Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

Capturing coalescence and break-up processes in vertical gas–liquid flows: Assessment of population balance methods

Gas–liquid flows are commonly encountered in industrial flow systems. Numerical studies have been performed to assess the performances of different population balance approaches – direct quadrature method of moments (DQMOMs), average bubble number density (ABND) model and homogeneous MUlti-SIze-Group (MUSIG) model – in tracking the changes of gas void fraction and bubble size distribution under complex flow conditions and to validate the model predictions against experimental measurements from medium- and large-sized vertical pipes. Subject to different gas injection method and flow conditions, bubble size evolution exhibited a coalescence dominant trend in the medium-sized pipe; while bubble break-up was found to be dominant in large-sized pipe. The two experiments were therefore strategically selected for carrying out a thorough examination of existing population balance models in capturing the complicated behaviour of bubble coalescence and break-up. In general, predictions of all the different population balance approaches were in reasonable agreement with experimental data. More importantly, encouraging results have been obtained in adequately capturing the dynamical changes of bubbles size due to bubble interactions and transition from wall peak to core peak gas void fraction profiles. As a compromise between numerical accuracy and computational time, DQMOM has performed rather well in capturing the essential two-phase flow structures within the medium- and large-sized vertical pipes when compared to those of ABND and homogeneous MUSIG models. From a practical perspective, the ABND model may still be considered as a more viable approach for industrial applications of gas–liquid flow systems.     

川渝裂缝性地层自动压井环空多相压力波速特性研究北大核心CSCD

考虑虚拟质量力、环空沿程压力、气液相间阻力、气体滑脱、环空空隙率等因素,基于小扰动理论,提出了裂缝性地层自动压井环空多相压力波速数学模型,结合半显式差分方法,以彭州PZ-5-3D井(垂深5827 m)为实例,对模型编程求解.结果表明:裂缝性地层出气具有段塞流特点,随空隙率增大,压力波速呈现先减小后增大趋势;空隙率在0%至16%区间,压力波速以液弹为主,压力波速呈急剧下降趋势;空隙率在16%至40%区间,压力波速趋于平缓恒定值;空隙率在42%至100%区间,压力波速呈现增大趋势,压力波速以气弹为主;随环空井深减小,环空空隙率减小,压力波速整体呈现减小趋势;随压井循环排气井口回压增大,压力波速整体呈现增大趋势;环空空隙率在0%至13%区间内,气体滑脱速度对压力波速影响不大;环空空隙率在13%至85%区间内,随气体滑脱速度增大,压力波速呈现减小趋势;节流阀调阀时间间隔与井底压力响应时间具有跟随性,随井底压力响应时间增大,调阀时间间隔增大.     

Numerical modelling of EHD effects on heat transfer and bubble shapes of nucleate boiling

In this article, mathematical and numerical models are developed to study pure electrohydrodynamic (EHD) effects on heat transfer and bubble shapes when an initial bubble attached to a superheated horizontal wall in nucleate boiling. In the modelling of EHD effects on heat transfer, an undeformed bubble is considered; the electric body force and Joule heat are added to the momentum and energy equations; governing equations for heat, fluid flow and electric fields are coupled numerically and solved using a non-orthogonal body-fitted mesh system with necessary interfacial treatments at the gas–liquid boundary. While, to study the pure effect of EHD on the deformation of the bubble, the evaluation of a deformable bubble without heat transfer is simulated by volume of fluid (VOF) method based on an axial symmetric Cartesian coordinate system. The simulations indicate that EHD can effectively enhance heat transfer rate of nucleate boiling by influencing the motion of the ring vortex around the bubble and that bubble can be elongated due to the pull in axial direction and push in the negative radial direction by the electric field force.     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

壁面效应对剪切稀化流体内气泡上浮特性的影响

数值研究了壁面效应对剪切稀化流体内气泡上浮运动特性的影响,气液两相的界面捕捉采用流体体积(VOF)法,剪切稀化流体流变特性和气液相间表面张力的计算分别采用Carreau模型和连续表面张力模型.详细研究了不同流变指数下,壁面效应对气泡形状、液相流场和气泡终端速度的影响.结果表明,强的壁面效应或弱的剪切稀化程度会限制气泡的变形和尾涡的形成,使气泡的终端速度减小;气泡终端速度最易受壁面效应的影响;强的壁面效应和强的剪切稀化程度会导致高剪切速率区域出现在壁面附近,引起壁面附近液相表观黏度大幅度的下降.     

