A two-fluid model for pulsatile flow in catheterized blood vessels

The pulsatile flow of blood through a catheterized artery is analyzed, assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the peripheral region of plasma as a Newtonian fluid. The resulting non-linear implicit system of partial differential equations is solved using perturbation method. The expressions for shear stress, velocity, flow rate, wall shear stress and longitudinal impedance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio, amplitude, pulsatile Reynolds number ratio and peripheral layer thickness are discussed. It is observed that the velocity distribution and flow rate decrease, while, the wall shear, width of the plug flow region and longitudinal impedance increase when the yield stress increases. It is also found that the velocity increases, but, the longitudinal impedance decreases when the thickness of the peripheral layer increases. The wall shear stress decreases non-linearly, while, the longitudinal impedance increases non-linearly when the catheter radius ratio increases. The estimates of the increase in the longitudinal impedance are considerably lower for the present two-fluid model than those of the single-fluid model.     

Scaling laws for gas-solid riser flow through two-fluid model simulation

Scale up of gas-solid circulating fluidized bed (CFB) risers poses many challenges to researchers.In this paper,CFD investigation of hydrodynamic scaling laws for gas-solid riser flow was attempted on the basis of two-fluid model simulations,in particular,the recently developed empirical scaling law of Qi,Zhu,and Huang (2008).A 3D computational model with periodic boundaries was used to perform numerical experiments and to study the effect of various system and operating parameters in hydrodynamic scaling o...     

A two-fluid model of non-Newtonian blood flow induced by peristaltic waves

The problem of blood flow induced by peristaltic waves in a uniform small diameter tube has been investigated. Blood has been represented by a two-fluid model consisting of a core region of suspension of all the erythrocytes, assumed to be a Casson fluid, and a peripheral layer of plasma as a Newtonian fluid. The expressions for dimensionless pressure drop and friction force have been obtained. The results obtained in the analysis have been evaluated numerically and discussed briefly. The significance of the present model over the existing models has been pointed out by comparing the results with other theories both analytically and numerically.     

Analysis of time integration methods for the compressible two-fluid model for pipe flow simulations

In this paper we analyse different time integration methods for the two-fluid model and propose the BDF2 method as the preferred choice to simulate transient compressible multiphase flow in pipelines. Compared to the prevailing Backward Euler method, the BDF2 scheme has a significantly better accuracy (second order) while retaining the important property of unconditional linear stability (A-stability). In addition, it is capable of damping unresolved frequencies such as acoustic waves present in the compressible model (L-stability), opposite to the commonly used Crank–Nicolson method. The stability properties of the two-fluid model and of several discretizations in space and time have been investigated by eigenvalue analysis of the continuous equations, of the semi-discrete equations, and of the fully discrete equations. A method for performing an automatic von Neumann stability analysis is proposed that obtains the growth rate of the discretization methods without requiring symbolic manipulations and that can be applied without detailed knowledge of the source code.The strong performance of BDF2 is illustrated via several test cases related to the Kelvin–Helmholtz instability. A novel concept called Discrete Flow Pattern Map (DFPM) is introduced which describes the effective well-posed unstable flow regime as determined by the discretization method. Backward Euler introduces so much numerical diffusion that the theoretically well-posed unstable regime becomes numerically stable (at practical grid and timestep resolution). BDF2 accurately identifies the stability boundary, and reveals that in the nonlinear regime ill-posedness can occur when starting from well-posed unstable solutions. The well-posed unstable regime obtained in nonlinear simulations is therefore in practice much smaller than the theoretical one, which might severely limit the application of the two-fluid model for simulating the transition from stratified flow to slug flow. This should be taken very seriously into account when interpreting results from any slug-capturing simulations.     

One-dimensional two-fluid equations for horizontal stratified two-phase flow

Advanced computer codes for water reactor loss-of-coolant analysis are based on the use of the two-fluid model of two-phase flow, in which conservation equations are solved for the gas and liquid phases separately. The standard two-fluid equations, however, sometimes predict the growth of instabilities in the flow, and occasionally become improperly posed. These difficulties have in the past led to the proposal of several different forms for the conservations equations.To help resolve these uncertainties a widely accepted form of the one-dimensional two-fluid equations is used to calculate wave propagation speeds, and stability limits, for the illustrative case of a frictionless horizontal stratified gas-liquid flow. Calculated propagation velocities are shown to agree with the appropriate limit of an exact solution, and the predicted stability limits are found consistent with available observations on the stability of the stratified flow regime.These comparisons help improve confidence in the ability of the two-fluid equations to analyse more complex problems in transient two-phase flow.     

