Andrea Prosperetti

Linear pressure waves in bubbly liquids: Comparison between theory and experiments

Recent work has rendered possible the formulation of a rigorous model for the propagation of pressure waves in bubbly liquids. The derivation of this model is reviewed heuristically, and the predictions for the small‐amplitude case are compared with the data sets of several investigators. The data concern the phase speed, attenuation, and transmission coefficient through a layer of bubbly liquid. It is found that the model works very well up to volume fractions of 1%–2% provided that bubble resonances play a negligible role. Such is the case in a mixture of many bubble sizes or, when only one or a few sizes are present, away from the resonant frequency regions for these sizes. In the presence of resonance effects, the accuracy of the model is severely impaired. Possible reasons for the failure of the model in this case are discussed.

Computational methods for multiphase flow

Preface 1. Introduction: a computational approach to multiphase flow A. Prosperetti and G. Tryggvason 2. Direct numerical simulations of finite Reynolds number flows G. Tryggvason and S. Balachandar 3. Immersed boundary methods for fluid interfaces G. Tryggvason, M. Sussman and M. Y. Hussaini 4. Structured grid methods for solid particles S. Balachandar 5. Finite element methods for particulate flows H. Hu 6. Lattice Boltzmann methods for multiphase flows S. Chen, X. He and L. S. Luo 7. Boundary integral methods for Stokes flows J. Blawzdziewic 8. Averaged equations for multiphase flows A. Prosperetti 9. Point particle methods for disperse flows K. Squires 10. Segregated methods for two-fluid models A. Prosperetti, S. Sundaresan, S. Pannala and D. Z. Zhang 11. Coupled methods for multi-fluid models A. Prosperetti References Index.

Nonlinear bubble dynamics

The standard approach to the analysis of the pulsations of a driven gas bubble is to assume that the pressure within the bubble follows a polytropic relation of the form p=p0(R0/R)3κ, where p is the pressure within the bubble, R is the radius, κ is the polytropic exponent, and the subscript zero indicates equilibrium values. For nonlinear oscillations of the gas bubble, however, this approximation has several limitations and needs to be reconsidered. A new formulation of the dynamics of bubble oscillations is presented in which the internal pressure is obtained numerically and the polytropic approximation is no longer required. Several comparisons are given of the two formulations, which describe in some detail the limitations of the polytropic approximation.

Dynamics of bubble growth and detachment from a needle

Several aspects of the growth and departure of bubbles from a submerged needle are considered. A simple model shows the existence of two different growth regimes according to whether the gas flow rate into the bubble is smaller or greater than a critical value. These conclusions are refined by means of a boundary-integral potential-flow calculation that gives results in remarkable agreement with experiment. It is shown that bubbles growing in a liquid flowing parallel to the needle may detach with a considerably smaller radius than in a quiescent liquid. The study also demonstrates the critical role played by the gas flow resistance in the needle. A considerable control on the rate and size of bubble production can be achieved by a careful consideration of this parameter. The effect is particularly noticeable in the case of small bubbles, which are the most difficult ones to produce in practice.

Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids

A linearized theory of the forced radial oscillations of a gas bubble in a liquid is presented. Particular attention is devoted to the thermal effects. It is shown that both the effective polytropic exponent and the thermal damping constant are strongly dependent on the driving frequency. This dependence is illustrated with the aid of graphs and numerical tables which are applicable to any noncondensing gas–liquid combination. The particular case of an air bubble in water is also considered in detail.

Bubble entrainment by the impact of drops on liquid surfaces

The impact of a drop on the plane surface of the same liquid is studied numerically. The accuracy of the calculation is substantiated by its good agreement with available experimental data. An attempt is made to explain the recent observation that, in a restricted range of drop radii and impact velocities, small air bubbles remain entrained in the liquid. The implications of this process for the underwater sound due to rain are considered. The numerical approach consists of a new formulation of the boundary-element method which is explained in detail. Techniques to stabilize the calculation in the presence of strong surface-tension effects are also described.

Averaged equations for inviscid disperse two-phase flow

Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Walliss ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10. As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.

The thermal behaviour of oscillating gas bubbles

Several aspects of the oscillations of a gas bubble in a slightly compressible liquid are discussed by means of a simplified model based on the assumption of a spatially uniform internal pressure. The first topic considered is the linear initial-value problem for which memory effects and the approach to steady state are analysed. Large-amplitude oscillations are studied next in the limit of large and small thermal diffusion lengths obtaining, in the first case, an explicit expression for the internal pressure, and, in the second one, an integral equation of the Volterra type. The validity of the assumption of uniform pressure is then studied analytically and numerically. Finally, the single-bubble model is combined with a simple averaged-equation model of a bubbly liquid and the propagation of linear and weakly nonlinear pressure waves in such a medium is considered.

Bubble phenomena in sound fields: part one

This paper presents a mainly theoretical review of the physical aspects of the behaviour of bubbles in sound fields. Firstly, an equation for the radial motion, including the effects of liquid compressibility, is critically presented. The equilibrium radius and its stability are then considered, followed by a presentation of results concerning the small-amplitude radial oscillations of gas and vapour bubbles.

A theoretical study of sonoluminescence

The production of OH radicals by dissociation of water vapor in oscillating argon bubbles is studied theoretically to examine a possible mechanism for the emission of the 310‐nm line observed in sonoluminescence experiments. Accurate models are used for the calculation of the temperature field in the gas and for the description of the associated chemical kinetics. Heat transfer between the bubble and the liquid is found to play a dominant role in the process. At the low excitation amplitudes considered, the bubble radius is also an important parameter.