Modifying the cavitation bubble collapse in the erosive regime with a surface bar structure

Abstract

The collapse of cavitation bubbles near surfaces can cause severe erosion, posing significant risks to fast-rotating turbines. Previous studies suggest that the energy concentration of shock waves during non-spherical collapse is a potential mechanism behind this erosion. An effective strategy to mitigate shock wave focusing and subsequent erosion could involve modifying the boundary structure. In an initial effort to influence shock wave focusing, we introduce a symmetry-breaking boundary structure, specifically a slender bar, to quantitatively investigate how asymmetry affects cavitation bubble dynamics during the final collapse stage and the self-focusing process of shock waves. Using two high-speed cameras to capture the behavior of a single laser-induced cavitation bubble near this structure, we identify two distinct regimes based on the characteristic morphologies of the bubble during the final collapse: the Island-Bridge regime and the Asymmetric Torus regime. We analyze how the dynamic and geometric characteristics of the toroidal bubble evolve with increasing distance between the bubble and the structure, revealing distinct trends in each regime. Furthermore, we examine the first collapse location and collapse propagation velocity of the toroidal bubble, which are likely to affect the shock wave energy focusing. The findings of this study provide insights into the role of structures attached to the boundary in cavitation bubble dynamics and may offer a potential methodology for designing surface microstructures to mitigate cavitation erosion or to concentrate cavitation bubble energy for engineering applications.

Highlights

• •A slender bar structure is introduced on the boundary to quantitatively modify the asymmetry in the surrounding environment of the cavitation bubble.
• •The final stage of cavitation bubble collapse exhibits distinct dynamics, categorized into two regimes.
• •The system asymmetry affects the self-focusing of shock waves emitted during the collapse.

1. Introduction

Cavitation refers to the physical process of rapid growth and collapse of the cavitation bubbles formed by a phase transition due to a sudden pressure decrease in the liquid [1], [2]. Cavitation bubbles exhibit complex behaviors close to particles, droplets, and walls [3], [4], [5], [6], [7], [8]. When cavitation bubbles collapse near a solid boundary, the non-spherical collapse of bubbles causes serious erosion to the boundary, which has been observed in high-speed rotating turbines, spillway tunnels, transient processes in pipelines, and traumatic brain injury [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. The energy concentrated during the cavitation bubble collapse can also be utilized in many fields, such as breaking up calculi [19], [20] and cleaning surfaces [21], [22], [23], [24]. To learn more about the mechanisms responsible for erosion, research has focused on model experiments that investigate the details of a single cavitation bubble collapse close to a boundary. The common findings can be summarized that two factors may contribute to the plastic deformation of the boundary: the impact of a liquid jet and the emission of shock waves [25], [26], [27], [28], [29], [30], [31]. During the collapse of a bubble near a solid boundary, the part of the bubble farther from the boundary collapses faster than the part closer to it. This non-spherical deformation of the bubble eventually results in the formation of a high-speed liquid jet directed towards the boundary, with velocities reaching hundreds of meters per second. It was speculated that these velocities may be sufficient to generate stagnation pressures to damage the material [32]. Because of this non-spherical collapse, rather complex compressible fluid focusing occurs during the last stage of the bubble collapse. In a first careful single cavitation bubble study utilizing high-speed streak imaging, Ref. [33] reported the direct and indirect importance of jets in combination with shock waves for the plastic deformation of a soft metal boundary made of indium. However, as indium is an extraordinarily soft metal that, for example, can even be easily scratched with a fingernail, damage mechanism on engineering metals may differ. When the bubble is close to the boundary, damage primarily occurs during the first collapse. In contrast, for bubbles positioned further away, only the second collapse phase interacts significantly with the boundary. The importance of the closeness of the bubble collapse was identified by [33] and later confirmed by [34]. The latter work also revealed that the damaged shape strongly correlates with the shape of the bubble during the near-boundary collapse. This finding was quantified by measuring the volume of the plastic deformation as a function of the non-dimensional stand-off distance $\gamma$—defined as the ratio between the bubble center and the boundary, normalized by the maximum radius of the cavitation bubble, which revealed a maximum for $\gamma \le 0.3$ [34]. However, smaller stand-offs were not investigated in that work.

Already in 1998, Ref. [34] revealed a strong increase in the damage with decreasing $\gamma$ down to 0.3. This small stand-off regime attracted recently increased research after it was predicted through simulations, that a thin and super-fast jetting within a novel regime would reach speeds of 1000 m/s and more [35], [36]. This regime was confirmed experimentally by [37] and the jet was named as the needle jet (NJ). Following the penetration of the needle jet through the bubble, the bubble evolves into a deformed torus. Minor asymmetries in the system or instability factors can cause part of the vapor cavity to implode earlier than others [38], [39]. The shock waves emitted from this partial collapse can drive the implosion of the remaining bubble fragments. This process results in self-focusing of the emitted shock waves, with the collapse of the final toroidal fragments coinciding precisely with the erosion pit location [38]. On the other hand, no damage from jet impact was detected on any of the studied metal specimens. These observations suggest a novel mechanism for cavitation erosion at very small stand-off distances, driven by self-focusing shock waves during the asymmetric collapse of the toroidal structure formed in the final stage of bubble collapse.

With this understanding, it becomes straightforward to explore means to influence the collapse dynamics of the torus, and thereby mitigate cavitation erosion. Previous approaches have focused on preventing bubble collapse at the boundary, either with compliant boundaries [40], [41], [42], [43], or by introducing gas bubbles and oil droplets trapped at the boundary [7], [44], [45], [46], [47]. Both methods effectively repel the bubble from the boundary before its collapse. However, altering the properties of the boundary may conflict with its intended functionality. For instance, compliant boundaries are generally unsuitable for pumping or ship propulsion applications. Recognizing the critical role of toroidal collapse in energy focusing, an alternative approach could be the modification of the structure rather than the mechanical properties of the boundary.

