Numerical Investigation on Collapse Dynamics of Near-Wall Bubbles under Elliptical Surface Boundaries Conditions

1  Introduction

Bubble dynamics plays a pivotal role in a wide range of scientific and engineering applications, including underwater explosions (UNDEX) [1–3], cavitation collapse [4–6], mono and hybrid nanofluids [7–9], and ocean engineering [10,11]. The behavior of bubbles—particularly their expansion, collapse, and jetting processes—has been a subject of intense research due to its fundamental complexity and practical significance. When a bubble is generated in a liquid, it typically undergoes rapid expansion followed by a violent collapse [12]. This collapse can produce high-speed liquid jets, shock waves, and localized high-pressure zones, all of which can significantly affect nearby structures or biological tissues [12–14]. Understanding these phenomena is crucial for predicting and controlling the impact of bubble activity in both natural and engineered systems. In underwater explosion events, the detonation of an explosive charge generates a high-pressure shock wave and a subsequent gas bubble that oscillates in the surrounding water [15–18]. This bubble pulsation is accompanied by complex physical phenomena such as bubble migration, jet formation, and interaction with boundaries like the free surface or rigid walls. These interactions are not only of theoretical interest but also have direct implications for naval architecture, marine engineering, and defense applications.

The study of bubble dynamics dates back to the early 20th century, with the foundational work of Lord Rayleigh, who modeled the collapse of a spherical cavity in an infinite liquid [19]. Since then, numerous theoretical, experimental, and numerical studies have extended this work to account for the effects of boundaries, compressibility, viscosity, and multiphase interactions. One of the earliest and most influential models is the Rayleigh-Plesset equation, which describes the radial oscillations of a spherical bubble in an incompressible liquid [20]. While this model provides valuable insights into the basic dynamics of bubble oscillation, it assumes spherical symmetry and neglects the influence of boundaries, which are critical in real-world applications.

To address these limitations, more advanced models have been developed. For example, the Keller-Miksis equation incorporates weak compressibility and allows for more accurate predictions of bubble oscillations in acoustic fields [21]. Similarly, the doubly asymptotic approximation (DAA) model developed by Geers and Hunter accounts for the effects of fluid compressibility and boundary interactions, enabling the prediction of bubble migration and pressure wave propagation [16]. Despite these advancements, theoretical models based on spherical symmetry are inherently limited in their ability to predict non-spherical phenomena such as jet formation and bubble splitting, which are common in the presence of boundaries.

Numerical simulations have become an indispensable tool for studying bubble dynamics, particularly in complex geometries and under extreme conditions. Various numerical methods have been developed, including boundary element methods (BEM), finite volume methods (FVM), and diffuse interface methods (DIM). BEM is highly accurate for incompressible and inviscid flows but struggles with compressibility and large deformations [22]. On the other hand, FVM-based solvers can handle compressible multiphase flows and are well-suited for simulating underwater explosions, but they often suffer from numerical diffusion at interfaces [23]. To overcome these challenges, hybrid methods and interface-sharpening techniques have been proposed. One notable example is the THINC (Tangent of Hyperbola for INterface Capturing) method, which enhances the accuracy of interface reconstruction in multiphase flows [24,25]. Liu et al. applied the THINC scheme to a six-equation compressible multiphase model to simulate underwater explosion bubbles, demonstrating significant improvements in capturing jet formation and bubble morphology [2]. Their results showed that the interface-sharpening model could accurately predict bubble pulsation periods, jet velocities, and the direction of bubble migration under various boundary conditions.

