Acoustic cavitation dynamics of bubble clusters near solid wall: A multiphase lattice Boltzmann approach

Abstract

Understanding the behavior of cavitation bubble clusters in an acoustic field is crucial for advancing the study of acoustic cavitation. This study uses the multi-relaxation time lattice Boltzmann method (MRT-LBM) to simulate the dynamics of cavitation bubble clusters near a wall, offering new insights into complex cavitation phenomena. The effectiveness of MRT-LBM was verified through thermodynamic consistency, mesh independence, and comparison with the K-M equation solution. The study focuses on the effects of bubble cluster position, acoustic frequency, amplitude, and bubble number on cavitation dynamics. The results found that the impact of bubble cluster proximity to solid boundaries, where smaller offsets result in stronger cavitation effects, significantly increasing wall pressure and jet velocity. The analysis also reveals that low frequencies promote complete bubble collapse, while high frequencies enhance jet velocity but weaken pressure waves. Additionally, higher amplitudes increase jet velocity but disperse energy, reducing wall pressure. Frequency spectrum analysis of wall pressure $p_w$ and velocity $u_w$ further uncovers significant differences in their spectra and how they influence cavitation intensity, finding that frequency and amplitude are key factors in balancing pressure and jet velocity. These findings underscore the importance of optimizing frequency and amplitude to enhance cavitation effects, which can improve applications relying on acoustic cavitation.

1. Introduction

Acoustic cavitation refers to the formation, growth, expansion, and eventual collapse of bubbles in liquids induced by high-intensity ultrasonic excitation [1]. It is a critical phenomenon in various engineering and industrial applications, including ultrasonic cleaning [2], [3], biomedical treatment [4], [5], [6], and fluid machinery [7]. Particularly, the behavior of bubble clusters near solid surfaces is of great interest, as their collapse can generate high-speed microjets and strong shock waves, exerting severe pressure on the solid surface. This process not only causes localized damage to the material surface but also accelerates corrosion and degradation, playing a pivotal role in the material erosion process. Therefore, understanding the dynamics of near-wall bubble clusters is essential for controlling and mitigating their destructive effects.

Over the past few decades, the dynamics of acoustic cavitation near solid surfaces have been extensively studied, particularly focusing on bubble collapse behavior, jet formation, and bubble-wall interactions. Research has highlighted the potential risks associated with cavitation bubble clusters, particularly their capacity to cause localized damage to solid surfaces [8], [9]. Investigations into micro-jets generated by bubble collapse near walls have provided insights into the mechanical impacts on different materials, while studies using high-speed visualization techniques have deepened our understanding of the lifecycle of acoustically excited bubbles, from nucleation to collapse [10], [11]. Furthermore, experiments have explored methods for controlling cavitation-induced erosion, emphasizing ways to mitigate destructive effects through strategies like surface structuring and boundary modifications [12].

Despite these advancements, accurately modeling the non-spherical deformation of cavitation bubbles near solid surfaces remains a challenge. Jet formation, bubble splitting, and asymmetric collapses are inherently difficult to capture through analytical models due to the nonlinear nature of the problem. Experimental studies, while valuable, are often costly, complex, and difficult to reproduce, making numerical methods essential tools for advancing our understanding of bubble dynamics.

Early numerical approaches relied heavily on finite element methods (FEM) and boundary element methods (BEM) to simulate microbubble behavior near boundaries. These methods have proven effective in reducing problem dimensionality and providing precise solutions [13], [14], [15]. For instance, the boundary integral equation method has been widely used to study bubble collapse speed, bubble-bubble interactions, and key parameters affecting collapse dynamics [16], [17]. More recent studies have employed volume-of-fluid (VOF) methods to explore asymmetries in bubble expansion and collapse, particularly under pressure pulses [18]. However, traditional computational fluid dynamics (CFD) methods face limitations when dealing with non-standard complex fluids and irregular grid geometries. These methods often suffer from grid distortion, convergence issues, and high computational costs when applied to strong nonlinear conditions typical of cavitation phenomena.

In contrast, the lattice Boltzmann method (LBM) has emerged as an alternative numerical approach for studying multiphase flows and fluid-solid interactions in cavitation research [19], [20], [21], [22], [23]. Unlike conventional CFD approaches, which rely on macroscopic governing equations, LBM operates at a mesoscopic scale, representing fluid behavior through particle distribution functions on a discrete lattice grid. This approach inherently handles complex boundaries and fluid-solid interactions without requiring explicit interface tracking, thereby improving computational efficiency and numerical stability [24], [25], [26], [27], [28].

Since its introduction to cavitation research by Sankaranarayanan et al. [29], the Shan–Chen LBM model has been applied to simulate bubble nucleation, growth, and collapse under various conditions. Subsequent studies have expanded the application of LBM to include bubble clusters, chemical reactions coupled with bubble collapse, and thermodynamic effects [30], [31], [32]. For instance, Mishra et al. calculated the fluid dynamics of cavitation bubble collapse while coupling this process with chemical reactions involving solutes [31]. Yang et al. explored the thermodynamics of collapsing bubbles using thermal lattice Boltzmann methods (TLBM), providing valuable insights into the role of temperature effects [32]. Further investigations by Peng et al. focused on simulating weak and strong interactions between adjacent cavitation bubbles, as well as interactions within bubble clusters, using an improved single-component multiphase LBM approach [33]. More recent studies have examined the effects of wall wettability, fractal boundary geometries, and bubble-bubble interactions on cavitation behavior using hybrid LBM models [34], [35], [36], [37].

Research has also expanded to include the dynamic properties of acoustic cavitation, where acoustic fields play a significant role in influencing bubble behavior. For example, Ezzatneshan et al. explored cavitation bubble cluster dynamics influenced by acoustic fields, demonstrating the complexity of these multiphase systems [38]. Beom-Jin Joe et al. developed a stable LBM scheme for simulating acoustic scaling under high shear flow conditions, enhancing the understanding of cavitation bubble growth [39]. Recently, Our team utilized a double distribution function LBM model to investigate interactions between single and multiple bubbles within an acoustic field, highlighting the complexity of bubble oscillations, jet deflections, and pressure wave propagation in nonlinear cavitation phenomena [40].

These studies highlight the importance of numerical methods to study the growth and collapse of bubble clusters, significantly improving our understanding of cavitation dynamics. Ongoing studies aim to address the complexity of bubble dynamics, promote innovation, and rationally regulate cavitation effects. In summary, the generation and collapse of bubble clusters is fundamental to cavitation research, generating important data and deepening our understanding of this complex phenomenon. Compared to previous studies that primarily focused on single bubbles or multi-bubble systems, this work provides a comprehensive analysis of cavitation bubble cluster dynamics near solid boundaries under varying acoustic conditions. This study utilizes the MRT-LBM to simulate the dynamics of cavitation bubble clusters driven by acoustic waves near solid walls, which investigates the effects of various factors, including the position of the bubble cluster, acoustic frequency, amplitude, and the number of bubbles, on cavitation dynamics. To better understand the behavior of cavitation bubble clusters, the study also introduces a novel frequency spectrum analysis of the pressure and velocity signals collected on the wall, which provides new insights into the complex dynamics of bubble clusters, jet formation, and shock wave propagation. The results of this research are expected to contribute valuable knowledge for optimizing acoustic cavitation in engineering applications, offering a powerful computational tool for simulating complex cavitation phenomena and enhancing future research in this field.

