
==== Front
Ultrason Sonochem
Ultrason Sonochem
Ultrasonics Sonochemistry
1350-4177
1873-2828
Elsevier

S1350-4177(24)00314-6
10.1016/j.ultsonch.2024.107066
107066
Fundamentals of AC and HC
Influence of distribution parameters on acoustic radiation from bubble clusters
Deng Fuqiang
Zhang Lingxin zhanglingxin@zju.edu.cn
⁎
Wang Peng
Wu Yizhe
Zhao Di
Li Yang
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
⁎ Corresponding author. zhanglingxin@zju.edu.cn
12 9 2024
12 2024
12 9 2024
111 1070661 5 2024
29 8 2024
11 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Cavitation noise is the major noise in underwater, and the study of acoustic radiation from bubble clusters is the primary means to reveal the mechanism of cavitation noise. In this study, direct numerical simulation (DNS) of bubble clusters with volume fractions of 20–40 % with different bubble sizes and bubble position distributions are performed, and the far-field sound pressure is calculated using the Ffowcs Williams–Hawkings (FW-H) method. Then, we compare the collapse and acoustic radiation of bubble clusters with equivalent bubble. The results show that the collapse times of bubble clusters at the same volume fraction are identical and close to equivalent bubble, despite the different bubble sizes and positions in the bubble cluster. Further, in terms of acoustic radiation, the layered arrangement of bubble positions results in bubble clusters exhibiting layer-by-layer collapse and emitting multiple sound pressure pulses. In contrast, a random arrangement of bubble positions lacks this feature, resulting in the collapse of the bubble cluster without a layered phenomenon and radiating only a single primary sound pulse, which is consistent with the equivalent bubble. Additionally, the distribution of bubble sizes in the bubble cluster has almost no effect on the acoustic radiation of the bubble cluster. Notably, when the volumetric fraction exceeds 25 %, the sound pressure levels of bubble clusters with different distributions in the frequency domain are nearly identical, with differences from the equivalent bubble within 5 dB.

Keywords

Cavitation noise
Bubble cluster
FW-H method
Direct numerical simulation
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pmc1 Introduction

Cavitation noise, emerging when local pressure in a fluid falls below the vapor pressure threshold, leads to significant environmental concerns including pressure loss, erosion, vibration, and intense noise pollution [1], [2]. This noise is a primary source of underwater disturbance from ships, severely disrupting marine ecosystems by interfering with the natural communication of animal communities, thus posing a threat to their survival [3], [4], [5], [6]. Experiments and numerical simulations have demonstrated that the pressure fluctuations and the surge of the flow-induced noise are mainly induced by the shedding of the periodic cavitation cloud induced by the ejector flow and the collapse of the bubbles in the cloud [7].

Cavitation clouds consist of a large number of bubbles whose collapse induces broadband noise [8]. Empirical modeling of broadband noise from cavitation was investigated by Brown [9], who proposed a simple equation to describe the upper limit of broadband noise in the frequency range of 100 Hz to 10 kHz, based on thruster measurements. Although the method has yielded satisfactory results in predicting noise in open propellers, it lacks insight into the underlying physical phenomena and is not universally applicable. Firenze and Valdenazzi et al [8], Aktas et al. [10] suggested that the high-frequency portion of the cavitation noise originates from sheet cavitation shedding into a cloud and then the collapse and rebound of a large number of small bubbles in the cavitation cloud. Therefore, the understanding of the physical mechanisms of broadband noise from the perspective of bubble dynamics is beneficial. Baiter [11] predicts propeller cavitation noise by superimposing the noise generated by the collapse of a single cavitation bubble. Matusiak [12] proposed that the high-frequency portion of cavitation bubble noise originates from the non-constant dynamic properties of cavitation bubbles. For propeller cavitation noise prediction, the process of cavitation bubble shedding needs to be modeled, and then the single-bubble acoustic radiation model of Fitzpatrick and Strasberg [13] as well as Baiter's superposition method of independent single-bubble sounding are applied. However, this method for predicting cavitation noise lacks stability and universal applicability across varying conditions due to its excessive reliance on parameter adjustment [8], [14]. Consequently, accurate calculation of broadband noise requires a fine-grained simulation of bubble clusters.

In recent years, the development of computer technology has made it possible to finely simulate the dynamical behavior of bubble clusters. Research led by Bremond et al. [15], Quinto et al. [16], and Chahine et al. [17] has experimentally and numerically demonstrated a shielding effect where outer bubbles collapse before inner ones, indicating inter-bubble interactions within the cluster. This effect was further substantiated through direct numerical simulations by Tiwari et al. [18], Bui et al. [19], and Rasthofer et al. [20] Our previous works have also revealed that the collapse of central bubbles in a cluster generates significantly higher pressure than individual bubbles, contributing to intense noise and erosion [21], [22]. Moreover, the research of Wang and Brennen [23], as well as Tda et al. [24] and Ochiai et al. [25], has extended these findings by showing how bubble interactions within a cluster can lead to shock wave formation, accelerating collapse and amplifying its effects. This collective body of work underscores the complexity of bubble cluster dynamics and their critical role in understanding and predicting cavitation noise.

