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.

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.

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.

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 p_b = 1 atm and an initial density ρ_b = 1.0 kg/m³. The ambient water is set to a pressure of p_w = 120 atm and a density of ρ_w = 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 E_y 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.

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.

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.