Bubbles in liquids with phase transition—part 2: on balance laws for mixture theories of disperse vapor bubbles in liquid with phase change

We study averaging methods for the derivation of mixture equations for disperse vapor bubbles in liquids. The carrier liquid is modeled as a continuum, whereas simplified assumptions are made for the disperse bubble phase. An approach due to Petrov and Voinov is extended to derive mixture equations for the case that there is a phase transition between the carrier liquid and the vapor bubbles in water. We end up with a system of balance laws for a multi-phase mixture, which is completely in divergence form. Additional non-differential source terms describe the exchange of mass, momentum and energy between the phases. The sources depend explicitly on evolution laws for the total mass, the radius and the temperature of single bubbles. These evolution laws are derived in a prior article (Dreyer et al. in Cont Mech Thermodyn. doi:10.1007/s00161-0225-6, 2011) and are used to close the system. Finally, numerical examples are presented.     

On the modeling and simulation of a laser‐induced cavitation bubble

Single cavitation bubbles exhibit severe modeling and simulation difficulties. This is due to the small scales of time and space as well as due to the involvement of different phenomena in the dynamics of the bubble. For example, the compressibility, phase transition, and the existence of a noncondensable gas inside the bubble have strong effects on the dynamics of the bubble. Moreover, the collapse of the bubble involves the occurrence of critical conditions for the pressure and temperature. This adds extra difficulties to the choice of equations of state. Even though several models and simulations have been used to study the dynamics of the cavitation bubbles, many details are still not clearly accounted for. Here, we present a numerical investigation for the collapse and rebound of a laser‐induced cavitation bubble in liquid water. The compressibility of the liquid and vapor are involved. In addition, great focus is devoted to study the effects of phase transition and the existence of a noncondensable gas on the dynamics of the collapsing bubble. If the bubble contains vapor only, we use the six‐equation model for two‐phase flows that was modified in our previous work [A. Zein, M. Hantke, and G. Warnecke, J. Comput. Phys., 229(8):2964‐2998, 2010]. This model is an extension to the six‐equation model with a single velocity of Kapila et al. (Phys. Fluid, 13:3002‐3024, 2001) taking into account the heat and mass transfer. To study the effect of a noncondensable gas inside the bubble, we add a third phase to the original model. In this case, the phase transition is considered only at interfaces that separate the liquid and its vapor. The stiffened gas equations of state are used as closure relations. We use our own method to determine the parameters to obtain reasonable equations of state for a wide range of temperatures and make them suitable for the phase transition effects. We compare our results with experimental ones. Also our results confirm some expected physical phenomena. Copyright © 2013 John Wiley & Sons, Ltd.     

Propagation of nonlinear waves in a gas-liquid medium. Exact and approximate analytical solutions of wave equations

A number of exact and approximate analytical solutions of the equations for one-dimensional and weakly non-one-dimensional waves propagating in a liquid with gas bubbles are presented for the case where the bubble distribution density is a continuous function of the bubble radius and spatial coordinates.     

Evolution of finite perturbations in a viscoelastic relaxing liquid with gas bubbles

The propagation of one-dimensional perturbations in a viscoelastic relaxing liquid containing gas bubbles is investigated within the framework of the homogeneous model of the medium when the wavelength of the perturbation is much larger than the distance between the bubbles and the bubble radius. The evolution of stationary and nonstationary waves is investigated analytically and with the use of numerical integration; shock waves are also investigated. The results are compared with the behavior of perturbation waves in a Newtonian liquid with gaseous inclusions. The models of the gas-liquid medium [1, 2] are generalized to the case when the liquid phase is a viscoelastic liquid, for example, a weak aqueous solution of polymers. The propagation of longwave perturbations of finite amplitude in such a mixture is investigated using the technique developed in [3].     

可压缩流场中气泡脉动数值模拟

在应用边界元方法对气泡动力学的研究中, 绝大多数模型是建立在不压缩势流理论基础之上, 针对可压缩流场中气泡运动特性的研究很少. 从波动方程出发, 分别在气泡运动前期和后期对波动方程进行简化, 得到气泡运动局部和全局简化方程, 采用双渐进方法对简化方程进行匹配, 提出了考虑流场可压缩性的非球状气泡运动模型. 该模型的计算结果与Prospertti 等的解析结果吻合很好, 气泡脉动最大半径和内部最大压力随气泡脉动逐渐减小. 基于该模型对比了自由场中药包爆炸考虑可压缩性与不考虑可压缩性的计算结果, 发现考虑可压缩性气泡射流速度较小, 随后基于该模型计算了刚性边界下气泡的运动特性.     

