Comparison of the Rayleigh–Plesset and Gilmore Equations and Additional Aspects for the Modelling of Seismic Airgun Bubble Dynamics

Seismic airguns are commonly used in geophysical exploration. More recently, they are also being used as an alternative to underwater explosions for the shock testing of defence vessels. The study of the dynamics of the bubble produced by a seismic airgun is beneficial in understanding the resultant pressure field and shockwave. The Rayleigh–Plesset and Gilmore equations for modelling spherical bubble dynamics are compared for the expansion of an initially highly pressurised gas bubble. The relationship between initial gas pressure and both the first maximum bubble radius and the first period of oscillation are presented. The initial gas pressure is non-dimensionalised against hydrostatic pressure and studied over a range of 1 – 50. The separate contributions of presence of the airgun body, mass throttling, effective viscosity and heat diffusion to the first maximum radius and period are modelled and discussed. The effects of evaporation and condensation at the bubble wall are also considered. Introduction The Royal Australian Navy is currently investigating the feasibility and advantages of employing seismic airguns for shock testing naval craft. Shock testing with seismic airguns, rather than high explosives, is less expensive, safer, and more environmentally friendly. To perform shock testing effectively, an array of airguns must be used and the interactions between the bubbles can alter the pressure fields produced. Several methods exist for calculating the interactions between bubbles in an array [9, 14], but all rely on a basic understanding of the parameters affecting a single airgun bubble and the pressure field and shockwave produced. The Gilmore equation for bubble dynamics is commonly used as the underlying basis for seismic airgun bubbles and underwater explosions. Comparisons exist of this equation with other bubble models, including the well known Rayleigh–Plesset equation; however, they consider a bubble’s collapse from its maximum radius rather than expansion from its minimum radius. In modelling seismic airgun bubbles it is more practical to consider the initial bubble pressure and radius, rather than the conditions at the first maximum. The present work compares the Gilmore equation to the Rayleigh–Plesset equation to confirm the use of the Gilmore equation as the basic bubble model. Several contributions have been made to improve the numerical modelling of individual seismic airgun bubbles by considering additional factors to the basic bubble dynamics. Ziolkowski [17] used Gilmore’s equation and found a polytropic index of 1.13 gave good results for the first period of oscillation; this value was also obtained by Dragoset [2] for a range of gun sizes. Shulze–Gatterman [13] emphasised the effect of the actual airgun body on the period of oscillation. Safar [12] compared the equation of a bubble to an electrical circuit and developed a model for the rise time, amplitude of the initial pulse, and period of the airgun. Johnston [7] and Dragoset [2] considered the effect of the shuttle motion and choked flow rate on the chamber pressure, with Dragoset allowing for the actual port size. Ziolkowski [18] proposed that heat transfer occurs through the latent heat released by evaporation and condensation at the bubble wall. This concept is repeated by Langhammer and Landro [8]. Laws et al. [9] consider mass transfer due to evaporation and condensation, classical heat diffusion, flow throttling and an ‘effective viscosity’ induced by the turbulent nature of the bubble. It is claimed that this turbulent nature also has an amplifying effect on the heat transfer across the bubble wall. Li et al [10] includes the effect of mass throttling (but not choked flow) through ports, the airgun body, heat transfer and hydrostatic pressure changes as the bubble rises through the water. There appears to be no work that considers all of these parameters together and provides values for coefficients with a summary of the impact of each parameter on the bubble behaviour. The present work uses the Gilmore equation as the basic bubble model and considers the individual effects of the presence of the airgun body, mass throttling, effective viscosity, heat diffusion and condensation and evaporation, providing a summary of each contribution. Comparison of Rayleigh–Plesset and Gilmore Equations The Rayleigh–Plesset equation describes the motion of a spherical bubble in an incompressible liquid [3]. When considering bubble velocities of an appreciable order of magnitude compared with the speed of sound in water, compressibility of the liquid cannot be ignored. The Gilmore equation includes second-order compressibility terms, accounting for the loss of bubble energy due to the radiated pressure waves [5]. Both equations are commonly used to model bubble dynamics, with the Gilmore equation often used in underwater explosion applications. The Rayleigh–Plesset equation is given by: