One-dimensional Two-fluid Model Research Articles

Thermodynamics of Alfvénic slow solar wind produced by Alfvénic-turbulence

Abstract Alfvén-wave turbulence is known as a plasma heating mechanism associated with the acceleration of fast solar wind, found emanating from open magnetic fields adjacent to coronal holes. In this study, we expand the scope of this mechanism to investigate the thermodynamics of Alfvénic slow solar wind, a phenomenon originating from open fields near a streamer, as observed in recent inner heliospheric missions. We demonstrate a one-dimensional two-fluid model that incorporates three components: (1) low-frequency Alfvén-wave turbulence, serving as the primary dissipation mechanism, (2) a curved magnetic field that reproduces the streamer’s boundary, and (3) the kinetic instabilities to address proton temperature anisotropy. Our findings suggest that this dissipation mechanism can be applied in common to both fast and Alfvénic slow solar winds. We identify the proton-cyclotron instability near the Sun and the oblique and parallel firehose instabilities occurring close to 1AU as crucial factors governing temperature anisotropy. This study contributes to our understanding of the complex thermodynamics of solar winds and provides valuable insights for future space missions.

A novel numerical approach for assessing the gas-liquid flow characteristics in pipelines utilizing a two-fluid model

The accumulation of liquid condensate in the wet gas pipelines gives negative effects on the piping efficiency and flow managements. Accurate prediction of the condensate behavior is of crucial importance in the pipeline engineering. In this study, the one-dimensional two-fluid model with the stratified modeling in its source term is used to describe the gas-liquid kinetic motion along the pipe and the flow fields obtained from the newly proposed numerical approach are utilized to analyze the flow characteristics. Firstly, four cases with varying inlet liquid fractions, outlet pressures, and different meshes are utilized to investigate the gas-liquid steady-state variations along the pipe. The numerical results indicate that the outlet pressure influences the working pressure of the inlet gas-liquid, subsequently altering the equilibrium state and momentum exchange direction near the inlet, resulting in different condensate distributions in the pipeline. In the transient analysis with initial empty pipe, the inlet gas-liquid mainly impacts the pressure wave propagations in pressure spread stage, the gas kinetic motion in gas filling stage and the motion of the dividing interface between different regions in liquid filling stage, which is validated by the proposed numerical approach through comparison with the simulation tool OLGA designed by Schlumberger. Furthermore, when the liquid condensates appear in the initial state, the transition zones have two physical forms between different flow regions, which include the sudden dividing interface due to the high liquid amount carried by the inlet gas and the transition region resulted from the liquid accumulations.

A comprehensive examination of the linear and numerical stability aspects of the bubble collision model in the TRACE-1D two-fluid model applied to vertical disperse flow in a PWR core channel under loss of coolant accident conditions

The one-dimensional Two-Fluid concept uses an area-average approach to simplify the time and phase-averaged Two-Fluid conservation equations, making it more suitable for addressing difficulties at an industrial scale. Nevertheless, the mathematical framework has inherent weaknesses due to the loss of details throughout the averaging procedures. This limitation makes the conventional model inappropriate for some flow regimes, where short-wavelength perturbations experience uncontrolled amplification, leading to solutions that need to be physically accurate. The critical factor in resolving this problem is the integration of closure relations. These relationships play a crucial function in reintroducing essential physical characteristics, thus correcting the loss that occurs during averaging and guaranteeing the stability of the model. To improve the accuracy of predictions, it is essential to assess the stability and grid dependence of one-dimensional formulations, which are particularly affected by closure relations and numerical schemes. The current research presented in the text focuses on improving the well-posedness of the TFM, specifically within the TRACE code, which is widely utilized for nuclear reactor safety assessments. Incorporating a bubble collision model in the momentum equations is demonstrated to enhance the TFM's resilience, especially in scenarios with high void fractions where conventional TFMs may face challenges. The analysis presents a linear stability analysis performed for the transient one-dimensional Two-Fluid Model of system code TRACE within the framework of vertically dispersed flows. The main emphasis is on evaluating the stability characteristics of the model while also acknowledging its susceptibility to closure relations and numerical techniques.