Another important aspect of bubble dynamics is the influence of nearby boundaries, such as free surfaces and rigid walls. The presence of these boundaries can drastically alter the behavior of bubbles through the generation of pressure gradients and the imposition of kinematic constraints. For example, when a bubble collapses near a rigid wall, the pressure gradient induced by the wall can cause the formation of a high-speed jet directed toward the wall [12]. Conversely, when a bubble is located near a free surface, the jet is typically directed away from the surface due to the attraction-repulsion effect known as the Bjerknes force [13]. These phenomena have been extensively studied using both experimental and numerical approaches, leading to the development of predictive criteria such as the Blake criterion, which estimates the direction of bubble jets based on the relative strength of buoyancy and boundary-induced forces [26]. In more complex scenarios, such as when a bubble interacts with multiple boundaries or other bubbles, the dynamics become even more intricate. For instance, when two bubbles are placed in close proximity, their interaction can lead to phenomena such as bubble coalescence, jet redirection, or even bubble splitting [27]. Han et al. investigated the dynamics of a near-wall bubble influenced by an adjacent bubble using both experiments and numerical simulations [28]. Their results showed that the presence of a second bubble could significantly alter the direction and velocity of the jet formed by the first bubble, depending on the relative positions and sizes of the bubbles. They also developed a theoretical model based on the Kelvin impulse to predict the jet behavior, demonstrating the utility of theoretical frameworks in interpreting complex bubble interactions. Liu et al. developed a flow field separation model to simulate the interaction between a bubble and a rigid plate with a circular opening [29]. Their approach allowed for the accurate prediction of jet behavior even when the bubble threaded through the opening, demonstrating the importance of adaptive numerical techniques in handling complex geometries.

Numerous research efforts have sought to investigate the impacts of bubbles on boundary surfaces as well as the effects of boundary conditions on the collapse process of bubbles induced by lasers and sparks in the vicinity of boundaries. Andrews et al. [30] and Trummler et al. [31] focused their investigations on the collapse behaviors of cavities and bubbles adjacent to slot and crevice structures, illustrating how surface topographies affect the generation of jets and the concentration of pressure. Cao et al. [32] expanded upon this research by exploring the function of acoustic impedance in shock-triggered bubble collapse, underscoring the energy transfer process that is dependent on material characteristics. Lin et al. conducted studies on the dynamic behaviors of single bubbles under the influence of mesoscale surfaces, in addition to examining the cavitating flow around a flexible hydrofoil [33]. Zhang et al. proposed a theoretical model for compressible bubble dynamics that incorporates phase transition and migration processes, while also taking into account the influences of rigid walls [34]. Kadivar et al. [35] experimentally demonstrated that micro-structured riblets with heights of approximately 200 μm substantially alter single bubble collapse dynamics, mitigating toroidal ring formation post-collapse. This finding aligns with broader investigations into surface texturing effects on cavitation bubble energy dissipation [36]. Hu & Yao [37] further elucidated that superhydrophobic riblet coatings synergistically reduce turbulent skin friction through combined mechanisms of slip velocity enhancement and bubble-induced turbulence modification. Kadivar et al. [38] established that cavitation bubbles collapsing near cylindrical rods generate mushroom-shaped morphologies at small standoff distances (γ0 < 1), producing significantly different microjet characteristics than planar boundaries. Subsequent investigations by Pan et al. [39] revealed that rigid filaments induce three distinct collapse morphologies—flat necking, teardrop-shaped, and spherical collapse—depending on the dimensionless distance γ0 and filament radius r*. While research specifically addressing spherical boundaries remains limited, investigations into bubble–particle interactions and curved wall geometries provide relevant insights. Wang et al. [40] developed Kelvin impulse theoretical models for bubbles collapsing near spherical particles, validated through experimental correlation between maximum bubble radius and non-circular deformation metrics. Studies on oscillating boundaries [41] and right-angle wall configurations [42] reveal that boundary curvature and motion significantly modulate collapse times and jet formation mechanisms. Collectively, these investigations highlight the crucial role of boundary conditions in determining bubble collapse dynamics, providing valuable guidance for simulations related to oscillatory motions and cavitation phenomena.

To further elaborate on the influence of complex boundaries on the bubble collapse process, this study investigates the collapse dynamics of bubbles in close proximity to elliptical surfaces. For this purpose, three distinct boundary conditions are specifically selected: a planar surface, an elliptical convex surface, and an elliptical concave surface. A computational model for multiphase compressible fluid dynamics is employed to reveal the effects of these complex boundaries on the bubble oscillation period, water jet evolution, and wall pressure induced by the jet. The findings of this work are expected to provide a foundation for a deeper understanding of the dynamic characteristics of near-wall bubbles. For the reader’s convenience, the remainder of the paper is structured as follows: Section 2 describes the physical model and numerical methods for simulating multiphase compressible flow. The validation of the method is presented in Section 3. Section 4 details the schematic of the problem investigated in this study. Section 5 presents the numerical results and discussion. Finally, concluding remarks are provided.

