Multidimensional Two-fluid Model Research Articles

On the global well-posedness and optimal large-time behavior of strong solution for a multi-dimensional two-fluid plasma model

On the global well-posedness and optimal large-time behavior of strong solution for a multi-dimensional two-fluid plasma model

Effect of Collision Force on Well-Posedness and Stability of the Two-Fluid Model for Vertical Bubbly Flows

It is well-known that an incomplete two-fluid model (TFM) leads to imaginary roots of the characteristic polynomial, thus rendering the model ill-posed. A common approach to fix this problem has been to introduce sufficient numerical/artificial diffusion or nonphysical hyperbolizing terms to stabilize the model. The disadvantage of this approach is that the physical instabilities that can be accurately predicted by the TFM either get severely dampened or disappear entirely. The preferred alternative is to introduce appropriate physics that may stabilize the TFM at short wavelengths while preserving the physical long-wavelength instabilities. For instance, in near-horizontal stratified flows, the appropriate physical mechanism is surface tension. However, it is not apparent what such a mechanism should be in dispersed bubbly flows.Researchers in the past have demonstrated that the inclusion of the momentum transfer due to interfacial pressure along with virtual mass force makes the model conditionally well-posed up to a gas volume fraction of 26%. However, in practice, one may observe bubbly flows having gas concentrations beyond this theoretical limit. Hence, it is important to make the behavior of the TFM well-posed for the entire range of gas volume fractions that is physically permissible. In this paper, the often-neglected phenomenon of bubble collisions is considered. The colliding bubbles generate a dispersed-phase pressure that is resistive to increased compaction. The inclusion of bubble pressure in the TFM renders the model well-posed up to the maximum packing limit. Furthermore, it is also shown that the collision force is necessary to predict the wave propagation velocities for bubbly flows over the entire range of void fractions observed in reality. Comparisons are made with the data, and a reasonable agreement is seen. Finally, it is demonstrated with computational fluid dynamics calculations that the addition of appropriate physical mechanisms (i.e., interfacial pressure and collision) makes the multidimensional TFM well-posed.

Two-Dimensional Two-Fluid Two-Phase Flow Simulation Using an Approximate Jacobian Matrix for HLL Scheme

In this article, a numerical method to solve the two-set, eight-equation, compressible, two-fluid, two-phase flow model is developed in two dimensions as an extension of the earlier one-dimensional version. The multidimensional two-fluid model can be effectively solved by a finite-volume method in a rotated reference frame. In order to estimate the fastest wave speeds in the hyperbolic equation system for the Harten–Lax–van Leer (HLL) scheme, we first regard the liquid phase as compressible by taking the stiffened-gas equation of state. Then we derive the two-dimensional approximate Jacobian matrix and obtain the associated eight analytic eigenvalues. Using the HLL scheme, we solve a few two-phase flow problems including shape cavitation and underwater explosion, demonstrating application of the present numerical method to meaningful problems.

Volumetric interfacial area prediction in upward bubbly two-phase flow

In two-phase flow studies, a volumetric interfacial area balance equation is often used in addition to the multidimensional two-fluid model to describe the geometrical structure of the two-phase flow. In the particular case of bubbly flows, numerous works have been done by different authors on the subject. Our work concerns two main modifications of this balance equation: (1) new time scales are proposed for turbulence induced coalescence and breakup, (2) modeling of the nucleation of new bubbles on the volumetric interfacial area. The 3D module of the CATHARE code is used to evaluate our new model, in comparison to three other models for interfacial area found in the literature, on two different experiments. First, we use the DEBORA experimental data base for the comparison in the case of boiling bubbly flow. The comparison of the different volumetric interfacial area models to the DEBORA experimental data shows that even though the theoretical values of the coefficients are adopted in our modified model, this model has a quite good capability to predict the local two-phase geometrical parameters in the boiling flow conditions. Secondly, we compare the predictions obtained with the same models to the DEDALE experimental data base, for the case of adiabatic bubbly flow. In comparison to the other models tested, our model also gives quite good predictions of the bubble diameter in the case of adiabatic conditions.