The integration of direct numerical simulations of cavitation bubbles with the Ffowcs Williams-Hawkings (FW-H) method has enabled accurate calculations of the sound radiated by cavitation bubbles. Research by Jamaluddin et al. [26] and Ye et al. [27], [28] has shown that this approach is effective in predicting the noise from non-spherical bubble collapses, facilitating accurate noise assessments for bubble clusters. Further studies using the acoustic analogy method, such as those by Leighton et al. [29] and Ye et al. [30], have modeled the collapse of single and multiple bubbles, employing FW-H methods to compute far-field acoustic pressures. These methods have proven to be precise in understanding the complex dynamics of bubble cluster collapse and their acoustic emissions, offering a promising avenue for accurate cavitation noise prediction.

In our previous study [31], we conducted direct numerical simulations on a bubble cluster containing 352 vapor bubbles and predicted its far-field sound pressure using the FW-H method. We observed that for layered bubble clusters, as the volume fraction exceeds 20 %, the increase in volume fraction leads to collapse and sound radiation behaviors that increasingly resemble those of an equivalent bubble (whose initial volume equal to the bubble cluster). Furthermore, measurements by Liu et al. [32] on the volume fraction of actual cavitation clouds indicate that within a larger region, the volume fraction of cavitation clouds tends to concentrate between 20 %-40 %. However, due to the complexity of the internal structure of cavitation clouds at this volume fraction, traditional numerical methods and experimental techniques are difficult to accurately capture their internal characteristics, thereby limiting the exploration of their collapse and acoustic emission patterns. Direct numerical simulations of bubble clusters within this volume fraction range can effectively overcome this limitation and provide a theoretical foundation for understanding the collapse and acoustic emission behaviors of cavitation clouds. Additionally, existing research indicates that in real cavitation clouds, the position of bubbles is usually randomly distributed, and the bubble size often follows a log-normal distribution [32], [33], [34]. Studies by Du [24], Naoya [25], show that the different distributions of bubble positions and sizes within a cluster affect its dynamic behavior. Therefore, we studied the collapse and sound radiation characteristics of three bubble clusters with different distributions of position and size. These three types of clusters are: a cluster with equal-sized bubbles and layered positions; a cluster with equal-sized bubbles and random positions; and a cluster with log-normally distributed bubble sizes and random positions. For ease of comparison, we refer to these three types of clusters as layered cluster, random cluster, and log-normal cluster.

2 Methods

2.1 Numerical method for bubble clusters

Utilizing the two-phase homogeneous model within the OpenFoam open-source software, we incorporate continuity, momentum, and phase fraction equations. The continuity and momentum governing equation are as follows:(1) ∂ρ∂t+∇·ρui=0

(2) ∂ρui∂t+∇·ρuiuj=-∂p∂xi+∂τij∂xj+fσi

where ρ is the density, p is pressure and ui is the velocity. τij represent the viscose stress and fσi is the surface tension force. The mixing density ρ is given by(3) ρ=αρl+1-αρv

The scalar α, representing the volume fraction within a computational cell, varies from 0 to 1, whereα = 1 denotes a cell fully occupied by liquid andα = 0 indicates complete vapor presence. The subscripts l and v correspond to the liquid and vapor phases, respectively. The vapor–liquid interface is captured using the volume of fluid (VOF) method [35]. The value of α is obtained by solving the transport equation.(4) ∂αl∂t+∇·αlui+∇·urαl1-αl=αlαvdgdt+αl∇·ui

Eq. (4) maintains the boundedness of the phase fraction, becoming zero when either phase fraction is zero, with dgdt=1ρvDρvDt-1ρlDρlDt. To mitigate numerical diffusion at the vapor–liquid interface and preserve its sharpness, an additional term, representing the artificial rate of compression ur, has been incorporated into the left-hand side of Eq. (4).(5) ur=cui∇α∇α

The surface tension force fσi is calculated by(6) fσi=σ∇·∇α∇α∇α

where σ is the surface tension coefficient.

