Anisotropy in the sound field generated by a bubble chain
Introduction
Sounds emitted by bubbles have been studied quantitatively since the time of Rayleigh [1]. A good review of bubble acoustics was made by Leighton [2]. Bubbles produce an acoustic signal owing to compression of the gas in the bubble. The ‘spring’ of the compressible gas and the mass of liquid around the bubble create a natural oscillator, sending a pressure fluctuation through the liquid. For a bubble detaching from a parent body of gas, the initial compression or rarefaction of the gas may be caused by the recoil of the bubble neck on formation, or by deformation of the bubble during its motion, although the mechanism of either process is still the subject of research [3], [4]. Under adiabatic conditions, the natural frequency of a single, millimetre-sized, linearly oscillating bubble is given by [5]where ω0 is the radian frequency, γ is the ratio of specific heats, P0 is the absolute liquid pressure, ρ is the liquid density and R0 is the equivalent spherical radius of the bubble. The Minnaert equation (1) is a simplified solution to the general equation for spherical bubble oscillations usually called the Rayleigh–Plesset equation [2]. A diameter bubble, such as the ones studied here, has a natural frequency of about .
The acoustic behaviour of pairs of bubbles, mostly under the influence of an applied sound field, has been the subject of significant research (e.g. [6], [7], [8], [9], [10], [11]). Much of this work is directed to predicting the relative motion of pairs of acoustically driven bubbles.
The acoustics of clouds or ‘swarms’ of bubbles, where the number of bubbles is large enough for the ensemble to be treated by a continuum approximation, has likewise been extensively researched (e.g. [12], [13], [14], [15], [16]).
The fluid dynamics of a single chain of rising bubbles has been studied by several authors (e.g. [17], [18]), without considering its acoustics. As the number of bubbles in the chain becomes large, and certainly once multiple chains are introduced side-by-side, the system becomes more like a ‘cloud’ than a ‘chain’. In the limit of large numbers of bubbles, a continuum acoustic theory (e.g. Ref. [15]) should become valid.
Considering briefly the continuum limit, a small quantity of gas has a dramatic influence on the ‘averaged’ speed of sound of the medium. It is easy to show that when the void fraction reaches 50%, the speed of sound cm of an ‘averaged’ medium drops to only , lower than the speed of sound in air. A laboratory tank inevitably places top and bottom boundaries on a bubble cloud, permitting wave modes of the entire cloud with wavelengths λ on the order of the tank height. Although individual millimetre-sized bubbles have acoustic frequencies in the order of , a bounded ‘averaged-medium’ cloud in a tank a quarter-metre high would have frequencies of its lower modes in the order of cm/λ, which could be as low as . Thus, when there are a very large number of bubbles in a laboratory tank, an anisotropy in the sound field at frequencies of order should be expected. Such an anisotropy would correspond to the mode structure of the bounded cloud and was clearly demonstrated by Nicholas et al. [15]. However, the sound distribution around a chain of a finite number of bubbles has not been studied.
The acoustic behaviour of a bubble chain is of particular interest for feedback measurements of industrial aerators [19], [20], [21] and for the emission of sound by submarines [22].
This paper investigates a transition from individual bubble-acoustic behaviour to continuum behaviour, using a single chain of bubbles as the paradigm. Frequencies in the bandwidth of the individual bubble resonances, of order , were investigated. Experiments presented in Section 2 demonstrate an anisotropic sound distribution for a low number of bubbles in a chain, at frequencies around . In 3 Coupled-oscillator theory, 4 Numerical method, settings and issues a simple coupled-oscillator model is developed to try to explain the data from the experiment. In Section 5 the model and experiment are compared and contrasted.
Section snippets
Experiments
The experimental set-up is shown in Fig. 1. A chain of air bubbles was produced in water by a highly repeatable system similar to that of Manasseh et al. [4]. The number of bubbles in the chain was varied systematically. Further details are in Ref. [23]. The nozzle had a internal orifice diameter; it was supplied with air via a precision pressure regulator (CompAir Maxam type A216) at about pressure. Small variations of about were sufficient to establish steady and
Formulation
The bubbles are assumed to lie on the z-axis of an axisymmetric cylindrical co-ordinate system (r,z) with z pointing downwards from the free surface. This assumption creates a fundamental inconsistency in co-ordinate systems, since the bubbles are assumed to be spherically symmetric (monopole) oscillators, while generating motions in the cylindrical frame. Provided the bubbles do not get too close, the inconsistency can be tolerated. As a consequence, this coupled-oscillator model, like some
Numerical method, settings and issues
The experimental signals show interesting phenomena inherently observed in the time domain, such as the modulation of the pulse produced on bubble formation, its decay, and the variation in pulse waveform at different heights up the chain. Thus, it is of interest to make a comparison with the model's predictions in the time domain. A set of N eigenmodes from a linear solution to (4) would not be useful, since the initial condition of each mode would have to be known.
In reality, the physical
Comparison of model and experiment
Fig. 9, Fig. 10 show the coupled-oscillator model's prediction of the pressure that would be measured by a hydrophone placed at the bottom and top of the chain, respectively. The model measured pressure pm(t) is the sum of the pressures pi(t) in the liquid at each bubble wall, reduced according to the monopole assumption,where (r,z) is the measurement point of interest. Fig. 9, Fig. 10 should be compared with Fig. 4, Fig. 6, respectively, for the experiment. Given the
Conclusion
Experiments on the distribution of acoustic pressure around a bubble chain revealed a strong anisotropy in the acoustic field. Sound appeared to propagate much more efficiently along the chain than normal to it. The phenomenon was present in the resonant frequency band of individual bubbles. This may have relevance to the distribution of underwater sound generated by bubbles streaming from a vessel's hull or caught in its wake. It may also be relevant to the acoustic measurement and control of
Acknowledgements
We are grateful to Li Chen at the Australian Defence Science and Technology Organisation (DSTO) and John Davy, Anh Bui, Ian Shepherd and Frank LaFontaine at CSIRO for discussions, and to Lachlan Graham, Tony Swallow and Anthony Antzakas at CSIRO for help with the experiments. AN is supported by The University of Melbourne through MIRS and MIFRS grants (00-1456) and by DSTO/CSIRO through the SPS grant.
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CSIRO: Commonwealth Scientific and Industrial Research Organisation.