Oscillations of a gas-vapor bubble in an acoustic field

It is shown that at large vapor contents, as a result of the combined action of phase transitions and capillary effects, the small radially symmetric oscillations of gas-vapor bubbles in an acoustic field are unstable in amplitude. The critical vapor concentration in the bubble separating regions of qualitatively different bubble behavior in the acoustic field is determined. Expressions are obtained for the decay rate of the radial oscillations of the gas-vapor bubble and the growth rate characterizing the rate of increase of oscillation amplitude in the region of instability. It is shown that adding only a slight amount of gas to the vapor bubble leads to a marked decrease in the growth rate. It is found that in the particular case of a vapor bubble the tine growth rate characterizing the development of the instability is of the same order as the second resonance frequency of the vapor bubble. This may serve to explain why in the case of vapor bubble oscillations the second resonance effect, which has been established in a number of theoretical studies and is widely discussed in the literature, has not yet been experimentally confirmed. The problem of spherically symmetrical processes around gasvapor bubbles was posed in [1], and their small oscillations are investigated in detail in [2–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–33, May–June, 1986.The authors are grateful to R. I. Nigmatulin for useful discussions.     

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.     

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.     

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed. The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line. The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically. Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence.     

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.     

Finite amplitude oscillations of a hyperelastic spherical cavity

We present numerical results for the finite oscillations of a hyperelastic spherical cavity by employing the governing equations for finite amplitude oscillations of hyperelastic spherical shells and simplifying it for a spherical cavity in an infinite medium and then applying a fourth-order Runge-Kutta numerical technique to the resulting non-linear first-order differential equation.The results are plotted for Mooney-Rivlin type materials for free and forced oscillations under Heaviside type step loading. The results for Neo-Hookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the problem is also discussed in length.     

Explosive boiling of water in parallel micro-channels

The objective of this study is to visualize the flow pattern and to measure heat transfer coefficient during explosive boiling of water in parallel triangular micro-channels. Tests were performed in the range of inlet Reynolds number 25–60, mass flux 95–340 kg/m²s, and heat flux 80–330 kW/m².The flow visualization showed that the behavior of long vapor bubbles, occurring in a micro-channel at low Reynolds numbers, was not similar to annular flow with interposed intermitted slugs of liquid between two long vapor trains. This process may be regarded as explosive boiling with periodic wetting and dryout.In the presence of two-phase liquid–vapor flow in the micro-channel, there are pressure drop oscillations, which increase with increasing vapor quality.This study shows strong dependence of the heat transfer coefficient on the vapor quality. The time when liquid wets the heated surface decreases with increasing heat flux. Dryout occurs immediately after venting of the elongated bubble.     

Stereo-PIV measurements of vapor-induced flow modifications in confined jet impingement boiling

A single subcooled jet of water which undergoes boiling upon impingement on a discrete heat source is studied experimentally using time-resolved stereo particle image velocimetry (PIV). The impinging jet issues from a 3.75 mm diameter sharp-edged orifice in a confining orifice plate positioned 4 orifice diameters from the target surface. The behavior at jet Reynolds numbers of 5,000 and 15,000 is compared for a constant jet inlet subcooling of 10 °C. Fluorescent illumination allows for simultaneous imaging of both the flow tracers and the vapor bubbles in the flow. Flow structure, time-averaged velocities, and turbulence statistics are reported for the liquid regions within the confinement gap for a range of heat inputs at both Reynolds numbers, and the effect of the vapor generation on the flow is discussed. Vapor generation from boiling is found to modify the liquid velocities and turbulence fluctuations in the confinement gap. Flow in the confinement gap is dominated by vapor flow, and the vapor bubbles disrupt both the vertical impinging jet and horizontal wall jet flow. Moreover, vapor bubbles are a significant source of turbulence kinetic energy and dissipation, with the bubbly regions above the heated surface experiencing the most intense turbulence modification. Spectral analysis indicates that a Strouhal number of 0.023 is characteristic of the interaction between bubbles and turbulent liquid jets.     

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.     

Radial oscillations of soluble vapor-gas bubbles in a liquid

Soluble vapor-gas bubbles performing small radial oscillations in a liquid are considered. The heat- and mass-transfer processes and temperature and concentration inhomogeneities in the vaporgas mixture are taken into account. Expressions for the damping rate of radial oscillations of soluble vapor-gas bubbles are obtained. In [1–3] the dynamics of vapor-gas bubble oscillations were considered for a gas insoluble in liquid.     

Decay rates of oscillations and effective heat transfer coefficients for bubbles pulsating radially in a liquid

Vapor (gas) bubbles executing free radial oscillations in a liquid are considered. Expressions are obtained for the frequency and decay rate of small free oscillations of the bubbles. The effective coefficients of heat transfer between the radially pulsating bubbles and the liquid are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 80–87, November–December, 1980.     

An investigation on primary resonance phenomena of elastic medium based single walled carbon nanotubes

In this paper, the geometrically nonlinear free and forced oscillations of simply supported single walled carbon nanotubes (SWCNTs) are analytically investigated on the basis of the Euler–Bernoulli beam theory. The nonlinear frequencies of SWCNTs with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The numerical results reveal that the nonlinear free and forced vibration of nanotubes is effected significantly by both surrounding elastic medium and CNT aspect ratio.     

