Bubble dynamics: a review and some recent results

Several aspects of small-amplitude oscillations of bubbles containing gas, vapor, or a gas-vapor mixture are discussed. An application to pressure-wave propagation in a bubbly liquid is described. Nonlinear forced oscillations are considered in the light of recent research on forced oscillations of nonlinear systems. The growth of vapor bubbles, an extension of the Rayleigh-Plesset equation to non-Newtonian liquids and appreciable mass transfer at the interface, and a boundary integral numerical method for nonspherical cavitation bubble dynamics are also briefly discussed.     

Dynamics of soluble gas bubbles

The problem of the mass, thermal and dynamic interaction between a bubble containing a soluble gas and a liquid is considered. It is shown that this problem can be reduced to the problem of the behavior of a vapor bubble with phase transitions investigated in detail in [1–3]. Expressions are obtained for the rate of decay of the radially symmetric oscillations of the bubbles due to the solubility of the gas in the liquid. The effective coefficients of mass transfer between the radially pulsating bubbles and the liquid are determined. A numerical solution is obtained for the problem of the radial motion of a bubble created by a sudden change of pressure in the liquid which, in particular, corresponds to the behavior of the bubbles behind the shock front when a shock wave enters a bubble screen.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 52–59, November–December, 1985.     

Radial oscillations of vapor bubbles in solutions

The dynamics and heat and mass transfer of vapor bubbles in binary systems are investigated. An anomalous effect of the component composition on the bubble dynamics in boiling nonideal solutions is established. It is shown that in some binary systems the value of the logarithmic decay rate for small free radial oscillations does not lie within the limiting values calculated for the pure components, which is associated with the cardinal importance of the effect of diffusion in the liquid phase on the intensity of the phase transitions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1989.     

Modeling and computation of cavitation in vortical flow

The paper presents an investigation of Euler–Lagrangian methods for cavitating two-phase flows. The Euler–Euler methods, widely used for simulations of cavitating flows in ship technology, perform well in regions of moderate flow changes but fail in zones of strong, vortical flow. Reasons are the strong approximations of cavitation models in the Euler concept. Alternatively, Euler–Lagrangian concepts enable more detailed formulations for transport, dynamics and acoustic of discrete vapor bubbles. Test calculations are performed to study the influence of different parameters in the equations of motion and in the Rayleigh–Plesset equation for bubble dynamics. Results confirm that only Lagrangian models are able to describe correctly the bubble behavior in vortices, while Eulerian results deviate strongly. Lagrangian formulations enable additionally the determination of acoustic pressure of cavitation noise. Two-way coupling between the phases is required for large regions of the vapor phase. A new coupling concept between continuous fluid flow and discrete bubble phase is developed and demonstrated for flow through a nozzle. However, the iterative coupling between the phases via volume fractions is computationally expensive and should therefore be applied only in regions where Eulerian treatment fails. A corresponding local concept for combination with an Euler–Euler method is outlined and is in progress.     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.     

On the dynamics and instability of bubbles formed during underwater explosions

Dynamics of explosion bubbles formed during underwater detonations are studied experimentally by exploding fuel (hydrogen and/or carbon monoxide)–oxygen mixture in a laboratory water tank. Sub-scale explosions are instrumented to provide detailed histories of bubble shape and pressure. Using geometric and dynamic scaling analyses it has been shown that these sub-scale bubbles are reasonable approximations of bubbles formed during deep sea underwater explosions. The explosion bubble undergoes pulsation and loses energy in each oscillation cycle. The observed energy loss, which cannot be fully explained by acoustic losses, is shown here to be partly due to the excitation of instability at the interface between the gaseous bubble and the surrounding water. Various possible mechanisms for the dissipation of bubble energy are addressed. The analysis of the experimental data gives quantitative evidence (confirmed by recent numerical studies) that the Rayleigh–Taylor instability is excited near the bubble minimum. The dynamics of the bubble oscillation observed in these experiments are in good agreement with experimental data obtained from deep sea explosions     

