Virtual Mass Force Research Articles

Numerical investigation of closures for interface forces acting on single air-bubbles in water using Volume of Fluid and Front Tracking models

Closures for the drag and virtual mass forces acting on a single air bubble rising in initially quiescent pure water have been numerically investigated using direct numerical simulation techniques. A 3D Front Tracking model was used and the results were compared with simulation results obtained with a 2D Volume of Fluid model to assess the influence of the third dimension. In the simulations realistic values were taken for the physical properties, i.e., a density ratio of 800. The computed time-averaged terminal rise velocity and mean aspect ratio for individual air bubbles ranging in equivalent diameter from 1 to 10 mm rising in pure water compare well with available experimental data.

Simulation of SBLOCA based on an Improved Choked Flow Model for RELAP5/MOD3 Code

This paper is a continuation of the present author’s previous publication dealing with a new choked flow model for two-phase flow. The model based on a hyperbolic one-dimensional two-fluid model, where in the momentum equations the terms representing the interfacial pressure difference has been included in lieu of the virtual mass force terms. The new choked flow model is an improvement upon the choked flow model of the current RELAP5/MOD3 code, which itself is based on the Trapp–Ransom method. The author compares the predictions of this improved model with Trapp–Ransom model and Henry–Fauske model, for an assumed flow in a vertical pipe. The author simulates a typical PWR system with a hypothetical SBLOCA as well, and compares the system behaviors predicted by RELAP5/MOD3, based on the aforementioned choked flow models. He shows that the improved choked flow model leads to better predictions.

Drag and non-drag force influences in numerical simulations of metallurgical ladles

The modeling of metallurgical gas stirred ladles involves the interaction between several different phases with effects that, in many cases, have been considered in the literature. However, the way in which the different non-drag effects influence the behavior of this system is not clear. In this paper we analyze how numerical results are affected when the parameters used in modeling non-drag forces (such as virtual mass, lift, and turbulent dispersion force) are varied. Other effects such as the increment in turbulence due to the bubbles, a deformable free surface, and the influence of the bubble size were also considered. The knowledge of the way in which each physical phenomenon modifies the behavior of the main variables of the system allowed the fitting of experimental data in a variety of conditions. We conclude that a gas stirred ladle can be properly represented when the non-drag effects analyzed here are included, and give suitable values for the different parameters considered.

A mathematical model and numerical simulation of pressure wave in horizontal gas-liquid bubbly flow*

Abstract By using an ensemble-averaged two-fluid model, with valid closure conditions of interfacial momentum exchange due to virtual mass force, viscous shear stress and drag force, a model for pressure wave propagation in a horizontal gas-liquid bubbly flow is proposed. According to the small perturbation theory and solvable condition of one-order linear uniform equations, a dispersion equation of pressure wave is induced. The pressure wave speed calculated from the model is compared and in good agreement with existing data. According to the dispersion equation, the propagation and attenuation of pressure wave are investigated systemically. The factors affecting pressure wave, such as void fraction, pressure, wall shear stress, perturbation frequency, virtual mass force and drag force, are analyzed. The result shows that the decrease in system pressure, the increase in void fraction and the existence of wall shear stress, will cause a decrease in pressure wave speed and an increase in the attenuation co...

Multi-scale modeling of dispersed gas–liquid two-phase flow

In this work the concept of multi-scale modeling is demonstrated. The idea of this approach is to use different levels of modeling, each developed to study phenomena at a certain length scale. Information obtained at the level of small length scales can be used to provide closure information at the level of larger length scales. In this work the front tracking model is used to simulate the rise of a single bubble in a quiescent liquid and in a shear field. The results of the front tracking model are used to obtain coefficients for the drag, virtual mass and lift forces that are required in the Euler–Lagrange model used to study the behavior of dispersed gas–liquid flow on a larger scale. Finally, the Euler–Lagrange model, equipped with the correct closure information is used to simulate the flow in a square cross-sectioned bubble column. It is found that the results of the Euler–Lagrange model are in good agreement with experimental data.