A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems

The aim of this paper is to describe a simple Eulerian interface-capturing approach for the efficient numerical resolution of a hybrid barotropic and non-barotropic two-fluid flow problem in more than one space dimension. We use the compressible Euler equations as a model system with the thermodynamic property of each of the barotropic and non-barotropic fluid components characterized by the Tait and Noble–Abel equations of state, respectively. The algorithm is based on a volume fraction formulation of the equations together with an extended equation of state that is devised to give an approximate treatment for the mixture of more than one fluid component within a grid cell. A standard high-resolution wave propagation method is employed to solve the proposed two-fluid model with the dimensional-splitting technique incorporated in the method for multidimensional problems. Several numerical results are presented in one and two space dimensions that show the feasibility of the algorithm as applied to a reasonable class of practical problems without the occurrence of any spurious oscillation in the pressure near the smeared material interfaces. This includes, in particular, solutions for a study on the variation of the jet velocity with the incident shock pressure arising from the collapse of an air cavity in water under a shock wave.     

Two-group drift-flux model for closure of the modified two-fluid model

In an effort to improve the prediction of void fraction and heat transfer characteristics in two-phase systems, closure relations to the one-dimensional modified two-fluid model are addressed. The drift-flux general expression is extended to two bubble groups in order to describe the void weighted mean gas velocities of spherical/distorted (group-1) bubbles and cap/slug/churn-turbulent (group-2) bubbles. Therefore, correlations for group-1 and group-2 distribution parameters and drift velocities are proposed and evaluated with experimental data. Furthermore, the covariance in the convective flux of the one-dimensional two-fluid model is addressed and interpreted with the available database. The dataset chosen for evaluation of the two-group drift-flux general expression contains 126 total data points taken in an annulus geometry. The proposed distribution parameters show an agreement within ±4.9% and ±1.2% for group-1 and group-2 data, respectively. The overall estimation of group-1 and group-2 void weighted mean gas velocity calculated with the newly proposed two-group drift-flux general expression shows an agreement of ±11.8% and ±17.7%, respectively.     

A mechanistic critical heat flux model for subcooled flow boiling based on local bulk flow conditions

The critical heat flux (CHF) mechanisms for subcooled flow boiling are reviewed. Based on experimental observations reported by previous investigators, the authors have developed a new mechanistic CHF model for vertical subcooled flow at high pressure and high mass velocity. This model is based on the dryout of a thin liquid layer (sublayer) beneath an intermittent vapor blanket due to a Helmholtz instability at the sublayer-vapor interface. The parametric trends of CHF have been explored qualitatively and quantitatively with respect to variations in pressure, mass velocity, subcooling and tube diameter. Comparisons of the model predictions with experimental data for water show good agreement in the simulation of subcooled flow conditions of pressurized water reactors (PWRs).     

A two-fluid model of gas-assisted atomization including flow through the nozzle,phase inversion,and spray dispersion

This paper describes a new approach to modelling compressible gas–liquid flows that undergo change of the continuous phase. The presented model includes the system of the ensemble averaged Navier–Stokes equations together with the particle number density equation for each phase. The constitutive equations that depend on the flow regime are obtained from many sub-models that have been developed alongside the main model. Droplet size is allowed to vary in the flow field but is considered constant within a control volume. Bubbles and droplets break-up and coalescence models are adapted to the flow conditions. The proposed model for atomization treats it as a catastrophic phase inversion that takes place over the surface determined by the local values of phase volume fractions. The model is applied to simulate the premixed air-assisted atomization of water in a nozzle-type device. The computational domain includes the nozzle and the surrounding area of the spray dispersion. The model performance has been verified by comparing the predicted and measured liquid flow rates in the spray as well as the pressure values along the nozzle wall. Computational results are analysed, and the main flow features are presented.     

Numerical modeling of slug flow initiation in a horizontal channels using a two-fluid model

This paper presents a methodology for modeling slug initiation and growth in horizontal ducts. Transient two-fluid equations are solved numerically using a class of high-resolution shock capturing methods. The advantage of this method is that slug formation and growth in a stratified regime can be calculated directly from the solutions to the flow field differential equations. In addition, by using high-resolution shock capturing methods that do not contain numerical diffusion, the discontinuity generated by slugging in the flow field can be modeled with good accuracy. The two-fluid model is shown to be well-posed mathematically only under certain conditions. Under these circumstances, the two-fluid model is capable of correctly predicting and modeling the flow physics. When ill-posed, an unbounded instability occurs in the flow field solution, and the instability amplitude increases exponentially with decreasing mesh sizes. This work shows that there are three zones associated with slug formation. In addition, long wavelength slugs are shown to initiate from short wavelength waves. These short waves are generated at the interface of the two phases by the Kelvin-Helmholtz hydrodynamic instability. The results obtained through numerical modeling show good agreement with experimental results.     

High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model, which cannot describe certain nonequilibrium phenomena by finite collisions of particles, decreasing the fidelity of the solution. Based on an alternative formulation of the targeted essentially non-oscillatory(TENO) scheme, a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems. The numerical flux is constructed by the TENO interpolation of the solution and its deri...     