Numerical simulation of shock propagation in a polydisperse bubbly liquid

The effect of distributed bubble nuclei sizes on shock propagation in a bubbly liquid is numerically investigated. An ensemble-averaged technique is employed to derive the statistically averaged conservation laws for polydisperse bubbly flows. A finite-volume method is developed to solve the continuum bubbly flow equations coupled to a single-bubble-dynamic equation that incorporates the effects of heat transfer, liquid viscosity and compressibility. The one-dimensional shock computations reveal that the distribution of equilibrium bubble sizes leads to an apparent damping of the averaged shock dynamics due to phase cancellations in oscillations of the different-sized bubbles. If the distribution is sufficiently broad, the phase cancellation effect can dominate over the single-bubble-dynamic dissipation and the averaged shock profile is smoothed out.     

The interaction of background ocean air bubbles with a surface ship

A two-fluid model suitable for the calculation of the two-phase flow field around a naval surface ship is presented. This model couples the Reynolds-averaged Navier–Stokes (RANS) equations with equations for the evolution of the gas-phase momentum, volume fraction and bubble number density, thereby allowing the multidimensional calculation of the two-phase flow for monodisperse variable size bubbles. The bubble field modifies the liquid solution through changes in the liquid mass and momentum conservation equations. The model is applied to the case of the scavenging of wind-induced sea-background bubbles by an unpropelled US Navy frigate under non-zero Froude number boundary conditions at the free surface. This is an important test case, because it can be simulated experimentally with a model-scale ship in a towing tank. A significant modification of the background bubble field is predicted in the wake of the ship, where bubble depletion occurs along with a reduction in the bubble size due to dissolution. This effect is due to lateral phase distribution phenomena and the generation of an upwelling plume in the near wake that brings smaller bubbles up to the surface. © 1998 John Wiley & Sons, Ltd.     

自由场中液氮单空泡动力学特性的实验研究

本文专门设计搭建了低温介质空泡演化实验测试平台,对液氮单空泡非定常演化过程和动力学特性开展了实验研究.实验中利用电火花瞬态放电激发液氮汽化形成单空泡,通过高速摄影系统对单空泡的瞬态特征进行了精细化捕捉.为了进一步揭示低温介质独特的物理性质以及强热力学效应对单空泡演化过程的影响机制,对比分析了在相同环境压力下, 77.41 K液氮和298.36 K水单空泡的演化过程和动力学特性.基于实验得到空泡半径与界面速度等定量数据,阐明了液氮单空泡球形与非球形演化阶段的非定常特性.研究结果表明:(1)在相同输入电压下,液氮单空泡的整体尺寸比常温水更小,当输入电压为400 V时,液氮空泡的最大半径约为常温水空泡的0.69倍;同时,液氮单空泡经历了膨胀阶段-收缩阶段-振荡阶段以及上升阶段的演化过程.(2)液氮空泡的收缩过程主要由相界面的热传导主导,没有明显的塌陷现象,收缩阶段液氮空泡的最小收缩半径约为常温水的5.5倍.(3)在液氮空泡振荡初期,空泡相界面传热增强, Rayleigh-Taylor不稳定与热力学效应共同引起了空泡界面的表面粗化效应;在整个振荡阶段,空泡界面附近存在破碎的小泡.当输入电压较高...     

Experimental investigation of horizontal air–water bubbly-to-plug and bubbly-to-slug transition flows in a 3.81 cm ID pipe

The present study seeks to investigate horizontal bubbly-to-plug and bubbly-to-slug transition flows. The two-phase flow structures and transition mechanisms in these transition flows are studied based on experimental database established using the local four-sensor conductivity probe in a 3.81 cm inner diameter pipe. While slug flow needs to be distinguished from plug flow due to the presence of large number of small bubbles (and thus, large interfacial area concentration), both differences and similarities are observed in the evolution of interfacial structures in bubbly-to-plug and bubbly-to-slug transitions. The bubbly-to-plug transition is studied by decreasing the liquid flow rate at a fixed gas flow rate. It is found that as the liquid flow rate is lowered, bubbles pack near the top wall of the pipe due to the diminished role of turbulent mixing. As the flow rate is lowered further, bubbles begin to coalesce and form the large bubbles characteristic of plug flow. Bubble size increases while bubble velocity decreases as liquid flow rate decreases, and the profile of the bubble velocity changes its shape due to the changing interfacial structure. The bubbly-to-slug transition is investigated by increasing the gas flow rate at a fixed liquid flow rate. In this transition, gas phase becomes more uniformly distributed throughout the cross-section due to the formation of large bubbles and the increasing bubble-induced turbulence. The size of small bubbles decreases while bubble velocity increases as gas flow rate increases. The distributions of bubble size and bubble velocity become more symmetric in this transition. While differences are observed in these two transitions, similarities are also noticed. As bubbly-to-plug or bubbly-to-slug transition occurs, the formation of large elongated bubbles is observed not in the uppermost region of bubble layer, but in a lower region. At the beginning of transitions, relative differences in phase velocities near the top of the pipe cross-section to those near the pipe center become larger for both gas and liquid phases, because more densely packed bubbles introduce more resistance to both phases.     