The organization of this paper proceeds as follows. In Section 2, we establish the theoretical backdrop of the MRT LBM. Section 3 delves into the numerical viability of the model. Section 4 introduces the findings and discussions, emphasizing the impacts of position offset, frequency, amplitude, and the number of bubbles in proximity to the wall. Finally, Section 5 presents our conclusions.

2. The pseudopotential LBM method

The MRT collision operator proposed by Higuera et al. is an important generalization of LBM [41]. The standard equation for MRT-LBM can be expressed as follows [42], [43]

$$f_i(\mathbf{x}+{\mathbf{c}}_i\Delta t,t+\Delta t)-f_i(\mathbf{x},t)=-\hat{\mathbf{\Lambda }}(f_j(\mathbf{x},t)-f_j^{eq}(\mathbf{x},t))+F_{\alpha }^{\prime },$$ (1)

where $f_i$ represents the discrete density distribution function, $\mathbf{x}$ refers to the space position, $\mathit{t}$ refers to the time step, $F_{\alpha }^{\prime }$ is the external force term in the velocity space, and ${\mathbf{c}}_i$ represents the velocity vector with the $i$th velocity direction. In this work, the D2Q9 lattice model is adopted, the specific speed configuration is as follows

$${\mathbf{c}}_i=\left\{\begin{array}{ccc}\hfill (0,0)\hfill & \hfill \hfill & \hfill i=0\hfill \\ \hfill c(cos{\theta }_i,sin{\theta }_i)\hfill & \hfill {\theta }_i=(i-1)\pi /2\hfill & \hfill i=1,2,3,4\hfill \\ \hfill \sqrt{2}c(cos{\theta }_i,sin{\theta }_i)\hfill & \hfill {\theta }_i=(i-5)\pi /2+\pi /4\hfill & \hfill i=5,6,7,8\hfill \end{array}\right)$$ (2)

where $c=\Delta x/\Delta t$, $\Delta x$ is the unit lattice spacing in space, $\Delta t$ is the unit time step, thus, c is taken as 1.

In Eq. (1), $\hat{\mathbf{\Lambda }}={\mathbf{M}}^{-\mathbf{1}}\mathbf{\Lambda }\mathbf{M}$ represents the collision matrix, $\mathbf{\Lambda }$ is a diagonal matrix, M is the orthogonal transformation matrix, ${\mathbf{M}}^{\mathbf{-1}}$ is inverse matrix of M. For the D2Q9 model, the transformation matrix M can be given as [44]

$$\mathbf{M}=\left[\begin{array}{ccccccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -4\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill -2\hfill & \hfill -2\hfill & \hfill -2\hfill & \hfill -2\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right].$$ (3)

By applying the transformation matrix, both $f_i$ and its equilibrium distribution function $f_i^{eq}$ can be represented in the momentum space. For the D2Q9 model, the resulting expressions can be derived as follows

$$\mathbf{m}=\mathbf{Mf}=M_{ij}{f}_j={\left(\rho ,e,\varsigma ,j_x,q_x,j_y,q_y,p_{xx},p_{xy}\right)}^T$$ (4)

where $\mathbf{m}$ refers to the velocity moment, $\mathbf{f}=(f_0,f_1,\dots ,f_8),{\mathbf{f}}^{eq}=(f_0^{eq},f_1^{eq},\dots ,f_8^{eq})$. $\rho$ is the density, $e$ is the squared energy, ($j_x,j_y$) are the components of momentum, ($q_x,q_y$) correspond to the energy flux, and ($p_{xx},p_{yy}$) relate to the diagonal and off-diagonal components of the stress tensor.

For the D2Q9 model, the diagonal matrix $\mathit{\Lambda }$ consisting of relaxation times is written as

$$\mathit{\Lambda }=diag({\tau }_{\rho }^{-1},{\tau }_e^{-1},{\tau }_{\zeta }^{-1},{\tau }_j^{-1},{\tau }_q^{-1},{\tau }_j^{-1},{\tau }_q^{-1},{\tau }_v^{-1},{\tau }_v^{-1}),$$ (5)

where ${\tau }_{\rho }={\tau }_j=1.0$ is the relaxation time of the conservation momentum, ${\tau }_e={\tau }_{\zeta }=1.25$, and ${\tau }_q=1/1.1$, ${\tau }_v$ determines the kinematic viscosities are $\mu =(1/{\tau }_v-0.5)c_s^2\Delta t$.

The collision process of Eq. (1) can be obtained by multiplying the transformation matrix in momentum space

$${\mathbf{m}}^{\ast }=\mathbf{m}-\mathbf{\Lambda }(\mathbf{m}-{\mathbf{m}}^{\mathbf{eq}})+\mathbf{\Delta }\mathbf{t}(\mathbf{I}-\mathbf{\Lambda }/2)\overline{\mathbf{F}},$$ (6)

where ${\mathbf{m}}^{\ast }=(m_0^{\ast },m_1^{\ast },\dots ,m_8^{\ast })$, $f_i^{\ast }={\mathbf{M}}^{\mathbf{-1}}{\mathbf{m}}^{\ast }=M_{ij}^{-1}{m}_j^{\ast }$, ${\mathbf{m}}^{eq}$ represents

$${\mathbf{m}}^{eq}=\rho {(1,-2+3{\left|\mathbf{u}\right|}^2,1-3{\left|\mathbf{u}\right|}^2,u_x,-u_x,u_y,-u_y,u_x^2-u_y^2,u_x{u}_y)}^T$$ (7)

where $u_x$ and $u_y$ are the $x$ and $y$ components of macroscopic velocity u, and the superscript $T$ represents the transposition operator. Besides, I in Eq. (6) is the unit tensor, and $\overline{\mathbf{F}}=\mathbf{M}{F}^{\prime }$ represents the forcing term vector in the momentum space. In this paper, Li’s modified forcing term is adopted, as followed [45]

$$\overline{\mathbf{F}}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 6v\cdot \mathbf{F}+\frac{0.75\varepsilon {\left|F_m\right|}^2}{{\psi }^2{\delta }_t({\tau }_e-0.5)}\hfill \\ \hfill -6v\cdot \mathbf{F}-\frac{0.75\varepsilon {\left|F_m\right|}^2}{{\psi }^2{\delta }_t({\tau }_{\zeta }-0.5)}\hfill \\ \hfill F_x\hfill \\ \hfill -F_x\hfill \\ \hfill F_y\hfill \\ \hfill -F_y\hfill \\ \hfill 2(v_x{F}_x-v_y{F}_y)\hfill \\ \hfill (v_x{F}_y+v_y{F}_x)\hfill \end{array}\right],$$ (8)

where the adjusted parameter $\varepsilon$ proposed by Li et al. [45] can enhances thermodynamic consistency and improves mechanical stability. $\mathbf{F}=({\mathit{F}}_{\mathit{x}},{\mathit{F}}_{\mathit{y}})$ is the total force. Strictly speaking, the interparticle interaction force F is composed of the pseudopotential interaction force $F_m$ and other possible volume forces $F_b$ (such as buoyant, gravity force, etc.). $F_b$ can be introduced according to the selected multiphase flow model. Since cavitation bubbles are micrometer-level, the influence of buoyant or gravity can be ignored. The pseudopotential interaction force $F_m$ is written as [46], [47]

