Interactions between a central bubble and a surrounding bubble cluster

The interaction of multiple bubbles is a complex physical problem. A simplified case of multiple bubbles is studied theoretically with a bubble located at the center of a circular bubble cluster. All bubbles in the cluster are equally spaced and own the same initial conditions as the central bubble. The unified theory for bubble dynamics (Zhang et al. arXiv:2301.13698) is applied to model the interaction between the central bubble and the circular bubble cluster. To account for the effect of the propagation time of pressure waves, the emission source of the wave is obtained by interpolating the physical information on the time axis. An underwater explosion experiment with two bubbles of different scales is used to validate the theoretical model. The effect of the bubble cluster with a variation in scale on the pulsation characteristics of the central bubble is studied.

Bubble dynamics is a topic of great interest due to its extensive applications in underwater explosions [1][2][3] , air-gun exploration [4][5][6] , cavitation [7][8][9] , and ultrasonic cleaning [10,11] . The dynamic characteristics of a single bubble under different environments have been well developed in the past few decades [12][13][14][15][16][17] . In recent years, the dynamics of multi bubbles have been drawing increasing interest [18][19][20][21][22] , because proper control of the characteristic parameters can promote the utilization of bubble dynamics. Experimental and numerical methods are adopted in most studies on multiple bubble dynamics. In terms of experimental research, Tomita et al. [23] quantitatively studied the coupling effect between two simultaneously generated bubbles by laser-induced bubble experiments, and found that the bubble dynamics are sensitive to the ratio of bubble scale and distance parameters.
The effect of the time difference in bubble generation was systematically summarized in the works of Tomita and Sato [24] , where the jet impact was found enhanced under certain conditions. Fong et al. [25] also discovered some interesting phenomena in the double-bubble interaction through sparkinduced bubble experiments, such as extremely thin liquid jets. However, it is hard to accurately generate and control a bubble array in the experiments, and obtaining the pressure in the flow field requires high-precision experimental instruments of high cost. In terms of numerical research, Han et al. [26] used the boundary element method to study the coalescence of multiple bubbles. Li et al. [27] and Liu et al. [28] studied the characteristics of double-bubble interaction with the finite volume method and the finite element method. Huang et al. [29] developed a dual fast multipole boundary element method to quickly simulate the dynamic characteristics of a bubble array. Nevertheless, numerical methods have limitations in stability and accuracy due to the large deformation and splashing of bubble surfaces, especially when the number of bubbles is large. In addition, theoretical methods are also involved in some works. Bremond et al. [30] studied the mechanical properties of a hexagonal bubble cluster formed by dozens of laser bubbles, and simulated the bubble radius using the improved Rayleigh-Plesset equation. Qin et al. [31] derived analytical solutions for the pulsation characteristics of bubble clusters in different forms in incompressible fluids. In practical engineering, a large number of applications of the bubble-array exist in air-gun exploration [32][33][34] , since an optimized arrangement for the array can fully utilize the energy of the bubbles. Zhang et al. [35] applied the particle swarm optimization method to the design of bubble-array in different environments.
The above research indicates that the dynamic characteristics of multiple bubbles are complicated, and it is hard to completely clarify the underlying mechanisms using experimental and numerical methods, especially for the pressure characteristics induced by bubbles. In terms of theoretical research, the propagation of pulsation pressure under the influence of a bubble cluster is significantly affected by the propagation time, which needs to be studied with the consideration of fluid compressibility. Thus, the research on the pressure characteristics of the flow field considering the propagation process is relative rare. In this study, the unified theory established by Zhang et al. [36] is applied to study a simple case of multiple bubbles: a bubble is located at the center of a circular bubble cluster, and all bubbles are generated synchronously with the same initial conditions. The purpose behind this study is to, by deeply understanding the dynamics of the bubble affected by surrounding bubbles in all directions, gain insight into the pulsation characteristics of bubbles inside a bubble array and to develop new technologies in the fields of cavitation, marine exploration and underwater explosion. Figure 1   The fluids around the bubble are considered to be weakly compressible, and the gas inside the bubble obeys the adiabatic assumption. The bubble always maintains a spherical shape during the pulsation process, so according to the work of Zhang et al. [36] , the unified equation of the bubble pulsation is: where c is the sound speed in water; R , R and R are respectively the bubble radius, pulsation velocity and pulsation acceleration; u is the value of migration velocity of the bubble; h is the enthalpy [36,37] on the bubble surface: where b P is the internal pressure of the bubble, (the subscript '0' denotes the physical quantity at the initial moment;  is the specific heat ratio of gas); a P is the pressure of background flow field of the bubble, and its value equals to the hydrostatic pressure at infinity P  when the bubble is in the free field;  is the fluid density around the bubble.