Model Development for Bubble Turbulent Diffusion and Bubble Diameter in Large Vertical Pipes

Multi-dimensional analyses have been expected recently with expanding computation resources for gas-liquid two- phase flow analyses of advanced nuclear systems such as passive safety systems and natural-circulation-type reactors. However, the applicability of previous constitutive equations for multi-dimensional analyses has not been fully investigated especially for the effects of flow path scale because the equations have been assessed for small-scale experiments. In this study, we analyzed the scale effects by the multi-dimensional two-fluid model code using data in 38 mm and 200 mm diameter pipes. We clarified a key-parameter to model the scale effects and developed models for the effects on phase distribution. The scale effects can be classified by the relative relationship between bubble diameter db and turbulent length scale lT . Bubble-induced turbulence is increased under that db is smaller than lT and bubble coalescence is predominated rather than breakup under that lT is about three times larger than db and under higher void fraction. Based on these findings, we established new models for bubble turbulent diffusion and bubble diameter. The applicability was promising through assessments against the 38 mm and 200 mm pipes under different flow rates and against databases for 60 mm, 100 mm and 480 mm pipes.

Numerical Simulation of Two-Phase Boiling Flows and Prediction of DNB under PWR Conditions with a Multidimensional Two-Fluid Model.

The present paper reports numerical modeling of multi-dimensional two-phase boiling flows with a multi-dimensional two-fluid model and prediction of the Departure from Nucleate Boiling (DNB) under PWR accidental conditions. The conventional two-fluid model is adopted and various constitutive equations are incorporated to develop a computer code capable of predicting multi-dimensional void fraction distribution in a flow-boiling channel. The numerical results are compared with the data in literature of the measured profile of void fraction, with the agreement being satisfactory. To predict the occurrence of DNB, the Weisman-Pei model is adopted, which predicts the occurrence of DNB when the local void fraction exceeds a predetermined value in the near-wall buble layer. The predicted DNB power is compared with the results from an experiment of DNB power in an annular channel under PWR flow and pressure conditions, and the agreement is encouraging.

Numerical Investigation of Heat Transfer Enhancement Phenomenon during the Reflood Phase of PWR-LOCA

The heat transfer in higher power bundles was enhanced in large-scale reflood tests at Japan Atomic Energy Research Institute. The heat transfer enhancement in the core under a radial power distribution is very important to quantify the safety margin in PWR-LOCA. In this study, we analyzed the physical mechanism by numerical simulations with a multi-dimensional two-fluid model code, REFLA/TRAC, using data from the large scale reflood test. The heat transfer enhancement is caused by the increase of local upward liquid velocity resulting from the formation of flow circulation in the core. The flow circulation is generated by a radial difference of waterhead below quench front under a radial power distribution. The upward liquid velocity depends on the bundle power and the cross flow resistance. The higher power and the smaller cross flow resistance give the higher upward liquid velocity, which increases the magnitude of the heat transfer enhancement. Through the present study, some guidelines were obtained for the multi-dimensional analyses to predict the heat transfer enhancement phenomenon with high accuracy.

Best Estimate BWR Transient Analysis with TRACG Assessment using Data from BWR Startup Tests and LOCA Integral Tests

TRACG is a new version of the best estimate BWR transient analysis code, which utilizes a multi-dimensional two-fluid model for the thermal hydraulics and a three-dimensional neutron kinetics model. A three-dimensional neutronics, a fully implicit integration scheme and models for advanced BWR components have been implemented in the code upon TRAC-BF1.Assessment of TRACG has been performed in this study for the predictive capability of plant transients, which include thermal-hydraulic and neutronic interactions, as affected by responses of the plant control system. Simulations were presented for BWR representative transient tests, which were done as part of a series of BWR5 startup tests. As for the capability to predict thermal hydraulics during the design basis LOCAs, simulations were presented for the LOCA integral tests conducted in the ROSA-III at JAERI and the Hitachi TBL, which had been used for assessment of the TRAC former version.Consequently, (1)the space-dependent power flow transitions in a BWR were confirmed by TRACG simulations in which the module coupled with neutronics and thermal hydraulics during transients has been newly introduced, and (2) the characteristic thermal-hydraulic phenomena including multi-channel effects during the design basis LOCAs were confirmed, as well as the TRAC former version, by TRACG simulations on which the influence due to a fully implicit integration scheme has not extended. Capability of TRACG to predict BWR transients ranging from simple plant operational transients to design basis LOCAs was successfully demonstrated.