Energy-stable discretization of the one-dimensional two-fluid model

In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy.We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative.The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

The study on the partition method of two-phase wall drag in the one-dimensional two-fluid model

The interfacial area transport equation(IATE) is being investigated to replace the flow regime map in the one-dimensional two-fluid model (1-D TFM). The partition of two-phase wall drag is a part of flow regime-dependent constitutive relations in the code using the 1-D TFM such as RELAP5. Hence, to develop the partition method for two-phase wall drag that is consistent with 1-D TFM using the IATE, the mechanism for the partition was investigated and clarified in this work. And the partition method for two-phase wall drag that can capture the dynamic change of two-phase flow structure was developed with the clarified mechanism. Furthermore, the method was preliminarily validated with the experimental data, which shows that the two-phase wall drag computed by the method can accurately capture the variation of flow structure and relative velocity in transient two-phase flow. Accordingly, the developed method is able to take advantage of the IATE.

Remedy for ill-posedness and mass conservation error of 1D incompressible two-fluid model with artificial viscosities

The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.

The evaluation of interfacial area concentration weighted area-averaged gas velocity based on the extended drift-flux model

The interfacial area transport equation (IATE) is a more accurate and stable way to compute the interfacial area concentration than the empirical correlation, and the interfacial area concentration weighted area-averaged gas velocity is an important parameter to close the one-dimensional IATE. However, there has been no theoretical model for it in open literature until now. To fill this gap, the theoretical model of interfacial area concentration weighted area-averaged gas velocity based on the drift-flux model is developed in this work, which is named the extended drift-flux model. Meanwhile, interfacial area concentration weighted area-averaged gas velocity is evaluated based on the extended drift-flux model. And the evaluation is validated with the experimental data of vertical bubbly flow and vertical slug flow in the adiabatic pipe. This work is meaningful for the implementation of IATE in the one-dimensional two-fluid model (1-D TFM) and the computation of the area-averaged interfacial area concentration.

Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

We show that the one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the model. This result extends upon earlier work on the shallow water equations (SWE), with the important difference that we include non-conservative pressure terms in the analysis, and that we propose a formulation that holds for ducts with an arbitrary cross-sectional shape, with the 2D channel and circular pipe geometries as special cases.The second novel result of this work is the formulation of a finite volume scheme for the TFM that satisfies a discrete form of the continuous energy equation. This discretization is derived in a manner that runs parallel to the continuous analysis. Due to the non-conservative pressure terms it is essential to employ a staggered grid, which requires careful consideration in defining the discrete energy and energy fluxes, and the relations between them and the discrete model. Numerical simulations confirm that the discrete energy is conserved.

RELAP5 application for water hammer analysis in different thermal-hydraulic systems

• Causes of water hammers: rapid valve operation, void-induced, condensation-induced. • RELAP5 code may represent water hammer due to valve operation and appearance of void. • The developed modelling recommendations are based on comparative analysis. • Water hammer in nuclear, fusion and district heating systems possible due to fast valve actions. • Integrity of pipelines in nuclear, fusion and district heating systems remain not violated. The water hammer phenomenon investigations employing RELAP5 code with one-dimensional two-fluid six-equation models of water and steam flow were performed and the main results shared in this paper. The capability of RELAP5 code to simulate transients leading strong pressure pulses was investigated employing the measurement results of three test facilities for different water hammer types: PPP at UMSICHT, CWHTF at FZR Dresden and PMK-2 at AEKI. Based on the benchmarking results the recommendations for the modelling of piping systems, which could be affected by different types of water hammer were developed and tested. The recommendations cover selection of thermal-hydraulic system conditions, parameters of modelling and RELAP5 code correlations. These recommendations have enabled to develop a detail RELAP5 models for the simulation of water hammer effects in further research works. The applications of RELAP5 code for water hammer analysis in different thermal-hydraulic systems (RBMK-1500 reactor main circulation loop, targets cooling/heating circuit in Wendelstein 7-X experimental stellarator facility and pipelines in district heating system) are presented in this paper. The performed analyses of transients demonstrated that the pressure pulsations could appear in all these systems as results of check valve closures. However, pressure surges are not so strong due to specific of such systems and integrity of pipelines remains not violated.

A new two-fluid model for flow rate measurement of annular flow in horizontal pipe

Non-separation measurement is a research difficulty and hotspot in multiphase flow. In this paper, the multimodal ultrasonic sensors and differential pressure transmitter combined with two-fluid model are used to measure the wet gas flow rates under non-separation state. Firstly, a modeling measurement system is built. The liquid film thickness and pressure drop are obtained by using ultrasonic sensor and differential pressure transmitter. Based on the steady-state one-dimensional two-fluid model and the limitations of existing models, a new prediction model is proposed. Comparing the pressure drop data obtained by new two-fluid model with the experimental data, the errors are within ±30%. Then, wet gas flow rates measurement system is constructed. The liquid film thickness and gas velocity are measured by using multimodal ultrasonic sensors. Combined this new two-fluid model, the wet gas flow rates are obtained. The measurement errors of gas and liquid flow rate are within ±10% and ±15%.