In this study, we investigate the influence of a simple structural modification – a slender bar positioned on the boundary – on the final stage of bubble collapse and energy focusing. Our goal is to quantitatively analyze how varying degrees of system asymmetry, induced by changing the distance between the bubble and the structure, affect energy focusing. This research aims to provide guidance for designing microstructures on boundaries that eventually regulate cavitation erosion.

2. Method

A schematic diagram of the experimental setup is depicted in Fig. 1(a). A cavitation bubble is induced near a vertical plate by focusing a pulsed frequency-doubled Nd:YAG laser (Litron Nano S, U.K., pulse length $\sim 5\mathrm{ns}$, wavelength $=532\mathrm{nm}$) into deionized (DI) water in a transparent glass container ($50\mathrm{mm}\times 50\mathrm{mm}\times 50\mathrm{mm}$). A thin flat plate ($20\mathrm{mm}\times 13\mathrm{mm}\times 1\mathrm{mm}$) with a single horizontal slender bar is used as a modified boundary for the cavitation bubble in the experiments, see Fig. 1(b). The plate is 3D-printed using a resin printer (Forms 3+, Formlabs, US, resin Clear V4). The cross-sectional shape of the bar structure is approximately semicircular with a radius of $a\sim \text{200}\text{µm}$. Fig. 1(c) depicts the profile recorded with an X-ray microscope (Carl Zeiss, US, Xradia 610 Versa). For precise control of the position of the plate with respect to the laser focus, the plate is translated with a motorized three-axis stage (x,y,z-direction, Physik Instrumente M-410. DG).

Fig. 1.

Experimental set-up and notation. (a) Schematic of the experimental configuration for a single cavitation bubble near the boundary with the slender bar structure. (b) Enlarged schematic of the side view from camera 1. (c) Surface morphology of the plate sample with the slender bar structure.

The laser beam is focused using an in-house sealed submerged objective lens (Mituyoyo $50\times$, numerical aperture NA$=0.42$, nominal working distance $20.5\mathrm{mm}$). Half of the laser beam is blocked using a semicircular aperture mask to prevent it from direct interaction with the specimen. We define the distance between the laser focus and the farthest point of the bubble from the boundary as the vertical radius $R_v$, and an equivalent radius of a circle with the same area observed from the front view as the lateral radius $R_l$, to characterize the dimensions of the bubble. The maximum radius the bubbles reach during their oscillation is $R_{l,max}\sim 0.65\mathrm{mm}$. As shown in Fig. 1(b), the induced bubble is located very close to the plate, i.e., with a very small stand-off distance between the bubble center and the plate: $\gamma =d/R_{v,max}\sim 0.15$. Under this condition, the needle jet (NJ) and the self-focusing effect are prominent features of the cavitation bubble collapse [38], [48]. To quantify the influence of the slender bar structure on the cavitation bubble, we introduce the dimensionless distance that connects the laser focus with the center of the slender bar, i.e., $ϵ=z/(a+R_{l,max})$, where $z$ is the coordinate of the bubble seeding center, see Fig. 1(b). The case $ϵ=1$ corresponds to the case in which the cavitation bubble is just in contact with the slender bar structure when it grows to its maximum volume. As $ϵ$ becomes smaller, the asymmetry in the bubble’s environment during both growth and collapse becomes more pronounced.

The dynamics of the bubble are recorded with two high-speed cameras from two perpendicular views both equipped with the Canon MP-E 65 mm lens. Camera 1 (Photron Fastcam Mini AX, operated at 100,000 fps, resolution=5.4 µm/pixel) captures the overall growth and collapse of the bubble from the side to measure the stand-off distance $\gamma$. A second, faster high-speed camera 2 (Shimadzu HPV-X2, operated at 5,000,000 fps, resolution=4.4 µm/pixel) captures the detailed bubble dynamics and the emission of shock waves during the final stage of the bubble collapse. For this purpose, this front view is illuminated with an expanded femtosecond laser pulse train (Ekspla Femtolux, wavelength $515\mathrm{nm}$, pulse duration $220\mathrm{fs}$). The laser light is brought to the experiment via a glass fiber. All the cameras and lasers are synchronized using a pulse generator (BNC 525, Berkeley Nucleonics, CA).

3. Results and discussion

3.1. Bubble collapse in the absence of the slender bar structure

As a base case, we first take a look at the behavior of the cavitation bubble in the absence of the slender bar structure, as illustrated in Fig. 2. The time $t=0$ is the instant of bubble generation. The cavitation bubble grows to its maximum volume ($R_{l,\max }=\text{0.64}\text{µm}$), attaining a nearly hemispherical shape at $\text{50}\text{µs}$. Subsequently, the cavitation bubble gradually shrinks, displaying a symmetric kink at $t=\text{100}\text{µs}$ in the side view. The overall growth and collapse of the cavitation bubble reveal axisymmetry, even after the first collapse, and the remaining bubble fragments concentrate close to the location where the bubble is seeded.

Fig. 2.

High-speed imaging of cavitation bubble dynamics near the boundary without the slender bar. (a) Side view and the front view of the overall bubble dynamics. Times are shown with respect to the bubble seeding frame $t=\text{0}\text{µs}$. The red dashed line indicates the surface of the sample. (b) presents the final stage of the collapse dynamics. Times are given with respect to the time of the first collapse $t_c$. The green and yellow arrows in (b) indicate the position of the NJ and the shock wave (SW), respectively. The shock fronts are clearly visible in the supplementary videos. In the still images here they are sketched for better visibility. The corresponding movies S1 and S2 are available in supplementary materials at https://doi.org/10.1016/j.ultsonch.2025.107439.