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.     

A novel numerical scheme for the investigation of surface tension effects on growth and collapse stages of cavitation bubbles

In the present study the effects of surface tension on the growth and collapse stages of cavitation bubbles are studied individually for both spherical and nonspherical bubbles. The Gilmore equation is used to simulate the spherical bubble dynamics by considering mass diffusion and heat transfer. For the collapse stage near a rigid boundary, the Navier–Stokes and energy equations are used to simulate the flow domain, and the VOF method is adopted to track the interface between the gas and the liquid phases. Simulations are divided into two cases. In the first case, the collapse stage alone is considered in both spherical and nonspherical situations with different conditions of bubble radius and surface tension. According to the results, surface tension has no significant effects on the flow pattern and collapse rate. In the second case, both the growth and collapse stages of bubbles with different initial radii and surface tensions are considered. In this case surface tension affects the growth stage considerably and, as a result, the jet velocity and collapse time decrease with increasing surface tension coefficient. This effect is more significant for bubbles with smaller radii.     

Heat and mass transfer characteristics of air bubbles and hot water by direct contact

 This paper has dealt with direct contact heat and mass transfer characteristics of air bubbles in a hot water layer. The experiments were carried out by bubbling air in the hot water layer under some experimental conditions of air flow rate, inlet air temperature and humidity as a dispersion fluid, and hot water temperature and hot water layer depth as a continuous fluid. Heat transfer and evaporation of water vapor from hot water to air bubbles occurred during air bubbles ascending into the hot water. Air bubble flow patterns were classified into three regions of independent air bubble flow, transition and air bubble combination growth. Non-dimensional correlation equations of direct contact heat and mass transfer between air bubbles and hot water were derived by some non- dimensional parameters for three regions of bubble flow pattern. Received on 14 July 2000 / Published online: 29 November 2001     

Hydrodynamic and mass transfer characteristics of slug flow in a vertical pipe with and without dispersed small bubbles

In this work, we present a numerical study to investigate the hydrodynamic characteristics of slug flow and the mechanism of slug flow induced CO₂ corrosion with and without dispersed small bubbles. The simulations are performed using the coupled model put forward by the authors in previous paper, which can deal with the multiphase flow with the gas–liquid interfaces of different length scales. A quasi slug flow, where two hypotheses are imposed, is built to approximate real slug flow. In the region ahead of the Taylor bubble and the liquid film region, the presence of dispersed small bubbles has less impacts on velocity field, because there are no non-regular intensive disturbance forces or centrifugal forces breaking the balance of the liquid and the dispersed small bubbles. In the liquid slug region, the strong centrifugal forces generated by the recirculation below the Taylor bubble lead to the effect of heterogeneity, which makes the profile of the radial liquid velocity component sharper with higher volume fraction of dispersed small bubbles. The volume fraction has a maximum value in the range of r/R = 0.5–0.6. Meanwhile, it is usually higher than 0.35, which means that larger dispersed bubbles can be formed by coalescences in this region. These calculated results are in good agreement with experimental results. The wall shear stress and the mass transfer coefficient with dispersed small bubbles are higher than those without dispersed small bubbles due to enhanced fluctuations. For short Taylor bubble length, the average mass transfer coefficient is increased when the gas or liquid superficial velocity is increased. However, there may be an inflection point at low mixture superficial velocities. For the slug with dispersed small bubbles, the product scales still cannot be damaged directly despite higher wall shear stress. In fact, the alternate wall shear stress and the pressure fluctuations perpendicular to the pipe wall with high frequency are the main cause for breaking the product scales.     

Evolution of distortions of the spherical shape of a cavitation bubble in acoustic supercompression

The evolution of small perturbations of the spherical shape of a vapor bubble in the process of its single strong expansion and compression in deuterated acetone is studied. In the mathematical model used the motion of vapor and liquid is broken down into the spherical component and its small nonspherical perturbation. The spherical component is described by the fluid dynamics equations with account for time-dependent heat conduction and evaporation and condensation on the liquid-vapor interface using equations of state constructed from experimental data. In describing the nonspherical component the liquid viscosity and the surface tension are taken into account, while the effect of the bubble content is disregarded. Certain simple analytical formulas are presented which describe the bubble radius at the moment of maximum expansion, its variation in the compression stage, and the evolution of the bubble sphericity distortion in compression.     

