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.     

Analysis of coupled thermo-hydro-mechanical behavior of unsaturated soils based on theory of mixtures I

This paper analyzes a coupled thermo-hydro-mechanical behavior of unsaturated soils based on the theory of mixtures. Unsaturated soil is considered as a mixture composed of soil skeleton, liquid water, vapor, dry air, and dissolved air. In addition to the mass and momentum conservation equations of each component and the energy conservation equation of the mixture, the system is closed using other 37 constitutive （or restriction） equations. As the change in water chemical potential is identical to the change in vapor chemical potential, a thermodynamic restriction relationship for the phase transition between pore water and pore vapor is formulated, in which the impact of the change in gas pressure on the phase transition is taken into account. Six final govern- ing equations are given in incremental form in terms of six primary variables, i.e., three displacement components of soil skeleton, water pressure, gas pressure, and temperature. The processes involved in the coupled model include thermal expansions of soil skeleton and soil particle, Soret effect, phase transition between water and vapor, air dissolution in pore water, and deformation of soil skeleton.     

Studies on two-phase co-current air/non-Newtonian shear-thinning fluid flows in inclined smooth pipes

In this work, co-current flow characteristics of air/non-Newtonian liquid systems in inclined smooth pipes are studied experimentally and theoretically using transparent tubes of 20, 40 and 60 mm in diameter. Each tube includes two 10 m long pipe branches connected by a U-bend that is capable of being inclined to any angle, from a completely horizontal to a fully vertical position. The flow rate of each phase is varied over a wide range. The studied flow phenomena are bubbly flow, stratified flow, plug flow, slug flow, churn flow and annular flow. These are observed and recorded by a high-speed camera over a wide range of operating conditions. The effects of the liquid phase properties, the inclination angle and the pipe diameter on two-phase flow characteristics are systematically studied. The Heywood–Charles model for horizontal flow was modified to accommodate stratified flow in inclined pipes, taking into account the average void fraction and pressure drop of the mixture flow of a gas/non-Newtonian liquid. The pressure drop gradient model of Taitel and Barnea for a gas/Newtonian liquid slug flow was extended to include liquids possessing shear-thinning flow behaviour in inclined pipes. The comparison of the predicted values with the experimental data shows that the models presented here provide a reasonable estimate of the average void fraction and the corresponding pressure drop for the mixture flow of a gas/non-Newtonian liquid.     

Numerical simulation of circulation in gas-liquid column reactors: isothermal, bubbly, laminar flow

The gas-liquid flow inside a circular, isothermal column reactor with a vertical axis has been studied using numerical simulations. The flow is assumed to be in the laminar, bubbly flow regime which is characterized by a suspension of discrete air bubbles in a continuous liquid phase such as glycerol water. The mathematical formulation is based on the conservation of mass and momentum principle for the liquid phase. The gas velocity distribution is calculated via an empirically prescribed relative velocity as a function of void fraction. The interface viscous drag forces are prescribed empirically. For some cases a profile shape is assumed for the void ratio distribution. The influence of various profile shapes is investigated. The results are compared with those where the void ratio distribution is calculated from the conservation of mass equation. The mathematical model has been implemented by modifying a readily available computer code for single-phase newtonian fluid flows. The numerical discretization is based on a finite volume approach. The predictions show a good agreement with measurements. The circulation pattern seems not to be so sensitive to the actual shape of the void fraction profiles, but the inlet distribution of it is important. A significantly different flow pattern results when the void fraction distribution is calculated from the transport equation, as compared to those with a priori prescribed profiles. When the void fraction is uniformly distributed over the whole distributor plate, no circulation is observed. Calculations also show that even the two-phase systems with a few discrete bubbles can be simulated successfully by a continuum model.     