To formulate a complete set of governing equations, we require equations of state for both vapor and liquid phases. Given the intricate thermodynamics of bubble collapse, where numerous variables are uncertain, Johnson's findings that thermal and mass diffusion near cavitation bubbles are significantly lesser than viscosity allows us to neglect these aspects in our analysis [36]. Moreover, the pressures from bubble rebound and secondary collapses are minor compared to the initial collapse [37]. For simplicity, our approach adopts a fixed pressure condition within the bubble, a technique consistently applied in our earlier work [38], [39]. We employ the Tait equation of state to accurately represent the liquid phase's compressibility under the intense pressures of bubble collapse and to describe the propagation of high amplitude sound waves [40].(7) plρl=Bρlρl0m-1+A

where ρl0=1×103kg/m3,A=1.013×10-4Gpa,B=0.331Gpa,m=7.1. The parameter m is obtained from a best-fit analysis of pressure–volume data across a specific temperature range. B is associated with the ambient sound speed, leading to its constancy, and A represents the ambient pressure.

2.2 FW-H method

Ffowcs Williams & Hawkings et al. [41] developed the FW-H equation, an inhomogeneous wave equation for modeling acoustic propagation around moving bodies, based on the conservation of mass and momentum. Direct solutions to this equation are complex, prompting Farassat [42], [43] and Di Francescantonio [44] to propose crucial mathematical modifications that broadened its usability. This flexibility is especially beneficial for using a permeable integration surface, which can be placed anywhere in the flow field to facilitate far-field noise predictions from events such as cavitation or bubble collapse. The equation enables the calculation of sound pressure p′ produced by bubble cluster collapse, measurable at an observer x and emission time τ.(8) p′(x,τ)=pT′(x,τ)+pL′(x,τ)+pQ′(x,τ)

where pT′(x,τ),pL′(x,τ) and pQ′(x,τ) represent the thickness noise, loading noise and quadrupole noise, respectively.

Solving the FW-H equation involves surface integrals that account for monopoles, dipoles, and quadrupoles within the integration surface, and volume integrals for external quadrupole sources. However, given the minimal and rapidly diminishing impact of quadrupole sources in low subsonic regions, coupled with the high computational demands of volume integrals, this work excludes the contribution from external quadrupoles. We utilize the Farassat 1A integral solution for permeable surfaces to determine the far-field sound pressure [45].(9) 4πpT′x,τ=∫f=0ρl0U˙n+Unr1-Mr2retdS+∫f=0ρl0UnrM˙r+cl0Mr-M2r21-Mr3retdS

(10) 4πpL′x,τ=1cl0∫f=0L˙rr1-Mr2retdS+∫f=0Lr-LMr21-Mr2retdS1cl0∫f=0LrrM˙r+cl0Mr-M2r21-Mr3retdS

In our framework, c10=1450m/s and r=|x-y|, where the permeable surface is defined by f(x,t)=0, making n^=∇f the unit outward normal. The Mach number vector M of the source has components Mi=vi/c0 and Mr=Mir^i, with vi representing the surface velocities. Given that bubble clusters exhibit no migratory motion, our chosen integration surface remains stationary, implying vi=0. The dot superscript indicates differentiation with respect to emission time τ, and the 'ret' subscript signifies the influence of retarded time.

Ui and Li are defined as(11) Ui=1-ρlρl0vi+ρlρl0ui

(12) Li=Pijn^j+ρluiun-vn

Here ui are the fluid velocities on f, and Pij is given by(13) Pij=pδij-μ∂ui∂xj+∂uj∂xi-23∂uk∂xkδij

The integral terms in Eq. (10) and Eq. (11) are evaluated at the emission time τ, defined as τ=t+r/c0. Due to the finite speed of sound in compressible liquids, there is an inherent time delay r/c0. Given the brief collapse duration of bubble clusters and the complexity that the time delay r/c0 introduces in post-processing and sound pressure reconstruction, this study simplifies by neglecting r/c0 and setting τ=t. This approximation aligns with methodologies previously employed in related research [46].

2.3 Setup and validation

2.3.1 Set up

This study conducts direct numerical simulations on three types of bubble clusters: layered, random, and log-normal clusters. Our previous work [38] has already detailed the setup of layered clusters, which have a distinct structure with 4, 5, or 6 layers, corresponding to 165, 352, and 646 bubbles, respectively. As the number of bubbles increases, the demand for computational resources also rises. For example, calculating a cluster with 352 bubbles requires approximately 90 million grids, while 646 bubbles necessitate about 190 million grids. Since our earlier research [38] demonstrated that clusters with varying numbers of layers show similar collapse and acoustic radiation behaviors, we chose a cluster with 352 bubbles as the subject of this study to balance computational efficiency and accuracy. Additionally, it's important to note that in these clusters, each bubble has the same initial radius R0=2mm, and the distance between adjacent layers and bubbles is kept essentially uniform.

