Standard Two-fluid Model Research Articles

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.

Enhanced Magnetoresistance Effect in Graphene Coupled to a Ferromagnetic Oxide with Charge Orbital Ordering

In this paper, we fabricated graphene/Fe3O4 heterostructure devices by stacking monolayer graphene on magnetite (Fe3O4) substrate and investigated their magneto-transport properties. Interestingly, graphene/Fe3O4 heterostructure devices exhibit a giant magnetoresistance (MR) of 70% at a low magnetic field of 0.65[Formula: see text]T and at 11[Formula: see text]K, which is three times greater than that of graphene on SiO2. Based on standard two-fluid model and LDA[Formula: see text]U simulation, we showed that the observed enhanced MR effect is due to the increased disorder in graphene induced through the charge polarization via the alignment of C atoms of graphene over the charge ordered B-site cations of Fe3O4. Our results demonstrate a potential way to enhance graphene MR effect through coupling graphene with a suitable substrate with charge orbital ordering.

Estimation of the spatial discretization error in numerical simulations of bubbly flows

Accurate spatial discretization error estimates are essential for some CFD applications. Notwithstanding, this subject has been systematically disregarded in the literature on bubbly flow simulation. This work evaluated the suitability of three error estimators (EEMs) in simulations of a benchmark bubble column. The employed EEMs were: Direct Taylor series (DTS), Richardson Extrapolation (RExt), and Eça-Hoekstra Least-Squares (EH-LS). The modeling approaches for the bubbly flow were: standard two-fluid model (TFM), hyperbolic TFM (hTFM), and mixture model (MixtM). Geometric means of the estimated error and of the difference between the solution obtained with the considered mesh and a nearly mesh-independent solution were used as comparison metrics. DTS resulted in inaccurate estimates for coarse meshes. Both EH-LS and RExt generated accurate results for MixtM, the latter presenting lower computational cost. RExt could not deal adequately with the oscillatory convergence pattern of hTFM. EH-LS was the only method suitable for both MixtM and hTFM.

Exact conservative solutions of fluid models for the scrape-off layer as the ancestors of plasma blobs?

New exact, analytical solutions are presented for the conservative part of a standard two-fluid (density plus vorticity) model of the scrape-off layer (SOL) which are of the travelling-wave type and describe the transport of large, machine-scale structures across a plasma cross-section (radially and/or poloidally). It is conjectured that amongst these conservative solutions (some extended throughout space, others much more localised) might be the ancestors of propagating coherent structures, known as blobs, often seen in experiments and numerical simulations of SOL turbulence. Several types of solutions can be obtained, capable of mimicking not only high-density blobs propagating outwards, but also inwardly moving plasma holes (structures with densities lower than the backrgound’s). Besides their fundamental interest as conservative solutions of the equations describing SOL turbulence, these exact solutions have the added value of providing benchmarks for the verification of numerical algorithms, as is here illustrated. Having thus verified one’s numerical implementation, a more realistic SOL model (including diffusion, parallel losses and a source of core plasma) is solved (by gradually adding the extra terms to the conservative part) to check whether these conservative solutions survive the full dynamics or not. They actually do survive (albeit enduring some degree of distortion, ending up by being eventually lost) for parameters representative of SOL plasmas in present-day fusion devices, thus somewhat vindicating one’s original conjecture that they might be the ancestors of blobs. In addition, being actual solutions of the conservative part of the fluid SOL equations (and with the possibility of taking essentially Gaussian forms), there is no need to further justify their use as seeds with which to initialise (so-called seeded) simulations. A further characteristic that these Gaussian, blob-like conservative solutions possess is that they have intrinsic net vorticity (or spin), which is also believed to be the case for turbulence-created blobs.

Large Time Step Roe scheme for a common 1D two-fluid model

We present the Large Time Step (LTS) extension of the Roe scheme and apply it to a standard two-fluid model. Herein, LTS denotes a class of explicit methods that are not limited by the CFL (Courant–Friedrichs–Lewy) condition, allowing us to use very large time steps compared to standard explicit methods. The LTS method was originally developed in the nineteen eighties (LeVeque, 1985), where the Godunov scheme was extended to the LTS Godunov scheme. In the present work, the relaxation of the CFL condition is achieved by increasing the domain of dependence. This might lead to difficulties when it comes to boundary and source terms treatment. We address and discuss these difficulties and propose different ways to treat them. For a shock tube test case, where there are neither source terms nor difficulties associated with the boundaries, the method increases both accuracy and efficiency. For a water faucet test case that includes a source term, the method increases the efficiency, while the accuracy strongly depends on the appropriate treatment of boundary conditions and source terms.

