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.     

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.     

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.     

Numerical evaluation of turbulence models for dense to dilute gas–solid flows in vertical conveyor

A two-fluid model (TFM) of multiphase flows based on the kinetic theory and small frictional limit boundary condition of granular flow was used to study the behavior of dense to dilute gas–solid flows in vertical pneumatic conveyor. An axisymmetric 2-dimensional, vertical pipe with 5.6 m length and 0.01 m internal diameter was chosen as the computation domain, same to that used for experimentation in the literature. The chosen particles are spherical, of diameter 1.91 mm and density 2500 kg/m³. Turbulence interaction between the gas and particle phases was investigated by Simonin's and Ahmadi's models and their numerical results were validated for dilute to dense conveying of particles. Flow regimes transition and pressure drop were predicted. Voidage and velocity profiles of each phase were calculated in radial direction at different lengths of the conveying pipe. It was found that the voidage has a minimum, and gas and solid velocities have maximum values along the center line of the conveying pipe and pressure drop has a minimum value in transition from dense slugging to dilute stable flow regime. Slug length and pressure fluctuation reduction were predicted with increasing gas velocity, too. It is shown that solid phase turbulence plays a significant role in numerical prediction of hydrodynamics of conveyor and the capability of particles turbulence models depends on tuning parameters of slip-wall boundary condition.     

Interactions of particles and microbubbles with turbulence

Dilute, dispersed two-phase flows arise in many contexts ranging from solid particles or droplets in gas flows to bubbles in liquids. Many of the flows of interest are turbulent, which presents a complex problem to analyze or to determine the dominant physical processes contributing to the observed phenomena. Advances in experimental techniques have made it possible to measure directly turbulent and particle velocity fluctuations in dilute systems. This has provided a counterpart to advances in computational and analytical models and a basis on which to test these models. Three specific areas are considered: the fluctuating forces on an individual particle in an unsteady flow, the response of a solid particle to a turbulent air flow, and the corresponding response of a small bubble in turbulent liquid flows. Results from direct numerical simulations are presented for each of these, including the nonuniform distribution of particles generated by local instantaneous features of the flow. The issue of turbulence modulation at low to moderate void fractions is discussed.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

Application of the ultrasonic technique and high-speed filming for the study of the structure of air–water bubbly flows

Multiphase flows are very common in industry, oftentimes involving very harsh environments and fluids. Accordingly, there is a need to determine the dispersed phase holdup using noninvasive fast responding techniques; besides, knowledge of the flow structure is essential for the assessment of the transport processes involved. The ultrasonic technique fulfills these requirements and could have the capability to provide the information required. In this paper, the potential of the ultrasonic technique for application to two-phase flows was investigated by checking acoustic attenuation data against experimental data on the void fraction and flow topology of vertical, upward, air–water bubbly flows in the zero to 15% void fraction range. The ultrasonic apparatus consisted of one emitter/receiver transducer and three other receivers at different positions along the pipe circumference; simultaneous high-speed motion pictures of the flow patterns were made at 250 and 1000 fps. The attenuation data for all sensors exhibited a systematic interrelated behavior with void fraction, thereby testifying to the capability of the ultrasonic technique to measure the dispersed phase holdup. From the motion pictures, basic gas phase structures and different flows patterns were identified that corroborated several features of the acoustic attenuation data. Finally, the acoustic wave transit time was also investigated as a function of void fraction.     

Novel shear modulus equations for concentrated emulsions of two immiscible elastic liquids with interfacial tension

Starting from the shear modulus equation for a dilute emulsion system of two immiscible liquids with interfacial tension, four new equations have been developed for the shear modulus of concentrated emulsions using a differential scheme. The continuous phase and the dispersed droplets are treated as elastic liquids in the derivation. Out of the four models developed in the paper, two models predict the relative shear modulus (ratio of emulsion modulus to continuous phase modulus) to be a function of three variables: elastocapillary number, modulus ratio (dispersed phase modulus to continuous phase modulus) and volume fraction of dispersed phase. The remaining two models include an additional parameter, i.e. the maximum packing volume fraction of droplets. The proposed models are evaluated using three sets of experimental data on high frequency shear modulus of concentrated polymer-thickened oil-in-water emulsions.     