Modeling and simulation of severe slugging in air-water systems including inertial effects

A mathematical model and numerical simulations corresponding to severe slugging in air-water pipeline-riser systems are presented. The mathematical model considers continuity equations for liquid and gas phases, with a simplified momentum equation for the mixture. A drift-flux model, evaluated for the local conditions in the riser, is used as a closure law. In many models appearing in the literature, propagation of pressure waves is neglected both in the pipeline and in the riser. Besides, variations of void fraction in the stratified flow in the pipeline are also neglected and the void fraction obtained from the stationary state is used in the simulations. This paper shows an improvement in a model previously published by the author, including inertial effects. In the riser, inertial terms are taken into account by using the rigid water-hammer approximation. In the pipeline, the local acceleration of the water and gas phases are included in the momentum equations for stratified flow, allowing to calculate the instantaneous values of pressure drop and void fraction. The developed model predicts the location of the liquid accumulation front in the pipeline and the liquid level in the riser, so it is possible to determine which type of severe slugging occurs in the system. A comparison is made with experimental results published in literature including a choke valve and gas injection at the bottom of the riser, showing very good results for slugging cycle and stability maps. Simulations were also made assessing the effect of different strategies to mitigate severe slugging, such as choking, gas injection and increase in separation pressure, showing correct trends.     

Numerical simulation of the interaction between vortex ring and bubble plume

In present paper a three-dimensional Vortex-In-Cell method with two-way coupling effect was developed to study the bubble plume entrainment by a vortex ring. In this method the continuous flow was calculated by the three-dimensional Vortex-In-Cell method and the bubbles are tracked through bubble motion equation. Two-way coupling effect between continuous flow and dispersed bubbles is considered by introducing a vorticity source term, which is induced by the change of void fraction gradient in each computational cell. After validated by the comparison between experimental measurements and simulation results for the motion of vortex rings and the rising velocity of bubble plume, present method is implemented to simulate the interaction between an evolving vortex ring and a rising bubble plume. It was found that there is little effect of the bubble entrainment to the total circulation of vortex ring while the effect of bubble entrainment to the vortex ring structure is quite obvious. The bubble entrainment by the vortex ring not only changed the vorticity distribution in the vortex structure, but also displaced the positions of the vortex cores. The vorticity in the lower vortex core of the vortex ring decreases more than that in the upper vortex core of the vortex ring while the vortex core in the upper part of the vortex ring is displaced to the center of vortex ring by the entrained bubbles. Smaller bubbles are easier to be entrained by the large scale vortex structure and the transportation distance is in inverse proportion to bubble diameter.     

Simulation of wave processes in a vapor-liquid medium

Numerical methods for the simulation of nonlinear wave processes in a vapor-liquid medium with a model two-phase spherical symmetric cell, with a pressure jump at its external boundary are considered. The viscosity and compressibility of the liquid, as well as the space variation of pressure in the vapor, are neglected. The problem is described by the heat equations in the vapor and liquid, and by a system of ODEs for the velocity, pressure, and radius at the bubble boundary. The equations are discretized in space by an implicit finite-volume scheme on a dynamic adaptive grid with grid refinement near the bubble boundary. The total time derivative is approximated by a method of backward characteristics. “Nonlinear” iterations are implemented at each time step to provide a specified high accuracy. The results of numerical experiments are presented and discussed for the critical thermodynamic parameters of water, for some initial values of the bubble radius and pressure jump.     

CFD modeling of hydrodynamic characteristics of a gas–liquid two-phase stirred tank

A three-dimensional CFD model was developed in this work to simulate hydrodynamic characteristics of a gas–liquid two-phase stirred tank with two six-bladed turbines and four baffles, coupling of the Multiple Size Group model to determine bubble size distribution. Important hydrodynamic parameters of the multi-phase system such as volume-averaged overall and time-averaged local gas holdups and axial liquid velocities along time and transversal courses were simulated and analyzed in detail, under varied operating conditions (inlet air flow rate and impeller rotation speed). Model predictions of local transient gas holdup and liquid velocity distributions on vertical and horizontal sections of the tank were also carried out. The overall flow patterns were discussed in detail to assess the mixing. Bubble size distributions were further predicted to reveal the unique properties of gas phase. Experimental measurements of overall gas holdups and local axial liquid velocities were used to validate the developed model.     