For random clusters, many studies [18], [20], [24] typically used a method of randomly generating bubble positions for arrangement. However, this approach often results in large gaps within the clusters that cannot accommodate any bubbles, making the arrangement of dense clusters challenging. Our tests show that the densest cluster achievable by this method has a volume fraction of only 25 %, while Liu et al. [47] have shown that the volume fraction of cavitation clouds can exceed 40 %. Currently, there is a lack of research on clusters with high volume fractions, making it crucial to explore the mechanisms of collapse and acoustic radiation of high-volume fraction clusters. To address the difficulty of arranging dense clusters, this study proposes a new strategy for bubble cluster arrangement using the Discrete Element Method (DEM), as depicted in Fig. 1. The process is as follows: initially, a sparse cluster consisting ofN = 352 vapor bubbles is generated, followed by the placement of a gradually contracting rigid boundary around the cluster. When bubbles come into contact with the rigid boundary, they experience an inward normal force fwall, preventing them from crossing the boundary. To avoid overlaps between bubbles, two repulsive forces fb1 and fb2 are generated upon contact between neighboring bubbles, pushing them in opposite directions. As the rigid boundary continues to contract, the cluster becomes more denser, and once the volume fraction reaches 40%, the positions of all bubbles are recorded. Additionally, based on the volume fraction formula αc=NR03/Rc03, we calculate the ratio of initial radii of bubble clusters at different volume fractions to the initial radius of the 40% volume fraction cluster. This ratio is then used to scale the positions of the 40% volume fraction cluster, thus determining the distribution of clusters at various volume fractions.Fig. 1 Schematic diagram of the arrangement of random clusters (with equal bubble radii and randomly distributed bubble positions).

The bubble radius Rl0 in log-normal cluster follows a log-normal distribution:(14) PRl0=1Rl0σl2πexp-(lnRl0-μl22σl2

where σl is the standard deviation and μl is the mean. Given that both the layered and random clusters consist of 352 bubbles with an initial radius R0, it implies that their initial total volume Vvapor0 is equivalent. To maintain comparability with the layered and random clusters, the log-normal cluster also contains a total of 352 bubbles, ensuring the overall initial volume is consistent across the three types. According to research by Liu et al. [32], bubble diameters within cavitation clouds follow a log-normal distribution with σl typically around 0.6; however, a larger σl indicates a greater discrepancy between the maximum and minimum bubble diameters, which can lead to insufficient grid resolution for smaller bubble radii during simulations. Therefore, we opted for a smaller σl value, adopting theσl = 0.3 bubble size distribution used in Rasthofer's [20] simulation of bubble clusters. Taking these factors into account, we selectedμl = 0.6 andσl = 0.3 as the distribution parameters for the log-normal bubble cluster, as illustrated in Fig. 2. The maximum and minimum initial bubble radii within the cluster, R0max and R0min, are 3mm and 1.2mm respectively. It is worth noting that the initial average bubble radius Ra0=3Vvapor0/4Nπ3 for all three bubble clusters is equal to 2 mm.Fig. 2 Distribution of bubble radii for log-normal clusters (with a lognormal distribution of bubble radii and a random distribution of bubble positions).

This study employs direct numerical simulations to investigate three distinct bubble cluster distributions: layered, random, and log-normal, spanning volume fractions from 20% to 40%. Notably, the initial total volumes of these three types of clusters remain constant across all volume fractions. Fig. 3 depicts the comparative distributions of these clusters at volume fractions of 20% and 40%. Additionally, for any specific volume fraction, the initial radius Rc0 of the bubble cluster is consistent across all three cluster.Fig. 3 Distribution of bubble clusters for (a) layered cluster, (b) random cluster and (c) log-normal cluster for volume fractions of 20% and 40%.

The FWH acoustic analogy was utilized to post-process direct numerical simulation data of the flow field, enabling the computation of far-field sound pressure. To enhance the accuracy of capturing the far-field sound pressure emitted by spherical bubble clusters, a spherical surface, centered at the bubble cluster's central location point O(000), was employed as the acoustic integration surface. The acoustic monitoring point is positioned at x(0 0 500 Ra0). The time step used for calculating sound pressure matches that of the flow field simulation. By conducting a Fourier transform on the time-domain sound pressure data p′, we obtain its frequency domain amplitude, Afftp′. This amplitude is then transformed into the Sound Pressure Level (SPLs) using the formula:(15) SPL=20logAfftp′pref

where pref=10-6 Pa serves as the reference sound pressure, with the assumption that the medium of the free radiating sound field is water.

2.3.2 Numerical simulation conditions

In this study, we conducted direct numerical simulations on bubble cluster consist of 352 vapor bubbles, using a cubic computational domain with sides of length L=500Ra0, and centered the bubble cluster at coordinate O(0 0 0). This setup is shown in Fig. 4, with the cluster's initial radius denoted as Rc0. Inside the bubbles, vapor was maintained at a saturated pressure ofPv = 3541pa, contrasted against an external pressure ofp∞ = 101325pa. We adopted boundary conditions that included a zero-gradient for velocity and a non-reflective condition for pressure to minimize prevent sound waves from reflecting at the boundary. The simulation employed the compressible two-phase solver in OpenFoam, discretizing the governing equations through the finite volume method and using the PIMPLE algorithm for iterative solutions. The discretization process involved the second-order upwind scheme for the convection term, central difference for the Laplacian term, and the first-order implicit Euler method for time.Fig. 4 The computational domain and conditions of the bubble cluster, where the computational domain is a cube, the bubble cluster is located at the center O(0 0 0) of the cube, and the pressure boundary condition is a non-reflecting boundary.