2  Physical Model and Numerical Methods

2.1 Physical Model

This study uses a compressible 5-equation multi-component flow model [15,23] that conserves mass, momentum and total energy to investigate interface motion and pressure propagation during the collapse of a cavitation bubble near an elliptical surface. For the cases considered herein, the effects of viscosity and surface tension are insignificant when compared to inertial effects, and so they are not included in the model. The governing equations are given as follows:

∂q∂t+∂f(q)∂x+∂g(q)∂r=S(q)(1)

q=(ρ1z1ρ2z2ρuρvρE),f(q)=(ρ1z1uρ1z1uρu2+pρuv(ρE+p)u),g(q)=(ρ1z1vρ1z1vρuvρv2+p(ρE+p)v),S(q)=−n−1r(ρ1z1vρ1z1vρuvρv2(ρE+p)v)(2)

and

∂z1∂t+u∂z1∂x+v∂z1∂r=0(3)

where q represents the vector of unknown variables in the governing equations; f(q) and g(q) correspond to the physical flux vectors along the x-coordinate and r-coordinate directions, respectively; u and v denote the flow velocity components in the x- and r-directions, respectively; ρE is the total energy per unit volume, which is defined as ρE=ρe+0.5ρ(u2+v2). The parameter n is a dimensionless system constant, with a value restricted to either 1 or 2. Given that a two-dimensional axisymmetric model is predominantly employed in the present study, the value of n is specified as 2.

The stiffened equation of state can be used to each phase in mixture [43]

p=ρe(γ−1)−γp∞(4)

where γ and p∞ are parameters for different materials. For perfect gases, γ is the ratio of specific heats and p∞=0. For water medium, γ and p∞ are determined based on the shock Mach number, which is a measure of the stiffness of the liquid. In the literature [43], γ=5.5,p∞=4.92×108 Pa is proposed for water and γ = 1.4 for air.

2.2 Numerical Methods

The governing equations mentioned above are solved using the finite volume method. Specifically, the fifth-order accuracy Weighted Essentially Non-Oscillation (WENO) reconstruction [44] combined with the HLLC approximate Riemann solver [45] is adopted for spatial discretization, while the third-order TVD Runge-Kutta scheme is employed for temporal discretization. For the interface diffusion problem, the interface compression technology is introduced and discretized via the finite difference method [46,47].

3  Method Validation

In our validation test, we simulate a shock wave propagating in air and interacting with a helium cylinder. This configuration was originally investigated by Haas and Sturtevant [48], whose benchmark experimental data are available for numerical validation and comparison [49,50]. The computational domain is a two-dimensional rectangle with a length of 445 mm and a width of 89 mm. Solid boundary conditions are imposed on the upper and lower boundaries. The flow field is initially filled with air, and a helium bubble with a radius of 25 mm is placed at the center of the domain. A shock wave with a Mach number of 1.22 in air propagates from right to left and impinges on the helium bubble. Time is initialized at the instant the shock wave first contacts the bubble surface. The entire flow field is discretized using a uniform Cartesian grid consisting of 2000 × 600 cells. Figs. 1 and 2 respectively show comparisons between the computed and experimental bubble interface morphologies over two time intervals, with numerical results presented in the form of density contours. Fig. 1 displays results within the time range of 32–72 μs, while Fig. 2 covers 82–427 μs. From the comparison, it can be seen that the numerical method employed in this study effectively captures the key dynamic features during the shock–bubble interaction, including the shock front, reflected rarefaction waves, and the complex deformation and motion of the gas–gas interface.

Figure 1: Validation results for 2D air-helium shock bubble interaction: comparison between experimental and numerical results (right part of each panel). (a) 32 μs, (b) 52 μs, (c) 62 μs, (d) 72 μs. The numerical results are represented by density contours.

Figure 2: Validation results for 2D air-helium shock bubble interaction: comparison between experimental and numerical results (right part of each panel). (a) 82 μs, (b) 102 μs, (c) 245 μs, (d) 427 μs. The numerical results are represented by density contours.