Flow regime transition criteria for two-phase flow in a vertical annulus

In this work, a new flow regime transition model is proposed for two-phase flows in a vertical annulus. Following previous works, the flow regimes considered are bubbly (B), slug (S) or cap-slug (CS), churn (C) and annular (A). The B to CS transition is modeled using the maximum bubble package criteria of small bubbles. The S to C transition takes place for small annulus perimeter flow channels and it is assumed to occur when the mean void fraction over the entire region exceeds that over the slug–bubble section. If the annulus perimeter is larger that the distorted bubble limit the cap-slug flow regime will be considered since in these conditions it is not possible to distinguish between cap and partial-slug bubbles. The CS to C transition is modeled using the maximum bubble package criteria. However, this transition considers the coalescence of cap and spherical bubbles in order to take into account the flow channel geometry. Finally, the C to A transition is modeled assuming two different mechanisms, (a) flow reversal in the liquid film section along large bubbles; (b) destruction on liquid slugs or large waves by entrainment or deformation. In the S to C and C to A flow regime transitions the annulus flow channel is considered as a rectangular flow channel with no side walls. In all the modeled transitions the drift-flux model is used to obtain the final correlations. The final equations for every flow regime transition are easy to be implemented in computational codes and not experimental input is needed. The prediction accuracy of the newly developed model has been checked against air–water as well as boiling flow regime maps. In all the cases, the new developed model shows better predicting capabilities than the existing correlations most used in literature.     

Acoustics of a Bubbly Liquid with a Weak Chemical Reaction in the Gaseous Phase

The influence of the composition and thermophysical properties of gas-liquid bubbly systems with a dissociating component in the gaseous phase on the laws of small-disturbance propagation and attenuation is investigated. It is found that the reacting gas component in the bubbles significantly affects the sonic-wave attenuation coefficient in the bubbly liquid. This follows from the fact that when a gas bubble is compressed isothermally, a recombination reaction occurs which prevents pressure growth in the bubble.Small-disturbance propagation in bubbly liquids was investigated in a number of publications discussed in review [1]. The acoustics of a bubbly liquid with a gas phase containing active admixtures are of both methodical and practical interest. The dynamics of such multicomponent bubbles were investigated in [2].     

Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation

Equations describing the propagation of waves of small but finite amplitude in a liquid with gas bubbles are derived. The bubble distribution density is a continuous function of bubble size and spatial coordinates. It is found that, for a uniform bubble distribution, the obtained equations become the Korteweg-de Vries, Kadomtsev-Petviashvili and Khokhlov-Zabolotskaya equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 188–197, March–April, 2009.     

Numerical investigation of unsteady shock waves in vapor-gas-droplet mixtures

The results of mathematical modeling of the evolution of unsteady shock waves in two-phase mixtures of inert gas, vapor and suspended liquid droplets with allowance for dynamic, thermal and mass phase interaction processes are presented. The influence of interphase mass transfer effects (droplet breakdown and evaporation, vapor condensation) on the structure of unsteady shock waves in vapor-gas-droplet mixtures is analyzed. The important influence of phase mass transfer and, in particular, droplet breakdown as a result of surface layer stripping by the gas flow on the distribution of the parameters of the carrier and dispersed components of the mixture behind the shock front is demonstrated. The effect of the principal governing parameters of the two-phase mixture on the unsteady shock wave propagation process is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 67–75, July–August, 1992.     

Parametric Analysis for a Single Collapsing Bubble

This work presents a sensitivity analysis for cavitation processes, studying in detail the effect of various model parameters on the bubble collapse. A complete model (Hauke et al. Phys Rev E 75:1–14, 2007) is used to obtain how different parameters influence the collapse in SBSL experiments, providing some clues on how to enhance the bubble implosion in real systems. The initial bubble radius, the frequency and the amplitude of the pressure wave are the most important parameters determining under which conditions cavitation occurs. The range of bubble sizes inducing strong implosions for different frequencies is computed; the initial radius is the most important parameter characterized the intensity of the cavitation processes. However, other parameters like the gas and liquid conductivity or the liquid viscosity can have an important effect under certain conditions. It is shown that mass transfer processes play an important role in order to correctly predict the trends related with the effect of the liquid temperature, which translates into the bubble dynamics. Moreover, under some particular circumstances, evaporation can be encountered during the bubble collapse; this can be profitably exploited in order to feed reactants when the most extreme conditions inside the bubbles are reached. Thus, this paper aims at providing a global assessment of the effect of the different parameters on the entire cycle of a single cavitating spherical bubble immersed in an ultrasonic field. This work has been partially supported by Ministerio de Ciencia y Tecnologia, under grant number CTM2004-06184-C02-02.