2.3.3 Validation

we employed the refineHexMesh tool from the OpenFoam third-party library to multi-level subdivide the flow field grid within the computational domain, achieving the highest grid density in the bubble cluster area. Our previous research has thoroughly verified the reliability of the steam bubble model used [31], [39]. To ensure the accuracy of bubble cluster simulations, it was essential to perform convergence tests on the grid resolution of the flow field. Given the necessity to evaluate the behavior of bubble clusters with volume fractions ranging from 20 % to 40 %, it is impractical to conduct exhaustive grid convergence tests for every condition. Therefore, we selected a bubble cluster with a 30 % volume fraction, representing a medium porosity, for our grid convergence testing. Due to the potential variability in sensitivity to grid resolution among different bubble cluster distributions, we also specifically tested three distinct distributions of bubble clusters at 30 % volume fraction. We established grid resolutions in the initial average bubble diameter 2Ra0 direction at 20, 24, 28, 32, and 36 cells, respectively, and calculated the volume evolution and time-domain acoustic pressure at these resolutions, as illustrated in Fig. 5. To improve the readability and identifiability of the simulation results, we nondimensionalized time using tc=0.915Ra0√(ρl0/(p∞-pv)). The results indicate that when the grid count in the initial average bubble diameter 2Ra0 direction reaches 32, the time-domain and frequency-domain acoustic pressures closely match those obtained with a 36-grid configuration. Thus, to balance accuracy and computational efficiency, we opted for a setup with 32 grid cells in the initial average bubble diameter 2Ra0 direction.Fig. 5 Convergence validation of flow field computational grids for (a) layered cluster, (b) random cluster, (c) Log-normal cluster. Volume evolution and time-domain sound pressure for a volume fraction of 40 % at different flow field grid resolutions. (Case 1: Ncell/2Ra0 = 24; Case 2: Ncell/2Ra0 = 28; Case 3: Ncell/2Ra0 = 32; and Case 4: Ncell/2Ra0 = 36;). The acoustic monitoring point is located at x(0 0 500 Ra0).

Within the existing research framework, a consensus on the optimal location for penetrable FW-H integration surfaces has yet to be established. Accordingly, our study introduced spherical integration surfaces at radii of 1.1 Rc0, 2Rc0, and 3 Rc0, and assessed the time-domain and frequency-domain sound pressures of three distinct bubble clusters for each radius, as illustrated in Fig. 6. The analysis of sound pressures across these varied radii showed negligible differences, especially in spectral analyses where the calculated outcomes were nearly identical. To reduce numerical dissipation and maintain mesh density near the integration surface, we opted for a radius of 1.1 Rc0 for further analyses of sound pressures.Fig. 6 The choice of acoustic integration surfaces for (a) layered cluster, (b) random cluster, (c) Log-normal cluster. Time-domain sound pressure and frequency-domain sound pressure radiated by bubble cluster at acoustic monitoring point x(0 0 500 Ra0) corresponding to different radii of the integration surface. The radii of the integrating surfaces are 1.1 Rc0, 2 Rc0 and 3 Rc0.

3 Results and discussion

3.1 Bubble dynamics

Fig. 7 (a)-(c) presents the collapse morphology and pressure field evolution around bubble clusters with a 30 % volume fraction, comprising layered, random, and log-normal distributions. The collapse details of these clusters exhibit distinct characteristics. In the layered cluster, bubbles of uniform diameter are arranged in a stratified manner with rotational symmetry around the cluster center within each layer, leading to an almost synchronous collapse process within each layer and a pressure field around the cluster that also exhibits rotational symmetry. For the random cluster, although the bubbles have uniform diameters, their positions are randomized, resulting in an absence of symmetry in the arrangement and a non-synchronous collapse process, with the surrounding pressure field lacking symmetry as well. In the log-normal cluster, bubbles vary in initial radius, leading to asynchronous collapse events, where smaller-radius bubbles in the outer layers collapse sooner than larger-radius ones, and the distribution of the surrounding pressure field also lacks symmetry. Despite differing in collapse details, all three distributions demonstrate a similar collapse process, with outer bubbles collapsing before inner ones, highlighting a significant shielding effect of the outer layers on the inner bubbles. Notably, the overall collapse rate of the three different bubble distributions remains fundamentally consistent.Fig. 7 Pressure field and morphology evolution of (a) layered clusters, (b) random clusters and (c) Log-normal clusters with a volume fraction of 30% during collapse.