Simulation of 3D freely bubbling gas–solid fluidized beds using various drag models: TFM approach

In this article, 3D modeling and simulation of bubbling fluidized beds has been conducted using various drag models, and the model predictions were validated against reported experimental data and 2D simulation results. In this regard, different drag models reported in the literature including Gidaspow, Syamlal–O’Brien, Hill–Koch–Ladd, and Wen–Yu were applied. A standard Two-Fluid Model (TFM) closed by the Kinetic Theory of Granular Flows (KTGF) was used to simulate bubbling gas–solid fluidized beds. Excellent agreements between the simulation results and experimental data, concerning bed expansion ratio, gas volume fraction, and time-averaged particles velocity, were found over a wide range of particle size, static bed height, and fluidization velocity. Moreover, comparison of 2D and 3D simulation results with experimental data shows that overpredictions attributed to 2D simulations can be a direct result of neglecting frictional stresses. In addition, it was found that the Wen–Yu drag model can provide better predictions for the bed expansion ratio and solids velocity relative to other drag models.

Modeling average gas–solid heat transfer using particle-resolved direct numerical simulation

The purpose of this work is to develop gas–solid heat transfer models using Particle-resolved Direct Numerical Simulations (PR-DNS). Gas–solid heat transfer in steady flow through a homogeneous fixed assembly of monodisperse spherical particles is simulated using the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). PR-DNS results are obtained over a range of mean slip Reynolds number (1–100) and solid volume fraction (0.1–0.5). Fluid heating is important in gas–solid heat transfer, especially in dense low-speed flows, and the PUReIBM formulation accounts for this through a heat ratio which appears in the thermal self-similarity boundary condition that ensures thermally fully-developed flow. The average volumetric interphase heat transfer rate (average gas–solid heat transfer) that appears in the average fluid temperature evolution equation is quantified and modeled using PR-DNS results. The Nusselt number corresponding to average gas–solid heat transfer is obtained from PR-DNS data, and compared with Gunn’s Nusselt number correlation (Gunn, 1978). A new Nusselt number correlation is proposed that fits the PR-DNS data more closely and also captures the Reynolds number dependence more accurately. It is shown that the use of Nusselt number correlations based on the average bulk fluid temperature in the standard two-fluid model for gas–solid heat transfer is inconsistent, and results in up to 35% error in prediction of the average gas–solid heat transfer. Using PR-DNS data, a consistent two-fluid model is proposed that improves the predicted average gas–solid heat transfer.

PICKUP ION MEDIATED PLASMAS. I. BASIC MODEL AND LINEAR WAVES IN THE SOLAR WIND AND LOCAL INTERSTELLAR MEDIUM

Pickup ions (PUIs) in the outer heliosphere and the local interstellar medium are created by charge exchange between protons and hydrogen (H) atoms, forming a thermodynamically dominant component. In the supersonic solar wind beyond >10 AU, in the inner heliosheath (IHS), and in the very local interstellar medium (VLISM), PUIs do not equilibrate collisionally with the background plasma. Using a collisionless form of Chapman-Enskog expansion, we derive a closed system of multi-fluid equations for a plasma comprised of thermal protons and electrons, and suprathermal PUIs. The PUIs contribute an isotropic scalar pressure to leading order, a collisionless heat flux at the next order, and a collisionless stress tensor at the second-order. The collisionless heat conduction and viscosity in the multi-fluid description results from a non-isotropic PUI distribution. A simpler one-fluid MHD-like system of equations with distinct equations of state for both the background plasma and the PUIs is derived. We investigate linear wave properties in a PUI-mediated three-fluid plasma model for parameters appropriate to the VLISM, the IHS, and the solar wind in the outer heliosphere. Five distinct wave modes are possible: Alfven waves, thermal fast and slow magnetoacoustic waves, PUI fast and slow magnetoacoustic waves, and an entropy mode. The thermal and PUI acoustic modes propagate at approximately the combined thermal magnetoacoustic speed and the PUI sound speed respectively. All wave modes experience damping by the PUIs through the collisionless PUI heat flux. The PUI-mediated plasma model yields wave properties, including Alfven waves, distinctly different from those of the standard two-fluid model.