Fig. 2(b) presents the dynamics of the final stage of the cavitation bubble collapse from the front view, where $t_c$ is defined as the time of the minimum bubble volume, i.e., the starting time of the collapse frame. This front view onto the solid plate also shows a grainy structure of the plate as a result of the 3D printing. During this stage, the outline of the cavitation bubble initially appears as a nearly ellipse shape elongated vertically and is contoured by a purple dashed line. This elongation may be attributed to the initial mildly elliptical shape of the plasma that nucleated the cavitation bubble, see, for example, $t=0$ in Fig. 2(a). Subsequently, a thin NJ is observed that penetrates the bubble towards the boundary, which is marked with a green arrow in frame 3 of Fig. 2, forming a toroidal bubble. The occurrence of the NJ is consistent with the numerically and experimentally determined stand-off distance conditions $\gamma <0.24$ provided in Refs. [35], [37], [49]. The toroidal bubble then collapses, and shock waves emitted by the top and bottom vertices of the torus are observed, marking the final collapse points of the toroidal bubble.

3.2. Bubble collapse near the slender bar structure

We now study the effect of the slender bar surface structure on the bubble dynamics. According to the definition of the key parameter $ϵ$ in Section 2, we can distinguish our experimental observations between $ϵ\le 1$ and $ϵ>1$: in the former cases, the bubble contacts the slender bar during expansion, and in the latter, no contact occurs. In the following sections, we will discuss the cavitation bubble behaviors for both regimes in detail.

Let us start with $ϵ>1$, where the cavitation bubble does not contact the structure during its growth and collapse. For $ϵ=1.42$ the bubble in the side view and for most of the expansion and collapse phase exhibits only mild differences compared to the case without the structure, see Fig. 3(a). However, just after the first bubble collapse, $t=\text{110}\text{µs}$, the center of mass of the remaining microbubbles has migrated towards the structure.

Fig. 3.

High-speed imaging of cavitation bubble dynamics near the boundary with the structure. (a) The side view and the front view of the entire bubble dynamics and (b) the dynamics of the collapse stage of the bubble for $z=\text{1200}\text{µm}$, $ϵ=1.42$. (c) Side view and the front view of the entire bubble dynamics, and (d) dynamics of the last stage of the bubble collapse $z=\text{500}\text{µm}$, $ϵ=0.58$. Times are shown with respect to the bubble seeding frame $t=\text{0}\text{µs}$ and $t_c$ is defined as the instant of the first bubble collapse. The red dashed lines in (a)(c) indicate the boundary of the sample, including the slender bar. The colored arrows indicate different features of the dynamic process: the NJ position (green), the beginning points of IB or AT structure’s collapse (pink), the distal and the proximal focusing points (DFP and PFP, blue), and the shock wave (SW, yellow). The green lines in (c) mark the shape of the cavitation bubble. The corresponding movie S3-S6 are available in supplementary materials at https://doi.org/10.1016/j.ultsonch.2025.107439

The detailed view of the final stage of the cavitation bubble collapse is shown in Fig. 3(b). The projected shape of the bubble evolves into an ellipse. The NJ appears below the center of the bubble in $t_c-\text{800}\text{ns}$ due to the slower shrinkage of the part of the bubble that is closer to the structure. In the absence of the surface structure, the NJ forms at the geometric center of the bubble. The NJ penetrates the cavitation bubble, and the bubble takes the shape of an Asymmetric Torus (AT), where the part closer to the structure is larger. At $t_c-600\mathrm{ns}$ the radial expansion of the jet and the contraction of the cavitation bubble’s outline lead to the collapse of the part distal to the structure, marking the start of the AT collapse, as indicated by the pink arrows. Subsequently, the collapse propagates along the AT starting at these two points of first collapse, towards both distal and proximal parts of the bubble (with respect to the bar structure). The distal part finally collapses to the distal focusing point (DFP) in frame $t_c-400\mathrm{ns}$. shock waves (SW) emitted from the proximal focusing point (PFP) are marginally visible in high-speed videos and marked in frames $t_c$ and $t_c+200\mathrm{ns}$ of Fig. 3(b).

For $ϵ\le 1$, the cavitation bubble during its growth and collapse comes into contact with the structure. As exemplified in Fig. 3(c), $ϵ=0.58$, the cavitation bubble grows to its maximum volume rather symmetrically, only the bar structure somewhat confines the growth of the bubble, see the kinks at $\text{30}\text{µs}$. To be more specific, from the front view, the left and the right sides of the cavitation bubble, whose height is lower than the structure, are blocked by the slender bar. While the central part extends atop the structure and grows continuously. During shrinkage, however, the bubble asymmetry quickly increases. This can be well perceived in the side view between $t=\text{70}\text{µs}$ and $t=\text{100}\text{µs}$ where the shape of the cavitation bubble is shown with green lines. The proximal part of the bubble is indented from the top, indicating that the part of the bubble further from the solid boundary shrinks faster than the part that is attached to the bar. The last $\text{2.4}\text{µs}$ of the collapse are further detailed at higher temporal resolution in Fig. 3(d). Times are given now with respect to the collapse time $t_c=\text{107}\text{µs}$. This series reveals complex dynamics with a pear-shaped bubble. It has a narrower top and a broader bottom. As shown at $t_c-2400\mathrm{ns}$ in Fig. 3(d), the proximal and distal parts of the bubble develop as distinct objects, i.e., start to split, into an upper more circular and a lower crescent-shaped, which are marked with purple dashed lines. During the splitting process, the two bubble parts are connected by gas bridges. Considering this distinctive shape of the cavitation bubble at the initial stages of collapse, we term it Island-Bridge (IB) collapse, where the bridge refers to the connection between both parts. Again, an NJ forms and penetrates the bubble through the edge of the upper island and bridge, as indicated with the green arrow at $t_c-2200\mathrm{ns}$. Subsequently, the liquid of the NJ spreads radially on the substrate outward, while the proximal and distal islands shrink independently. The proximal island shrinks radially, whereas the distal island keeps its crescent shape and contracts in thickness. The locations where the two islands split are marked with pink arrows at $t_c-1200\mathrm{ns}$. The details of these rather complex dynamics are analyzed in greater detail in the following sections. The distal part of the cavitation bubble collapses completely at the location denoted as DFP, $t_c-600\mathrm{ns}$. During its rebound, the proximal part reaches minimum volume at time $t_c$ and location PFP in Fig. 3(d).