Our prior research [31] has indicated that the collapse morphology evolution of dense layered clusters is similar to that of equivalent bubble (a bubble with the same initial volume as the bubble cluster). To examine the differences and connections in the collapse morphology evolution among bubble clusters with different distributions at volume fractions of 20 %, 30 %, and 40 %, we analyzed the evolution of the volume Vvapor during collapse. By converting the volume to an equivalent radius using the formula Re = (3Vvapor/4π)3, and comparing it with the radius evolution of the equivalent bubble, the results are shown in Fig. 8 (a)-(c). In terms of the evolution details of the equivalent radius, the three distributed bubble clusters demonstrated distinctive characteristics related closely to the distribution pattern: the layered cluster showed a noticeable fluctuation in the rate of decrease of the equivalent radius due to sequential layer collapses; the random cluster, lacking synchronous collapse among bubbles, exhibited fluctuations in the decreasing rate of the equivalent radius, albeit to a lesser extent than the layered cluster; the log-normal cluster, with bubbles of varying initial radii, almost showed no layered collapse phenomenon, resulting in minimal fluctuation in the rate of decrease of the equivalent radius, rendering the equivalent radius-time curve smoother, akin to the evolution of the equivalent bubble. Notably, despite the differences in the details of equivalent radius evolution among the three different distributions, we found that their collapse times are exactly the same at volume fractions of 20 %, 30 %, and 40 %. Moreover, the collapse times of bubble clusters at different volume fractions are close to that of the equivalent bubble.Fig. 8 Evolution of the equivalent radii of (a) layered cluster, (b) random cluster, and (c) log-normal cluster with volume fractions of 20%, 30%, and 40% during the collapse process and comparison with equivalent bubble.

The findings above reveal that bubble clusters of different distributions with volume fractions of 20 %, 30 %, and 40 % exhibit identical collapse times tcollapse at the same volume fraction, and these times are closely comparable to the collapse time of an equivalent bubble. To validate the universality of this conclusion, we computed the collapse scenarios for three differently distributed bubble clusters within the volume fraction range of 20 % to 40 %. Subsequently, we recorded the collapse times of these clusters and compared them with the collapse time of an equivalent bubble, as depicted in Fig. 9. The results confirmed that the collapse times for bubble clusters of different distributions within the 20 % to 40 % volume fraction range are consistent and closely approximate the collapse time of bubbles with an equivalent radius.Fig. 9 Collapse times of layered cluster, random cluster and log-normal cluster with volume fractions of 20%-40% and compared with equivalent bubble.

3.2 Acoustic radiation

During the collapse of bubble clusters, the outer bubbles collapse before the inner ones, continuously emitting sound waves throughout the process, which results in higher pressure values depicted on pressure cloud diagrams. Fig. 10 displays the pressure cloud diagrams induced by the collapse of the outer bubbles and the complete collapse of bubble clusters with a 30 % volume fraction, featuring three different distributions: layered, random, and log-normal clusters. The sound radiation from the collapse of the outer bubbles of these clusters exhibits unique characteristics: due to its stratified arrangement, the layered cluster experiences a synchronized collapse of its outer bubbles, uniformly radiating sound waves in all directions upon completion. Conversely, the random and log-normal clusters, with their arbitrarily placed bubbles lacking a clear stratification, show varying collapse sequences, resulting in a sequential emission of sound waves as each bubble implodes. Despite these differences in sound radiation during the collapse of the outer bubbles, the sound emissions at the end of the complete collapse process demonstrate consistency across all three types of clusters, as evidenced by the shape and magnitude of the pressure values on the pressure cloud diagrams. Notably, the sound pressure radiated by the final collapse of the three different distribution bubble clusters is much larger than that radiated by the collapse of the outer bubbles.Fig. 10 Pressure clouds of the outer bubbles and the complete collapse of (a) layered cluster, (b) random cluster, and (c) log-normal cluster with a volume fraction of 30% for characterizing the strength of acoustic radiation.