$${\mathbf{F}}_m\left(\mathbf{x}\right)=-G\psi \left(\mathbf{x}\right)\sum _i\omega \left({\left|{\mathbf{c}}_i\right|}^2\right)\psi (\mathbf{x}+{\mathbf{c}}_i\delta t){\mathbf{c}}_i,$$ (9)

where $\omega \left({\left|{\mathbf{c}}_i\right|}^2\right)$ represents the weights, with $\omega \left(1\right)=1/3$ and $\omega \left(2\right)=1/12$. The pseudopotential, $\psi \left(\mathbf{x}\right)$, is defined by Yuan et al. [48]

$$\psi \left(\mathbf{x}\right)=\sqrt{2(p_{EOS}-\rho c_s^2)/G{c}^2},$$ (10)

where $c_s$ is the lattice speed of sound. To ensure that the expression within the square root on the right-hand side of Eq. (10) remains positive, the interaction strength between particles, denoted as G, is set to −1. The non-ideal equation $p_{EOS}$ is introduced into the pseudopotential model, allowing for a reduction in virtual velocity, an increase in the density ratio, and an improvement in thermodynamic consistency [49], [50]. In the present work, the Peng-Robinson (P-R) equation of state is adopted, which is written as

$$p_{EOS}=\frac{\rho R_{g}T}{1-b\rho }-\frac{a\vartheta \left(T\right){\rho }^2}{1+2b\rho -b^2{\rho }^2},$$ (11)

where $a=0.45724{\left(R{T}_c\right)}^2/p_c$, $b=0.0778R{T}_c/p_c$, $\vartheta \left(T\right)={[1+(0.37464+1.54226\omega -0.26992{\omega }^2)(1-\sqrt{T/T_c})]}^2$. $a=2/49,b=2/21,R_g=1$.

The streaming process of particles, that is, updating the density distribution function, is as follows

$$f_i(\mathbf{x}+{\mathbf{c}}_i\Delta t,t+\Delta t)=f_i^{\ast }(\mathbf{x},t),$$ (12)

where $f_i^{\ast }={\mathbf{M}}^{\mathbf{-1}}{\mathbf{m}}_i^{\ast }$. The macroscopic density $\mathit{\rho }$ and macroscopic velocity $\mathbf{u}$ are calculated using the following equations

$$\rho =\sum _i{f}_i,$$ (13)

$$\rho \mathbf{u}=\sum _i{\mathbf{c}}_i{\mathit{f}}_i+\frac{{\delta }_t}{2}\mathbf{F},$$ (14)

Note that this study employs lattice units within the LBM framework, as shown in Table 1.

Table 1.

Basis parameter units.

Name	Variable	Units	Dimensions
Time	$\delta t$	$ts$	$T$
Length	$\delta x$	$lu$	$L$
Velocity	$u$	$lu$/$ts$	$L/T$
Density	$\rho$	mu/$l{u}^3$	$M$/$L^3$
Pressure	$p$	mu/($t{s}^2$ lu)	$M$/$L$$T^2$
Viscosity	$\mu$	$l{u}^2/ts$	$L^2/T$
frequency	$f$	$t{s}^{-1}$	$1/T$

3. Model validation

3.1. Thermodynamic consistency verification

To ensure the reliability of the proposed LB model, it is essential to verify its thermodynamic consistency. This involves comparing the numerical results obtained from the pseudo-potential LBM combined with the P-R EOS against analytical solutions derived from Maxwell’s equal area method. The process of saturated cavitation bubble formation in a saturated liquid is utilized to establish the equilibrium coexistence density. The computational grid domain is 400 × 400, with periodic boundary conditions applied to all boundaries of the flow field. The fluid density field is initialized to

$$\rho (x,y)=\frac{{\rho }_l+{\rho }_v}{2}+\frac{{\rho }_l-{\rho }_v}{2}\times \left[tanh\left(\frac{2(\sqrt{{(x-x_0)}^2+{(y-y_0)}^2}-r_0)}{W}\right)\right],$$ (15)

where $(x_0,y_0)$ represents the domain center coordinate point, and $r_0=60$ is the initial bubble radius. The width of the vapor–liquid phase interface $W=5$. The hyperbolic tangent function $tanh\left(x\right)=(e^{2x}-1)(e^{2x}+1)$. In addition, the adjusted parameter $\varepsilon$ (1.0 $<\varepsilon$ $<$ 2.0) studied by Li et al. makes the pseudopotential model mechanical stability meet the thermodynamic consistency requirements as much as possible. In this paper, the numerical solution of the coexistence density corresponding to the five values of $\varepsilon$, 1.0, 1.22, 1.46, 1.86, and 2.0, is obtained and compared with the analytical solution of the coexistence density corresponding to the thermodynamic consistency requirements. Fig. 1 shows a comparison of the density curve obtained by the numerical solution corresponding to different parameters $\varepsilon$ through LBM simulation and the density curve obtained by the analytical solution of Maxwell equation.

Fig. 1.

Verification of thermodynamic consistency.

In Fig. 1, the horizontal axis represents the gas-liquid phase density, and the vertical axis represents the dimensionless temperature $T/T_c$. As the dimensionless temperature of the vertical axis decreases, the density ratio of the gas-liquid phase gradually increases. From the analysis of Fig. 1, it can be seen that as the parameter $\varepsilon$ increases, the numerical density curve tends to the same direction. This suggests that there exists an optimal value of $\varepsilon$ that ensures the LBM simulations align with the analytical solution based on Maxwell’s equation, thereby satisfying thermodynamic consistency and enhancing the model’s stability. For gas phase density and liquid phase density, the LBM numerical solution and analytical solution are consistent at higher $T/T_c$. When the dimensionless temperature decreases, the deviation between the numerical solution and analytical solution of gas phase density increases. Moreover, the numerical density curve corresponding to the parameter $\varepsilon =1.86$ still has a good coupling degree with the thermodynamic consistency requirement when the dimensionless temperature is $T/T_c=0.5$.

3.2. Grid independence verification

In order to verify the grid independence of the adjustable parameter $\varepsilon$, this work simulates the change process when the grid size corresponds to the bubble radius to observe whether the value of the optimal parameter $\varepsilon$ will be affected. Three different fluid regions of 200 × 200, 400 × 400, and 600 × 600 are set respectively, and the initial bubble radius also changes proportionally with the size of the calculation region, that is, 40, 60, and 80 respectively, as shown in Fig. 2.

Fig. 2.

Verification of grid independence.

The results in Fig. 2 show that no matter how the region changes relative to the initial radius, the coexistence density curves completely overlap, which means that the optimal parameter does not change with the size of the calculation region, thus verifying that the adjustable parameter $\varepsilon$ satisfies the grid independence.