Oscillations of rapidly rotating superfluid stars

Using time evolutions of the relevant linearized equations, we study non-axisymmetric oscillations of rapidly rotating and superfluid neutron stars. We consider perturbations of Newtonian axisymmetric background configurations and account for the presence of superfluid components via the standard two-fluid model. Within the Cowling approximation, we are able to carry out evolutions for uniformly rotating stars up to the mass-shedding limit. This leads to the first detailed analysis of superfluid neutron star oscillations in the fast rotation regime, where the star is significantly deformed by the centrifugal force. For simplicity, we focus on background models where the two fluids (superfluid neutrons and protons) corotate, are in β-equilibrium and co-exist throughout the volume of the star. We construct sequences of rotating stars for two analytical model equations of state. These models represent relatively simple generalizations of single fluid, polytropic stars. We study the effects of entrainment, rotation and symmetry energy on non-radial oscillations of these models. Our results show that entrainment and symmetry energy can have a significant effect on the rotational splitting of non-axisymmetric modes. In particular, the symmetry energy modifies the inertial mode frequencies considerably in the regime of fast rotation.

Charge transport and inhomogeneity near the minimum conductivity point in graphene

The magnetic-field-dependent longitudinal and Hall components of the resistivity ${\ensuremath{\rho}}_{\mathrm{xx}}(H)$ and ${\ensuremath{\rho}}_{\mathrm{xy}}(H)$ are measured in graphene on silicon dioxide substrates at temperatures $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{K}\ensuremath{\le}T\ensuremath{\le}300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. At charge densities near the minimum conductivity point ${\ensuremath{\rho}}_{\mathrm{xx}}(H)$ is strongly enhanced and ${\ensuremath{\rho}}_{\mathrm{xy}}(H)$ is suppressed, indicating nearly equal electron and hole contributions to the current. The data are inconsistent with the standard two-fluid model but consistent with the prediction for inhomogeneously distributed electron and hole regions of equal mobility. At low $T$ and high $H$, ${\ensuremath{\rho}}_{\mathrm{xx}}(H)$ saturates to a value $\ensuremath{\sim}h∕{e}^{2}$, with Hall conductivity $\ensuremath{\ll}{e}^{2}∕h$, which may indicate a regime of localized $v=2$ and $v=\ensuremath{-}2$ quantum Hall puddles.

Couplage interfacial instationnaire de modèles diphasiques

The primary coolant circuit in a nuclear power plant contains several distinct components (vessel, core, pipes,...). To these components correspond one or several codes which have been developed to solve specific systems of partial differential equations. In order to perform two-phase flow simulations over the whole primary circuit, the coupling of these codes has to be considered. The approach that has been examined within this work consists in coupling the codes by providing information through the coupling interface, taking unsteady effects into account. For that purpose, numerical fluxes calculated at the interface are used as boundary conditions for each code. The numerical technique relies on the use of an interface model, which is combined with the basic strategy that was introduced by Greenberg and LeRoux in order to compute approximations of steady solutions of non-homogeneous hyperbolic systems. Three different coupling cases have been examined : the coupling of a one-dimensional Euler system with a two-dimensional Euler system ; the coupling of two distinct homogeneous two-phase flow models ; the coupling of a four-equation homogeneous model with the standard six-equation two-fluid model.

Analysis of a 1D incompressible two-fluid model including artificial diffusion

This article examines a 1D incompressible two-fluid model including artificial tensor diffusion. The aim is to obtain a formulation that provides convergent numerical solutions for all flow conditions within the stratified and the stratified wavy flow regime. With appropriate simplifications, the two-fluid model reduces to one momentum balance, one mass conservation and two algebraic equations. It has previously been established that a formulation that is well posed in possessing exclusively real characteristics can be obtained by including an axial diffusion term in the momentum balance. In this article, however, we demonstrate that this is not sufficient to obtain a system suitable for numerical simulations. Although the unbounded growth rates of the standard two-fluid model are eliminated, linear stability theory predicts that infinitesimal wavelengths still experience finite growth. This entails that grid refinement always will result in new unstable wavelengths being resolved. On the other hand, if artificial axial diffusion is added to both the mass and the momentum equations as suggested here, a cut-off wavelength is established below which all wavelengths are stable. Thus, a numerically converging model is formed, which retains the long-wavelength properties of the standard two-fluid model. The conclusions of the mathematical analysis are substantiated by numerical simulations of ID gravity waves.