Our previous studies [31] have shown that for layered clusters, the radiated sound pressure consists of multiple acoustic pulses, and when the volume fraction exceeds 30 %, these pulses closely resemble those from an equivalent bubble. To further investigate the differences and similarities in sound radiation between three types of distributed bubble clusters and their comparison with an equivalent bubble, we calculated the time-domain sound pressure for bubble clusters with volume fractions of 20 %, 30 %, and 40 %, and compared them to that of an equivalent bubble, as illustrated in Fig. 11 (a)-(c). The time-domain sound pressure of the three distributions exhibits unique characteristics. Due to the layered structure of bubble positions and the layer-by-layer collapse of each layer radiating sound waves outward, the layered bubble cluster exhibits multiple sound pulses. At a 20 % volume fraction, these pulses have similar peak values, but as the volume fraction increases to 30 % and 40 %, the variance in peak values between the pulses grows, with the final pulse significantly surpassing the others in peak value. In contrast, the bubble position arrangement in random clusters and lognormal clusters lacks a layered structure, resulting in the absence of the layer-by-layer collapse characteristic. Consequently, these clusters do not produce multiple distinct sound pressure pulses and are ultimately dominated in the time domain by a single acoustic pulse with similar peak value and pulse width. Compared to an equivalent bubble, the time-domain sound pressure pulses' peak values and shapes of the three distributions at a 40 % volume fraction are significantly more similar to those of an equivalent bubble, particularly at a 20 % volume fraction where the random and log-normal clusters' pulses more closely resemble those of an equivalent bubble than those from the layered cluster.Fig. 11 Time-domain sound pressures of (a) αc = 20 %, (b) αc = 30 %, and (c) αc = 40 % with different distributions, and compared with equivalent bubble. The acoustic monitoring point is located at x(0 0 500 Ra0).

To verify the universality of the aforementioned conclusions, we calculated the time-domain sound pressure radiated by three differently distributed bubble clusters within a volume fraction range of 20 % to 40 %, comparing it with that of an equivalent bubble. The results are presented as two-dimensional contour plots in Fig. 12 (a)-(c). The analysis revealed that the layered cluster exhibits multiple acoustic pulses within the 20 %-40 % volume fraction range, whereas the random and log-normal clusters show a single acoustic pulse. Notably, at a 40 % volume fraction, the layered cluster's final pulse peak significantly surpasses the other pulses, closely resembling the equivalent bubble. As the volume fraction decreases, the peak of the final acoustic pulse gradually reduces, and the difference between the peaks of the pulses diminishes, increasing the disparity with the equivalent bubble. Between the 20 %-40 % volume fraction range, the single pulse peak of the random and log-normal clusters decreases more gradually with decreasing volume fraction, making their sound pressure more akin to that of the equivalent bubble.Fig. 12 Two-dimensional cloud maps of the time-domain sound pressure for (a) layered cluster, (b) random cluster, and (c) log-normal cluster with volume fractions of 20%-40%, respectively, and compared with equivalent bubble. The acoustic monitoring point is located at x(0 0 500 Ra0).

To delve into the differences and connections in frequency-domain SPLs among three differently distributed bubble clusters and an equivalent bubble, we applied Fourier transform techniques to convert time-domain sound pressure signals from bubble clusters with volume fractions of 20 %, 30 %, and 40 %, and from an equivalent bubble into frequency-domain sound pressure signals. These were subsequently quantified as SPLs, as depicted in Fig. 13(a)-(c). It was observed that both bubble clusters and the equivalent bubble exhibit a trend of SPLs decay with increasing frequency. The frequency-domain SPLs curve of the equivalent bubble is relatively smooth, with a slower rate of SPLs decrease compared to the bubble clusters. The rate of SPLs decay in the frequency domain increases with decreasing volume fraction for the three different distributions. It is noteworthy that at volume fractions of 30 % and 40 %, the SPLs across the frequency domain are almost identical for the three different distributions of bubble clusters, and the fluctuations of SPLs remain within 5 dB of the equivalent bubble. However, at a volume fraction of 20 %, the SPLs of the random and log-normal clusters are nearly the same in the frequency domain, whereas the layered cluster significantly deviates from the other two types. In addition, the layered cluster shows fluctuations in SPLs within the frequency domain, with minimal fluctuations for bubble clusters at 30 % and 40 % volume fractions within a 30 KHz range, staying within 5 dB of the equivalent bubble. Nevertheless, the 20 % volume fraction cluster shows significantly more pronounced fluctuations in SPLs, diverging greatly from the equivalent bubble. The random and log-normal clusters exhibit less fluctuation in SPLs compared to the layered cluster, with small fluctuations within a 30 KHz range for 20 %, 30 %, and 40 % volume fractions, also staying within 5 dB of the equivalent bubble.Fig. 13 Frequency domain SPLs of (a)αc = 20 %, (b)αc = 30 %, and (c)αc = 40 % with different distributions, and compared with equivalent bubble. The acoustic monitoring point is located at x(0 0 500 Ra0).