We hereby distinguish the two types of collapse dynamics as IB and AT, corresponding to $ϵ<1$ and $ϵ>1$, respectively. In the following discussions, we systematically analyze the parametric behavior of the bubble for each type.

Fig. 4 presents high-speed photography of the final collapse of three IB cases, for $ϵ$-values of 0.47(a), 0.74(b), and 0.87(c). With increasing $ϵ$ the distal island, showing a crescent shape, expands in the vertical direction, causing the curvature at the structure’s lowest point to increase progressively. A second observation is that the position of the NJ moves away from the bubble seeding center, leading to a shift of the first collapse location of the IB structure (indicated by the pink arrows). Notably, for $ϵ=0.87$, the location at which the collapse begins nearly aligns with the structure’s lowest point, as shown in frame 2 of Fig. 4(c). Lastly, the time between the impact of the NJ onto the solid and the cavitation bubble collapse increases. The time of NJ impact can be read from the front view imaging as it shapes the bubble toroidally by piercing completely through it. The NJ impact is $-1000\mathrm{ns}$, $-1600\mathrm{ns}$ and $-1800\mathrm{ns}$, respectively, before the bubble collapse.

Fig. 4.

Collapse dynamics during the final stage of the cavitation bubble in the IB regime (front view)$:$ (a) $z=\text{400}\text{µm}$, $ϵ=0.47$; (b) $z=\text{600}\text{µm}$, $ϵ=0.74$; (c) $z=\text{700}\text{µm}$, $ϵ=0.87$. Times are indicated in each frame with respect to the collapse frame for each case. The corresponding movie S7 is available in supplementary materials at https://doi.org/10.1016/j.ultsonch.2025.107439.

For $ϵ>1$ we find a collapse of an asymmetric torus (AT). The details of the dynamics can be observed in the high-speed frames shown in Fig. 5 for (a) $ϵ=1.22$, (b) $ϵ=1.94$, and (c) $ϵ=3.55$. With increasing $ϵ$ the bubble volume is smaller at the instance of NJ impact and the time between NJ impact and minimum bubble volume decreases. The NJ impact times are $-1200\mathrm{ns}$, $-600\mathrm{ns}$, and $-400\mathrm{ns}$, respectively. This observation is totally different from the case in the IB regime. Secondly, the position where the NJ occurs approaches the torus center, accompanied by a shift of the first collapse location of the AT structure towards the center (pink arrows). In the case $ϵ=3.55$ where the cavitation bubble is a considerable distance away from the bar structure, its impact on the asymmetry can be considered nearly negligible. Here, the AT collapse displays near the left–right symmetry (front view), and the left and right vertices of the ellipse (the outline of the bubble) are the collapse beginning points.

Fig. 5.

The collapse dynamics of the final stage of the cavitation bubble for AT regime (front view)$:$ (a) $z=\text{1000}\text{µm}$, $ϵ=1.22$; (b) $z=\text{1600}\text{µm}$, $ϵ=1.94$; (c) $z=\text{3000}\text{µm}$, $ϵ=3.55$. Times are indicated in each frame with respect to the collapse frame for each case. The corresponding movie S8 is available in supplementary materials at https://doi.org/10.1016/j.ultsonch.2025.107439.

To quantitatively describe the influence of the structure on cavitation bubble dynamics, we first focus on the overall process of bubble growth and collapse. Therefore, we plot the variation of dimensionless vertical radius $R_v/R_{v,max}$ (as defined in Fig. 1(b)), and dimensionless lateral radius $R_l/R_{l,max}$ over time (with timescale $R_{max}\sqrt{\rho /\Delta p}$, $\Delta p=p_{\infty }-p_v$ and $p_v$ is the vapor pressure [1], $R_{max}$ is obtained from a curve fit to the experimental observations) for different $ϵ$. We also compare the results with Rayleigh equation [50], which describes the spherical bubble dynamics in an infinite liquid. For bubble growth and collapse in the vertical direction relative to the plate, the experimental results show good agreement with the Rayleigh equation during the expansion stage but demonstrate a delay in the collapse process, as illustrated in Fig. 6(a). To explore the discrepancy of the bubble collapse delay, we calculate the Rayleigh prolongation factor $k=T_L/2T_c^{\mathrm{Rayleigh}}$ [51], [52], where $T_L$ is the cavitation bubble’s lifetime for the first cycle and $T_c^{\mathrm{Rayleigh}}=0.91468R_{v,max}\sqrt{\rho /\Delta p}$ is the Rayleigh collapse time. The Rayleigh prolongation factor $k$ represents the bubble’s lifetime deviation from a spherical bubble collapse. We further compare our experimental prolongation factors under different dimensionless distances $ϵ$ with those reported by Reuter et al. [48], where the cavitation bubble is very close to an unmodified plate (i.e., without the slender bar structure). As shown in Fig. 6(c), our experimental results align well with Reuter et al. [48] within the margin of error, demonstrating that the delay is mainly attributed to the solid boundary while the slender bar structure almost has no effect on the bubble’s lifetime with the present temporal resolution.

Fig. 6.

