubble collapse and wall pressure impact

After the jet has pierced the bubble, it impacts on the wall and the remaining torus bubble continues to collapse. The size/position of the toroidal bubble is an important characteristic in the context of surface cleaning as the maximum cleaning radius corresponds well to the position of the torus.^{13} In Fig. 17, the radius of the torus center, $Rtorus$, is given as a function of the standoff distance at the time the minimum bubble volume is reached. Our results are compared with experimental data of Tomita and Shima,^{113} Lindau and Lauterborn,^{64} and Reuter, Deiter, and Ohl.^{17}

For decreasing standoff distances, the torus radius increases linearly and reaches a maximum at $\gamma \u22480.6$ with $Rtorus/Rmax\u22480.6$. At this point, the delay time between jet impact and bubble collapse is maximal and the inward motion of the jet is counteracted by the outward flow after jet impact.^{113} When the bubble moves further toward the wall, the torus radius decreases again. At around $\gamma \u22480.1\u22120.15$, a local minimum is reached where the torus ring is only around 10% of the maximum expansion radius. This correlates well with the position of maximum erosion found in Reuter, Deiter, and Ohl.^{17} Until a standoff distance of $\gamma >0.2$, simulation and experiment of Tomita and Shima^{113} show the same trends and agree quantitatively. Only for very small standoff distances, a deviation is visible which we attribute to complex jetting phenomena and bubble shapes in the transition zone between regular and needle jets following recent experiments of Reuter, Deiter, and Ohl.^{17} Compared with the results of Lindau and Lauterborn,^{64} we observe a shift of the maximum torus radius toward lower standoff distances and larger radii. We attribute this observation to the driving force^{80} of the collapse and splashing after jet impact. For example, Koch *et al.*^{117} found a dependency of the minimum collapse radius on the laser energy. Another important factor is the disintegration of the torus bubble after jet impact. As the delay time is large in the range $0.4<\gamma <0.9$, the formation of nanojets at the jet tip and subsequent splashing effects lead to a disintegration of the bubble into multiple smaller ones.^{50,53,64} This makes the measurement of the torus radius more difficult and sometimes even ambiguous in the simulations and experiments.^{64} Hence, we believe that the maximum torus radius lies somewhere in the range $0.6<\gamma <0.7$ and its size can be approximated by $Rtorus/Rmax\u22480.5\u22120.6$. This is confirmed by data of Reuter and Mettin,^{13} who found that the maximum cleaning radius lies around $\gamma \u22480.6\u22120.7$, which directly correlates with the size of the torus. Recently, Reuter, Deiter, and Ohl^{17} have investigated in detail the effect of the torus radius and collapse on erosion patterns. They hinted that for very close standoff distances $\gamma <0.2$ the torus radius might increase again, but no clear statement was possible from the experimental data. We can confirm such an increase in the torus radius in the needle jet regime. Since the radial inflow meets at the axis of symmetry further away from the wall, the jet has more time to develop leading to wider jets and therefore larger torus radii. The minimal torus radius in the needle jet regime corresponds well to the point, where the jet volume is minimal, see Fig. 14.

When the torus bubble collapses, a strong shock wave is emitted that impacts on the wall. In general, during the collapse phase two major pressure peaks are generated, one after jet impact and one at bubble collapse.^{118} In Fig. 18, the maximum wall pressure of these two peaks is plotted as a function of the standoff distance.

When only the collapse of the bubble is investigated, the maximum wall pressure occurs at *γ* = 0 and decreases with increasing standoff distances.^{39,42,51,119} This trend also holds when the expansion of the bubble is included, where we observe a drop of the maximum wall pressure by two orders of magnitude between *γ* = 0 and *γ* = 1. Similar observations apply to the Rayleigh collapse case, where higher absolute peak pressures are obtained as the bubble is directly attached to the solid wall. In a detached scenario, the pressure load will decrease due to the finite size of the liquid layer between bubble and wall, and the formation of nanojets at the jet tip.^{50}