With the fourth-order Runge Kuta method, the time history of bubble radius can be obtained by solving the expanded differential form of Eq. (1): Compared with the Keller-Mixis equation [38] the effect of bubble migration on the pulsation characteristics is considered in Eq. (3); thus, the bubble migration needs to be solved separately. Base on the conservation of momentum for the bubble [36] , the migration of the bubble is obtained by the following equation: where a C and d C are the additional mass coefficient and drag coefficient of the bubble, respectively; a P  represents the pressure gradient of background flow field of the bubble, and it is the hydrostatic pressure gradient in the free field; () is a special defined symbol, ( ) ( ) | |  =   .
Combine Eqs. (1) and (4), the time history of bubble radius and displacement can be obtained.
The pressure of flow field [36] induced by bubble pulsation can be calculated by the following equation: where r is the position vector of the measuring point in the flow field. Note that we interpolate the physical information of the bubble on the time axis to take into account the propagation process of the pulsating pressure, that is, the physical information in the flow field at t is induced by the bubble To model the interaction of multiple bubbles, the background flow field of the bubble needs to be corrected, that is, the pressure field induced by other bubbles are superimposed on Pa. The pressure of the background flow field of the bubble [36] induced by bubble N are given as below: (6) where o is the position vector of the studied bubble;  [39][40][41] , as shown in where  is velocity potential;  is the solid angle; G is Green function; s is the bubble surface; DPM v denotes the velocity of nodes on the bubble surface obtained by the density optimization method [42] ;  is the dimensionless inner bubble pressure at inception; 0 V and V are the bubble volume at the initial and current moment, respectively.  forward according to the maximum radius of the bubble in the experiment. Detailed implementation details can be found in the works of Wang [13] . Here, we directly give the initial conditions of Bubble    a negative pressure, similar to the situation of the bubble near a free surface [43] . The negative pressure is even lower than the saturated vapor pressure, as illustrated by the blue dotted line. We accounted for this negative pressure using the truncated model [44] , letting the saturated vapor pressure be the lower limit of the pressure value. The characteristics of the pressure wave at the end of the second period are similar to that at the end of the first period, and would not be repeated here.   Figure 8, does not increase with the increasing spatial scale of the bubble cluster. More bubbles in the circular cluster lead to stronger ghost reflections, which weaken the pressure wave induced by bubble o to a greater extent. It is worth mentioning that the pulsation pressure peak for three-bubble circular cluster (P = 42.5) is smaller than four-bubble cluster (P = 43.2). Thus, it is concluded that the peak of pulsation pressure is highest as the circular cluster is composed by four bubbles among three to ten bubbles. In this paper, the unified theory established in the work of Zhang et al. [36] is applied to model the dynamics of a circular bubble cluster and a bubble placed at the center. All bubbles are generated synchronously and own the same initial conditions. An experiment of two underwater explosion bubbles was compared with the theoretical results and great agreement was obtained. Under the influence of the circular bubble cluster, the period of the central bubble is obviously prolonged and the contraction is accelerated at the end of the collapse. At the measuring point directly below the central bubble, a high pulsating pressure peak is observed, which is much larger than that induced by a bubble in a free field. As the spatial scale of the bubble cluster increases, the size and pulsation period of the bubbles become larger, but the pulsation pressure induced by the central bubble is weakened due to the stronger ghost reflection, resulting in the decrease in the pressure peak. Among the circular clusters consisting of three to ten bubbles, the pulsation pressure peak of the central bubble is the highest for the cluster with four bubbles.