Evolution of the cavitation bubble growth and collapse. (a) The evolution of the normalized radius of the cavitation bubble in the vertical direction, $R_v/R_{v,max}$ with the dimensionless time $t/\left(R_{v,max}\sqrt{\rho /\Delta p}\right)$ for different $ϵ$. (b) The evolution of the normalized lateral radius of the cavitation bubble from the front view $R_l/R_{l,max}$ with the dimensionless time $t/\left(R_{l,max}\sqrt{\rho /\Delta p}\right)$ for different $ϵ$. The gray lines in (a) and (b) are the solution of the Rayleigh equation for a spherical bubble in an infinite liquid. (c) The change of the Rayleigh prolongation factor $k$ with $ϵ$ (the dimensionless distance from the slender bar structure).

In contrast, for bubble growth and collapse in the direction parallel to the plate, the evolution of the lateral radius follows the Rayleigh equation during the first cycle, as illustrated in Fig. 6(b). However, differences in bubble dynamics for various $ϵ$ values are observed after the first cycle. Specifically, as $ϵ$ increases, the maximum lateral radius of the cavitation bubble during the second cycle decreases, while the period of the second cycle initially increases and then diminishes. Combining observations from high-speed frames and bubble radius evolution, we conclude that the structure has minimal effect on the overall process of bubble growth and collapse during the first cycle. Instead, the structure primarily impacts the final stage of bubble collapse, which in turn affects the dynamics of the cavitation bubble during the second cycle. Although the overall radius evolution process appears similar, the detailed dynamics during the final stage of collapse differ significantly with $ϵ$. Therefore, we have to use the ultra high-speed camera with a framerate of 5,000,000 fps to explore the detailed dynamics of the final stage of the cavitation bubble collapse, which constitutes the core content of this paper.

Next, we focus on the final stage of the cavitation bubble collapse, particularly the dynamics of the toroidal bubble formed after the NJ penetration. To analyze the entire movie in one single 2D image, we emulate streak imaging, i.e., we select only one line of the movie and plot the respective line for all times one below another. Due to the left–right symmetry of the system (in front view), we select a vertical line with a width of only one pixel through the center of the bubble, see Fig. 7(a). Each streak image consists of 30 lines covering $\text{6}\text{µs}$ of the bubble collapse for the cases $ϵ=0.467,1.094,3.548$ (from left to right). The dark areas in the frames indicate the bubble’s presence, as light from the back of the plate is reflected and scattered by the wrinkled bubble surface. Horizontal stripes are formed due to the roughness of the 3D-printed plate. The three frames reveal that the bubble’s outline first continuously shrinks inward. Additionally, at the moment $t_{nj}$ which is defined as the instant of the NJ’s impact onto the solid, the center of the bubble appears brighter due to the absence of the gas phase in that region from the penetration of the NJ. Subsequently, the inner hole of the toroidal bubble gradually expands outward. The times $t_c^D$ and $t_c^P$ denote the meeting of the contraction of the outline and the expansion of the inner torus hole for the distal part and the proximal part, respectively, indicating the collapse of the toroidal bubble. In our results, $t_c^D$ always precedes $t_c^P$ because the proximal part collapses earlier in the presence of the slender bar structure. After the collapse, the cavitation bubble disintegrates into multiple microbubbles and enters the second cycle, corresponding to the black area that increases over time after $t_c^P$. The tilted stripes in the lower part of Fig. 7(a) reveal the migration of microbubbles that do not merge. Comparing the evolution of streak images for these three cases, we observe significant differences in asymmetry. For the case $ϵ=1.094$, the NJ formed in the distal part of the bubble, resulting in the bubble and its fragments after collapse re-expanding mostly on the proximal side, while hardly any microbubbles are found at the bottom. In contrast, for the case $ϵ=3.548$, the NJ position is located almost at the center of the bubble, and the times $t_c^D$ and $t_c^P$ are close to equal. As a result, the re-expansion of the bubble’s distal and proximal parts is rather symmetric.

Fig. 7.

Evolution of the toroidal bubble in the final stage of the cavitation bubble collapse. (a) The change of the centerline of the bubble with time for different $ϵ=0.467,1.094,3.548$. The red and blue arrows indicate the contraction process of the outlines and the expansion process of the inner holes, respectively. The purple arrow indicates the migration process of microbubbles. (b)(c) Radial expansion velocity of the central torus hole $v_{D,exp},v_{P,exp}$ and the outline contraction velocity $v_{D,\mathrm{con}},v_{P,\mathrm{con}}$ with $ϵ$ for the distal part (b) and the proximal part (c).

The streak images allow measuring the average contraction speed of the outlines ($v_{D,\mathrm{con}},v_{P,\mathrm{con}}$) and the average expansion speed of the inner holes ($v_{D,exp},v_{P,exp}$), respectively. We measure the average contraction velocities and the expansion velocities of the central hole between the time of needle jet impact on the solid $t_{nj}$ and the instance of collapse $t_c^D$ or $t_c^P$. In Fig. 7(b) and (c), these speeds are plotted as a function of the distance from the structure $ϵ$ for the distal and proximal parts of the toroidal bubble, respectively. The blue lines depict the expansion speeds and the red lines depict the contraction speeds. Note, the vertical axes for the contraction speeds are reversed to more intuitively represent the collision speed of the toroidal bubble at collapse, which is the sum of the expansion speed and the contraction speed of the torus [39], i.e., $v_{D,exp}+v_{D,\mathrm{con}}$ and $v_{P,exp}+v_{P,\mathrm{con}}$, respectively. The results indicate that as $ϵ$ increases, the expansion speed first increases and then remains nearly constant once $ϵ>2$. The contraction speeds follow a similar trend. Additionally, the proximal part of the bubble expands slightly faster and contracts slightly slower than the distal part in the presence of the symmetry-breaking structure, leading to nearly identical collision speeds. For large $ϵ$ such as the case $ϵ=3.548$, the speeds for the proximal and distal parts converge.