We identify two different regimes. In the needle jet regime, $\gamma <0.2$, high jet velocities impose a significant load on the wall, which exceeds the pressure induced by the bubble collapse. As the radial inflow generates a cusp at the bubble head, which is finally separated from the main bubble, the compression of the torus reduces. At the same time, ultra-fast needle jets are generated containing a high amount of kinetic energy. The opposite holds for the regular jet regime, where the collapse wall pressure is higher than the water-hammer pressure of the jet due to lower jet velocities, thicker liquid layer between bubble and wall, and the intensified collapse of the larger torus bubble.^{64} Since the torus radius slightly increases and the delay time remains constant, the wall-induced peak pressures level out or even decrease for $\gamma <0.2$. Similar trends have recently been observed experimentally by Reuter, Deiter, and Ohl.^{17}

In the range $0.7<\gamma <0.9$, we observe a drop of the peak pressure values. This phenomenon has already been observed experimentally by Shima *et al.*^{63,120} and Tomita and Shima^{113} and was attributed to positive and negative delay times of jet impact. Since in our simulations the jet always impacts before the bubble collapses, $\Delta Tjet>0$, we propose a different explanation. When the bubble moves closer to the wall, the collapse pressure reduces due to larger torus volumes and stronger disintegration of the torus by the jet.^{17,18,60} At the same time, the bubble centroid moves toward the wall reducing the traveling distance of the collapse shock wave. As the centroid move sufficiently fast toward the wall, despite the reduction of the collapse pressure, an overall increase in the collapse wall pressure can be observed. The local maximum correlates well with the point of maximum migration at $\gamma \u22480.9\u22121$. For smaller standoff distances, the bubble remains close to the wall, but the collapse pressure has not reached its minimum.^{17,18} Hence, for $\gamma <0.9$, the maximum wall pressure follows the evolution of the collapse pressure and drops before it increases again. In the experimental data of Tomita and Shima^{113} the drop occurs at around $\gamma \u22481$, which is in reasonable agreement with our findings.

Recently, Boyd and Becker^{58} have compared numerically the maximum wall pressure for a Rayleigh and energy-deposit scenario. Interestingly, they found a similar drop of the peak pressure between $0.5<\gamma <0.9$ for the Rayleigh case, which has not been observed by others,^{39,42,119} but could not find such behavior in the energy-deposit scenario. This might be attributed to the strong numerical diffusion of the applied computational model, which is even more pronounced when the expansion is included into the bubble evolution.^{38,58} Park, Phan, and Park^{121} have observed numerically a plateau in the maximum wall pressure between $0.65<\gamma <1$ using a homogeneous mixture model. As they do not provide additional data in this regime, no final conclusion can be drawn about the occurrence of a drop, but the trends outside this regime agree well.

### G. Prolongation factor

The different sizes of the bubble and corresponding movement of the centroid position also translates into the global time scales of the bubble evolution. In Fig. 19, the prolongation factor, $kc=TVmin/Tc,ub$, is plotted as function of the standoff distance. The simulations are compared with experimental data of Vogel and Lauterborn^{104} and numerical results of Lechner *et al.*^{53}

For large standoff distances, the collapse time is fairly well approximated by the analytical model of Rattray,^{79} Eq. (8), and agrees with the experimental data of Vogel and Lauterborn.^{104} The prolongation factor asymptotically approaches $kc,ub=1$ in an unbounded liquid and increases for lower standoff distances. Similarly to the Rayleigh collapse scenario, the analytical correlation overpredicts the retarding effect for larger standoff distances and underpredicts it when the standoff distance approaches unity.^{51,104}

Below a standoff distance of *γ* = 1, the analytical correlation of Rattray tends toward infinity as it is only first order accurate, whereas the collapse time of the simulations remains finite. This is mainly attributed to the presence of the wall that prevents an expansion of the bubble in wall-normal direction and leads to the formation of flat bubbles. At $\gamma \u22480.8$, the collapse time reaches a first local maximum at $kwallmax\u224821/3$. A similar peak value can be observed when the standoff distance tends to zero, $\gamma \u21920$. In between, a local minimum can be found at $\gamma \u22480.3$. Similar trends have already been observed by Lechner *et al.*,^{53} but further explanations are missing.