Radiation and scattering of sound from a vortex ring

The mechanisms of generation and scattering of sound by a vortex ring are investigated on the basis of fluid dynamics. The vortex ring can serve as a simple dynamic model of the large-scale structures observed in shear flows. Moreover, it is probably the most easily studied vortex element that can be created experimentally. The sound scattering investigation also served to determine the extent to which the vortex is affected by sound, its selectivity with respect to such parameters as the acoustic frequency, the angle of incidence of the wave, etc. The perturbed motion is considered against the background of the steady-state motion of the ring. The perturbed motion in the vortex core is determined on the basis of linear incompressible fluid dynamics. Two terms of the expansion in the M number of the far acoustic field generated by the perturbations in the core are found in accordance with Lighthill's theory. The acoustic power and directivity of the radiation and the acoustic instability growth rate are calculated. It is shown that the scattering of sound by the vortex ring is a resonance effect, and the scattering amplitude near resonance is determined. The acoustic action on the hydrodynamic structure of the flow in the core of the ring is especially intense near the resonances and extends over a period short as compared with the characteristic time of the acoustic instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 83–95, May–June, 1987.     

Pressure Waves in a Gas–Liquid Medium with a Stratified Liquid–Bubbly Mixture Structure

Evolution and decay of pressure waves of moderate amplitude in a vertical shock tube filled by a gas–liquid medium with a nonuniform (stepwise) distribution of bubbles over the tube cross section are studied experimentally. The gas–liquid layer has the form of a ring located near the tube wall or the form of a gas–liquid column located in the center of the tube. It is shown that the nonuniformity of bubble distribution over the tube cross section increases the attenuation rate of pressure waves.     

Numerical simulations of the primary Bjerknes force experienced by bubbles in a standing ultrasonic field: Nonlinear vs. linear

The primary Bjerknes force experienced by a population of multiple bubbles in a liquid set in a nonlinear ultrasonic standing field and their translation are calculated and analyzed by numerical simulations. The force field is evaluated by considering the nonlinear bubble oscillations as well as the nonlinear character of the ultrasonic pressure field (both variables are unknown in the coupled nonlinear differential system). The results at small amplitudes agree with the classical theory on bubble translation, depending on the driving frequency in relation to the bubble resonance. It is shown that, when amplitudes are raised, the force field exhibits important modifications that strongly affect the motion of the bubbles and the way they form agglomerates. An analysis is performed on the importance of the terms in the differential system that provoke (a) the nonlinearity of the bubble oscillations and (b) the nonlinearity of the acoustic wave. This study reveals that both features should be considered to better approximate the primary Bjerknes force field. Simulations of the nonlinear ultrasonic field after the bubbles form agglomerates under the influence of this force are also performed.     

Current topics in the dynamics of gas and vapor bubbles

Summary A review of topics of current interest in the dynamics of gas and vapor bubbles in liquids under conditions of no relative motion is presented. The aspects of gas bubble dynamics considered include: radiation pressure (Bjerknes) forces, analytical and numerical studies of transient forced oscillations, connection of the subharmonic resonance with the onset of acoustic cavitation, thermal behaviour of the gas in small-amplitude and large-amplitude oscillations. For vapour bubbles a detailed discussion of the growth process is given. A scaled formulation of this process is also presented and phenomena connected to the forced oscillations are discussed. Some aspects of the dynamics of vapor bubbles in «cold» liquids (i.e. cavitation bubbles) are also included. Finally several phenomena encountered in the dynamics of non-spherical bubbles are reviewed, namely the effect of viscosity on the stability of the spherical shape, a variational formulation, and phase-change effects.

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Sommario Si presenta una rassegna di argomenti attualmente oggetto di ricerca nella dinamica di bolle di gas e di vapore nei liquidi. La discussione è limitata al caso in cui non esista moto di traslazione relativo fra le fasi. Gli aspetti considerati della dinamica di bolle di gas includono: forze di pressione di radiazione (o di Bjerknes), studi analitici e numerici dei transitori nelle oscillazioni forzate, rapporto fra la risonanza subarmonica e l'apparizione di fenomeni di cavitazione acustica, comportamento termico del gas contenuto nella bolla durante oscillazioni di piccola ampiezza e di grande ampiezza. Per le bolle di vapore si presenta una discussione dettagliata del processo di crescita, una formulazione di similarità di tale processo, e alcuni fenomeni connessi alle oscillazioni forzate. Vengono inoltre discussi alcuni aspetti della dinamica di bolle di vapore in liquidi «freddi» (cioè bolle di cavitazione). Infine si considerano alcune questioni connesse alla dinamica di bolle non sferiche, e cioè l'effetto della viscosità sulla stabilità della forma sferica, una formulazione variazionale, e fenomeni relativi al cambiamento di fase.