Besides the velocities, the location of the NJ impact point and the two focal points of the torus are strongly affected by the lateral distance $ϵ$ of the cavitation bubble from the slender bar structure. The locations of these three features are depicted in Fig. 8(a). We distinguish here between the proximal (PFP) and the distal focus point (DFP) of the torus with respect to the structure. These three locations are sketched in Fig. 8(b). Here the outline of the toroidal bubble is approximated with an ellipse, where the NJ is located beneath the ellipse’s center and the distal focusing occurs before the proximal focusing. The ellipse contracts while the inner hole expands, as analyzed above. Particularly, the two focal points PFP and DFP may be important to assess the potential effect of the structure on the resulting damage to the surface. Unfortunately, in the present study, the surface finish does not allow for assessing this damage, we nevertheless report the positions of the likely regions of damage in Fig. 8(a). The positions stated are given relative to the bubble’s seeding position, indicated with the horizontal black dashed line. Within the IB regime, $0.45<ϵ<1$, the PFP shifts rapidly upwards relative to the bubble seeding center, moving from approximately $\text{0.05}\text{mm}$ to $\text{0.3}\text{mm}$ away from the seeding position. The positions of the NJ and DFP transition from below the seeding position towards it. For $ϵ\approx 1$, the NJ is close to the seeding position, yet the asymmetry between the distance of the PFP and the DFP remains. For larger $ϵ$, thus in the AT regime, this asymmetry is gradually reduced. As $ϵ$ increases, the PFP shifts back towards the bubble seeding center from $\text{0.3}\text{mm}$ to $\text{0.1}\text{mm}$ away from it. We find for $ϵ\approx 3.6$ as a symmetric collapse, thus indicating that the distances of the PFP and DFP from the seeding position are the same, and the NJ impacts at the seeding position, also.

Fig. 8.

Geometry of the toroidal bubble collapse. (a) The position of the NJ, the DFP, and the PFP in the two regimes as a function of the non-dimensional distance from the structure, $ϵ$. (b) A schematic diagram of the toroidal bubble’s evolution and the definition of $l_{\mathrm{pro}}$ and $l_{\mathrm{dis}}$. (c) Quantification of the asymmetry of the toroidal bubble $A_t$ with the asymmetry of the system $A_s$ for different regimes.

Here, we provide an explanation for the position variation of the NJ, DFP, and PFP. Prior works explain the NJ formation with the convergent radial flow that occurs during the cavitation bubble’s shrinkage [36], [37]. In our experimental system, the outgoing flow from the proximal part of the expanding bubble is hindered by the structure. Therefore, during the shrinkage of the bubble, we expect a stronger focusing of the flow and thus the formation of the NJ shifting away from the structure. This influence is particularly evident at small values of $ϵ$, where the asymmetry is more pronounced. As for the position change of the DFP, the growth and the shrinkage of the distal part are less affected by the slender bar structure. As a result, the DFP shows a similar trend as the NJ, shifting closer to the bubble’s seeding position with increasing $ϵ$ within the IB regime.

For the position change of the PFP, the slender bar structure’s influence on the bubble dynamics of the proximal part can be explained with two competing effects: in the IB regime, the growing proximal side of the bubble comes into contact with the bar, limiting its expansion compared to the distal side. This asymmetric growth leads to a PFP located closer to the seeding position. The second effect occurs during bubble collapse once the size of the bubble is smaller than to the slender bar structure. Then the structure hinders the liquid inflow to the proximal part of the bubble, resulting in a slower shrinkage than the distal part, see also Fig. 7(b). Consequently, the PFP moves away from the bubble’s seeding position. For $ϵ\leqq 0.46$ the first effect dominates, placing the PFP very close to the seeding position, but diminishes with increasing $ϵ$, and the second effect dominates, and the PFP moves away from the seeding position. For $ϵ>1$ both effects weaken, resulting in a shift of the PFP back to the seeding position. Then the influence of the slender bar structure becomes negligible, not affecting the position of the PFP.

Furthermore, we define the ratio of the distance between NJ impact position and PFP $l_{\mathrm{pro}}$ and the distance between NJ and DFP $l_{\mathrm{dis}}$, see Fig. 8(b), as a quantification of the resulting bubble torus asymmetry

$$A_t=\frac{l_{\mathrm{pro}}}{l_{\mathrm{dis}}}.$$ (1)

A $A_t$ value of 1 indicates the impact of the NJ located at the center between the PFP and DFP. This is the result of the presence of the structure and the distance of the bubble from the structure. We use the reciprocal of the dimensionless distance $1/ϵ$ as a measure for the system environmental asymmetry $A_s$ during the bubble growth and collapse.

In the limit of $A_s$ the bubble does not “see” the structure and a symmetric collapse is expected, while for larger $A_s$ the structure gains influence. Fig. 8(c) presents the variation of $A_t$ as a function of $A_s$. Within the AT regime where $ϵ>1$, $A_t$ increases monotonically with $A_s$, which corresponds to the contribution of the slender bar structure in hindering the liquid flow at the final collapse stage as mentioned in above. However, in the IB regime where $ϵ<1$, as $A_s$ continues to increase, $A_t$ decreases sharply. This reminds us that when the bubble is very close to the structure, due to the direct contact with the structure, the pattern of morphology during bubble collapse is probably different from that for the AT regime, as seen in Fig. 3(d).