When the bubble moves closer to the surface, the retarding effect of the wall leads to an increase in the collapse time as already predicted by the analytical model. The first maximum is reached, when the torus bubble after jet penetration reaches its minimal compression at $\gamma \u22480.8$; that is, the volume of the torus bubble has a maximum. Furthermore, the bubble centroid moves closer to the wall after jet formation leading to less contraction on the wall-side of the bubble, Fig. 16. Once the flattening of the bubble becomes more important and as the liquid film decreases linearly, $\gamma <0.6$, the collapse time reduces mainly due the smaller maximal bubble volume at lower standoff distances, Fig. 15. Accompanying effects, such as the Blake splash,^{25,26} see Fig. 5(a), further enhance the destructive behavior of the reentrant jet on the remaining torus bubble. The minimal collapse time at $\gamma \u22480.3$ corresponds well to the point of minimal bubble volume and where the centroid movement starts to align with the point of maximal expansion. For very low standoff distances, $\gamma <0.3$, the regular jet formation is suppressed by a significant increase in annular inflow, leading to the formation of the needle jets and the formation of bubble cusps.^{53} The jet volume significantly drops in this regime, and splashing effects are prevented by the thin liquid layer between bubble and wall. Finally, the second peak collapse times are reached once the bubble initially is attached to the wall.

In the simulations, a small and highly energetic bubble initially expands into a surrounding liquid at rest. When the standoff distance of the initial bubble is reduced below $\gamma 0$, a cap is cut from the bubble by the wall. Hence, the initial deposited energy is reduced, and the collapse time follows again the wall correlation of Eq. (11) with a scaled standoff distance, $\gamma /\gamma 0$, see the inset in Fig. 19. The maximum observed prolongation of the collapse is found to match $kwallmax$, and for fully attached bubbles, *γ* = 0, the collapse time of an unbounded liquid is recovered.^{42} The overshoot of the collapse time slightly above $kmaxwall$ is attributed to the asymmetry of expansion and collapse. As will be shown below, the prolongation of the full bubble cycle matches $kmaxwall$. With decreasing initial bubble radius, this transition zone would reduce, and for vanishing initial radii, $R0\u21920$, it limits in a collapse time of $kwallmax$. This observation has been exploited in Lechner *et al.*^{53} by increasing the initial bubble radius with the same factor to recover a higher collapse time for *γ* = 0. Yet, when bubbles are created experimentally, the initial bubble radius/plasma will have a finite size.

When we compare the different prolongation factors of the bubble evolution, we see that only the collapse shows the stated oscillating behavior, see Fig. 20. The expansion and cycle time decrease monotonically. In the initial transition zone, $0\u2264\gamma \u2264\gamma 0$, all time scales increase toward the theoretically predicted value of $21/3$ due to the increase in the energy content inside the bubble.

Once the initial bubble is detached from the wall, $\gamma >\gamma 0$, the expansion time decreases monotonically. For increasing standoff distances, the flattening of the bubble near the wall reduces. Therefore, wall-normal expansion is faster leading to an overall reduction of the expansion time. We identify two different regions, where the expansion reduces approximately with constant slope. For $\gamma 0<\gamma <\gamma \u2009exp\u2009=0.3$, the smaller maximal bubble volume leads to a reduced expansion time, even though the wall prevents an expansion toward it. Above $\gamma >\gamma \u2009exp$, the wall allows significant growth in wall-normal direction overcoming the increased bubble volume.