4  Problem Description

Fig. 1 displays the schematic of the problem investigated in this paper, which includes three boundary conditions: planar, elliptical convex and elliptical concave. The initial gas bubble is spherical with a radius R0 and a stand-off distance L from the center of the left wall surface. In the numerical simulation, the spherical bubble is initialized at the state where the cavitation bubble expands to its maximum volume. The stand-off distance L is defined as the distance from the bubble center to the left planar wall (as shown in Fig. 3). In this numerical simulation, a two-dimensional axisymmetric model was employed, where the actual computational domain is the upper half area shown in Fig. 3. The computational domain extends 50R0 along the x-axis and 30R0 along the y-axis. A rigid solid wall condition is imposed on the left boundary, an axisymmetric boundary condition is applied to the bottom boundary, and non-reflecting boundary conditions are assigned to all other boundaries. The immersed boundary method is adopted for the numerical treatment of the left-side region adjacent to the elliptical surface. For the elliptical concave and convex surfaces in Fig. 1, their surface boundary conditions are ultimately imposed as rigid solid wall conditions.

Figure 3: Schematic view of a gas bubble near three types of surface conditions. (a) planar wall boundry, (b) elliptical convex boundry, (c) elliptical concave.

In this study, a gas bubble with a radius of R0 = 0.1 mm is adopted, which is filled with non-condensable gas with an initial pressure pb = 1 atm and an initial density ρb = 1.0 kg/m3. The ambient water is set to a pressure of pw = 120 atm and a density of ρw = 1000 kg/m3 to replicate the pressure condition of a deep-water environment. In Fig. 3, the semi-minor axis Ex and semi-major axis Ey of the elliptical profile are set to R0 and 4R0, respectively. Under each type of boundary condition, three operating conditions corresponding to the stand-off distance ratios L/R0 = 1.0, 1.5 and 2.0 are adopted in the present simulation. Table 1 presents the controlled parameters of 9 simulation cases conducted in this paper to investigate the influence of elliptical surface on the collapse of gas bubbles. Fig. 4 presents the schematic illustrations of the overall and local mesh division for the computational domain. Uniform fine meshing is applied to the region near the bubble, whereas a progressively refined mesh is used for the ambient fluid field. The interior of the initial bubble is discretized into 30 uniform meshes over its radius R0, and an adaptive mesh technique is introduced to perform three-level mesh refinement for the whole computational domain based on the density and pressure gradients.

Figure 4: Axisymmetric computational mesh and close-ups of the bubble-adjacent zone for three boundary-surface configurations. The full domain is shown in (a); enlarged views of the plane (b), elliptical-convex (c), and elliptical-concave (d) wall treatments appear alongside.

5  Numerical Results and Discussion

5.1 Bubble Collapsed near a Plane Surface Condition

Three types of stand-off distances of L/R0 = 1.0, 1.5 and 2.0 in the case of the plane surface condition are shown in Fig. 5. Fig. 6 visualizes the flow field of the collapsing bubble in terms of density and pressure distributions at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0, over the time range from 0 to 1.2 μs. It is evident from Fig. 6 that the initially spherical stationary bubble initiates its collapse process under the influence of the ambient high-pressure fluid environment. The existence of the rigid solid wall on the left boundary renders the bubble collapse process highly nonlinear. During the evolution of the bubble interface, the wall-induced effect exerts a significant impact, resulting in intense tensile stretching on the portion of the bubble wall that is proximal to the rigid wall. Quantitative observations reveal that at the time point of 0.9 μs, the bubbles experience different levels of morphological distortion, including interface indentation and jetting phenomena, across the three test cases with distinct stand-off distance ratios L/R0 = 1.0, 1.5 and 2.0. In detail, when L/R0 = 1.0, the bubble is in the initial phase of jet generation triggered by the indentation of the bubble interface; when L/R0 = 1.5, the developed jet is about to penetrate the inner surface of the bubble wall adjacent to the rigid wall; when L/R0 = 2.0, the jet has successfully pierced through the corresponding bubble wall. Furthermore, the density and pressure contour plots clearly demonstrate the presence of a prominent high-pressure zone in the external flow field of the bubble wall on the side distal to the rigid wall. This high-pressure zone serves as the direct driving force for the directional propagation of the jet toward the rigid wall, and the generation of this high-pressure zone is directly caused by the constraint imposed by the left rigid solid wall boundary condition.