Modeling, numerical methods, and simulation for particle-fluid two-phase flow problems

In this paper, we study the issues of modeling, numerical methods, and simulation with comparison to experimental data for the particle-fluid two-phase flow problem involving a solid-liquid mixed medium. The physical situation being considered is a pulsed liquid fluidized bed. The mathematical model is based on the assumption of one-dimensional flows, incompressible in both particle and fluid phases, equal particle diameters, and the wall friction force on both phases being ignored. The model consists of a set of coupled differential equations describing the conservation of mass and momentum in both phases with coupling and interaction between the two phases. We demonstrate conditions under which the system is either mathematically well posed or ill posed. We consider the general model with additional physical viscosities and/or additional virtual mass forces, both of which stabilize the system. Two numerical methods, one of them is first-order accurate and the other fifth-order accurate, are used to solve the models. A change of variable technique effectively handles the changing domain and boundary conditions. The numerical methods are demonstrated to be stable and convergent through careful numerical experiments. Simulation results for realistic pulsed liquid fluidized bed are provided and compared with experimental data.

A Finite Element Method for an Averaged Multiphase Flow Model

The paper discusses a numerical method for solving a two-phase flow model based on the interpenetrating continual hypothesis. The model incorporates terms to account for the effects of virtual mass force, different pressures for the two phases and the viscous dissipation. Our numerical scheme extends the incremental projection scheme for the incompressible Navier–stokes equations towards multiphase flows. An optimal stability is obtained by slightly modifying the Galerkin formulation. The stabilized Galerkin technique is based on a two-level hierarchical decomposition of the approximation space. Numerical simulations of three-dimensional bubbly flows in a periodic domain are presented. These simulations are compared with experimental data. The stability of a periodic flow with respect to 3D perturbations is studied numerically, and a discussion of the results is presented.

Liquid–solid separation phenomena of two-phase turbulent flow in curved pipes

The present study is to contribute some knowledge of phase separation phenomena of liquid–solid two-phase turbulent flow in curved pipes and provide a basis for the invention and development of a new type of curved pipe separator. Firstly, the solid–liquid two-phase flows in two-dimensional (2D) curved channels were numerically simulated using a two-way coupling Euler–Lagrangian scheme. Phase distribution characteristics of 2D curved channel two-phase flow were examined under conditions of different particle size, liquid flowrate and coil curvature. Based on the numerical results, the dynamic effects and contributions to phase separation of particle-subjected forces, including centrifugal force, drag force, pressure gradient force, gravity force, buoyancy force, virtual mass force and lift force, were exposed by kinematic and dynamic analysis along particle trajectories. Secondly, measurement of particle size and concentration profiles in helically coiled tube two-phase flow was conducted using a nonintrusive Malvern 2600 particle sizer based on laser diffraction. Particle size and concentration distribution characteristics of helically coiled tube two-phase flow and the effect of secondary flow on phase separation were analyzed based on experimental data.

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method

We present closures for the drag and virtual mass force terms appearing in a two-fluid model for flow of a mixture consisting of uniformly sized gas bubbles dispersed in a liquid. These closures were deduced through computational experiments performed using an implicit formulation of the lattice Boltzmann method with a BGK collision model. Unlike the explicit schemes described in the literature, this implicit implementation requires iterative calculations, which, however, are local in nature. While the computational cost per time step is modestly increased, the implicit scheme dramatically expands the parameter space in multiphase flow calculations which can be simulated economically. The closure relations obtained in our study are limited to a regular array of uniformly sized bubbles and were obtained by simulating the rise behaviour of a single bubble in a periodic box. The effect of volume fraction on the rise characteristics was probed by changing the size of the box relative to that of the bubble. While spherical bubbles exhibited the expected hindered rise behaviour, highly distorted bubbles tended to rise cooperatively. The closure for the drag force, obtained in our study through computational experiments, captured both hindered and cooperative rise. A simple model for the virtual mass coefficient, applicable to both spherical and distorted bubbles, was also obtained by fitting simulation results. The virtual mass coefficient for isolated bubbles could be correlated with the aspect ratio of the bubbles.

Large eddy simulation of the Gas–Liquid flow in a square cross-sectioned bubble column

In this work the use of large eddy simulations (LES) in numerical simulations of the gas–liquid flow in bubble columns is studied. The Euler–Euler approach is used to describe the equations of motion of the two-phase flow. It is found that, when the drag, lift and virtual mass forces are used, the transient behaviour that was observed in experiments can be captured. Good quantitative agreement with experimental data is obtained both for the mean velocities and the fluctuating velocities. The LES shows better agreement with the experimental data than simulations using the k– ε model.