3.3. Young–Laplace law verification

Simulate a static bubble using the Shan–Chen model, obtaining bubbles of different radii and corresponding pressure differences between the inside and outside of the bubble by adjusting the initial radius to verify the Young–Laplace law. In the absence of external forces, the stable pressure difference between the inside and outside of the bubble is inversely proportional to the radius, i.e.,

$$\Delta p=p_{in}-p_{out}=\frac{\sigma }{r}$$ (16)

where $\Delta p$ is the pressure difference between the inside and outside of the bubble, $r$ is the radius, and $\sigma$ is the surface tension. Therefore, surface tension can be obtained by simulating a series of bubbles with different initial radii and measuring the pressure difference between the inside and outside when equilibrium is reached. A simulation of a two-dimensional static bubble was conducted in a 400 × 400 grid fluid region without considering external forces such as gravity. Periodic boundary conditions were applied on all four edges of the square fluid domain. Initially, bubbles with different radii $r$ were centered at (200, 200), with initial radii set to $r=30,35,40,45,50,55,60,65,70$. The results, as shown in Fig. 3, demonstrate the relationship between the pressure difference across the bubble in the liquid and the reciprocal of the bubble radius. The linear fit of the discrete data points yields slopes of 0.0149 and 0.00822 for relative temperatures $T_r=T/T_c$ of 0.7 and 0.8, respectively, with the slope representing the surface tension.

Fig. 3.

Verification of Young–Laplace law.

Fig. 3 shows that as temperature increases, surface tension gradually decreases. It also clearly illustrates a linear relationship between the pressure difference and the reciprocal of the bubble radius, indicating that the simulation results are consistent with the Young–Laplace law, thereby validating the accuracy of the model.

3.4. Bubble collapse verification

First, a quantitative analysis of the LBM results is conducted. This work examines the acoustic cavitation dynamics of spherical bubbles by comparing the results of single spherical bubble radius changes simulated using LBM with the analytical results from the K-M equation. This comparison aims to validate the practicality of utilizing LBM for simulating acoustic cavitation phenomena within bubbles [51].

The K-M equation for the vibration of a single ultrasonic cavitation bubble is as follows

$$\left\{\begin{array}{c}\hfill (1-\frac{\dot{r}}{c})r\ddot{r}+\frac{3}{2}(1-\frac{\dot{r}}{3c}){\dot{r}}^2=(1+\frac{\dot{r}}{c})\frac{P_S}{{\rho }_l}+\frac{r}{\rho c}\frac{d{P}_S}{dt},\hfill \\ \hfill P_S=(P_0-P_v+\frac{2\sigma }{r_0}){\left(\frac{r_0}{r}\right)}^{3\gamma }-\frac{2\sigma }{R}-4\mu \frac{\dot{r}}{r}-P_0+P_v-P_A.\hfill \end{array}\right)$$ (17)

where $r_0$ is the initial bubble radius, $r$ is the instantaneous bubble radius, $\dot{r}$ and r̈ respectively represent the velocity and acceleration of the cavitation bubble wall motion. c is the speed of sound, ${\rho }_l$ is the density of liquid, $\sigma$ is the surface tension coefficient, $\mu$ is dynamic viscosity, and $\gamma$ is the specific heat ratio of the gas. $P_0$ is the static pressure in the liquid, $P_v$ is the vapor pressure of the gas in the bubble. $p_A=p_{a}sin(2\pi ft+\phi )$ describes the periodic excitation, with $P_A$ indicating the amplitude of the acoustic pressure.

In the LBM, pressure (acoustics) is typically implemented as a change in density at the boundary [52]. The applied sound wave can be characterized by the following density change relationship

$$\rho ={\rho }_l+\Delta \rho sin\left(2\pi ft\right),$$ (18)

where $\rho$ is the density variation at boundary, $\Delta \rho$ is the adjustable parameter corresponding to the change in sound pressure amplitude, $f$ represent frequency.

Table 2 shows the parameter values involved in LBM simulation of acoustic cavitation phenomenon.

Table 2.

Properties of parameter.

Quantity	Values (LBM unit)
$c_s$	$1/\sqrt{3}$
${\rho }_l$	7.205
${\rho }_v$	0.20435
$\mu$	0.1
$\gamma$	1.344
$\sigma$	0.0082
$\varepsilon$	1.86
$f$	$5.77\times 10^{-4}$
$\Delta \rho$	0.4
$T_r$	0.8

The validation involves comparing these results to the corresponding outcomes of the K-M equation, with both radius and time normalized. Fig. 4 presents a comparison of the radius evolution for initial bubble radii of 5 and 10. It is evident that a larger initial radius leads to a greater maximum radius during bubble expansion, and the numerical solution aligns well with the analytical solution.

Fig. 4.

A comparison is made of the bubble radius variation over time between the LBM and K-M models. All parameters are in LBM units.

Furthermore, we simulated the acoustic cavitation dynamics of single and multiple bubble collapse and compared them with experimental and other numerical simulation results, respectively, as shown in Figs. 5 and 6. We only qualitatively compared this study with the experimental results, specifically, the bubble shapes. The simulation results can effectively capture the dynamic process of cavitation bubble expansion, contraction, and collapse, which is crucial for practical engineering applications. In addition, to increase credibility, the present results are compared with the numerical 2D bubble dynamics data from the literature by Peng et al. [33] using SCMP LBM.

Fig. 5.

Evolution of a single acoustic cavitation bubble: (a) experimental results [53]; (b) density field of the bubble simulated in this study; (c) bubble simulation using the SCMP LBM by Peng et al. [33].

Fig. 6.

Evolution of multiple cavitation bubbles: (a) experimental results [53]; (b) density field of the bubbles simulated in this study; (c) bubble simulation using the SCMP LBM by Peng et al. [33].

The consistency of the above results fully demonstrates the effectiveness of the 2D LBM numerical simulation results adopted in this paper, and the current model has a higher density ratio compared with the SCMP LBM.

4. Cavitation effects of spherical bubble cluster collapsing near the wall in an acoustic field

4.1. Simulation initialization

In order to explore the dynamics of spherical bubble clusters, this work uses a spherical bubble cluster composed of five bubbles as a benchmark model for research, which can save computing resources while retaining key dynamic characteristics. Moreover, it is easier to obtain clear physical results, which helps analyze the coupling between the acoustic field and the bubble cluster in a multi-bubble system.

As shown in Fig. 7, the calculation domain is 400 × 400, periodic ultrasonic waves enter from the top of the area, and the bottom is a solid wall using a non-slip boundary format [54], [55]. Five bubbles of equal size are located in the center and labeled 1, 2, 3, 4, and 5, with an initial radius of $r_0$= 5. Simultaneously, it assumes that the parameter $d$ represents the distance separating the center of bubble 5 from the wall, and the parameter $\lambda =r_0/d$ is the offset parameter.

Fig. 7.

Computational domain.

4.2. Effect of position offsets

Firstly, the dynamic process of cavitation bubbles driven by acoustic waves $(f=1.92\times 10^{-4},\Delta \rho =0.4)$, and the cavitation effects on the solid wall were simulated under different offset parameters $\lambda$. To facilitate a clearer understanding of the relationship between pressure and velocity, the left half of the figure displays the pressure field distribution, while the right half presents the velocity field distribution.