Calculation of the momentum and heat transfer in turbulent gas-particle jet flows

The equations for the second moments of the dispersed-phase velocity and temperature fluctuations are used for calculating gas-suspension jet flows within the framework of the Euler approach. The advantages of introducing the equations for the second moments of the particle velocity fluctuations has previously been quite convincingly demonstrated with reference to the calculation of two-phase channel boundary flows [9–11]. The flows considered below have a low solid particle volume concentration, so that interparticle collisions can be neglected and, consequently, the stochastic motion of the particles is determined exclusively by their involvement in the fluctuating motion of the carrier flow. In addition to the equations for the turbulent energy of the gas and its dissipation, the calculation scheme includes the equations for the turbulent energy and turbulent heat transfer of the solid phase; however, the model constructed does not contain additional empirical constants associated with the presence of the particles in the flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 69–80, May–June, 1992.     

Spatial averaging in the mechanics of heterogeneous and dispersed systems

Spatial averaging of the equations describing two single-phase media is separately considered in this paper regarding the volumes occupied by either phase with allowance for the boundary conditions on phase interfaces. The equations obtained are specialized to describe monodispersed mixtures within a “cellular” scheme. It is shown that it is necessary to consider the average values both over the whole cells and over those intersected by the boundary of a selected mixture volume.The problem of motion in the cell is formulated. Fictitious parameters are introduced at “infinity” for the carrier phase to solve the problem. These parameters do not coincide with the average values for this phase. A closed system of equations is derived for two extreme cases: an ideal incompressible carrier fluid and an extremely viscous incompressible carrier fluid. These are correlative with the inertial and viscous motions in the cell.Various effects are discussed in this paper. These include the radial motion of bubbles, the oriented rotation of dispersed particles (the symmetry and asymmetry of stress tensor), viscosity, phase transitions and the finite volume content of dispersed particles. Some aspects of earlier studies are critically analyzed.     

Possibilities and Limitations of Computer Simulations of Industrial Turbulent Dispersed Multiphase Flows

We give an overview on the usage of computer simulations in industrial turbulent dispersed multiphase flows. We present a few examples of industrial flows: bubble columns and bubbly pipe flows, stirred tanks, cyclones, and a fluid catalytic cracking unit. The fluid catalytic cracking unit is used to illustrate the complexity of the physical phenomena involved, and the possibilities and limitations of the different approaches used: Eulerian–Lagrangian (particle-tracking) and Eulerian–Eulerian (two-fluid). In the first approach, the continuous phase is solved using either RANS simulations (Reynolds-Averaged Navier–Stokes simulations) or DNS/LES (Direct Numerical Simulations/Large-Eddy Simulations), and the individual particles are tracked. In the second approach, the dispersed phase is averaged, leading to two sets equations, which are quite similar to the RANS equations of single-phase flows. The Eulerian–Eulerian approach is the most commonly used in industrial applications, however, it requires a significant amount of modelling. Eulerian–Lagrangian RANS can be simpler to use; in particular in situations involving complex boundary conditions, polydisperse flows and agglomeration/breakup. The key issue for the success of the simulations is to have good models for the complex physics involved. A major weakness is the lack of good models for: the turbulence modification promoted by the particles, the inter-particle interactions, and the near-wall effects. Eulerian–Lagrangian DNS/LES can play an important role as a research tool, in order to get a better physical understanding, and to improve the models used in the RANS simulations (either Eulerian–Eulerian or Eulerian–Lagrangian).     

可侵蚀地表上方含尘大气运动的数值模型

在稀相气固两相流的双流体模型框架下，导出含尘大气运动的基本方程，其中的源项考虑相间双向耦合作用；通过对基本方程进行无量纲化，求得控制气相和弥散相动力学行为的相似准则，包括弗洛德数、罗斯贝数、颗粒雷诺数、颗粒惯性参数、颗粒剪切横移参数、颗粒旋转横移参数和沙尘质量载荷率等. 作为一个模型问题，研究可侵蚀地表上方充分发展湍流大气边界层流动对土壤颗粒的气动卷扬作用，其中计及气动阻力、萨弗曼升力和重力等因素的影响. 为了克服由于沙尘跃移运动轨迹交叉导致流动参数多值性所造成的困难，引进拉格朗日坐标下弥散相连续方程. 对两种不同风速和4种不同粒径条件下的沙尘运动进行数值模拟，给出相应情况中沙尘运动特性和密度分布剖面并讨论风速和粒径等参数的影响，还细致分析了含尘大气边界层中沙尘与气流之间的能量传递过程. 所得的方程、准则和方法可以为土壤风蚀和沙尘暴等自然灾害的预报提供理论基础.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle‘s governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.     