On the Wave Dispersion Relevant to the Virtual Mass Terms in the Two-Phase Two-Fluid Model

Interfacial pressure jump terms are introduced in the momentum equations of the two-fluid two-phase flow while at the same time we keep the conventional virtual mass force as a nonobjective formulation. The pressure discontinuity across a thin interface due to the surface tension is compactly represented by a function of the fluid bulk moduli. The governing equations with the interfacial pressure jump terms produced a hyperbolic equation system having real eigenvalues for the bubbly flow, regardless of whether the virtual mass terms are added or not. The mixture sound speed for the two-phase flow evaluated using the combined interfacial pressure jump and conventional virtual mass terms has shown increasing dispersion of the small-amplitude waves when the virtual mass coefficient is larger. When the virtual mass terms are added to account for the accelerating flow, care should be exercised therefore not to introduce nonphysical wave dispersion.

Particle Trajectory in Turbulent Boundary Layer at High Particle Reynolds Number

Particle Trajectory in Turbulent Boundary Layer at High Particle Reynolds Number

Phase Distribution Measurements during Transient Bubbly Two-Phase Flow in a Vertical Pipe

An experimental procedure for investigating transient bubble flow for an adiabatic air/water system with vertical upward flow in a pipe is presented. The results of the measured local transient two-phase flow parameters are shown along a pipe length of approx. four meters. From the measured radial phase distributions under steady and under transient conditions one can draw conclusions about the interfacial forces. Here, the effects indicate the action of forces such as a transverse lift force and a time dependent force like the virtual mass force during the transient. For modelling the transverse lift force which seems to play a dominant role for that flow regime the formulation of Zun was chosen and it was implemented into the commercial CFD-Code Fluent Release 4.4.4 via user-defined subroutines. Finally, results from the simulation of the steady states of start and end conditions of an experimental measured transient are shown

Basic design of a prototype liquid metal magnetohydrodynamic power generator for solar and waste heat

Liquid metal magnetohydrodynamic power conversion (LMMHD PC) systems have been recently proposed for electrical power generation for various heat sources. These systems are based on extension of the Faraday law of induction to liquid metal. They contain high density liquid metal and a suitable thermodynamic fluid. A unique solar and waste heat coupled demonstration plant has been designed for generation of hydrogen. A steady-state two-fluid model, consisting of a one-dimensional continuity equation for each phase, combined momentum equation, vapour momentum equation and appropriate drag force for multibubble, churn-turbulent and slug flow along with the virtual mass force and related transport and thermodynamic relations, have been used to determine various two-phase parameters in the riser. Electrodynamic equations have been solved for the MHD generator. End losses due to recirculating currents at the entrance and exit of the generator have been taken into account. Single fluid equations in the rest of the loop have been solved. Liquid metal lead and lead bismuth have been chosen as the electrodynamic fluid and steam as the thermodynamic fluid. Extensive optimization of various parameters, like geometry, fluid flow rates, electrical parameters etc., have been performed to design the LMMHD PC system. The optimized system consists of two loops of LMMHD—Loop 1 operating with lead at 350°C and loop 2 operating with lead–bismuth (25% of bismuth) at 215°C. Steam at 430°C enters loop 1 after receiving thermal energy of 2.5 MW from the regenerator, solar tower and incinerator waste heat. Net electrical power of 230 kW is generated by the system. The net efficiency of conversion from thermal to electrical is 9.2 % with a voltage of 3.4 V and current of 84 kA.

Effects of Source Terms and Virtual Mass Force on Mathematical Nature of a Two-Fluid Model. 2nd Report. Examination of Mathematical Nature.

As is by now well known, partial differential equations based on a one dimensional, one-pressure two fluid model are mathematically ill-posed as an initial-value problem. Up to the present, however, eigenvalue analysis of the two fluid model has been conducted only for a simplified two-fluid model without any source terms, since a necessary condition of well-posedhess can be derived from differential terms only. Consequently, effects of source terms have not been systematically investigated. In the previous report, we proposed the method to examine mathematical nature of the complete two-fluid model including source terms ; the effects of gravity were evaluated using this method. As a result, it was confirmed the two fluid model can be mathematically ill-posed due to source terms. In the present study, the effects of interfacial drag force and wall friction force were examined using the same method. The effects of virtual mass force were also evaluated, since its influence has not been clarified in spite of the fact that it can affect the numerical stability.