Fig. 8 illustrates the simulation results when the bubble cluster approaches the wall at $\lambda =0.1$. Figs. 8(a–d) show that the bubble expands under the initial negative pressure and contracts under the positive sound pressure. Due to the blocking effect of the solid wall, bubble 5 (the bottom bubble) is obviously elongated, and bubbles 2 and 4 deform into non-spherical shapes. Bubble 1 (the top bubble) is first affected by the sound wave, and the pressure and velocity at the top increase until it breaks, generating a high-pressure and high-speed microjet. Bubbles 2 and 4 are affected by the sound waves reflected back by the solid wall, showing non-spherical changes, and the velocity of the outer layer increases. Bubble 3 (the center bubble) remains nearly spherical due to the shielding effect of its neighboring bubbles. Subsequently, bubbles 2, 3, and 4 collapse under the pressure from both the sound wave and the shock wave generated by the collapsing of bubble 1. The multiple jet shock waves generated are superimposed on bubble 5 (Figs. 8(e–f)), leading to an increase in pressure and velocity at the top of bubble 5. This causes the bubble to collapse inward, ultimately resulting in the shock wave penetrating it and forming a high-pressure, high-speed jet that strikes the wall.

Fig. 8.

Evolution of bubble cluster dynamics at $\lambda =0.1$.

Fig. 9 is the dynamic results at $\lambda =0.08$. Compared with the previous case, the separation between the bubble cluster and the wall is greater, although their dynamic behaviors remain similar. Bubble 1 is the first to collapse due to the sound wave, and then the bubbles at the bottom collapse one by one. However, due to different offset parameters, the outline of bubble 5 is flatter, and the direction and intensity of the shock wave generated are also different, which will be analyzed later.

Fig. 9.

Evolution of bubble cluster dynamics at $\lambda =0.08$.

Furthermore, the case with $\lambda =0.05$ is simulated, as shown in Fig. 10. The bubble cluster is situated at the midpoint of the computational domain. Compared with the above two examples, in the case of $\lambda =0.05$, the bubble is less affected by the wall retardation effect, the degree of expansion is greater, bubble 5 is more round. As a result of the sound wave reflecting off the solid wall, the bottom of bubble 5 sinks inward and subsequently collapses completely due to the influence of the sound wave and shock wave from above, creating a downward impact jet.

Fig. 10.

Evolution of bubble cluster dynamics at $\lambda =0.05$.

Then, the influence of the cavitation effect caused by the bubble cluster is investigated. The pressure and velocity generated by the collapse of bubble 5 on the wall are quantitatively analyzed.

Fig. 11 displays the temporal evolution of wall pressure $p_w$ and wall velocity $u_w$. It is evident that the pressure and velocity are directly proportional. The greater the velocity, the greater the pressure. Fig. 11(a) shows the result of the example with $\lambda =0.05$. The pressure and velocity peaks on the solid wall are not obvious. Fig. 11(b) shows the result of the example with $\lambda =0.08$. At this time, it is observed that there are prominent peaks in both pressure and velocity. Furthermore, the superposition of shock waves stemming from the collapse of the upper bubble and the sound wave results in multiple fluctuating peaks along the wall. Fig. 11(c) presents the result for the case with $\lambda =0.1$. Compared with Figs. 11(a, b), the peaks of pressure and velocity generated on the wall are more intense. It indicate that the pressure and velocity of the jet are higher at this point, which further shows that as the bubble cluster moves closer to the wall, the jet pressure generated increases, leading to more severe damage to the wall.

Fig. 11.

Temporal evolution of $p_w$ and $u_w$ changes on the wall for different offset parameters. (a): at a offset parameter of $\lambda =0.05$; (b): at a offset parameter of $\lambda =0.08$; (c): at a offset parameter of $\lambda =0.1$.

In order to better understand the dynamic behavior of cavitation bubble clusters, the motion mechanism of jets and shock waves, and their impact on the solid wall, the changes in pressure $p_w$ and velocity $u_w$ signals collected on the wall are analyzed using frequency spectrum analysis. This method involves converting time-domain signals (i.e. the temporal changes in pressure and velocity) into frequency-domain signals, revealing the intensity and distribution of each frequency component in the signal. Specifically, the time-domain signals are processed using the Fast Fourier Transform (FFT), which transforms the pressure and velocity variations from the time domain into corresponding frequency components.

The analysis focuses on the positive frequency part of the spectrum, skipping the direct current (DC) component, which represents the average value of the signal. This helps isolate the frequency components that are most relevant to the dynamic processes of cavitation and the resulting interactions with the solid wall. The sampling rate is set to 1000 Hz, which ensures that the temporal resolution of the signals is high enough to capture the rapid changes associated with cavitation dynamics. The frequency range is limited to 200 Hz to focus on the most significant frequency components that influence cavitation behavior. The results from this frequency spectrum analysis provide crucial insights into the energy distribution across different frequencies, allowing for a more thorough understanding of how various frequencies contribute to the overall dynamics of the cavitation process and the resulting wall interactions.

As shown in Fig. 12, low-frequency components dominate in the pressure spectrum, especially for $\lambda =0.05$. This suggests that the pressure waves from the bubble collapse propagate more smoothly when the bubble cluster is farther from the wall. As $\lambda$ increases, the pressure spectrum shifts toward higher frequencies, indicating more intense transient pressure waves as the bubbles are closer to the wall. The high-frequency components in the pressure spectrum become more pronounced, particularly at $\lambda =0.1$, where the closer proximity to the wall results in higher intensity of shock waves and microjets.

Fig. 12.

Spectra of $p_w$ and $u_w$ under different $\lambda$. (a) Spectrum of pressure; (b) Spectrum of velocity.

The velocity spectrum reveals a similar trend. In contrast to the pressure spectrum, the low-frequency components are less pronounced, while the high-frequency components increase as $\lambda$ increases. This reflects the growing intensity of the jet impact and the resulting increase in velocity near the wall. Particularly at $\lambda =0.1$, the high-frequency spikes in the velocity spectrum highlight the sharp acceleration of the fluid caused by the collapse-induced jet.

These frequency spectrum results show a clear relationship between the pressure wave propagation and the fluid velocity during bubble collapse. As the distance between the bubble cluster and the wall decreases (with increasing $\lambda$), both the intensity and frequency of the pressure and velocity components increase.

Further analysis is conducted by extracting the maximum pressure $p_{wmax}$ and maximum velocity $u_{wmax}$ on the wall surface, and plotting them as shown in Fig. 13. The results reveal that as the offset parameter of the bubble cluster increases, the pressure generated by the jet impact on the wall becomes significantly larger. The cavitation effect on the wall is more intense when the bubble cluster is near the surface. In addition, The proportional increase in pressure with jet velocity suggests that velocity is a key factor in understanding the intensity of the cavitation phenomenon. As the jet speed rises, so does the energy transferred to the wall, resulting in a higher potential for damage.

Fig. 13.

Maximum values of $p_w$ and $u_w$ under different $\lambda$.

4.3. Effect of frequency

To investigate the influence of acoustic frequency on bubble cluster dynamics and cavitation effects, Fig. 14 presents the pressure (left) and velocity (right) fields of bubble clusters. The study begins at 800ts, omitting the growth phase of the clusters. The evolution differs at various frequencies.

Fig. 14.