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract

A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests are presented.     

A micromechanics based damage model for composite materials

The predictive capacity of ductile fracture models when applied to composite and multiphase materials is related to the accuracy of the estimated stress/strain level in the second phases or reinforcements, which defines the condition for damage nucleation. Second phase particles contribute to the overall hardening of the composite before void nucleation, as well as to its softening after their fracture or decohesion. If the volume fraction of reinforcement is larger than a couple of percents, this softening can significantly affect the resistance to plastic localization and cannot be neglected. In order to explicitly account for the effect of second phase particles on the ductile fracture process, this study integrates a damage model based on the Gologanu–Leblond–Devaux constitutive behavior with a mean-field homogenization scheme. Even though the model is more general, the present study focuses on elastic particles dispersed in an elasto-plastic matrix. After assessing the mean-field homogenization scheme through comparison with two-dimensional axisymmetric finite element calculations, an extensive parametric study is performed using the integrated homogenization-damage model. The predictions of the integrated homogenization-damage model are also compared with experimental results on cast aluminum alloys, in terms of both the fracture strain and overall stress–strain curves. The study demonstrates the complex couplings among the load transfer to second phase particles, their resistance to fracture, the void nucleation mode, and the overall ductility.     

Numerical study of the lift force influence on two-phase shock tube boundary layer characteristics

The importance of the lift force acting on the dispersed phase in the boundary layer of a laminar gas-particle dilute mixture flow generated by a shock wave is investigated numerically. The particle phase is supposed to form a continuum and is described by an Eulerian approach. The ability of the Eulerian model to simulate particle flows and the importance of the two-way coupling are proven by comparison with experimental data as well as with the numerical results from schemes based on a Lagrangian approach. The models used for the lift force are discussed through comparisons between numerical and experimental results found in the literature. Some results about the formation of a dust cloud are numerically reproduced and show the major role of the lift force. Simulations of two-dimensional two-phase shock tube flows are also performed including the lift force effects. Although the wave propagation is weakly influenced by the lift force, the force modifies substantially the dynamics of the flow near the wall. Received 17 February 2000 / Accepted 30 November 2000     

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.     

Study of two-phase flows in reduced gravity using ground based experiments

Experimental studies have been carried out to support the development of a framework of the two-fluid model along with an interfacial area transport equation applicable to reduced gravity two-phase flows. The experimental study simulates the reduced gravity condition in ground based facilities by using two immiscible liquids of similar density namely, water as the continuous phase and Therminol 59^® as the dispersed phase. We have acquired a total of eleven data sets in the bubbly flow and bubbly to slug flow transition regimes. These flow conditions have area-averaged void (volume) fractions ranging from 3 to 30% and channel Reynolds number for the continuous phase between 2,900 and 8,800. Flow visualization has been performed and a flow regime map developed which is compared with relevant bubbly to slug flow regime transition criteria. The comparison shows that the transition boundary is well predicted by the criterion based on critical void fraction. The value of the critical void fraction at transition was experimentally determined to be approximately 25%. In addition, important two-phase flow local parameters, including the void fraction, interfacial area concentration, droplet number frequency and droplet velocity, have been acquired at two axial locations using state-of-the-art multi-sensor conductivity probe. The radial profiles and axial development of the two-phase flow parameters show that the coalescence mechanism is enhanced by either increasing the continuous or dispersed phase Reynolds number. Evidence of turbulence induced particle interaction mechanism is highlighted. The data presented in this paper clearly show the marked differences in terms of bubble (droplet) size, phase distribution and phase interaction in two-phase flow between normal and reduced gravity conditions.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.