To corroborate the universality of our findings, we calculated the frequency-domain SPLs for three differently distributed bubble clusters within a volume fraction range of 20 % to 40 %, comparing these SPLs with those of an equivalent bubble. The results, illustrated in Fig. 14 (a)-(c), indicate that within the 20 %-40 % volume fraction range, the layered cluster exhibits significantly higher SPLs fluctuations in the frequency domain compared to the random and log-normal clusters. Particularly, when the volume fraction of the layered cluster is below 25 %, its frequency-domain SPLs fluctuations are markedly higher than those of clusters with a volume fraction above 25 %. Within the volume fraction range of 25 %-40 % for layered clusters, as well as 20 %-40 % for random and log-normal clusters, there is minimal fluctuation in SPLs, mirroring the behavior of an equivalent bubble.Fig. 14 Two-dimensional cloud plots of the frequency domain SPLs of (a) layered cluster, (b) random cluster, and (c) log-normal cluster with volume fractions of 20%-40%, respectively, and compared with equivalent bubble. The acoustic monitoring point is located at x(0 0 500 Ra0).

The analysis above indicates that a volumetric fraction of 25 % is a critical threshold for bubble clusters. However, the contour plots in Fig. 14 do not adequately reflect the specific differences in SPLs between the three different distributions of bubble clusters at this fraction, nor their differences with an equivalent bubble. To ascertain these specific differences, we have plotted the frequency domain sound pressure level curves for the three types of bubble clusters and the equivalent bubble at a volumetric fraction of 25 %, as shown in Fig. 15. The results reveal that at a volumetric fraction of 25 %, the SPLs of the three different distributions of bubble clusters are quite consistent, with the discrepancy from the equivalent bubble remaining within 5 dB, aligning with the findings at volumetric fractions of 30 % and 40 %. We therefore conclude that when the volumetric fraction exceeds 25 %, the SPLs of bubble clusters with different distributions in the frequency domain are nearly identical, and the differences compared to the equivalent bubble are within 5 dB.Fig. 15 Frequency domain SPLs of layered cluster, random cluster, and log-normal cluster with volume fractions of 25%, and compared with equivalent bubble. The acoustic monitoring point is located at x(0 0 500 Ra0).

4 Conclusions

In this study, we conducted direct numerical simulations on bubble clusters with volume fractions of 20 %-40 %, including layered clusters (with equal-sized bubbles and layered positioning), random clusters (with equal-sized bubbles and random positioning), and log-normal clusters (with log-normally distributed bubble sizes and random positions), as well as an equivalent bubble. The FW-H method was utilized to calculate the far-field noise of the bubble clusters. We examined the differences and connections in collapse and sound radiation behaviors among these three differently distributed bubble clusters and their comparison with an equivalent bubble. The primary conclusions include:

Despite variations in bubble sizes and positional arrangement across the three types of clusters, leading to distinct collapse details, the collapse time under the same volume fraction conditions is essentially consistent across all three clusters and closely approximates that of the equivalent bubble.

In terms of temporal sound pressure, layered clusters exhibit multiple acoustic pulses. At a volume fraction of 40 %, the peak of the last pulse is significantly higher than the others, approaching the behavior of equivalent bubble. However, as the volume fraction decreases, the difference between the peaks of each pulse gradually diminishes, increasing the deviation from equivalent bubble. Random and log-normal clusters display a single acoustic pulse in temporal sound pressure, and as the volume fraction increases, their behavior becomes more similar to that of an equivalent bubble.

For the frequency domain sound pressure level, all three types of bubble clusters and equivalent bubble show a decay in sound pressure level with increasing frequency. Frequency domain analysis shows that the SPLs of the three different distributions of bubble clusters are almost the same when the volume fraction exceeds 25 %, and the fluctuations of SPLs remain within 5 dB of the equivalent bubble. However, when the volume fraction is less than 25 %, the SPLs of the random and log-normal clusters are almost the same in the frequency domain and deviate from the equivalent bubble within 5 dB, while the layered clusters deviate significantly from the other two types and the equivalent bubble.

This study highlights the influence of various distribution parameters on the collapse and acoustic radiation of dense bubble clusters with volume fractions of 20 %-40, as well as their similarity to equivalent bubble behavior. These findings serve as a reference for future work aimed at understanding cavitation cloud noise mechanisms and developing theoretical models. However, our numerical analysis did not include the effects of bubble cluster inception, rebound, scale effects, non-uniform spatial distributions, or complex-shaped clusters on acoustic radiation. The gap between numerical simulations and experimental conditions remains, and this will be addressed in future research.

CRediT authorship contribution statement

Fuqiang Deng: Writing – original draft, Software, Methodology, Formal analysis, Conceptualization. Lingxin Zhang: Writing – review & editing, Resources, Funding acquisition, Conceptualization. Peng Wang: Methodology, Investigation. Yizhe Wu: Resources, Methodology. Di Zhao: Validation, Software. Yang Li: Visualization, Validation.

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.

Acknowledgements

This work was supported by the 10.13039/501100001809 National Natural Science Foundation of China (Grant No. 12272343 ). We express our gratitude to the anonymous reviewers for their valuable comments and insightful suggestions.
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