Bubbles in liquids with phase transition

In the forthcoming second part of this paper a system of balance laws for a multi-phase mixture with many dispersed bubbles in liquid is derived where phase transition is taken into account. The exchange terms for mass, momentum and energy explicitly depend on evolution laws for total mass, radius and temperature of single bubbles. Therefore in the current paper we consider a single bubble of vapor and inert gas surrounded by the corresponding liquid phase. The creation of bubbles, e.g. by nucleation is not taken into account. We study the behavior of this bubble due to condensation and evaporation at the interface. The aim is to find evolution laws for total mass, radius and temperature of the bubble, which should be as simple as possible but consider all relevant physical effects. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves and phase transition on the evolution of the bubble. We find ordinary differential equations that describe the bubble dynamics. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble. It is shown that a rebounding bubble is only possible, if it contains in addition an inert gas. In Part 2 of the current paper the equations derived are sought in order to close the system of equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change.     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

Modeling transient churn-annular flows in a long vertical tube

This work investigates the transient behavior of high gas fraction gas-liquid flows in vertical pipes (annular and churn flows). Hyperbolic balance equations for mass, momentum and entropy are written for the gas and liquid, which is split between a continuous film and droplets entrained in the gas core. Closure relationships to calculate the wall and interfacial friction and the rates of droplet entrainment and deposition were obtained from the literature. A finite-difference solution algorithm based on a coefficient matrix splitting method was implemented to deal with sharp variations in the spatial and temporal domains, such as pressure and phase holdup waves. The model results were compared with steady-state experimental data from eight different sources, totaling more than 1500 data points for pressure gradient, liquid film flow rate and void/core fraction. The absolute average deviation between the model and the data was 17% for the pressure gradient and 5.8% for the void fraction. A comparison of the model results with fully transient air-water data generated in a 49-mm ID, 42-m long vertical pipe is also presented. The experimental results consist of two outlet pressure-induced and two inlet mass flow rate-induced transient tests. Two main transient parameters are compared, namely the local void fraction and the pressure difference between selected points along the test section and the outlet (taken as a reference). The comparisons between the experiments and the numerical model indicate that the model was capable of describing the transient annular to churn flow transition with absolute average deviations of 14.5% and 7.9% for the pressure difference and void fraction, respectively.     

Experimental study on the influence of SO2 gas injection to pure liquids on pool boiling heat transfer coefficients

SO₂ gas is injected into the different pure liquids using new innovative method via meshed tubes. Many experiments have been performed to investigate the influence of gas injection process on the pool boiling heat transfer coefficient of pure liquids around the horizontal cylinder at different heat fluxes up to 114 kW m^?2. Results demonstrate that presence of SO₂ gas into the vapor inside the bubbles creates a mass transfer driving force between the vapor phase inside the formed bubbles and liquid phase and also between the gas/liquid interfaces. Local turbulences and agitations due to the gas injection process around the nucleation sites leads the pool boiling heat transfer coefficient to be dramatically enhanced. Besides, some of earlier well-known correlations were unable to obtain the reasonable values for the pool boiling heat transfer coefficients in this particular case. Therefore, the most accurate correlation among the examined correlations was modified to estimate the pool boiling heat transfer coefficient of pure liquids. Experimental data were in a good agreement with those of obtained by the new modified correlation with absolute average deviation of 10 %.     

Vapor pressure and void size effects on failure of a constrained ductile film

To achieve certain properties, semiconductor adhesives and molding compounds are made by blending filler particles with polymer matrix. Moisture collects at filler particle/polymer matrix interfaces and within voids of the composite. At reflow temperatures, the moisture vaporizes. The rapidly expanding vapor creates high internal pressure on pre-existing voids and particle/matrix interfaces. The simultaneous action of thermal stresses and internal vapor pressure drives both pre-existing and newly nucleated voids to grow and coalesce causing material failure. Particularly susceptible are polymeric films and adhesives joining elastic substrates, e.g. Ag filled epoxy. Several competing failure mechanisms are studied including: near-tip void growth and coalescence with the crack; extensive void growth and formation of an extended damaged zone emanating from the crack; and rapid void growth at highly stressed sites at large distances ahead of the crack, leading to multiple damaged zones. This competition is driven by the interplay between stress elevation induced by constrained plastic flow and stress relaxation due to vapor pressure assisted void growth.A model problem of a ductile film bonded between two elastic substrates, with a centerline crack, is studied. The computational study employs a Gurson porous material model incorporating vapor pressure effects. The formation of multiple damaged zones is favored when the film contains small voids or dilute second-phase particle distribution. The presence of large voids or high vapor pressure favor the growth of a self-similar damage zone emanating from the crack. High vapor pressure accelerates film cracking that can cause device failures.     