Figure 5: Overview of the simulation cases near a plane surface wall. The baffle on the left in each panel represents an infinitely long rigid solid wall boundary condition, and the blue sphere on the right denotes the initially stationary bubble.

Figure 6: Typical schlieren-type snapshots of density (upper part of each panel) and pressure (lower part of each panel) distributions for a bubble collapsing onto a planar rigid wall at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0.

At t = 0.96 μs as shown in Fig. 6, the jet tip is on the verge of penetrating the left-side bubble wall for the case with a stand-off distance ratio of L/R0 = 1.0. In contrast, for the other two cases, the toroidal bubbles formed by jet impact have already initiated their second expansion phase during their propagation toward the rigid wall. The secondary high-pressure waves generated by the high-velocity jet impact propagate outward and reflect off the rigid wall upon arrival at t = 1.0 μs.

Fig. 7 illustrates the time-history curves of the equivalent bubble radius, which are obtained by converting the statistically calculated total bubble volume in the numerical model. Under the stand-off distance ratio condition of L/R0 = 1.0, the duration of the bubble’s first contraction cycle is about 1.0 μs. With an increase in the stand-off distance ratio, the oscillation period of the bubble exhibits a gradual reduction trend. In particular, relative to the reference case of L/R0 = 1.0, the bubble oscillation periods for L/R0 = 1.5 and 2.0 show a decrease of 8.5% and 11.5%, respectively. It can also be observed that as the stand-off distance ratio increases, the equivalent radius of the bubble at the instant of minimum volume during collapse gradually decreases. Fig. 8 illustrates the time-history curves of pressure at the center point of the left rigid wall corresponding to various stand-off distance ratios L/R0. It is evident from the curves that under the condition of L/R0 = 1.0, the pressure peak induced by the water jet generated during the bubble collapse process reaches a relatively high value of 1.37 GPa; meanwhile, the effective duration of the pressure pulse is around 0.3 μs, which constitutes roughly 30% of the duration of the first bubble oscillation cycle. By comparison, the peak pressures of the collapse-induced jets for the cases of L/R0 = 1.5 and 2.0 are reduced to 0.29 and 0.22 GPa, corresponding to a reduction of about 79% and 84% relative to the reference case of L/R0 = 1.0.

Figure 7: The time history evolution of equivalent radius of the buble during the implosion of the bubble collapsed under plane surface conditions at the standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Figure 8: The time history evolution of pressure load on the center of the wall during the implosion of the bubble collapsed under plane surface conditions at three different standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

5.2 Bubble Collapsed near an Elliptic Convex Surface Condition

Fig. 9 presents an overview of the computational cases for bubbles adjacent to an elliptical convex surface under three stand-off distance ratio conditions. Herein, the rectangular box on the left still represents an infinitely long rigid wall, and the elliptical wall with a specified dimension is placed in close proximity to this left rigid wall. Fig. 10 presents typical schlieren-type snapshots that visualize the density and pressure distributions during the collapse of a bubble adjacent to an elliptical convex wall, corresponding to stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0. It is evident from Fig. 10 that the initially stationary spherical bubble located near the elliptical convex surface initiates its collapse process and undergoes a gradual leftward displacement driven by the wall-induced effect from the left boundary. At the instant of t = 0.90 μs, the bubble is at the incipient stage of jet generation for the L/R0 = 1.0 case. In contrast, for the remaining two cases with higher stand-off distance ratios, the jet has fully penetrated the bubble interface, and the entire bubble structure has transitioned into the expansion phase. When compared with the flow characteristics observed at t = 0.90 μs in Fig. 6, the switch from a planar rigid boundary condition to an elliptical convex boundary condition introduces a notable acceleration effect on the bubble collapse behavior to varying extents. At t = 0.96 μs, for the L/R0 = 1.5 and 2.0 configurations, the intense high-pressure load induced by the water jet under the elliptical convex boundary constraint propagates to the solid wall and undergoes reflective propagation. This pressure wave reflection then triggers the intricate dynamic behavior of the toroidal bubble during its subsequent expansion process.

Figure 9: Overview of the simulation cases near an elliptic convex surface wall.

