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Acoustic wave propagation in polydisperse bubbly liquids is relevant to diverse applications, such as ship propellers, underwater explosions, and biomedical applications. The simulation of bubbly liquids can date back to Foldy who presented a general theory. In the linear regime, two frequently used models for bubbly liquids are based on the continuum theory and on the multiple scattering theory. Under the homogenization-based assumption, models based on the volume-averaged equations or on the ensemble-averaged equations are designed to find the solutions of a given two-phase flow. The effective wave number is derived through the linearization of these equations. A second approach to the problem of linear wave propagation utilizes the multiple scattering theory. Bubbles are treated as point-like scatterers, and the total field at any location can be predicted by multiple scattering of scatterers. However, in most of experimental researches, the comparison between the approaches and the experimental results is not satisfactory for frequencies near the peak of phase speed and attenuation. In fact, the discrepancies between measurements and approaches are irregular, and the explanations of these discrepancies need further studying. We indicate that such a discrepancy should be attributed to an implicit assumption in these approaches:the bubbles are spatially uniform distribution and statistically independent of each other. In contrast, the complex bubble structures can be observed in many practical bubbly liquids which have important consequences for the acoustic wave propagation. In this paper, our intent is to model the effect of small bubble cluster on linear-wave propagation in bubbly liquids using the self-consistent method. The quasi-crystal approximation is applied to the self-consistent method, and the effective wave number is derived. According to the experimental results, the small clusters of bubbles often exist in bubbly liquids. Therefore, a three-dimensional random model, the Neyman-Scott point process, is proposed to simulate bubbly liquid with the cluster structure. Using this method, we study the influence of such a phenomenon on acoustic dispersion and attenuation relation. A formula for effective wavenumber in clustered bubbly liquid is derived. Compared with the results from the equation of Commander and Prosperetti[J. Acoust. Soc. Am. 85 732 (1989)], our results show that the clustering can suppress peaks in the attenuation and the phase velocity, each of which is a function of frequency. Further, we provide a numerical method. A clustered bubbly liquid is simulated with strict mathematical method and the statistical information is obtained through ratio-unbiased statistical approach. Using such a method, we quantificationally analyze the influence of estimated value on predictions.

With using the effective medium theory to describe acoustic scattering from bubble clouds, one of the underlying assumptions shows that the probability of an individual bubble located at some position in space is independent of the locations of other bubbles. However, bubbles within the clouds that naturally occur are usually influenced by the motion of the fluid, which makes them preferentially concentrated or clustered. According to Weber's method, it is a useful way of introducing the spatial correlation function to describe this phenomenon in bubble cloud. The spatial correlation function is involved in acoustic scattering and it is important to notice that the spatial correlation should be dependent on the position and radius of each bubble due to the “hole correction” or the effect of the dynamics of the fluid. Because of these reasons, it is hard to invert the spatial distribution of bubble clouds by using the spatial correlation function in acoustic scattering. A method is described here in which bubble clouds are separated into many small subareas and the conception, called effective spatial correlation function which is the statistic of spatial correlation function, is used to describe the correlation between subareas of bubble clouds. Since the effective spatial correlation function is independent of bubble radius and positions, the bubble clouddistribution and the trend of clustering can be inverted by using this function. The simulation indicates that the effective spatial correlation function can precisely trace the position of the clustering center, even the clustering center covered by other bubble clouds can be detected. With using the multi-bean sonar for measuring the bubbly ship wake generated by a small trial vessel, the method is used to invert the spatial distribution and clustering centers of bubble field in the ship wake. The results show that the effective spatial correlation function accurately inverts the distribution and clustering centers of bubbles in ship wake. Furthermore, the method presented in this paper could distinguish between the bubble clouds caused by different reasons and detect upper ocean bubble clouds covered by other bubbles generated by wave breaking as well.