Kinematic stability and simulations of the variational two-fluid model for slug flow

The two-fluid short-wave theory (TF-SWT) mode of the one-dimensional two-fluid model (TFM) [A. Clausse and M. Lopez de Bertodano, “Natural modes of the two-fluid model of two-phase flow,” Phys. Fluids 33, 033324 (2021)] showed that the incompressible kinematic and Kelvin–Helmholtz instabilities are the source of the long-standing ill-posed question. Here, the stability of the short wave mode is analyzed to obtain an unstable incompressible well-posed TFM for vertical slug flow, where inertial coupling and drag play the key role. Then, a computational method is implemented to perform non-linear simulations of slug waves. Linear stability analyses, i.e., characteristics and dispersion, of a variational TF-SWT for vertical slug flows are presented. The current TFM is constituted with a lumped-parameter model of inertial coupling between the Taylor bubble and the liquid. A characteristic analysis shows that this conservative model is parabolic, and it provides a base upon which other models can be constructed, including short-wave damping mechanisms, like vortex dynamics. The dispersion analysis shows that depending on the interfacial drag, the problem can be kinematic unstable. A new kinematic condition in terms of the inertial coupling and the interfacial drag is derived that is consistent with previous theoretical and experimental results. The material waves, which are predicted by linear stability theory, then develop into nonlinear slug waveforms that are captured by the numerical simulations. These and the horizontal stratified flow waves of previous research illustrate the TFM capability to model interfacial structures that behave like waves. Otherwise, when the physics of the TF-SWT waves is ignored, the model is ill-posed.

Investigation on the interphase drag and wall friction in vertically oriented upward and downward two-phase flows under accident conditions in light water reactors

In this study, applicability of the one-dimensional two-fluid model for predicting the vapor void fraction and the frictional pressure drop in the adiabatic upward and downward two-phase flows has been studied through comparing model predictions with experimental data of Tikhonenko (1973) taken in the 33 mm diameter pipe at system pressure ranging from 2 to 9 MPa, and rated velocity of 0.2 to 2 m/s (mass flux of 170 to 1400 kg⋅m−2 s−1). The experiments reproduce the conditions of the two-phase coolant flow which are expected to occur in the light-water-reactor systems under loss-of-coolant accidents and design extension conditions. It has been found that the existing empirical models for interfacial and wall friction used in the current thermal–hydraulic simulation codes allow adequate prediction of the vapor void fraction and the frictional pressure drop in vertical pipes only for the upward direction of two phase flow. Modeling deficiencies have been identified when these models are applied to the low velocity downward flow. For the downward two-phase flow at a rated velocity of 0.2 m/s (mass flux of 170 kg⋅m−2 s−1), the Lockhart-Martinelli model underpredicts the frictional pressure gradient of the liquid phase which leads to the over-prediction of the average velocity of the liquid phase. As a result, the calculated void fraction is higher compared to the experiment. The possible ways to improve the current models of the wall friction in two-phase flows have been discussed.

Partition method of wall friction and interfacial drag force model for horizontal two-phase flows

The improvement of thermal-hydraulic analysis techniques is essential to ensure the safety and reliability of nuclear power plants. The one-dimensional two-fluid model has been adopted in state-of-the-art thermal-hydraulic system codes. Current constitutive equations used in the system codes reach a mature level. Some exceptions are the partition method of wall friction in the momentum equation of the two-fluid model and the interfacial drag force model for a horizontal two-phase flow. This study is focused on deriving the partition method of wall friction in the momentum equation of the two-fluid model and modeling the interfacial drag force model for a horizontal bubbly flow. The one-dimensional momentum equation in the two-fluid model is derived from the local momentum equation. The derived one-dimensional momentum equation demonstrates that total wall friction should be apportioned to gas and liquid phases based on the phasic volume fraction, which is the same as that used in the SPACE code. The constitutive equations for the interfacial drag force are also identified. Based on the assessments, the Rassame-Hibiki correlation, Hibiki-Ishii correlation, Ishii-Zuber correlation, and Rassame-Hibiki correlation are recommended for computing the distribution parameter, interfacial area concentration, drag coefficient, and relative velocity covariance of a horizontal bubbly flow, respectively.