Figure 10: Typical schlieren-type snapshots of density (upper part of each panel) and pressure (lower part of each panel) distributions for a bubble collapsing near an elliptical convex wall at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Fig. 11 presents the time-history evolution curve of the equivalent bubble radius throughout the collapse process of bubbles subjected to the elliptical convex surface boundary condition, corresponding to stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0. As the stand-off distance ratio increases, the duration of the bubble’s first oscillation cycle decreases progressively, a trend that is consistent with the behavior of bubbles collapsing in the vicinity of a plane rigid wall. Furthermore, the minimum volume achieved by the bubble following its initial collapse diminishes with increasing stand-off distance ratio, which reflects the negative feedback effect exerted by the left-side rigid wall on the bubble collapse and contraction dynamics. Fig. 12 shows the time-history evolution of the pressure load acting on the central point of the wall during bubble collapse under the elliptical convex surface boundary condition at three distinct stand-off distance ratios L/R0 = 1.0, 1.5 and 2.0. The results demonstrate that for the cases of L/R0 = 1.0, 1.5 and 2.0, the peak pressure values generated by the bubble-induced water jets at the center of the left-side wall are 1.64, 0.30 and 0.22 GPa, respectively. A comparison with the data reported in Fig. 8 reveals that replacing the planar boundary with the elliptical convex surface boundary leads to a remarkable 20% increase in the water jet pressure peak for the L/R0 = 1.0 case, whereas the pressure peak values for the other two stand-off distance ratio cases are roughly comparable to their original counterparts.

Figure 11: The time history evolution of equivalent radius of the buble during the implosion of the bubble collapsed under epliptical convex surface conditions at the standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Figure 12: The time history evolution of pressure load on the center of the wall during the implosion of the bubble collapsed under epliptical convex surface conditions at three different standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

5.3 Bubble Collapsed near an Elliptic Concave Surface Condition

Fig. 13 provides a schematic overview of the computational model for simulation cases involving a bubble situated near an elliptical concave wall. As opposed to the configuration shown in Fig. 9, the key adjustment lies in altering the boundary condition of the wall adjacent to the bubble from an elliptical convex surface to an elliptical concave surface. Fig. 14 illustrates typical schlieren-type snapshots that visualize the density and pressure distributions during the collapse of a bubble near an elliptical concave wall, corresponding to stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0. A comparative analysis against the computational results from Figs. 6 and 8 reveals that the jet formation dynamics of bubbles near the elliptical concave wall exhibit varying degrees of temporal delay under the three different stand-off distance ratio conditions at the specific instant of t = 0.90 μs within the spherical bubble contraction stage. Furthermore, at t = 0.96 μs, the high-pressure loading generated by the water jet has not yet propagated to the left-side rigid wall boundary for the L/R0 = 1.5 and 2.0 cases.

Figure 13: Overview of the simulation cases near an eplliptic concave surface wall.

Figure 14: Typical schlieren-type snapshots of density (upper part of each panel) and pressure (lower part of each panel) distributions for a bubble collapsing near an elliptical concave wall at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Fig. 15 presents the time-history evolution curve of the equivalent bubble radius throughout the collapse process of cavitation bubbles under the elliptical concave surface boundary condition, corresponding to stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0. As demonstrated by the curve, for cavitation bubbles collapsing in the vicinity of an elliptical concave surface in an underwater environment, the duration of the bubble’s first oscillation cycle exhibits a decreasing trend with an increase in the stand-off distance ratio. Notably, a smaller stand-off distance leads to a more pronounced delay in the bubble contraction phase, accompanied by a smaller equivalent radius at the point of minimum bubble volume. Fig. 16 shows the time-history evolution of the pressure load acting on the central point of the rigid wall during the collapse of cavitation bubbles under the elliptical concave surface boundary condition at three distinct stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0. It can be seen that for the configurations with L/R0 = 1.0, 1.5 and 2.0, the peak pressure values generated by the impact of water jets on the left-side rigid wall are 1.23, 0.28 and 0.21 GPa, respectively. In addition, the effective loading time of the water jet load on the wall surface is around 0.3 μs, which constitutes roughly 30% of the duration of the first bubble oscillation period.

Figure 15: The time history evolution of equivalent radius of the buble during the bubble collapsed under epliptical concave surface conditions at the standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Figure 16: The time history evolution of pressure load on the center of the wall during the implosion of the bubble collapsed under epliptical concave surface conditions at three different standoff distance ratios of L/R0 = 1.0, 1.5 and 2.0.

