XXIII. The early stages of a submarine explosion

Optimization of an augmented Prosperetti-Lezzi bubble model

Three enhancements are introduced for predicting the violent collapse and rebound of a spherical bubble with the matched-asymptotic-expansion model of Prosperetti and Lezzi [(1986). J. Fluid Mech. 168, 457-478]. The first introduces spatial variation of the pressure field inside the bubble. It derives from the perturbation analysis of the interior Euler equations begun by Geers et al. [(2012). J. Appl. Phys. 112, 054910]. The second enhancement augments the Prosperetti and Lezzi equation with a term that accounts for the kinetic energy of the bubble gas, while the third provides an optimum value for the free variable appearing in that equation. The optimum value emerges from a comparison of peak pressures predicted by the augmented equation with corresponding results generated by finite-difference simulations based on transformed Euler equations for both the bubble gas and the surrounding liquid [Geers et al. (2012). J. Appl. Phys. 112, 054910]. The three enhancements considerably extend the range of applicability of a single-degree-of-freedom bubble model.

An integrated wave-effects model for an underwater explosion bubble

A model for a moderately deep underwater explosion bubble is developed that integrates the shock wave and oscillation phases of the motion. A hyperacoustic relationship is formulated that relates bubble volume acceleration to far-field pressure profile during the shock-wave phase, thereby providing initial conditions for the subsequent oscillation phase. For the latter, equations for bubble-surface response are derived that include wave effects in both the external liquid and the internal gas. The equations are then specialized to the case of a spherical bubble, and bubble-surface displacement histories are calculated for dilational and translational motion. Agreement between these histories and experimental data is found to be substantially better than that produced by previous models.

A model of sonoluminescence

A bubble of air, trapped at the centre of a spherical container of water on the surface of which spherical sound waves are maintained by transducers, may emit light, a phenomenon known as sonoluminescence. The surface of the bubble expands and contracts in obedience to the Rayleigh–Lamb equation, which requires knowledge of the gas pressure on the surface of the bubble. In many investigations of bubble pulsations, it is assumed that the air in the bubble moves adiabatically. To understand sonoluminescence, however, it is necessary to allow for the possibility that shocks are generated within the bubble. We couple the Rayleigh–Lamb equation governing the bubble radius to Euler’s equations governing the motion of air in the bubble, and solve the two equations simultaneously. The air is modelled by a van der Waals gas. Results are presented for a number of slightly different conditions of excitation, but in which the response of the system is widely different. If the frequency of the sound is high, the gas in the bubble moves adiabatically and no light is emitted. As the frequency is reduced (for the same ambient bubble radius and driving pressure), the incoming bubble surface acts as a piston that generates an ingoing shock wave that passes through the centre of the bubble, and then, when it strikes the bubble surface, halts and reverses its inward motion; a sequence of such inwardly and outwardly moving shocks occur. The shock waves generate such high temperatures that the air near the centre of the bubble is almost completely ionized, and emits light, which we attribute to bremmstrahlung. The light created by the second inwardly moving shock exceeds that created by the first but, as the frequency of the sound is further reduced, the energy from the first shock rises, and the overall luminosity of the bubble increases. When the sound frequency is further reduced, two shocks are launched successively by the inward moving bubble surface, the second colliding with the first after it has passed through the centre of symmetry but before it can collide with the bubble surface. Even higher temperatures are reached and the luminosity of the bubble continues to increase with decreasing frequency. Details of these solutions are presented, and estimates are made of the luminosity of the bubble in different conditions of excitation. Two other families of solutions are presented. In one, the frequency and ambient bubble radius are fixed and the driving amplitude is varied. In the other, the frequency and driving amplitude are fixed and the radius is varied. The effects of changing only the molecular weight of the trapped gas is also examined.