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This paper is based in part on an invited review lecture given at Euromech Colloquium 98 «Mechanics and Physics of Gas Bubbles in Fluids» held at Eindhoven, 22 – 24 November 1977. The study reported herein has been supported in part by Gruppo Nazionale di Fisica Matematica of C.N.R.     

Ellipsoidal Oscillations of a Gas Bubble with Periodic Variation of the Surrounding Fluid Pressure

Ellipsoidal linear and nonlinear oscillations of a gas bubble under harmonic variation of the surrounding fluid pressure are studied. The system is considered under conditions in which periodic sonoluminescence of the individual bubble in a standing acoustic wave is observable. A mathematical model of the bubble dynamics is suggested; in this model, the variation of the gas/fluid interface shape is described correct to the square of the amplitude of the deformation of the spherical shape of the bubble. The character of the air bubble oscillations in water is investigated in relation to the initial bubble radius and the fluid pressure variation amplitude. It is shown that nonspherical oscillations of limited amplitude can occur outside the range of linearly stable spherical oscillations. In this case, both oscillations with a period equal to one or two periods of the fluid pressure variation and aperiodic oscillations can be observed.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

Effect of homogeneous condensation on the dynamics of a hot vapor bubble in a cold liquid jet

The problem of initiating cavitation bubbles in a cold liquid jet by injecting hot steam into high-pressure zone specially organized at the nozzle outlet is considered. Previously, in [1], a plane flowfield in which vapor bubbles were formed at the cusp of the cavity (high-pressure zone) and propagated together with the liquid along the axis of symmetry was considered. In certain cases, in the bubble expansion process the vapor temperature drops below the saturation temperature. In the present paper, vapor condensation in the bubble volume (homogeneous condensation) is also taken into account.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 56–61, November–December, 1996.     

Forces exerted by a liquid on a vapor bubble growing within IT

Using the basic equations of hydromechanics and also the Lagrange equations of the second kind, expressions are derived for the force acting between a liquid and a vapor bubble growing within it. Cases studied include those of the growth of a bubble on a thin filament or plane surface in an ideal liquid and a liquid of low viscosity. The sign of the hydrodynamic forces depends on the particular law of growth of the vapor bubble.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–170, May–June, 1973.     

Dynamics of vapor-gas bubbles

The nonlinear problem of thermal, mass, and dynamic interaction between a vapor-gas bubble and a liquid is considered. The results of numerical solution of the problem of radial motion of the bubble caused by a sudden pressure change in the liquid, illustrating the behavior of vapor-gas bubbles in compression and rarefaction waves, are presented. The corresponding problem for single-component gas and vapor bubbles was considered in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 56–61, November–December. 1976.     

Acoustic cavitation in cryogenic and boiling liquids

Vapour bubble dynamics in cryogenic and boiling liquids affected by an acoustic field is considered. Linear pulsations and nonstationary growth of vapour bubbles in time due to linear effects of rectified heat and mass transfer are studied. The growth thresholds of vapour bubbles depending on thermodynamic parameters of liquid, static overcompression, and acoustic field frequency are presented. Essential influence of resonance properties of bubbles on the values of growth thresholds is shown. The results for different cryogenic liquids and boiling water are given.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Forced oscillations of two gas bubbles in a fluid in the vicinity of bubble contact

For a theoretical derivation of bubble coalescence conditions, nonlinear forced oscillations of two closely spaced spherical bubbles subjected to the action of a periodic external pressure field are considered. The equations, asymptotic with respect to a small distance between the bubble surfaces, are derived to describe the approach of the bubbles under the action of (i) the Bjerknes attraction force averaged over the oscillation period and (ii) the viscous drag. It is shown that due to nonlinear interaction of the viscous drag with the radial and translational oscillations of the bubbles a unidirectional repulsive force is generated, which prevents the approach of the bubbles. The coalescence of the bubbles is possible when the nondimensional parameter combined from the amplitude and frequency of the external pressure field, the bubble radius, and the fluid viscosity is greater than a certain critical value. The obtained coalescence condition is qualitatively confirmed by experiments.     

Effect of acoustic oscillations on the stability of a plane jet

The problem of the effect of acoustic oscillations on the stability of a compressible ideal-fluid jet flow is examined in the case of a plane jet with standing acoustic waves superimposed across it. The method of dividing the motion into fast and slow with allowance for nonlinear acoustic effects is employed. The acoustic oscillations are found to affect the growth rate of unstable hydrodynamic disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 54–60, July–August, 1991.