Finally, we concentrate on the self-focusing process of the shock wave in the AT regime, beginning with the determination of the points where the collapse of the toroidal bubble initiates. As mentioned earlier, we assume the shape of the toroidal bubble is elliptical. For a given case of $ϵ$, the shape parameters – specifically the semi-major axis $a_e$ and semi-minor axis $b_e$ – are obtained from experiments. From the position of the NJ, we can identify the location where the vapor torus collapses first, namely the two locations on the elliptic ring closest to the NJ, as sketched in the inset of Fig. 9(a). This position is measured with the angle $\varphi$ between the line connecting the center of the ellipse to the first collapse location and the horizontal axis. This angle can be obtained from the following equation:

$$\varphi =arctan{\left(b_e\sqrt{{\left(min\left(\delta {(1-\frac{\underset{e}{\overset{2}{b}}}{\underset{e}{\overset{2}{a}}})}^{-1},a_e\right)\right)}^{-2}-a_e^{-2}}\right)}^{-1}$$ (2)

Here, $\delta$ is the distance from the center of the toroidal bubble to the NJ. Fig. 9(a) depicts this angle $\varphi$ as a function of $ϵ$. As $ϵ$ increases, $\varphi$ decreases monotonically, indicating that the first collapse location moves towards the center of the ellipse. For very large $ϵ$, $\varphi$ approaches zero, suggesting that collapse begins at the short-axis vertices of the ellipse. Two selected high-speed images showing the bubble at the moment of collapse are shown as insets in Fig. 9(a) too, indicating the calculation results coincide with the experiments.

Fig. 9.

Some key factors during the self-focusing process. (a) The variation of the position $\varphi$ of the first collapse location with $ϵ$ and its comparison with experimental observation. The schematic diagram of the definition of $\varphi$ is also given. (b) The variations of the collapse time of the torus $T_{\mathrm{torus}}$ and the collapse propagation speed $v_{\mathrm{collapse}}$ with $ϵ$. The velocity is presented by the Mach number with the speed of sound $c$.

We also examine the ring collapse propagation velocity, i.e., the velocity at which the collapse progresses from the first collapse points along the collapsing torus. Effective shock wave self-focusing results in erosive cavitation and requires large ring collapse propagation velocities that are on the order of the speed of sound [38]. To measure the averaged ring collapse propagation velocity, the collapse time of the torus, $T_{\mathrm{torus}}$ is measured. It is obtained from high-speed imaging and represents the time interval from the start of the torus collapse to the moment it reaches the proximal focus point (PFP). Using the size of the toroidal bubble (i.e. the semi-major axis and the semi-minor axis), the initial location through the angle $\varphi$, and the collapse time of the torus $T_{\mathrm{torus}}$, we calculate the average collapse propagation speed $v_{\mathrm{collapse}}$. Fig. 9(b) presents the variation of the collapse time $T_{\mathrm{torus}}$ and the propagation speed $v_{\mathrm{collapse}}$ with $ϵ$. As $ϵ$ increases, the collapse time initially decreases rapidly before slowing its rate of decrease. The collapse speed rises initially before it eventually stabilizes at around $\text{0.6}\text{Ma}$. Here, the collapse propagation speed is represented by the Mach number, which is the ratio of the collapse propagation speed and the speed of sound in the medium, which we set here as $\text{1483}\text{m/s}$.

We pay close attention to the first collapse location and the collapse propagation velocity, as these factors are crucial in understanding the ultimate focusing strength of the shock wave. We take the self-focusing process of the proximal part as an example, as it typically focuses more energy than the distal part. For $\varphi =90°$, nearly all the energy of the toroidal bubble is directed towards the proximal focus, while for $\varphi =0°$, only half the energy is focused proximal and the remainder to the distal part. Consequently, as $ϵ$ increases, the energy supplied to the self-focusing process towards the proximal part is expected to decrease. Additionally, the propagation speed of the torus collapse affects the concentration of energy during the focusing process. As the collapse propagation speed approaches 1 Ma, energy tends to concentrate more in the small region around the PFP.

The final focusing strength of the shock wave near the PFP depends not only on the amount of energy involved but also on how concentrated that energy is. Here, we provide a simplified model to clarify how these factors influence the self-focusing of the shock wave. Fig. 10(a) illustrates with a sketch the wave emission and propagation along the elliptical ring for $ϵ=2.34$. The time indicated below each frame represents the interval from the onset of the torus collapse. For the present case the total collapse time is $T_{\mathrm{torus}}=\text{234}\text{ns}$. The collapse starts at $\varphi =29°$ and propagates along the torus at a constant speed. As the collapse progresses, each point on the torus can be seen as a shock wave emitter, generating spherical shock waves that propagate here for simplicity at the speed of sound, $c_0=\text{1483}\text{m/s}$. From the front view, these shock waves appear as circles of varying radii, with their centers corresponding to the collapse points, as shown in Fig. 10(a). As the collapse progresses from the first to the fourth image, we observe a focusing of the waves at the proximal side, which is also evident from the darker red color.

Fig. 10.

The self-focusing process of the shock wave. (a) The diagram illustrates the temporal sequence of the wave emission and propagation along the elliptical ring towards the proximal region, exemplified for $ϵ=2.34$. (b) The normalized focusing intensity as a function of $ϵ$.

This visual impression can be quantified by adding the pressure of all waves emitted during the collapse. This provides a focusing strength $I$, where the pressure is integrated over a small area ($\text{0.1}\text{mm}\times \text{0.1}\text{mm}$) around the PFP at the moment of the final torus collapse, i.e. $T_{\mathrm{torus}}$. Accounting for an inversely proportional pressure amplitude of a spherical wave, the focusing intensity, $I$, is calculated as

$$I\left(ϵ\right)={\int }_{s_0}^{\mathrm{PFP}}\frac{p_0\left(ϵ\right)}{c_{0}T\left(s\right)}l\left(s\right)\mathrm{d}s.$$ (3)

Here, $s_0$ denotes the location of the first collapse on the ellipse, and the integral is taken along the elliptical path from the $s_0$ to the PFP. $T\left(s\right)=T_{\mathrm{torus}}-\Delta t\left(s\right)$ represents the propagation time of the shock wave emitted from the point $s$, while $l\left(s\right)$ is the length of the circle in the certain area around the PFP corresponding to the shockwave released by point s. The pressure $p_0\left(ϵ\right)$ is the amplitude of the emitted spherical wave released per unit length, which is a function of $ϵ$, see below.