Evolution of bubble cluster. (a): at a frequency of $f=1.92\times 10^{-3}$; (b): at a frequency of $f=5.77\times 10^{-4}$; (c): at a frequency of $f=1.44\times 10^{-4}$.

Fig. 14(a) shows high-frequency conditions where the top bubble collapses first at t = 1360ts due to sound waves from the upper boundary. Reflected waves from the solid wall accelerate the contraction of bottom and side bubbles. However, at $t=1814ts$, bubbles rebound and oscillate instead of collapsing completely due to pressure fluctuations from wave interference. Remaining bubbles are periodically driven by the external field, resulting in oscillations without full collapse. The reasons for this phenomenon is that the interference effect of the sound waves reflected by the solid wall and the incident wave causes the pressure fluctuations around the bubble, causing the bubble to shrink and then expand. Besides, the remaining bubbles are unable to complete collapse within the high-frequency sound field cycle, and after each contraction, they are re-driven by the external sound field, resulting in periodic oscillations rather than complete collapse. Figs. 14(b,c) illustrate the dynamics of complete collapse of bubble clusters, indicating that in mid- to low-frequency acoustic fields, bubbles have sufficient time to expand and contract within the acoustic wave cycle until collapse. In Fig. 14(b), under a mid-frequency acoustic field, the oscillation response of the bubbles is fast, but due to the shorter cycle, the bubbles cannot reach their maximum volume in each cycle, resulting in a shorter collapse time for the bubble clusters. In Fig. 14(c), under a low-frequency acoustic field, the bubbles have sufficient time for significant expansion and contraction, allowing them to reach near-maximum volume, leading to a longer collapse time and a more pronounced cavitation effect.

In order to further verify the above results, the pressure and velocity caused by the cavitation effect of bubble cluster on the solid wall are quantified. Fig. 15 shows the temporal evolution of $p_w$ and $u_w$ changes when $f=1.44\times 10^{-4},5.77\times 10^{-4},1.92\times 10^{-3}$. The other parameter values are $(\lambda =0.08,\Delta \rho =1.0)$.

Fig. 15.

Temporal evolution of $p_w$ and $u_w$ changes on the wall for different frequencies. (a): at a frequency of $f=1.92\times 10^{-3}$; (b): at a frequency of $f=5.77\times 10^{-4}$; (c): at a frequency of $f=1.44\times 10^{-4}$.

At high frequencies (Fig. 15(a)), the period of the sound wave is short, preventing the bubbles from having sufficient time to fully grow and collapse, which results in the incomplete collapse of the bubble cluster. As a result, the shock waves generated are weak, exerting lower pressure and jet intensity on the solid wall, and the recorded pressure and velocity depend on the acoustic frequency, with no strong peaks due to high fluctuation. At lower frequencies (Fig. 15(b), relative to high frequency), the bubbles have more time to grow and collapse, allowing the bubble cluster to fully collapse. However, the results show that the pressure generated during the bubble collapse is relatively low, while the kinetic energy of the high-speed jets formed during the collapse is significantly enhanced. This is because the jets primarily generate kinetic effects, driving the fluid to move at high speeds, but with lower pressure peaks. In this case, the jet impacts on the wall produce higher flow speeds, while the pressure generated by the collapse is relatively small. At low frequencies (Fig. 15(c)), the bubbles have sufficient time to fully grow and collapse, and the energy released during the collapse is in the form of shock waves, resulting in high pressure peaks. Meanwhile, the kinetic energy of the jets is lower, so the velocity on the solid wall is reduced. In this scenario, pressure waves dominate the effects of bubble collapse.

To further understand the impact of acoustic frequency on bubble collapse dynamics, Fig. 16 presents the frequency spectra of pressure and velocity for bubble clusters under three different acoustic frequencies.

Fig. 16.

Frequency spectra of $p_w$ and $u_w$ for different acoustic frequencies. (a) Spectrum of pressure; (b) Spectrum of velocity.

At high frequency ($f=1.92\times 10^{-3}$), the pressure spectrum in Fig. 16(a) exhibits strong low-frequency components but weaker high-frequency components. This is due to the short acoustic period causing some bubbles in the cluster to not collapse completely. Therefore, the energy released during the collapse of the cluster is limited and the high-frequency jet is less prominent. Nevertheless, the rapid oscillation of the bubbles results in a relatively high jet velocity near the wall. This phenomenon is due to the rapid oscillation of the bubble cluster, which leads to rapid contraction and increased liquid velocity, resulting in a high-speed jet with a lower total energy release compared to the low frequency. Similarly, the velocity spectrum shows a weak high-frequency response, indicating reduced jet velocity near the wall but higher than what would be expected from the pressure spectrum alone.

Mid-frequency ($f=5.77\times 10^{-4}$) conditions result in a more balanced spectrum. The bubble clusters collapse more effectively than in the high-frequency case, leading to stronger shock waves and higher jet velocities. The velocity spectrum reveals enhanced high-frequency components, highlighting the increased kinetic energy of jets impacting the wall. The efficiency of bubble collapse increases with the mid-frequency conditions, leading to stronger shock waves and increased jet velocities, producing a more intense cavitation effect on the solid wall.

At low frequency ($f=1.44\times 10^{-4}$), the bubbles have sufficient time to grow and collapse within the acoustic wave cycle. The pressure spectrum demonstrates strong low-frequency components, corresponding to the energy released by the complete collapse of bubbles. However, the velocity spectrum indicates reduced high-frequency components, suggesting that while the pressure waves dominate, the jet velocity is lower due to the longer collapse cycle. Although the collapse is complete and the pressure is high, the longer collapse times result in lower jet velocities, indicating that the cavitation effect is more pressure-dominant compared to mid-frequency conditions.

The frequency spectrum analysis provides insights into the relationship between acoustic frequency and cavitation effects. Furthermore, the temporal evolution of $p_w$ and $u_w$ changes on the wall at different frequencies and the impact of different frequencies on the collapse time of the bubbles 1, 3, and 5 are explored.

From Fig. 17, Fig. 18, two important conclusions can be drawn: (1) The effect of acoustic frequency on bubble collapse time: When the frequency of the sound wave is high, the oscillation period of the bubbles is shorter, and the bubbles cannot fully grow and collapse within a complete cycle of the sound wave, resulting in partial bubble collapse (bubbles 3 and 5 did not collapse). As the frequency decreases, the bubbles have more time to fully expand and contract, which increases the collapse time. In this case, the bubble cluster releases more energy during the collapse, leading to a more intense cavitation effect, causing more significant pressure peaks and velocity changes on the solid wall (as illustrated in Figs. 15(b,c)).

Fig. 17.

Changes of $p_w$, $u_w$ at different frequencies.

Fig. 18.

Changes of collapse time at different frequencies.

(2) The effect of solid wall proximity on fluid kinetic energy: When the bubble cluster is near the solid wall, the high-speed jets formed during the collapse are reflected and confined by the wall, generating complex fluid dynamics. At medium frequencies, the interval between bubble collapses is shorter, and the shock waves generated by the collapse interfere. When the jet approaches the solid wall, the liquid flow speed increases, the incident flow and the reflected wave interact and produce a counteracting effect. Due to the viscosity effect of the fluid, the kinetic energy of the jet is gradually dissipated and converted into internal energy, making the momentum distribution in the jet impact area more dispersed, resulting in a relatively low local pressure near the wall. However, the jet velocity reaches its maximum at the moment it contacts the solid wall, as its momentum has not yet been fully dissipated. This explains why the wall velocity is higher but the pressure is lower under medium-frequency conditions (Fig. 15(b)). At low frequencies, the interval between bubble collapses is longer, and the energy during the final collapse is more concentrated. Part of the jet kinetic energy is converted into pressure, making the pressure effect on the wall more pronounced.