A model of critical boiling flow with account of wall boiling-up

A model for simulation of critical flows of boiling liquids is proposed, which takes into account the existence of two vapor phases: α-bubbles “attached” to the channel walls and β-bubbles moving in the flow. The model takes into account the possibility of breakdown of both groups of the bubbles due to both the Kelvin-Helmholtz instability caused by the flow around the bubbles and the Rayleigh-Taylor instability induced by the increase in the expansion rate of the vapor-water mixture. It is shown that the experiments on depressurization of high-pressure vessels can be explained by assuming that the initial boiling centers are located only on the walls, and in the flow core the bubbles appear only after the injection of the fragments of broken wall bubbles into the flow. The experimental oscillograms are compared with the calculated curves obtained using the model proposed and the models which take into account only one instability mode, i.e. the Kelvin-Helmholtz or Rayleigh-Taylor instability. The waves which accompany the vessel depressurization process are described.     

Separated two-phase flow model: application to critical two-phase flow

A separated flow model has been developed to allow the calculation of critical flow rates for steam-water mixtures. This model considers hydrodynamic as well as thermal non-equilibrium effects which are present due to rapid depressurization. Thus, this model incorporates interphase interaction terms for momentum, energy and mass. The mass transfer, evaporation or condensation rate, is coupled with the heat transfer between the two phases. Certain empiricisms are necessary to be included into this model, e.g. the size and number of nucleation sites at the onset of flashing. Transitions from one flow regime to the other are assumed to occur at certain void fractions. Justification of these assumptions is only possible by comparison with experimental results of different authors which in general shows good agreement.     

Particle mixing rates using the two-fluid model

In this work, a new methodology is introduced to calculate the solids mixing rate in dense gas-fluidized beds using the two-fluid model. The implementation of this methodology into an existing two-fluid model code was carefully verified. The solids phase continuity equation was satisfied using our method, and the sensitivity of the computational results to the time step, computational cell size, and discretization scheme was investigated to determine the optimal simulation settings. Using these simulation settings, the degree of solids mixing was observed to rapidly (exponentially) increase with increasing operating pressure and linearly decrease with increasing bed diameter. Our novel methodology can be applied to analyze mixing processes in large lab-scale beds as an alternative to existing time-consuming simulation techniques such as computational fluid dynamics combined with the discrete element model.     

Simulation of the two-fluid model on incompressible flow with Fractional Step method for both resolved and unresolved scale interfaces

In the present paper, the Fractional Step method usually used in single fluid flow is here extended and applied for the two-fluid model resolution using the finite volume discretization. The use of a projection method resolution instead of the usual pressure-correction method for multi-fluid flow, successfully avoids iteration processes. On the other hand, the main weakness of the two fluid model used for simulations of free surface flows, which is the numerical diffusion of the interface, is also solved by means of the conservative Level Set method (interface sharpening) (Strubelj et al., 2009). Moreover, the use of the algorithm proposed has allowed presenting different free-surface cases with or without Level Set implementation even under coarse meshes under a wide range of density ratios. Thus, the numerical results presented, numerically verified, experimentally validated and converged under high density ratios, shows the capability and reliability of this resolution method for both mixed and unmixed flows.     

Closure relations effects on the prediction of the stratified two-phase flow stability via the two-fluid model

The possibility of predicting the exact long wave linear stability boundary via the two-fluid (TF) model for horizontal and inclined stratified two-phase flow is examined. The application of the TF model requires the introduction of empirical closure relations for the velocity profile shape factors and for the wave induced wall and interfacial shear stresses. The latter are recognized as the problematic closure laws. In order to explore the closure relations effects and to suggest the necessary modifications that can improve the stability predictions of the TF model, the results are compared with the exact long wave solution of the Orr–Sommerfeld equations for the two-plate geometry. It is demonstrated that with the shape factors corrections and the inclusion of wave induced stresses effects, the TF model is able to fully reproduce the exact long wave neutral stability curves. The wave induced shear stresses in phase with the wave slope, which give rise to the so called “sheltering force”, were found to have a remarkable destabilizing effect in many cases of horizontal and inclined flows. In such cases, the sheltering effects must be included in the TF model, otherwise the region of smooth stratified flow would be significantly over predicted. Based on the results of the exact analysis, a simple closure relation for the sheltering term in the TF model is provided.     

Evaluation of the Darcy's law performance for two-fluid flow hydrodynamics in a particle debris bed using a lattice-Boltzmann model

In the present paper, multiphase flow dynamics in a porous medium are analyzed by employing the lattice-Boltzmann modeling approach. A two-dimensional formulation of a lattice-Boltzmann model, using a D2Q9 scheme, is used. Results of the FlowLab code simulation for single phase flow in porous media and for two-phase flow in a channel are compared with analytical solutions. Excellent agreement is obtained. Additionally, fluid-fluid interaction and fluid-solid interaction (wettability) are modeled and examined. Calculations are performed to simulate two-fluid dynamics in porous media, in a wide range of physical parameters of porous media and flowing fluids. It is shown that the model is capable of determining the minimum body force needed for the nonwetting fluid to percolate through the porous medium. Dependence of the force on the pore size, and geometry, as well as on the saturation of the nonwetting fluid is predicted by the model. In these simulations, the results obtained for the relative permeability coefficients indicate the validity of the reciprocity for the two coupling terms in the modified Darcy's law equations. Implication of the simulation results on two-fluid flow hydrodynamics in a decay-heated particle debris bed is discussed. Received on 1 December 1999