To better compare the focusing effects at different distances from the slender bar structure, we assume that the total energy released during the collapse of the toroidal bubble remains nearly constant across different $ϵ$ values, as the laser energy is unchanged. Therefore, we can relate the pressure as a function $ϵ$ with

$$p_0\left(ϵ\right)c\left(ϵ\right)=p_c.$$ (4)

Here, $c\left(ϵ\right)$ is the perimeter of the elliptical ring for a given $ϵ$, and $p_c$ is an arbitrary constant. Using this model, we calculate the focusing strength normalized by its maximum value, and plot it against $ϵ$, as shown in Fig. 10(b). This reveal that the normalized focusing strength is strongly reduced by the presence of the structure. For $ϵ<2$, the structure is mitigating the shock wave focusing. For large $ϵ$, the focusing strength as the expected levels. Interestingly, there is a local maximum around $ϵ\approx 2.3$ where the structure is enhancing the focusing.

Although we lack a quantitative measure for the severity of the damage as a function of distance from the structure, we expect that the slender bar structure reduces erosion as long as the bubble collapses within $ϵ<2$. In the present experiments, the structure is a hard plastic that is considerably softer than most metals commonly exposed to cavitation and demonstrating erosion. Unfortunately, the roughness of the structure from the 3D-printing process does not allow for a quantification of the erosion or damage to the surface.

4. Conclusion

We investigated experimentally the dynamics of a single cavitation bubble collapsing at a planar solid surface with a slender bar structure. A parametric study of the dimensionless lateral distance $ϵ$ of the location of bubble seeding to the bar structure was conducted. Particularly the torus collapse was analyzed in great detail as it is known to generate self-focusing shock waves. We distinguish two distinct bubble collapse dynamics depending on whether the bubble makes contact with the structure: the Island-Bridge (IB) regime for $ϵ<1$ and the Asymmetric Torus (AT) regime for $ϵ>1$. We further quantify how dynamic and geometric features, such as the contraction and expansion speeds of the toroidal bubble and the positions of the needle jet (NJ), the proximal focusing point (PFP), and the distal focusing point (DFP), vary with $ϵ$. When the bar structure is very close to the bubble, i.e., $ϵ\le 1$, direct blockage during the growth stage plays a crucial role. For larger distances, $ϵ>1$, the impeding effects on the flow focusing during the collapse stage become more significant.

Finally, we investigated the dynamics of the progressive torus collapse. The location of the first collapsing piece of the bubble, the duration of the collapse on the torus, and the collapse propagation velocity are strongly dependent on the distance to the bar structure $ϵ$. Thus, this distance is presumed to have a strong effect on the focus of the shock wave, and thus the maximum load on the substrate at the proximal focus point, which we analyze using a simplified model to show the trends qualitatively. While we have not studied the damage on the substrate or exerted pressures here, the size of the re-expansion of the bubble after the first collapse may give a good indication that the structure does inhibit the energy focusing. Fig. 6(b) reveals that the bubble after the first collapse expands much less when it is further away from the structure. This observation may explain why the bubble loses more energy from shock-wave emission as a result of higher energy focusing. While this is an indirect measure, it aligns with the other findings on the importance of the structure.

Future work should connect with the present findings and explore the shock wave focusing directly through imaging and the damage to the substrate. For the first, it would be helpful to have a suitable transparent structure. For the latter, a metal substrate with good surface polish would be ideal to directly measure the damage and quantify the focusing strength of the shock wave from the experiments. Additionally, numerical simulations could help to better understand the flow focusing and thus to optimize the surface structures for the best protection against cavitation erosion. Besides, the shape of the microstructures can also be adjusted to explore the effects of the compound and complex asymmetry of the system on the cavitation bubble dynamics and self-focusing process. Once the details are established, the structured surfaces should be tested in more complex cavitation conditions, i.e., acoustic or hydrodynamic cavitation erosion test scenarios. Our work makes the first step to explore the influence of the microstructure-modified boundary on the final collapse stage of the cavitation bubble and the self-focusing process of the shock wave within the very small stand-off distance regime ($\gamma =0.15$). Our findings offer valuable insights into reducing cavitation erosion or concentrating energy via the strategic placement of microstructures like slender bars, potentially leading to improved designs in applications such as pumping and ship propulsion.

CRediT authorship contribution statement

Jiajun Cui: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. Fabian Reuter: Writing – review & editing, Methodology, Investigation, Conceptualization. Zibo Ren: Writing – review & editing, Methodology, Investigation. Zhigang Zuo: Writing – review & editing, Supervision, Project administration, Investigation, Funding acquisition, Conceptualization. Shuhong Liu: Writing – review & editing, Supervision, Project administration, Investigation, Funding acquisition, Conceptualization. Claus-Dieter Ohl: Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.

Ethics statement

This research falls outside of human or animal studies, and institutional ethical approval was not required.

Funding

We acknowledge financial support from the Beijing Natural Science Foundation (No. QY23147), the National Natural Science Foundation of China (NSFC, Nos. U24A20140, 52076120 and 52079066), and the German Research Foundation (DFG, Nos. ME 1645/12-1 and OH 75/12-1).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following is the Supplementary material related to this article.

Video S1

Supplementary Video 1.

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Supplementary Video 2.

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Supplementary Video 3.

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Supplementary Video 4.

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Supplementary Video 5.

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Supplementary Video 7.

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Supplementary Video 8.