4.4. Effect of amplitude

The evolution of bubble cluster behavior under different amplitudes at a fixed frequency is shown in Fig. 19. It can be observed that the bubble clusters completely collapse, and the motion processes of the three cases are similar, with the bubbles closest to the sound pressure collapsing first and the bottom bubbles collapsing last. The difference lies in the higher amplitude case (Fig. 19(c), $t=1378ts$), where the reflected wave forms a lower pressure in the bottom region, causing the bubbles to stretch more significantly. Additionally, the higher the amplitude, the shorter the complete collapse time of the bubble cluster. Furthermore, the pressure and velocity on the wall surface are quantitatively analyzed to explore the influence of the amplitude on the cavitation effects.

Fig. 19.

Evolution of bubble cluster (a): at a density perturbation of $\Delta \rho =0.5$; (b): at a density perturbation of $\Delta \rho =0.85$; (c): at a density perturbation of $\Delta \rho =1.15$.

Fig. 18 is the time series diagram of $p_w$ and $u_w$ obtained by changing the boundary density perturbation $\Delta \rho$.

As shown in Fig. 20(a), when the amplitude $\Delta \rho =0.5$ (relatively low sound wave amplitude), the bubble cluster completely collapses, and an instantaneous peak in both pressure and velocity is recorded on the wall. When the amplitude increases to $\Delta \rho =0.85$ (Fig. 20(b)), the energy of the jet produced by bubble collapse increases, leading to a single, sharp peak being recorded on the wall. As the amplitude further increases to $\Delta \rho =1.15$ (Fig. 20(c)), multiple pressure peaks with fluctuations are recorded on the wall. This is likely caused by the bubble cluster collapse generating multiple shock waves.

Fig. 20.

Temporal evolution of $p_w$ and $u_w$ changes on the wall for different values of density perturbation: (a): at a density perturbation of $\Delta \rho =0.5$; (b): at a density perturbation of $\Delta \rho =0.85$; (c): at a density perturbation of $\Delta \rho =1.15$.

To further explore the influence of amplitude, the frequency spectra of pressure and velocity are analyzed for different density perturbations $\Delta \rho$, as shown in Fig. 21.

Fig. 21.

Spectra of $p_w$ and $u_w$ under different $\Delta \rho$. (a) Spectrum of pressure; (b) Spectrum of velocity.

The pressure spectrum shows a dominant low-frequency response at $\Delta \rho =0.5$, indicating that the collapse is more gradual and less intense, with limited energy released. The velocity spectrum displays a relatively weak high-frequency response, indicating a lower speed of the jets near the wall. The pressure spectrum reveals higher energy at $\Delta \rho =0.85$, especially in the mid-frequency range, with sharper peaks compared to the low amplitude case. This suggests that the bubble collapse is more pronounced, and more energy is transferred to the fluid, producing stronger shock waves. The velocity spectrum also shows a higher response, particularly in the high-frequency range, indicating stronger and faster jets impacting the wall. The pressure spectrum shows multiple peaks at $\Delta \rho =1.15$, reflecting the complex collapse dynamics where multiple shock waves interfere with each other. The velocity spectrum indicates a broad range of high-frequency components, signifying that the jets are faster and the collapse dynamics are more complicated. These results align with the observation that the collapse time decreases with increasing amplitude, and multiple shock waves are generated, resulting in more pressure peaks on the wall.

In summary, the frequency spectrum analysis confirms that as the amplitude increases, both the energy and complexity of the collapse increase, leading to stronger jets and multiple pressure peaks. However, when the amplitude reaches a certain threshold, the collapse dynamics become more complicated, and the energy is dispersed, leading to a decrease in the intensity of individual peaks despite the overall increase in energy.

To explore the specific reasons behind these observations, we extracted the $p_{wmax}$, $u_{wmax}$, and time under different amplitudes for further analysis.

As can be seen from Fig. 22, Fig. 23, both $p_{wmax}$ and $u_{wmax}$ increase with increasing amplitude. However, when the amplitude reaches a certain threshold (approximately $\Delta \rho =0.9$), both begin to decrease. In addition, the collapse time of each bubble in the bubble cluster decreases with increasing amplitude, that is, the larger the amplitude, the faster the bubble collapses. Then, we can infer some explanations for the above results.

Fig. 22.

Changes of $p_w$, $u_w$ at different amplitude.

Fig. 23.

Changes of collapse time at different amplitude.

(1) At low amplitudes, the collapse lasts for a long time, and a weak instantaneous pressure and velocity are collected on the wall. At medium amplitudes, the increase in the amplitude of the sound wave causes more energy to be transferred to the bubble, thereby enhancing the intensity of the collapse and producing a more powerful jet hitting the wall, resulting in a single but very sharp peak in both pressure and velocity. Besides, the collapse occurs in a shorter time due to the increased energy. At high amplitudes, the collapse of the bubble cluster becomes increasingly intense and intricate. The collapse time is shorter, the collapse interval between bubbles is shortened, and the superposition of multiple shock waves generated by the bubble cluster leads to multiple pressure peaks on the wall.

(2) When the amplitude exceeds a certain threshold, the bubble cluster collapses more violently and the collapse dynamics become more complicated. The collapse produces multiple shock waves and jets that interfere with each other, resulting in scattered and unfocused energy, which leads to a slight decrease in the peak pressure and velocity recorded on the wall. Although the energy of the jet is still high, it is dispersed over multiple wavefronts, reducing the intensity of a single peak.

This behavior reflects the nonlinear relationship between amplitude and bubble dynamics, where energy concentration and dispersion mechanisms play a role in determining the collapse intensity and its impact on the surrounding fluid.

4.5. Effect of bubble numbers

We investigated the impact of bubble quantity on cavitation effects, using a bubble cluster model composed of 37 bubbles $(f=1.92\times 10^{-4},\Delta \rho =0.4)$. Initially, the gas nuclei were uniformly set to 5, with sound waves entering from above and a solid wall at the bottom. In order to facilitate the understanding of the dynamic process inside the bubble cluster, we define each layer of bubbles (connected by black lines) as layer 1, layer 2, and layer 3. Fig. 24 illustrates the evolution of the density and pressure fields. In Figs. 24(a–c), the bubbles gradually expanded as the sound waves entered, with the uppermost bubble collapsing first due to its direct exposure to the sound wave, generating a shock wave that propagated outward. This shock wave affected nearby bubbles, accelerating their collapse. The bubbles at the bottom experienced greater hindrance from the solid wall. However, the reflection of sound waves off the wall caused irregularities in the bubble contours, increasing the pressure on the outermost bubbles and leading to inward concavity.

Fig. 24.

Evolution of bubble cluster density and pressure field.