5.4 Comparison and Discussion

Based on the analysis of the bubble collapse behaviors near the three aforementioned wall boundary conditions, it is evident that the elliptical convex and concave surface boundaries impose two distinct and opposite regulatory mechanisms on the overall bubble collapse process. Such mechanisms can be identified and validated through the characteristics of the first bubble oscillation period and the time-history curves of wall pressure.

Fig. 17 presents the time-history evolution curve of the equivalent bubble radius throughout the bubble collapse process under different surface boundary conditions and varying stand-off distance ratios. The results illustrated in the figure demonstrate that under the three test conditions with L/R0 = 1.0, 1.5 and 2.0, the elliptical convex surface boundary exerts a reducing effect on the bubble oscillation period, while the elliptical concave surface boundary leads to an increase in the bubble oscillation period. Fig. 18 illustrates the time history of the pressure load acting on the central point of the wall during bubble collapse under different wall boundary conditions and varying stand-off distance ratios. As indicated by the figure, for the configuration with L/R0 = 1.0, the elliptical convex surface boundary significantly increases the peak pressure of the water jet, while the elliptical concave surface boundary results in a decrease in the peak water jet pressure. For the cases with L/R0 = 1.5 and 2.0, however, the regulatory effect of the elliptical surface boundaries on the peak pressure is moderately attenuated. Notably, the elliptical surface boundary conditions exacerbate the oscillation of the time-history curve of the water jet pressure load at the wall center, especially in the latter stage of the water jet pressure loading process.

Figure 17: The time history evolution of the equivalent radius of the buble during bubble collapse under plane wall (a), elliptical convex surface (b), and elliptical concave surface (c) conditions, at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0.

Figure 18: Time history of pressure load on the wall center during bubble collapse under plane wall (a), elliptical convex surface (b), and elliptical concave surface (c) conditions, at stand-off distance ratios of L/R0 = 1.0, 1.5 and 2.0.

6  Conclusion

In this research, we numerically investigated the collapse dynamics of near-wall bubbles under three boundary conditions (planar, elliptical convex, and elliptical concave surfaces) at different standoff distance ratios L/R0 = 1.0, 1.5, and 2.0. Numerical simulations were conducted using a compressible five-equation multi-component flow model, combined with the finite volume method, fifth-order WENO reconstruction, and HLLC Riemann solver. The numerical approach has been validated against multiple classical test cases, demonstrating its accuracy. Some conclusions have been drawn as follows:

(1)   The boundary curvature exerts opposite regulatory effects on bubble collapse. The elliptical convex surface concentrates the flow field, accelerating bubble collapse, shortening the first oscillation period, and increasing the water jet pressure peak for L/R0 = 1.0. In contrast, the elliptical concave surface disperses the flow, delaying jet formation and bubble contraction, and reducing the pressure peak for L/R0 = 1.0.

(2)   The standoff distance ratio uniformly modulates bubble behavior across all boundaries. With increasing standoff distance ratios L/R0, the first oscillation period of bubbles decreases gradually, and the minimum equivalent radius during collapse reduces, indicating a weakened wall constraint. The regulatory effect of elliptical surfaces on pressure peaks attenuates.

(3)   Elliptical boundaries (both convex and concave) exacerbate the oscillation of the wall pressure time-history curve, especially in the latter stage of jet loading. This phenomenon arises from the reflection and superposition of pressure waves between the elliptical surface and the rigid wall, enhancing the unsteadiness of the pressure load on nearby structures.

Although the current study of bubble dynamics has not yet fully considered the influence of adjacent structural boundaries, which represents a minor limitation of the present work, the incorporation of a fluid–structure interaction model has been included in the future research plan to address this gap.

Acknowledgement: None.

Funding Statement: None.

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Jun Yu, Xian-Pi Zhang, Teng Xie; data collection: Jun Yu, Lun-Ping Zhang, Wei-Di Wu; analysis and interpretation of results: Jun Yu, Lun-Ping Zhang, Wei-Di Wu, Fang-Zhou Zhu; draff manuscript preparation: Jun Yu, Lun-Ping Zhang, Xian-Pi Zhang, Fang-Zhou Zhu. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The authors confirm that the data supporting the findings of this study are available from the corresponding author.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

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