As a combination of the expansion and the collapse, the time for a full cycle shows a plateau, $\gamma \u2009exp\u2009<\gamma <\gamma cyc=0.6$. In this regime, the fast expansion and slow collapse balance each other. In Kröninger,^{77} similar prolongation factors have been observed experimentally for the cycle time in the range of $\gamma \u2208[0.5,0.7]$. Several investigations both experimentally and numerically have studied the prolongation of the bubble evolution near a solid wall,^{55,62,75,76,122} in the combined vicinity of a wall and a free surface,^{74} and a wall entrapped with a gas hole.^{78} They found that the prolongation of the cycle time has a maximum at around $\gamma \u22480.5\u22120.7$ and decreases for smaller and larger standoff distances. On the contrary, Krieger and Chahine^{76} have found a plateau for $0<\gamma <0.6$. The different findings can be attributed to the formulations of the prolongation factors using different reference scales.^{55} We use a static formulation, where the time scales are normalized with the collapse time and radius in an unbounded liquid. Especially, the initial drop for wall attached bubbles, $\gamma <\gamma 0$, is only clearly visible with this formulation.

## VI. CONCLUSION

In this work, we have studied numerically the expansion and collapse dynamics of a single cavitation bubble near a solid wall using a high-resolution sharp–interface level-set method. We focused on characteristic length and time scales of the evolution including the formation of a liquid film between bubble and wall, the jet and collapse characteristics in the strong jetting regime, and the evolution of geometric shapes for varying the initial bubble standoff distances. The sharp–interface level-set method has been proven to be well suited for this problem in particular when the initial bubble standoff distance is small.

We have demonstrated the existence of three distinct thickness regions for the liquid film at jet impact: inner, outer, and transition regions. We find that the bubble does not attach the solid wall during its evolution and we observe a remaining finite separation until jet impact. For bubbles very close to the wall, contact is initiated from the beginning or at early stages of the bubble evolution.

We confirm the three jetting regimes proposed by Lechner *et al*.:^{53} fast needle jets for low standoff distances, the regular jets further away from the wall, and the transition zone in between. Compared with Lechner *et al.*,^{53} we observe two times lower needle jet velocities, which we attribute to the point of jet evaluation. In the needle jet regime, the high energetic jet leads to a strong pressure pulse at the wall that exceeds the force imposed by the collapse pressure. Furthermore, we find that the interaction between the jet delay time, nano-jet formation, and subsequent splashing phenomena, and a strong migration of the bubble toward the wall leads to a drop of the maximum wall pressure in a small range of intermediate standoff distances.

Finally, we have found that the prolongation factor of the bubble evolution increases toward lower standoff distances. Compared to the monotonically decreasing prolongation factor of the bubble expansion, the prolongation of the collapse shows an oscillating behavior and the overall prolongation of the full cycle a plateau. This can be attributed to the interaction between the bubble shape, and the size of the jet and resulting torus bubble. When the bubble is initially attached to the wall, a drop of all prolongation factors is observed resembling an unbounded evolution in the limit of vanishing standoff distances. This drop can be described accurately with a simple energy correlation solely depending on the standoff distance.

## ACKNOWLEDGMENTS

This research was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG; Project Nos. 440395856 and 445127149). The authors gratefully acknowledge the computational and data resources provided by the Leibniz Supercomputing Centre (www.lrz.de).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Alexander Bußmann:** Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead). **Farbod Riahi:** Investigation (supporting); Resources (supporting); Visualization (supporting). **Bilal Gökce:** Supervision (supporting); Writing – review & editing (supporting). **Stefan Adami:** Supervision (lead); Writing – review & editing (equal). **Stephan Barcikowski:** Conceptualization (supporting); Methodology (supporting). **Nikolaus A Adams:** Conceptualization (lead); Funding acquisition (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: GEOMETRIC SOURCE TERMS

In this paper, the simulations are three-dimensional but assume axi-symmetry. The transformation from Cartesian (*x*, *y*, *z*) to axi-symmetric cylindrical coordinates (*r*, *z*) is done by treating all singular terms of type $1/r$ that are introduced by the divergence operator as geometric source terms $S(U)$ on the right-hand side of the evolution equation

where the remaining fluxes $F(U)c$ and $F(U)\mu $ are treated as in the Cartesian space. Details on the full derivation of the geometric terms in the cylindrical space can be found in Meng.^{103} Here, we only give a summary of the final source terms

where *μ* is the shear viscosity of the fluid, *u _{r}* and

*u*are the velocities in radial and axial direction, and

_{z}*r*is the radial coordinate. A bulk or volume viscosity is neglected in all cases.