Acoustic Model of the Remnant Bubble Cloud from Un- derwater Explosion

A model of formation, development, and acoustic properties of the bubble cloud resulting from an underwater explosion is presented. The model includes several parts: explosion globe dynamics, initial break-up of the explosion globe, turbulence created by the explosion globe fragmentation, and further break-up of the bubbles by the turbulence. The time history of the bubble cloud properties is calculated under the assumption of the cloud being a collection of non-interacting bubbles. Model results are compared with the available experimental data. INTRODUCTION It is of interest for some defence applications to understand the dynamics and to model the acoustic properties of the bubble cloud remnant of the underwater explosion. There are not many papers published on the subject. We are only aware of one (Holt&Culver, 2011), in which the remnant bubble cloud is investigated experimentally by acoustic method. To the best of our knowledge, there were no attempts to develop a theoretical model of the remnant bubble cloud formation and dynamics. The present document reports the development of such a model. We apply a physics-based approach and avoid complex and computationally demanding numerical simulations. Such an approach will undoubtedly require certain approximations and the introduction of some empirical parameters into the model. These will have to be adjusted from the comparison with the existing or future experimental data or high-fidelity numerical simulations. We, however, are not aware of any such accurate numerical simulations of this problem. Taking into account the complexity of the phenomenon, it is unlikely that such simulations could be performed in the near future. MODEL OUTLINE As a result of an underwater explosion (UNDEX), a gas globe is formed, which experiences several expansions and contractions during short time after the detonation. We will use the term "globe" for the initial gas bubble formed by the underwater explosion to distinguish it from the bubbles as product of the initial globe disintegration. During these oscillations the explosion globe rises in water, especially quickly during contraction phases when the drag force is lower. After two to three oscillations the large explosion bubble disintegrates into a large number of smaller bubbles, which then rise towards the surface of water with various speeds depending on their size. The purpose of this model is to estimate the bubble size distribution, the bubble spatial distribution and the dynamics of their rise. The acoustic properties of the remnant bubble cloud can be easily derived from the known bubble size distribution. The steps of the currently suggested model are summarised below. The references to specific model elements will be given in the corresponding sections of the paper. The oscillations and rise of the explosion globe are based on the models available in the literature, with slight modifications, of spherical bubble shape dynamics and motion. There are more sophisticated numerical models describing the UNDEX globe oscillation and rise, but for the purpose of this research the current, computationally efficient approach will suffice. It is assumed that at the third minimum of the UNDEX globe oscillation, it disintegrates into smaller bubbles. The size distribution of the fragments is estimated from the growth rate of the modes of the Rayleigh-Taylor instability for spherical cavity. It is assumed that initial pressure of gas in the fragments is the same as in the explosion globe just before the break-up. Then the bubbles expand to bring the internal gas pressure into the balance with the ambient pressure. The resulting bubble cloud size is estimated as being proportional to the total gas volume. The coefficient of proportionality is treated as an empirical parameter in the current model. The explosion globe potential energy at the minimum of its radius goes into the kinetic energy of the turbulent spot in the fluid. The time-space distribution of the turbulent kinetic energy is obtained by solving the corresponding equation in approximation of spherical symmetry. The bubbles resulting from the initial fragmentation of the explosion globe are further broken-up by turbulence. To describe this, one of the bubble break-up models is used. It should be noted here that all of the bubble break-up models by turbulence are developed in the assumption of small gas volume fraction and isotropic nature of turbulence. In this problem, the gas volume fraction is high and the turbulence is not isotropic. However, in the absence of a better model, one of the existing models is applied with a note of addressing this problem in future. The unsteady equation of the bubble population balance is solved using the time-space distribution of the turbulence obtained in the previous step. The rise of the bubbles to the surface is then calculated from the balance of the buoyancy and drag forces. The spatial and size distribution of the gas volume fraction is then used to estimate the acoustical properties of the gas bubble cloud.