EULERIAN–LAGRANGIAN COMPUTATIONS ON PHASE DISTRIBUTION OF TWO-PHASE BUBBLY FLOWS

A comprehensively theoretical model is developed and numerically solved to investigate the phase distribution phenomena in a two-dimensional, axisymmetric, developing, two-phase bubbly flow. The Eulerian approach treats the fluid phase as a continuum and solved Eulerian conservation equations for the liquid phase. The Lagrangian bubbles are tracked by solving the equation of motion for the gas phase. The interphase momentum changes are included in the equations. The numerical model successfully predicts detailed flow velocity profiles for both liquid and gas phases. The development of the wall-peaking phenomenon of the void fraction and velocity profiles is also characterized for the developing flow. For 42 experiments in which the mean void fraction is less than 20 per cent, numerical calculations demonstrate that the predictions agree well with Liu's experimental data. © 1997 by John Wiley & Sons, Ltd.     

Effect of Interphase Heat Transfer on the Stability of a Turbulent Multicomponent Flow in a Vortex

The parameters of an axisymmetric turbulent two-phase swirling flow of a viscous heat-conducting gas containing a liquid dispersed phase in the presence of water vapor condensation on the particles are calculated. For the dispersed phase, a model taking into account the variation of the vapor concentration and the particle size due to condensation or evaporation is proposed. The distributions of the parameters of the basic unperturbed flow obtained numerically are used in the numerical solution of the linear problem of hydrodynamic stability within the time-dependent formulation. The parameters of small-amplitude harmonic perturbations propagating along the vortex axis are investigated in the linear formulation. A significant effect of heat release in the gas due to water vapor condensation on the parameters of the neutral perturbations and the neutral-stability curves is detected.     

Thermodynamic Correction for Salts in Variably Saturated Porous Media

Using thermodynamic principles, the general relationship describing the equilibrium vapor content in the gas phase above a saline liquid and across a curved liquid–gas interface is developed. Since high salt concentration affects the intensive and extensive liquid properties, it is also necessary to account for these effects in liquid water content/liquid water pressure relationship curves so that experimentally derived curves for pure water may be useful for elevated salt concentrations. The appropriate thermodynamic relationships are derived to describe the salt effects on liquid and vapor properties. The resulting equations are valid for salt concentrations between zero and saturation, and for any temperatures that nominally occur in nearsurface geologic materials.     

Method for calculating the dynamics of vapor bubbles when a liquid boils in a centrifugal force field

The specific feature of the study of the dynamics of vapor bubbles during boiling of a liquid in a centrifugal force field is the fact that the velocity of a bubble is much greater than the rate of change of its radius, and its movement occurs in fields of variable pressure and underheating that have to be determined in the solution of the problem. In addition, when investigating processes occurring when liquid helium boils in a centrifugal force field, its thermodynamic parameters may be close to the critical values, and the dependences of the thermophysical properties of the liquid and vapor on the temperature and pressure must be taken into consideration. The equation of state of a substance close to its critical thermodynamic point cannot be an approximation to the equation of state of an ideal gas, as has been suggested in a series of articles. The nonequilibrium nature of the phase transition must be taken into consideration in the case of substances existing at near-critical parameters and substances with a low coefficient of accommodation. A marked deformation of the bubbles, which also has to be taken into account, will be observed in strong centrifugal force fields. Such studies have not appeared in the specialist journals. Equations of the two-temperature and two-velocity hydrodynamics of two-phase media in a one-dimensional form for substances obeying the equation of state for an ideal gas were discussed in [1, 2] with allowance for the dependence of the thermophysical properties on the temperature and pressure. In strong centrifugal force fields the one-dimensional approach is unacceptable and the flow of liquid around a buoyant bubble must be taken into account. A joint examination of the change in the temperature field with time in the vicinity of a vapor bubble with changes in its dimensions and position was made for the first time in [3–8]. The present article is an extension of the latter work and takes the aforementioned factors into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–71, July–August, 1984.     

气液两相流泄漏扩散的数值模拟

采用Eulerian-Lagrangian方法数值模拟过热液体丙烷从喷嘴释放后射流的发展过程。其中,对丙烷蒸汽与空气的气相混合物采用欧拉法求解,利用拉格朗日法跟踪丙烷液滴的运动轨迹,并加入颗粒随机轨道模型来考虑液滴的湍流扩散效应,数值计算所得的物理参量与实验测量结果相吻合。在此基础上,本文将此数值模型和方法应用于开放空间、常温高压下储存液化气(丙烷)的容器发生小孔泄漏时,两相射流在大气中的扩散过程,计算结果符合实际情况。文中还讨论了影响两相流扩散过程的主要因素,为预测可能发生的火灾爆炸事故提供理论依据。     

Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.      