A suitability analysis of transient one-dimensional two-fluid numerical models for simulating two-phase gas-liquid flows based on benchmark problems

Gas-liquid two-phase flows in pipes are present in a variety of engineering applications. These applications require transient one-dimensional mathematical models coupled with robust numerical methods to simulate the flow behavior in time and space. As there exist several possible combinations of models and numerical methods, questions arise as to the suitability of these numerical models to best accommodate the problem features. Aiming to partially answer this question, we present in this work numerical simulations of two well-known gas-liquid benchmarks, the water faucet, and the shock tube problems, carried out with two two-fluid models – 4E1P (four equations and one pressure) and 5E2P (five equations and two pressures) – in association with three numerical methods – FCT (Flux-Corrected Transport method), FORCE (First-Order Centered Scheme) and ModFORCE. The numerical methods used herein are written in a general form and do not require the solution of the associated Riemann problem. The obtained results are sought to serve as a reference to choose the best approaches, among the ones studied, to tackle some transient one-dimensional two-phase flow problems, highlighting the advantages and disadvantages of each model and method combination regarding hyperbolicity, diffusive and dispersive effects. Overall, the combination of the 5E2P model with the ModFORCE numerical method provides the most accurate results compared to the analytical solutions available.

Effect of collisions on the plasma sheath in the presence of an inhomogeneous magnetic field

A low-pressure magnetized plasma is studied to find the dependency of sheath properties on ion-neutral collisions in presence of an inhomogeneous magnetic field. A self-consistent one-dimensional two-fluid hydrodynamic model is considered, and the system of equations is solved numerically. The study reveals that the width of the plasma sheath expands and space charge increases with collisions. The ion-neutral collisions and the inhomogeneous magnetic field restrict the ions to move towards the surface. The movement of the ions towards the wall can be controlled by choosing a suitable configuration of the magnetic field and ion-neutral collision frequency. A comparison between two different magnetic field configurations has been presented alongside to differentiate the commonly found scenarios in the field. The outcome of the study is supposed to help in understanding the complex dynamics of ions in plasma confinement and plasma processing of materials. Furthermore, the present work seeks to create a framework for two-fluid modeling of magnetized plasmas with any arbitrary magnetic field profiles. The analysis provided here is supposed to act as a basis for any future work in the respective field.

Density Wave Instability Verification of 1-D Computational Two-Fluid Model

A new dynamic verification of the one-dimensional (1-D) computational Two-Fluid Model (TFM) using the Type II density wave instability (DWI) theory of Ishii is presented. Verification requires convergence in the sense of the Lax Equivalence Theorem and dynamic comparison with the DWI theory. Rigorous verification of the computational TFM must be performed with a computational model that is well posed without regularization because, otherwise, since the theory of Ishii is well posed, regularization would make the TFM incompatible with it. Furthermore, since the TFM is well posed, it was possible to implement a second-order numerical method with a flux limiter that, together with a fine mesh, achieves numerical convergence. This is significant because numerical convergence and consistency, both of which are demonstrated, are prerequisites for the rigorous dynamic verification according to the Lax Equivalence Theorem. Thus, the apparent but previously unproven numerical verification of the 1-D TFM to simulate the two-phase long wave DWI instability is hereby performed.

Assessment of one-dimensional models used in the simulation of gas-liquid flows in vertical degassing columns

The present research aims to assess and analyze one-dimensional two-fluid models used in simulating gas-liquid dynamics and interphase transfers in degassing columns. The assessment of two-fluid models is based on benchmark measurement data carried out over a long period of time, both on dissolved gas concentrations in Lake Nyos at different depths and on the gas-liquid jet heights at the outlet of the degassing columns installed from 2001. The results of the numerical simulations with different two-fluid models were compared and their confrontation with the experimental data were discussed with respect to the sensitivity of the models to the different closures and hypotheses. The numerical analysis has led to a benchmark assessment of mono-dimensional two-fluid models in their capacity to account for the complexities associated with the different couplings between the dynamic aspects and interphase mass transfer phenomena that occur in degassing columns.