Modelling bubble rise and interaction with a glass surface

A theoretical model has been developed to analyse bubble rise in water and subsequent impact and bounce against a horizontal glass plate. The multiscale nature of the problem, where the bubble size is on the millimetre range and the film drainage process happens on the micrometre to nanometre scale requires the combined use of different modelling techniques. On the macro scale we solve the full Navier–Stokes equations in cylindrical coordinates to model bubble rise whereas modelling film drainage on the micro scale is based on lubrication theory because the film Reynolds number becomes much smaller than unity. Quantitative predictions of this model are compared with experimental data obtained using synchronised high-speed cameras. Video recording of bubble rise and bounce trajectories are combined with interferometry data to deduce the position and time-dependent thickness of the thin water film trapped between the deformed bubble and the glass plate. Bubble rise velocity indicated that the boundary condition at the bubble surface was tangentially immobile. Quantitative comparisons are presented for bubbles of different size to quantify similarities and differences.     

On the modelling of bubble plumes in a liquid pool

Turbulent, bubble plumes are investigated numerically using the commercial, Computational Fluid Dynamics (CFD) code CFX-F3D. A six-equation, two-fluid model approach is adopted, in which interphase momentum exchange models include buoyancy, drag, added mass, lift and turbulent dispersion effects. Particular attention is paid to turbulence modelling, in which generation and dissipation resulting from interaction between bubbles and liquid are specifically taken into account within the context of an extended k − ϵ turbulence model. Results from a number of calculations are presented and compared against published, experimental bubble plume data. It is suggested that existing bubble/liquid interaction models for plumes may be grouped into three categories: those which produce lateral bubble spreading, those which diffuse the ambient liquid velocity field, and those which couple the plume to the surrounding liquid and thereby ultimately govern the pool mixing behaviour.     

Numerical simulation of immersion quenching process of an engine cylinder head

In this article, we present the numerical simulations of a real cylinder head quench cooling process employing a newly developed boiling phase change model using the commercial CFD code AVL-FIRE v8.5. Separate computational domains constructed for the solid and liquid regions are numerically coupled at the interface of the solid–liquid boundaries using the AVL-Code-Coupling-Interface (ACCI) feature. The boiling phase change process triggered by the dipping hot metal and the ensuing two-phase flow is handled using an Eulerian two-fluid method. Multitude of flow features such as vapor pocket generation, bubble clustering and their disposition, are captured very effectively during the computation, in addition to the variation of the temperature pattern within the solid region. A comparison of the registered temperature readings at different monitoring locations with the numerical results generates an overall very good agreement and indicates the presence of intense non-uniformity in the temperature distribution within the solid. Overall, the predictive capability of the new boiling model is well demonstrated for real-time quenching applications.     

Influence of electromagnetic stirrer position on fluid flow and solidification in continuous casting mold

A computational study of the effect of stirrer position on fluid flow and solidification in a continuous casting billet mold with in-mold electromagnetic stirring has been carried out. The numerical investigation uses a full coupling method in which alternating magnetic field equations are solved simultaneously with the governing equations of fluid flow and heat transfer. An enthalpy-porosity technique is used for the solidification analysis while the magnetohydrodynamics technique is used for studying the fluid flow behavior under the electromagnetic field. The streamline, liquid fraction, and solid shell thickness at the mold wall have been predicted with and without EMS application at different positions along the length of the mold. Recirculation loops are seen to be formed above and below the stirrer position when fluid flow and electromagnetic field equations were solved, without incorporating the solidification model. Application of the solidification model interestingly resulted in the reduction of the size of the recirculation loops formed. The tangential component of velocity of the fluid near the solidification front, stirring intensity and the effective length of stirring below the stirrer decrease as the stirrer position is moved downwards. Significant changes in characteristics of solid shell formation like delay in initiation of solidification at the mold wall and formation of a gap in the re-solidified shell have been observed with change in stirrer position.     

Unstable Thermocapillary Flow at a Gas-Liquid Phase Boundary

We present results of thermocapillary experiments in the vicinity of a bubble under a heated wall. Thermocapillary convection is a flow along the interface of two fluids, which is caused by local gradients of the surface tensions σ. The aim of the present work is to study this flow phenomenon at higher Marangoni-numbers Mg. At sufficiently high values of Mg the flow exhibits oscillatory fluid motion and eventually becomes turbulent. For the detection of the flow velocity and temperature field, we applied a PIV method and a modified differential interferometer. In order to separate buoyancy effects from thermocapillary convection, we chose an experimental setup, where a bubble is positioned in a liquid matrix under a heated wall. We could observe various regimes of Marangoni flow, from steady flow over regular oscillating up to highly irregular oscillating flows at Mg up to 63,000. However, we also found unexpected stable flow regions and additional evaporation processes. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of a compressible multiphase cavitation approach for diesel spray modelling

