Anisotropy in the sound field generated by a bubble chain

Abstract

A vertical chain of rising bubbles represents a transition from individual to continuum behaviour in a compressible gas–liquid flow. Experiments on the distribution of acoustic pressure around a bubble chain revealed a strong anisotropy in the acoustic field in the frequency band generated by individual bubbles. Sound appeared to propagate much more efficiently along the chain than normal to it. A simple theoretical model using a linear coupled-oscillator approximation was developed to explain this result. Although comparison with experiments is inherently qualitative, the model clearly demonstrated the anisotropy. The model also reproduced the change in pulse waveform along the chain. The results suggest that the enhancement of sound intensity along the chain can to some extent be explained by bubbles acting as resonant amplifiers re-transmitting vibrations.

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]

$$\text{ω}{}_0\text{=}\text{3γP}{}_0\text{ρR}{}_0{}^2\text{,}$$

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 $\text{6}\text{mm}$ diameter bubble, such as the ones studied here, has a natural frequency of about $\text{1}\text{kHz}$.

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 c_m of an ‘averaged’ medium drops to only $\text{25}\text{m}\text{s}{}^{\mathrm{-1}}$, 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 $\text{1000}\text{Hz}$, a bounded ‘averaged-medium’ cloud in a tank a quarter-metre high would have frequencies of its lower modes in the order of c_m/λ, which could be as low as $\text{100}\text{Hz}$. Thus, when there are a very large number of bubbles in a laboratory tank, an anisotropy in the sound field at frequencies of order $\text{100}\text{Hz}$ 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 $\text{1000}\text{Hz}$, were investigated. Experiments presented in Section 2 demonstrate an anisotropic sound distribution for a low number of bubbles in a chain, at frequencies around $\text{1000}\text{Hz}$. 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.