Flow Control Characteristics of Gas-Magnetic Fluid Two-Phase Flow by the Application of a Magnetic Field.

The magnetic flow control characteristics of magnetic fluid two-phase slug flow in a vertical pipe was examined from measured void fractions and pressure drops together with the two-fluid model. The axial pressure distribution was primarily dominated by magnetic pressure rise and drop, and acceleration loss both of which depended on the axial distribution of void fraction. As the interaction between gas and liquid flows, the virtual mass force was remarkably large near the position of maximum magnetic field strength. The drag force acted on gas phase greatly due to large slip ratio which was caused by the magnetic force.

Effects of Source Terms and Virtual Mass Force on Mathematical Nature of a Two-Fluid Model. 1st Report. Effects of Gravity.

According to mathematical theorems on initial-value problem, real-valued characteristic roots are necessary for a system of partial differential equations. However, a one-dimensional, one-pressure two-fluid model has complex-valued characteristic root, so that it is called mathematically ill-posed. Up to the present, eigenvalue analysis of a two-fluid model has been conducted only for a simplified two-fluid model without any source terms. Although these simplified analyses can yield a necessary condition for well-posedness, they cannot give a sufficient condition. A necessary and sufficient condition can be obtained if and only if the eigenvalues of a complete two-fluid model with source terms are evaluated. Furthermore, numerical stability of the two-fluid model is also affected by source terms. The examination of source terms is, hence, indispensable to clarify the mathematical and numerical nature of a two-fluid model. In this report, the method to examine the mathematical nature of a complete two-fluid model with source terms is presented. Based on this method, it is revealed that the degree of ill-posedness of the two-fluid model is greatly affected by gravity.

Ensemble-Average Equations of a Particulate Mixture

A method of evaluating the transfer terms appearing in the ensemble-average fluid transport equation is developed and applied to obtain the transport equations describing flow of a dilute particulate mixture containing solid spherical particles. The equations apply in the limit where interactions between phases are both quasi-steady and viscous, which is defined as flows that meet the three criterion Ref(a/ℒ)2 ≪ 1, vfτ/a2 ≫ 1, and Rep ≪ 1. In this limit, the terms describing transfer of momentum between the two phases of the mixture are evaluated to O(1) in the particle radius and O(γp) in the particle phase density. The continuity equations and consistency principle are exact. When the first two conditions are not met, the transport equations require the terms that describe virtual mass forces; when the third is not met, the transport equations require terms that describe Oseen corrections to the drag term.

Theoretical analysis of the particle acceleration process in abrasive water jet cutting

In this paper, a general modelling for the acceleration process of abrasive particles in a high pressure water jet is presented. For this purpose, a new mathematical method is used which consists of the analytical resolution of the differential and non-linear equation of particle motion within a high speed waterjet flowing in a mixing nozzle (mixing tube). As a result, the equations of motion reveal that the particle velocity increases while the fluid jet velocity decreases as function of the distance within the mixing tube with respect to the assumption of conservation of momentum along the acceleration process. The original method of resolution used here enabled the authors to take account of all of the interfacial forces that act on the particle such as drag and virtual mass force. Moreover, it becomes more possible to predict the particle velocity at impact considering the real conditions of jet formation where air is entrained into the jet to feed abrasive particles by the Ventury phenomenon. The experimental validation of the present theoretical modelling has been conducted successfully by means of an experimental correlation which links the estimated particle velocity at impact and the experimental depth of cut. So the theoretical modelling of particle acceleration coupled with the experimental correlation provides a good estimate for control of cutting parameters such as hydraulic pressure of water, abrasive mass flow rate, traverse rate and depth of cut.

MODELING OF CONCENTRATED LIQUID-SOLIDS FLOW IN PIPES DISPLAYING SHEAR-THINNING PHENOMENA

This paper describes our work in modeling concentrated liquid-solids flows in pipes. Based on our previous analyses, some concentrated liquid-solids suspension flows display shear-thinning rather than Newtonian phenomena. Therefore, we developed a new two-phase non-Newtonian power-law model that includes the effect of solids concentration on solids viscosity. With this new two-phase power-law solids-viscosity model, and with constitutive relationships for interfacial drag, virtual mass and shear lift forces, and solids partial-slip boundary conditions at the pipe walls, COMMIX-M is capable of analyzing concentrated three-dimensional liquid-solids flows.