The influence of in-nozzle phenomena including cavitation on the morphology of the spray from a diesel injector with a sharp nozzle inlet is investigated numerically. A compressible, multi-phase Volume of Fluid Large Eddy Simulation is implemented in the OpenFOAM environment. The volume fraction transport equations for liquid and gas phases are reformulated to include mass transfer source terms. These source terms are modelled with two cavitation models by Schnerr and Kunz, which are extended to eliminate non-physical mass transfer rates. Validation is carried out only for the Schnerr cavitation model due to its independence of empirical parameters. The numerical method is validated by comparing the simulated mass flow rates, pressure and velocity profiles at different cavitation conditions against published experimental data obtained using a slightly converging square channel. Favourable comparison between simulations and experiments is achieved with minor discrepancies attributable to uncertainties in fuel properties, experimental artefacts and assumptions made in numerical models. Application of the method to calculation of in-nozzle phenomena and primary breakup of a diesel spray reveals that in-nozzle flow separation, wall shear and cavitation contribute greatly to the fragmentation of the jet. Comparison of the two cavitation models shows that after the onset of complete flow detachment, the Kunz implementation predicts higher air inflow at the nozzle outlet than the Schnerr model.     

Spatial Velocity and Temperature Measurements in Non‐Isothermal Liquid Flows

A surface tension driven flow in the liquid vicinity of an air bubble on a heated wall is studied experimentally. The liquid flow caused by the temperature gradient along the surface of the bubble is termed thermocapillary convection. The surface tension force and the buoyancy force oppose one another. The measurement technique is the 3D particle tracking velocity and thermometry, 3D PTV/T, using thermochromic liquid crystals and digital image processing. The paper describes the method in some detail and presents quantitative results for different Marangoni numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of multiply connected axisymmetric discontinuous incompressible potential flows

The ascend and evolution of an axisymmetric gas bubble are studied numerically using an inviscid incompressible potential flow model. The volume of the gas bubble varies adiabatically. The transition from a simply connected bubble to a doubly connected toroidal one and its interaction with the free surface are simulated. The change in connectedness is accompanied by a nonzero velocity circulation and a discontinuous velocity potential occurring over an arbitrary toroidal liquid surface enclosing the bubble.     

Multiphase transient flow model in wellbore annuli during gas kick in deepwater drilling based on oil-based mud

Transient-state gas and oil-based mud (OBM) two-phase flow in wellbore annuli will occur during gas kick. The phase behavior of influx gas and OBM will make the gas kick during OBM drilling more complicated. There are three possible cases in an annulus: only liquid flow in the entire annulus, gas and liquid two-phase flow in part of the annulus, and gas and liquid two-phase flow in the entire annulus. First, the phase behaviors of gas and OBM in wellbore annuli are studied based on the phase behavior of methane and diesel. A multiphase transient-flow model in annuli during gas kick based on OBM is then established based on gas–liquid two-phase flow theory and on flash theory in annuli. The influences of phase behavior in annuli and annular geometry are taken into account. The local flow parameters are predicted by the hydrodynamic models and the local thermodynamic parameters are predicted by the heat-transfer models in the corresponding flow pattern. The proposed model has a better performance, compared with two other models, against the published experimental data. Finally, the variation of pit gain, well-bottom hole pressure, and gas void fraction are obtained, leading to a better understanding of the occurrence and evolution mechanism of gas kick during deepwater drilling.     

Unsteady motion of a spherical bubble in a complex fluid: Mathematical modelling and simulation

The nonlinear response of an oscillatory bubble in a complex fluid is studied. The bubble is immersed in a Newtonian liquid, which may have a dilute volume fraction of anisotropic additives such as fibers or few ppm of macromolecules. The constitutive equation for the fluid is based on a Maxwell model with an extensional viscosity for the viscous contribution. The model is considered new in the study of bubble dynamics in complex fluids. The numerical computation solves a system of three first order ordinary differential equations, including the one associated with the solution of the convolution integral, using a fifth order Runge–Kutta scheme with appropriated time steps. Asymptotic solutions of governing equation are developed for small values of the pressure forcing amplitude and for small values of the elastic parameter. A study of the bubble collapse radius is also presented. We compare the results predicted by our model with other model in the literature and a good agreement is observed. The calculated asymptotic solutions are also used to test the results of the numerical simulations. In addition, the orientation of the additives is considered. The angular probability density function is assumed to be a normal distribution. The results show that the model based on the fully aligned additives with the radial direction overestimates the tendency of the additives to stabilize the bubble motion, since the effect of extensional viscosity occurs due to the particle resistance to the movement throughout its longitudinal direction.