Two fluid model for two-phase turbulent jets

Eulerian two-fluid models are widely used in nuclear reactor safety and CFD. In these models turbulent diffusion of a dispersed phase must be formulated in terms of the fluctuating interfacial force and the Reynolds stresses. The interfacial force is obtained using the probability distribution function approach by Reeks (1992). This paper is the first application of this force to a case of engineering interest outside homogeneous turbulence. An Eulerian multidimensional two-fluid model for a cylindrical two-phase dispersed particle jet is proposed and compared with experimental data. The averaged conservation equations of mass and momentum are solved for each phase and the turbulent kinetic energy equation is solved for the continuous phase. The turbulent diffusion force and the Reynolds stresses are constituted within the context of the k- ε model of turbulence. A dissipation term has been added to the k- ε model for the turbulence modulation by the particles. Once the constitutive relations have been defined, the two-fluid model is implemented in a computational fluid dynamics code. It is shown that when the particles are very small the model is consistent with a convection-diffusion equation for particle transport where the diffusivity is defined according to Taylor's model (Taylor, G.I., 1921. Diffusion by continuous movements. Proc. London Math. Society, A20, pp. 196–211). The two-fluid model is also compared against two experimental data sets. Good agreement between the model and the data is obtained. The sensitivity of the results to various turbulent mechanisms is discussed.

Best Estimate BWR Transient Analysis with TRACG Assessment using Data from BWR Startup Tests and LOCA Integral Tests.

TRACG is a new version of the best estimate BWR transient analysis code, which utilizes a multi-dimensional two-fluid model for the thermal hydraulics and a three-dimensional neutron kinetics model. A three-dimensional neutronics, a fully implicit integration scheme and models for advanced BWR components have been implemented in the code upon TRAC-BF1. Assessment of TRACG has been performed in this study for the predictive capability of plant transients, which include thermal-hydraulic and neutronic interactions, as affected by responses of the plant control system. Simulations were presented for BWR representative transient tests, which were done as part of a series of BWR5 startup tests. As for the capability to predict thermal hydraulics during the design basis LOCAs, simulations were presented for the LOCA integral tests conducted in the ROSA-III at JAERI and the Hitachi TBL, which had been used for assessment of the TRAC former version. Consequently, (1)the space-dependent power flow transitions in a...

Gas—liquid flow in bubble columns and loop reactors: Part I. Detailed modelling and numerical simulation

The state of the art of the modelling of gas—liquid flow in bubble columns and loop reactors is critically reviewed. Recent results obtained with dynamic multidimensional two-fluid models are compared with respective steady state simulations. It will be shown that no steady state solution for uniformly aerated bubble columns can be obtained above a certain gas velocity if a sufficiently fine space grid is used. A dynamic state is obtained instead where large and rising vortices are subsequently generated at the gas distributor. Only after a long time averaging is the well known flow structure with increased gas hold-up and liquid upflow in the column centre and reduced gas hold-up with downflow near the walls obtained. The above results are based upon a two-dimension Euler—Euler model. The underlying assumptions and simplifications are discussed and the advantages and drawbacks of a corresponding Euler—Lagrangian formulation are considered. The model is used in Part II of this contribution for a detailed comparison with the experimental results from a flat bubble column with various circulating flow patterns.

Turbulent bubbly two-phase flow data in a triangular duct

Multidimensional bubbly two-phase flow data was obtained in an isosceles triangular duct having D h = 40 mm and L/ D = 73. The data include local measurements of phase distribution, velocity and Reynolds stresses at 10,000 < Re 1 < 35,000 and Re bubble = 1250. The measurements were performed with single sensor and X-sensor cylindrical hot film probes of 0.025 mm diameter. These small probes have good bubble penetration characteristics so they are capable of the simultaneous measurement of the liquid phase instantaneous velocity and the phase indicator function. These data are useful for the validation of multidimensional two-fluid CFD models.