Continuum modeling of a porous solid with pressure-sensitive dilatant matrix

The pressure-sensitive plastic response of a material has been studied in terms of the intrinsic sensitivity of its yield stress to pressure and the presence and growth of cavities. This work focuses on the interplay between these two distinctly different mechanisms and the attendant material behavior. To this end, a constitutive model is proposed taking both mechanisms into account. Using Gurson's homogenization, an upper bound model is developed for a voided solid with a plastically dilatant matrix material. This model is built around a three-parameter axisymmetric velocity field for a unit sphere containing a spherical void. The void is also subjected to internal pressure; this can be relevant for polymeric adhesives permeated by moisture that vaporizes at elevated temperatures. The plastic response of the matrix material is described by Drucker–Prager's yield criterion and an associated flow rule. The resulting yield surface and porosity evolution law of the homogenized constitutive model are presented in parametric form. Using the solutions to special cases as building blocks, approximate models with explicit forms are proposed. The parametric form and an approximate explicit form are compared against full-field solutions obtained from finite element analysis. They are also studied for loading under generalized tension conditions. These computational simulations shed light on the interplay between the two mechanisms and its enhanced effect on yield strength and plastic flow. Among other things, the tensile yield strength of the porous solid is greatly reduced by the internal void pressure, particularly when a liquid/vapor phase is the source of the internal pressure.     

Numerical study of the dynamics of a vapor envelope round a heated solid particle in a liquid

A study is made of the radial motion of a vapor envelope surrounding an isolated spherical particle in an unbounded mass of liquid. It is assumed that the liquid is viscous and incompressible and that the temperature is distributed uniformly in the solid particle. A model of a calorifically perfect gas is used for the vapor phase. The same assumptions are made as in Rayleigh's formulation for the problem of the dynamics of a single bubble: that the process is spherically symmetric and that the pressure P₂ (t) in the vapor phase is homogeneous. The justification for making these assumptions in problems of the dynamics of gas, vapor, and vaporgas bubbles is discussed in [1–5]. In this paper, the collapse of the vapor layer and the boiling of the liquid on the surface of the heated particle are not considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 154–157, July–August, 1985.     

Volume of fluid approach of boiling flows in concentrated solar plants

The Computational Fluid Dynamics (CFD) model proposed in this paper allows the flow patterns that evolve during progressive boiling inside large scale horizontal tubes to be simulated from the initial vapor generation stage to large vapor slugs. The volume of fluid (VOF) model was employed in combination with relatively simple hypothesis. The aim of the present work is to improve the design of receiver tubes at concentrated solar power plants with direct steam generation by simulating the evolution of flow regimes within these tubes. Despite numerous studies conducted in the past years on convective boiling, only a few made use of the VOF model to simulate large flow regime transitions. This work presents a preliminary and relatively qualitative approach to address this problem. Heat and mass transfer at the tube inner wall and at the liquid-gas interface were solved with the additional transport of two scalars. One accounts for the enthalpy field and the other represents the dispersed vapor phase of the liquid. This new phase was created at the wall surface of the liquid phase and rises up to the liquid-vapor interface. Different phenomena linked to the boiling process were taken into account: vapor creation at the wall, its transport, recondensation and the creation of large structures. This model was validated with boiling flow in a bent tube at different mass flow rates and heat fluxes, which allowed us to observe the evolution of two-phase flow patterns. Finally, numerical simulation of direct steam generation inside a concentrated solar plant receiver clearly showed the apparition and evolution of various two-phase flow patterns.     

Measurements within unsteady cavitation

A current method used to characterize the phase distribution within gas (or vapor)/liquid two-phase flows consists of using local probes. Two sensors placed in series enable void fraction, velocity, and chord length measurements. The choice of the data treatment method depends strongly on the unsteady behavior of the flow: the higher the velocity fluctuations, the more complex and time consuming the data treatments. Three different methods have been developed to study various cavitating flow conditions. They have been applied to characterize two-phase flows within sheet cavities using a double optical probe. Received: 8 October 1999/Accepted: 29 December 1999     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.