During this process, the first layer of bubbles exhibited a larger expansion radius, oscillating continuously under alternating positive and negative sound pressure. When the positive sound pressure reached the second layer of bubbles, the interaction among bubbles increased the surrounding pressure, resulting in a smaller expansion radius for the second layer, which caused them to collapse first. The shock wave generated by this collapse mixed with the sound wave, accelerating the collapse of the third layer of bubbles, as shown in Fig. 24(d). The central bubble maintained a circular shape throughout, indicating that the second and third layers provided a shielding effect. However, as the third layer collapsed, the generated pressure wave propagated to the surrounding central bubble, eventually causing its complete collapse. The pressure waves resulting from the bubble collapse then spread outward (Fig. 24(e)). Under the influence of the sound wave, the first layer of bubbles began to collapse from top to bottom, while the bottom bubbles continued to oscillate until two bubbles completely collapsed (Figs. 24(f–h)). The shock waves produced impacted the wall, and the greater number of bubbles led to overlapping shock waves that generated multiple jet pressures against the wall (Fig. 24(i)). This phenomenon indicates that the quantity of bubbles significantly influences the cavitation effect, as a greater number of bubbles can enhance both the intensity and frequency of shock waves, leading to increased pressure on the solid wall.

Furthermore, the wall pressure and velocity following the complete collapse of the bubble cluster are quantified, as illustrated in Fig. 25. The data reveals multiple pressure peaks recorded on the wall. Due to the symmetrical distribution of the bubbles, the final two bubbles release distinct jets that strike the wall, causing the formation of two pressure peaks. A significantly higher pressure peak is observed when these shock waves converge at the center.

Fig. 25.

Time series of pressure and velocity changes on the wall. (a): Time series of pressure $p_w$; (b): Time series of velocity $u_w$.

Moreover, a comparison with the simulation results from a cluster of five bubbles shows that the cluster of 37 bubbles produced a greater pressure and velocity, indicating a more intense destructive force on the wall. The increase in bubbles enhances the cumulative energy release during the collapse, resulting in stronger jets and higher pressure peaks. This effect is due to the interactions among the bubbles, where the collapse of one bubble influences its neighbors, thereby amplifying the overall impact on the wall. The complexity of the flow dynamics and the overlapping shock waves generated by a larger bubble cluster contribute to a more significant cavitation effect, emphasizing the relationship between bubble quantity and the intensity of cavitation-induced pressure on solid surfaces.

The impact of bubble quantity on cavitation-induced pressure and velocity is further analyzed through the frequency spectra of pressure and velocity. Figs. 26 shows the pressure and velocity frequency spectra for a cluster of 37 bubbles.

Fig. 26.

Frequency spectrum of pressure and velocity for a cluster of 37 bubbles. (a) Spectra of pressure; (b) Spectra of velocity.

In Fig. 26(a), the pressure spectrum shows multiple high-amplitude peaks, corresponding to the shock waves generated by the collapsing bubbles. These overlapping shock waves from multiple bubbles enhance the overall pressure on the wall. The low to medium frequency range (50–150 Hz) is most prominent, which indicates the cumulative effect of the shock waves as they interact across multiple layers of bubbles.

Similarly, in Fig. 26(b), the velocity spectrum reveals a stronger high-frequency component compared to the case with fewer bubbles. This reflects the increased fluid velocity caused by the enhanced intensity of the shock waves from the collapse of the 37 bubbles. The sharper spikes in the high-frequency range indicate that the jets generated by the bubbles’ collapse are more intense, resulting in faster fluid movement.

The frequency spectrum analysis shows that a greater number of bubbles in the cluster results in higher intensity and frequency of shock waves, leading to stronger jets and higher pressures on the solid wall. These observations emphasize the role of bubble quantity in enhancing the cavitation effect and the potential for greater damage to the wall due to increased energy release during the collapse.

5. Conclusion

This study investigates the dynamics evolution of near-wall bubble clusters under acoustic wave control using the MRT-LBM. The feasibility and effectiveness of the pseudopotential LBM for simulating acoustic cavitation are verified by thermodynamic. The study highlights the significant influence of factors such as bubble cluster position, acoustic frequency, amplitude, and bubble number on cavitation dynamics. By introducing novel frequency spectrum analysis of pressure and velocity signals, this work offers deeper insights into the complex interactions within bubble clusters, jet formation, and shock wave propagation. Key findings include:

(1) As the bubble cluster moves closer to the solid wall, both the intensity of the jet and the cavitation effects increase. A higher offset parameter, corresponding to a smaller distance from the wall, leads to stronger jet pressure and velocity on the wall. Frequency spectrum analysis indicates that when the bubble cluster is closer to the wall, the pressure and velocity signals contain more high-frequency components, reflecting an increase in jet velocity and more dynamic bubble interactions. This enhanced cavitation effect underscores the importance of proximity to the solid boundary in the cavitation damage process.

(2) The acoustic wave frequency is a critical factor in bubble dynamics. At higher frequencies, bubbles do not have enough time to fully expand and collapse, resulting in weaker shock waves and reduced wall pressure and velocity. The frequency spectrum analysis shows that high-frequency sound waves contribute to increased kinetic energy in the higher frequency range, which corresponds to enhanced jet velocity but weaker pressure waves. On the other hand, lower frequencies allow for complete collapse, leading to stronger pressure waves and higher jet velocities, as evidenced by more energy concentrated in the lower frequency range in the spectrum.

(3) As the amplitude of the acoustic wave increases, the intensity of bubble collapse and the energy transferred to the bubble cluster also increase, leading to more powerful jets and sharper pressure peaks on the wall. However, at higher amplitudes, the frequency spectrum becomes more spread out across a wider range, leading to weaker but more frequent pressure peaks. This suggests that while the overall energy increases, it is more distributed across frequencies, which reduces the focus of the energy and results in lower pressure peaks but enhanced jet velocity.

(4) A larger number of bubbles in the cluster enhances the intensity and frequency of cavitation effects. The cascading shock waves from the collapse of more bubbles generate multiple pressure peaks on the wall, significantly increasing pressure on the solid surface. Frequency spectrum analysis shows that more bubbles lead to a wider distribution of energy across frequencies, with higher peak frequencies and more frequent oscillations, reflecting the increased complexity and intensity of the cavitation process.

These insights are valuable for optimizing acoustic cavitation in practical applications, where balancing frequency and amplitude is crucial for maximizing cavitation intensity and efficiency. The study not only advances our understanding of cavitation phenomena but also demonstrates the potential of MRT-LBM as a powerful tool for accurately simulating complex acoustic cavitation processes. This work lays the foundation for future studies aiming to optimize and control cavitation effects in engineering systems, offering new perspectives and methods for harnessing acoustic cavitation more effectively.

CRediT authorship contribution statement

Yu Yang: Writing – original draft, Validation, Software, Methodology, Investigation, Funding acquisition, Data curation. Juan Tu: Writing – review & editing, Supervision, Conceptualization. Minglei Shan: Writing – review & editing, Software, Methodology, Funding acquisition, Formal analysis, Conceptualization. Zijie Zhang: Validation, Software, Investigation, Data curation. Chen Chen: Writing – review & editing, Visualization, Validation, Investigation. Haoxiang Li: Writing – review & editing, Visualization, Validation, Funding acquisition, Data curation.

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.