Two-phase flow instabilities in a horizontal single boiling channel

A study of the stability of an electrically heated single channel, forced convection horizontal system was conducted by using Freon-11 as the test fluid. Two major modes of oscillations, namely, density-wave type (high frequency) and pressure-drop type (low frequency) oscillations have been observed. The steady-state operating characteristics and stable and unstable regions are determined as a function of heat flux, exit orifice diameter and mass flow rate. Different modes of oscillations and their characteristics have been investigated. The effect of the exit restriction on the system stability has also been studied.A mathematical model has been developed to predict the transient behavior of boiling two-phase systems. The model is based on homogenous flow assumption and thermodynamic equilibrium between the liquid and vapor phases. The transient characteristics of boiling two-phase flow horizontal system are obtained for various heat inputs, flow rates and exit orifice diameters by perturbing the governing equations around a steady state. Theoretical and experimental results have been compared.     

Flow boiling heat transfer characteristics of R123 and R134a in a micro-channel

Flow boiling heat transfer in a single circular micro-channel of 0.19 mm ID has been experimentally investigated with R123 and R134a for various experimental conditions: heat fluxes (10, 15, 20 kW/m²), mass velocities (314, 392, 470 kg/m² s), vapor qualities (0.2–0.85) and different saturation pressures (158, 208 kPa for R123; 900, 1100 kPa for R134a). The heat transfer trends between R123 and R134a are clearly distinguished. Whether nucleate boiling is suppressed at low vapor quality or not determines the heat transfer trend and mechanism in the flow boiling of micro-channels. High convective heat transfer coefficients in the two-phase flow of micro-channels enables nucleate boiling to be suppressed even at low vapor quality, depending on the wall superheat requirement for nucleate boiling. In the case of early suppression of nucleate boiling, specifically R123, heat transfer is dominated by evaporation of thin liquid films around elongated bubbles. In the contrary case, namely R134a, nucleate boiling is dominant heat transfer mechanism until its suppression at high vapor quality and then two-phase forced convection heat transfer becomes dominant. It is similar to the heat transfer characteristic of macro-channels except the enhancement of nucleate boiling and the short forced convection region.     

Prediction of inception of thermal oscillations and their waveforms in flow of heated subcritical liquids

A theoretical model is formulated to determine the instability of flow of subcritical liquids supplied from a reservoir to warm pipelines. The model predicts the waveform of the oscillations to get clipped for certain lengths of pipeline and a range of forced convective heating. Clipped oscillations are obtained when the ratio of residence time of flow to the characteristic thermal time varies between about 1 and 10. The role of residence time in the inception of oscillations and in modifying the waveform is ascertained. The ability of the model to predict the boundaries for onset of oscillations and the region of the clipped and full waveforms is discussed.     

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.     

On the non-linear vibration of a thin ring with steady pressurization

Large amplitude, flexural oscillations of an inextensible, linearly elastic, pressurized ring are analyzed. Non-linear governing equations describing the planar motion of a thin rod curved in its undeformed state and subject to a distributed load applied normal to the neutral axis are developed using Hamilton's extended principle. The equations are specialized to study the behavior of a circular ring, and approximate solutions are obtained for a single mode response by a perturbation technique. Free, undamped oscillations and forced response of the ring near resonance are discussed. The influence of the magnitude of pressurization on the non-linear character of the motion is investigated.     

Nonstationary one-dimensional gas flows in channels in the presence of periodic perturbations of the boundary conditions

A study is made of the propagation in channels of forced oscillations generated by harmonic variation of the boundary conditions at the entrance and exit sections. Linear theory is used to find classes of boundary conditions and frequencies of the forced oscillations corresponding to the greatest gain or attenuation of high-frequency oscillations in a channel of variable section and of oscillations of arbitrary frequency in a channel of constant section. The resonance phenomenon that arises in channels when the frequencies of the forced oscillations and the fundamental oscillations are equal is studied. The wave process in a channel of variable section is investigated numerically, its characteristics found, and a comparison made with the linear theory. It is shown that the results of the calculations and the data of the linear analysis agree well.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–135, September–October, 1980.We thank A. B. Vatazhin for helpful discussions.     

Effect of acoustic cavitation on boiling heat transfer

Boiling heat transfer on a horizontal circular copper tube in an acoustical field is investigated experimentally and the relation between the liquid cavitation, the boiling and the micro bubble radii are analyzed theoretically. The results show that cavitation bubbles have an important influence on the nucleation, growth and collapse of vapor embryo within cavities on the heat transfer surface and that the enhancement of boiling heat transfer by acoustic cavitation mainly depends on whether the vapor embryo is activated by the cavitation bubbles to initiate boiling.     

Time-scale for critical growth of partial and supercavitation development over impulsively translating projectiles

A physical mechanism is proposed to explain an experimentally observed critical time scale that governs the partial cavity development over blunt free flying cylindrical projectiles. The projectiles were ejected using a modified gas-gun mechanism consisting of a barrel and explosive charge. Upon ignition, high-pressure gases forced a projectile down the launch barrel and into quiescent water. Results indicate that initial small cavities created at the projectile forebody are convected downstream where they subsequently grow towards the forebody, partially enveloping the projectile in a vapor cavity. The time at which the initially stable bubbles rapidly expand signifies that the partial cavity development process has begun. When this time is quantified and is non-dimensionalized appropriately, a time-scale for the critical growth (CGTS) for the cavitation is revealed. A plausible explanation of the partial cavity development process observed in these experiments is that the process is due to the interaction between small cavitation bubbles shed from the projectile forebody and the vortex ring generated by the impulsively started projectile. This interaction mediates the destabilization and spontaneous growth of small unstable bubbles resulting in the formation of partial cavitation over the projectile. An additional supercavitation formation mechanism was observed and is attributed to the launch mechanism. This process is not due to pure hydrodynamic cavitation, but rather an effect we term “gas-leakage” whereby the driving gases contaminated the aft flow field near the projectile and thus facilitated supercavitation to occur on a reduced time scale. This mechanism is equivalent to off-body ventilated supercavitation.