Parametric study of a model for determining the liquid flow-rates from the pressure drop and water hold-up in oil–water flows

A flow-pattern-dependent model, traditionally used for calculation of pressure drop and water hold-up, is accustomed for calculation of the liquid production rates in oil–water horizontal flow, based on the known pressure drop and water hold-up. The area-averaged steady-state one-dimensional two-fluid model is used for stratified flow, while the homogeneous model is employed for dispersed flow. The prediction errors appear to be larger when the production rates are calculated instead of pressure drop and water hold-up. The difference in the calculation accuracies between the direct and inverse calculation is most probably caused by the different uncertainties in the measured values of the input variables and a high sensitivity of the calculated phase flow-rates on even small change of the water hold-up for certain flow regimes. In order to locate the source of error in the standard two-fluid model formulation, several parametric studies are performed. In the first parametric study, we investigate under which conditions the momentum equations are satisfied when the measured pressure drop and water hold-up are imposed. The second and third parametric studies address the influence of the interfacial waves and drop entrainment on the model accuracy, respectively. These studies show that both interfacial waves and drop entrainment can be responsible for the augmentation of the wall-shear stress in oil–water flow. In addition, consideration of the interfacial waves offers an explanation for some important phenomena of the oil–water flow, such as the wall-shear stress reduction.

A flux-conservative formalism for convective and dissipative multi-fluid systems, with application to Newtonian superfluid neutron stars

We develop a flux-conservative formalism for a Newtonian multi-fluid system, including dissipation and entrainment (i.e. allowing the momentum of one fluid to be a linear combination of the velocities of all fluids). Maximum use is made of mass, energy and linear and angular momentum conservation to specify the equations of motion. Also used extensively are insights gleaned from a convective variational action principle, the key being the distinction between each velocity and its canonically conjugate momentum (which is modified because of entrainment). Dissipation is incorporated to second order in the ‘thermodynamic forces’ via the approach pioneered by Onsager, which makes it transparent how to guarantee the law of increase of entropy. An immediate goal of the investigation is to understand better the number, and form, of independent dissipation terms required for a consistent set of equations of motion in the multi-fluid context. A significant, but seemingly innocuous detail is that one must be careful to isolate ‘forces’ that can be written as total gradients, otherwise errors can be made in relating the net internal force to the net externally applied force. Our long-range aim is to provide a formalism that can be used to model dynamical multi-fluid systems both perturbatively and via fully nonlinear 3D numerical evolutions. To elucidate the formalism we consider the standard model for a heat-conducting, superfluid neutron star, which is believed to be dominated by superfluid neutrons, superconducting protons and a highly degenerate, ultra-relativistic gas of normal fluid electrons. We determine that in this case there are, in principle, 19 dissipation coefficients in the final set of equations. A final reduction of the system is made by neglecting heat conduction. This leads to an extension of the standard two-fluid model for neutron star cores, which has been used in a number of previous applications, and illustrates how mutual friction is represented in our formalism.

Lagrangian perturbation theory of non-relativistic rotating superfluid stars

We develop a Lagrangian perturbation framework for rotating non-relativistic superfluid neutron stars. This leads to the first generalization of classic work on the stability properties of rotating stars to models that account for the presence of potentially weakly coupled superfluid components. Our analysis is based on the standard two-fluid model expected to be relevant for the conditions that prevail in the outer core of mature neutron stars. We discuss the implications of our results for dynamic and secular instabilities of a simple neutron star model in which the two fluids are allowed to assume different (uniform) rotation rates.

Experimental studies on the dual continuous flow pattern in oil–water flows

The dual continuous flow pattern (both phases retain their continuity at the top and bottom of the pipe while there is interdispersion), which occurs during the pipe flow of two immiscible liquid phases, was studied in detail, and pressure gradient, in situ volume fraction and phase distribution data were obtained. The experimental work was performed in a 38 mm diameter, horizontal, stainless steel test section, using water and oil (6 mPa s viscosity and 828 kg/m 3 density) as test fluids. The identification of the dual continuous flow pattern boundaries was achieved with the use of an impedance and a conductivity probe. Measurements were made for mixture velocities from 0.8 to 3 m/s and input oil volume fractions from 10% to 90%. Dual continuous flow appeared at intermediate mixture velocities between stratified and dispersed flows and resulted in pressure gradients less than those of single phase oil flow; the velocity ratio increased with increasing input oil fraction, and was above 1 at high oil fractions apart from the highest mixture velocities where it was reduced to values below 1. This behaviour was explained by the in situ phase distribution data and the shape of the oil–water interface. The standard two-fluid model was unable to predict the pressure gradient and hold-up during dual continuous flow.