Linear stability analysis of RELAP5 two-fluid model in nuclear reactor safety results

System thermal-hydraulic code RELAP5 is based on a two-fluid, non-equilibrium, and non-homogeneous hydrodynamic model for simulation of transient two-phase behavior. The code model includes six governing equations to incorporate the mass, energy, and momentum of the two fluids. In this paper, linear stability analysis is performed to check the ill-posedness of the RELAP5 specific two-fluid model (TFM) for all normal and accident conditions of a standard pressurized water reactor (PWR). The analysis gives information about the soundness of the model and identifies the range of parameters where the solutions obtained from the model will be numerically convergent. The linear two-phase fluid dynamic stability (by dispersion analysis) of the RELAP5 one-dimensional two-fluid model and numerical stability of the difference equation formulation (by Von Neumann method) is presented. The present analysis shows that the two-fluid model becomes ill-posed for some fluid conditions, where the results are less accurate, so sensitivity analysis plays an important role. The ill-posed nature implies that results thus obtained (by finite difference method) have to be interpreted carefully because of the sensitive nature of reactor safety analysis. It is also identified that the variation in various parameters (like slip ratio, system pressure, void fraction, and phasic velocities) can affect the error growth rates. It has been demonstrated that the basic system of one-dimensional two-phase flow equations, that possesses complex characteristics, exhibits unbounded instabilities in the short-wavelength limit and constitutes an improperly posed initial value problem. The semi-implicit numerical method, which is unconditionally stable for hyperbolic systems, becomes unstable for non-hyperbolic systems. For some of the fluid conditions, even after the introduction of artificial viscosity terms (in the difference equation formulation) that damp the high-frequency spatial components of the solution, are not sufficient for regularization of the two-fluid model. Thus, there is a need for the addition of newer terms, e.g. bubble collision, so that the existing incomplete model provides better results. It is also demonstrated that the basic TFM of RELAP5 with additional collision term makes the system unconditionally well-posed which are originally conditionally well-posed. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. It is found that the instability associated with the two-fluid model discretized system of equations is related to the instability associated with the ill-posedness of the two-fluid model but is quantitatively different.

Assessment of closure relations on the numerical predictions of vertical annular flows with the two-fluid model

In the present work, different closure relations in the one-dimensional Two-Fluid Model were investigated in order to assess the effect on the numerical simulation of vertical annular flows, including the formation and propagation of waves at the gas-liquid interface. Different interfacial friction formulations, momentum flux parameters and dynamic pressure terms were employed. The conservation equations were discretized and solved within the finite volume framework, with a first order time discretization, a second order TVD and first order Upwind schemes for the convective term. A rigorous mesh sensitivity analysis was performed, showing the advantages and shortcomings of each discretization scheme, as well as the impact of closure formulations and their importance for better physical representativeness and well-posedness. Flow and wave parameters were obtained with the different approaches for various test cases taken from the literature. A sensitivity analysis was performed with varying momentum flux parameter values, interfacial friction factors and dynamic pressure expressions. Liquid film thickness, pressure-drop and wave characteristics results were systematically compared to experimental data. Predictions for pressure gradient and mean liquid holdup with the best set of closures presented an average error of 9% and 22%. Comparison of the wave characteristics, given by the group speed and frequency with available measured values has shown that the model is capable to simulate the qualitative features of wave formation and propagation in vertical annular flow with a reasonable quantitative agreement, given the complexity of the physical phenomena involved. However, while it has been shown through the sensitivity analysis that the closure relations investigated here have an important effect on quantities of interest in annular flows, in particular that a momentum flux parameter value larger than unit seems beneficial, further investigations are suggested for a wider range of conditions, providing a robust physical support for the generalization of the closure model formulations.

Mass transfer modeling and simulation of a transient homogeneous bubbly flow in a bubble column

Bubble columns have wide applicability in the industry as contactors such as absorbers, strippers, and multiphase reactors. Despite their simple construction and operation, the success in the design of bubble columns requires deep knowledge of transport phenomena applied to multiphase flow. Modeling and simulating the fluid dynamics through numerical methods may circumvent the high cost of empirical methods when analyzing key parameters for the design and scale-up of bubble columns. The main objective of this work is to model and simulate the interfacial mass transfer in a bubble column, for both steady-state and transient analysis. The equations of mass, momentum, and species conservation are described in a Euler-Euler framework using a one-dimensional Two-Fluid Model (TFM), taking adequate closure relations for modeling interfacial transport phenomena. The equations are integrated by the Finite Volume Method (FVM) in an algorithm developed by the authors and the numerical results obtained are compared with experimental data found in the literature. The simulations are predictive and run with no need for matching experimental data, which is the main contribution of this work. The results obtained showed good agreement when compared with experimental data. Additionally, a high-order in space (TVD) and time (Crank-Nicolson) is developed to capture the gradients of gas fraction and composition, providing a remarkable tool for transient analysis.