Numerical Analysis of Bubble-Induced Natural Circulation based on Multidimensional Two-Fluid Model.

One of the next-generation reactors, SBWR (simplified boiling water reactor), is currently under development. It has a large upper plenum surrounded by the so-called "chimney" to keep a driving force of natural circulation. An accurate method for predicting bubble-induced natural circulation is therefore indispensable for the rational design of SBWR. In the present study, the applicability of the multidimensional two-fluid model to bubble-induced transient natural circulation was examined by comparing measured transient flow patterns with calculated ones. The calculation was conducted by coupling a two-dimensional incompressible two-fluid model with available constitutive equations for interfacial drag force, lift force, virtual mass force and viscous force. The 3rd-order new-class TVD scheme was adopted for the convection term of the mass equations to reduce the numerical diffusion of volumetric fractions. As the result, we could obtain good predictions for transient flow patterns. It was also confirmed that the large-scale turbulent flow field induced by the bubble plume can be predicted well without any turbulence model.

Transient simulation of 2–3D stratified and intermittent two‐phase flows. Part II: Applications

AbstractIn this paper a new type of transient multidimensional two‐fluid model has been applied to simulate intermittent or slug flow problems. Three different approaches to modelling interfacial friction, including an interfacial tracking scheme, have been investigated. The numerial method is based on an implicit finite difference scheme, solved directly in two steps applying a separate equation for the pressure. 2D predictions of Taylor bubble propagation in horizontal and inclined channels have been compared with experimental data and analytical solutions. The 2D model has also been applied to investigate a number of special phenomena in slug flow, including slug initiation, bubble turning in downflow and the bubble centring process at large liquid flow rates.

Numerical Solution Method for Incompressible Multidimensional Two-Fluid Model. A Method Suitable for the Evaluation of Constitutive Equations.

A number of accurate constitutive equations are required for reliable prediction of multidimensional two-phase flow. However, few constitutive equations have been established due to the lack of (1) simple numerical methods for solving governing equations of multidimensional two-phase flow and (2) reliable experimental data on multidimensional two-phase flow. In the present study, a simple and efficient numerical method for solving a multidimensional incompressible two-fluid model, which is based on the simplified marker and cell (SMAC) method, was proposed to facilitate the development of constitutive equations. This method enables us to calculate multidimensional two-phase transient phenomena with a personal computer. Details of the present method and a comparison of measured and predicted two-dimensional flow patterns of two-dimensional bubbly flow are described.

The analysis of phase separation and phase distribution phenomena using two-fluid models

It is the purpose of this paper to investigate the ability of multidimensional two-fluid models to predict the observed phase separation and lateral phase distribution phenomena. It will be shown that such phenomena can be predicted once all significant physical mechanisms have been adequately modelled. Indeed, this paper shows, for the first time, that closed-loop predictions are possible of multidimensional phase distribution and phase separation in two-phase flows.

An experimental study of 2-d phase separation phenomena

Experiments were performed to study phase separation and distribution phenomena in a 2-D test section. The conditions tested were intended to simulate the “chimney effect” that may occur during reflood in a pressurized water nuclear reactor (PWR) core after a hypothetical loss-of-coolant accident (LOCA). Such flow situations may enhance the cooling of the higher powered assemblies during reflood. The countercurrent two-phase flow conditions expected in a PWR were simulated using air/water in a special 2-D test section having two inlet and two outlet ports. The void fraction distribution across the test section was measured with a γ-ray densitometer for different flow conditions, including various flow rates, flow qualities and flow splits between the four ports. These flow conditions were run for cases with and without vertical rods mounted inside the test section. These rods were intended to simulate the effect of the lateral resistance to the flow due to fuel rods in a PWR core. It was found that the presence of the rods dramatically reduced the degree of recirculation observed. A variety of flow regimes were observed, including regions of single-phase water and air, and bubbly, slug and churn—turbulent flows. It is felt that these data form an excellent basis for the assessment of multidimensional two-fluid models of two-phase flow.