Velocity of the fourth sound in liquid helium II via extended thermodynamics

This work continues a study begun in previous works, where, using Extended Thermodynamics, a monofluid model of liquid helium II is formulated. The wave propagation in bulk liquid helium II is studied in the hypothesis that the thermal dilatation is not zero. The propagation of fourth sound, studied previously neglecting both the thermal dilatation and finite volume of the powder, is studied without these simplified hypotheses: a scattering correction n is introduced to take into account the porosity. The model is more general than the standard two-fluid model because it allows that a small amount of entropy is associated with helium when it flows through a very thin capillary or a porous medium. A comparison with experimental data is performed. From experimental values for velocities and attenuations of the two sounds in bulk liquid helium, the model provides the velocity of fourth sound in a porous medium. These values are determined at various temperatures and pressures and compared with fourth sound measurements in a packed powder.

Examination on the Applicability of Interfacial Area Transport Equation to the Prediction of Developing Bubble Flows.

Developing air-water bubble flows in a vertical pipe were calculated using the combination of a multi-fluid model and an interfacial area concentration (IAC) transport equation. Calculations were conducted by using (a) a standard two-fluid model, (b) a two-fluid model and a one-group IAC equation, (c) a two-fluid model and a two-group IAC equation and (d) a three-fluid model and a two-group IAC equation. As a result of comparisons between measured and calculated void fractions, we confirmed that (1) the use of IAC equations brings about the reconstruction or reconsideration of interfacial drag models, (2) the two-group IAC equation requires initial values for a large bubble group even for the condition of no large bubbles, (3) the combination of the two-group IAC equation and two-fluid model cannot give good predictions for flows consisting of large and small bubbles, and (4) a multi-fluid model and a reconstructed drag force model are indispensable to obtain good predictions for bubbly to slug flow transition.

The influence of the anomalous cosmic-ray component on the dynamics of the solar wind

It is well known that both the galactic and anomalous cosmic rays show positive intensity gradients in the outer heliosphere which are connected with corresponding pressure gradients. Due to an efficient dynamical coupling between the solar wind plasma and these highly energetic media by means of convected MHD turbulences, there exists a mutual interaction between these media. As one consequence of this scenario the enforced pressure gradients influence the distant solar wind expansion. Here we concentrate in our theoretical study on the interaction of the solar wind only with the anomalous cosmic-ray component. We use the standard two-fluid model in which the cosmic-ray fluid modifies the solar wind flow via the cosmic-ray pressure gradient. Then we derive numerical solutions in the following steps: first we calculate an aspherical pressure distribution for the anomalous cosmic rays, describing their diffusion in an unperturbed radial solar wind. Second, we then consider the perturbation of the solar wind flow due to these induced anomalous cosmic-ray pressure gradients. Within this context we especially take account of the action of a non-spherical geometry of the heliospheric shock which may lead to pronounced upwinddownwind asymmetries in the pressures and thereby in the resulting solar wind flows. As we can show in our model, which fits the available observational data, radial decelerations of the distant solar wind by between 5 to 11% are to be expected, however, the deviations of the bulk solar wind flow from the radialdirections are only slightly pronounced.

End-to-end modelling of He II flow systems

A practical computer code has been developed which uses the accepted two-fluid model to simulate He II flow in complicated systems. The full set of equations are used, retaining the coupling between the pressure, temperature and velocity fields. This permits modelling He II flow over the full range of conditions, from strongly or weakly driven flow through large pipes, narrow channels and porous media. The system may include most of the components used in modern superfluid flow systems: non-ideal thermomechanical pumps, tapered sections, constructions, lines with heated side walls and heat exchangers. The model is validated by comparison with published experimental data. It is applied to a complex system to show some of the non-intuitive feedback effects that can occur. This code is ready to be used as a design tool for practical applications of He II. It can also be used for the design of He II experiments and as a tool for comparison of experimental data with the standard two-fluid model.