Two-fluid Equations Research Articles

Comparison of the effect of hydrodynamic and hydrostatic models for pressure correction term in two-fluid model in gas-liquid two-phase flow modeling

This paper presents a numerical study using two-fluid model in order to compare the effect of hydrodynamic and hydrostatic models for pressure correction term in two-fluid model in gas-liquid two-phase flow modeling to provide a more accurate model. Two-fluid model is solved by Conservative Shock Capturing Method. The two-fluid model is applied using both hydrodynamic pressure correction term and hydrostatic pressure correction term for four test cases including Water Faucet Case, Water-Air Separation Case, Toumi's Shock Tube Case, and Large Relative Velocity Shock Tube Case.Hydrostatic pressure correction term is neglected for vertical geometry, therefore, in this geometry; two-fluid model cannot be hyperbolic. Thus, hydrostatic pressure correction term is not a stabilizing term. Also, in horizontal pipe and for atmospheric conditions, hydrostatic pressure correction term presents better results than hydrodynamic pressure correction term. But, in non-atmospheric conditions, hydrodynamic pressure correction term provides better results. Therefore, we should consider geometry (vertical or horizontal) and flow conditions (atmospheric or under-pressure) in order to select a proper pressure correction term for two-fluid model. Also, hydrodynamic pressure correction term in two-fluid equations system is hyperbolic in a boarder range than hydrostatic pressure correction term.

On the Partition Method of Frictional Pressure Drop for Dispersed Two-Phase Flows in the RELAP5/MOD3, TRACE V5, and SPACE Codes

Frictional pressure drop (also called wall drag) for a two-phase flow has been investigated for several decades. However, the two-phase frictional pressure drop models in the state-of-the-art thermal-hydraulic system codes are significantly different from each other, especially in the way to partition the wall friction force of liquid and vapor phases in the two-fluid momentum equations. This may lead to unphysical results in some flow conditions.In this technical note, the two-phase wall frictional pressure drop models in the RELAP5/MOD3, TRACE V5, and SPACE codes are discussed in terms of the wall friction partition into the liquid and vapor momentum equations. To show the effect of different partition methods in the three codes, we simulated air-water bubbly flows in a horizontal pipe. The results of the calculations show that the partition method has a direct effect on the relative velocity of the two phases, and it may lead to unphysical behaviors of dispersed bubbles and droplets. It is strongly recommended to revisit the two-fluid formulation and the partition method of two-phase wall drag in the state-of-the-art system codes.

Comparative study of the two-fluid momentum equations for multi-dimensional bubbly flows: Modification of Reynolds stress

Two-fluid equations are widely used to obtain averaged behaviors of two-phase flows. This study addresses a problem that may arise when the two-fluid equations are used for multi-dimensional bubbly flows. If steady drag is the only accounted force for the interfacial momentum transfer, the disperse-phase velocity would be the same as the continuous-phase velocity when the flow is fully developed without gravity. However, existing momentum equations may show unphysical results in estimating the relative velocity of the dispersephase against the continuous-phase. First, we examine two types of existing momentum equations. One is the standard two-fluid momentum equation in which the disperse-phase is treated as a continuum. The other is the averaged momentum equation derived from a solid/ fluid particle motion. We show that the existing equations are not proper for multi-dimensional bubbly flows. To resolve the problem mentioned above, we modify the form of the Reynolds stress terms in the averaged momentum equation based on the solid/fluid particle motion. The proposed equation shows physically correct results for both multi-dimensional laminar and turbulent flows.

Generation of auroral turbulence through the magnetosphere–ionosphere coupling

The shear Alfvén waves coupled with the ionospheric density fluctuations in auroral regions of a planetary magnetosphere are modeled by a set of the reduced magnetohydrodynamic and two-fluid equations. When the drift velocity of the magnetized plasma due to the background electric field exceeds a critical value, the magnetosphere–ionosphere (M–I) coupling system is unstable to the feedback instability which leads to formation of auroral arc structures with ionospheric density and current enhancements. As the feedback (primary) instability grows, a secondary mode appears and deforms the auroral structures. A perturbative (quasilinear) analysis clarifies the secondary growth of the Kelvin–Helmholtz type instability driven by the primary instability growth in the feedback M–I coupling. In the nonlinear stage of the feedback instability, furthermore, auroral turbulence is spontaneously generated, where the equipartition of kinetic and magnetic energy is confirmed in the quasi-steady turbulence.

Inelastic scattering of xenon atoms by quantized vortices in superfluids

We study inelastic interactions of particles with quantized vortices in superfluids by using a semi-classical matter wave theory that is analogous to the Landau two-fluid equations, but allows for the vortex dynamics. The research is motivated by recent experiments on xenon doped helium nanodroplets that show clustering of the impurities along the vortex cores. We numerically simulate the dynamics of trapping and interactions of xenon atoms by quantized vortices in superfluid helium and the obtained results can be extended to scattering of other impurities by quantized vortices. Different energies and impact parameters of incident particles are considered. We show that inelastic scattering is closely linked to the generation of Kelvin waves along a quantized vortex during the interaction even if there is no capture. The capture criterion of an impurity is formulated in terms of the binding energy.

Experimental Investigation and CFD Modeling of Hydrodynamic Parameters in a Pulsed Packed Column

ABSTRACTIn this work, a combination of computational fluid dynamics (CFD) and droplet population-balance model (DPBM) in the framework of Fluent was applied to simulate the drop-size distributions and flow fields in a pilot-plant liquid–liquid extraction pulsed packed column. The three-dimensional unsteady-state liquid–liquid flow was modeled using the Eulerian two-fluid equations in conjunction with the realizable k – ε turbulence model. The classes method (CM) was chosen for solving population-balance equations. Two models for breakage and coalescence, the models of Luo and Garthe, were used in the CFD code. The model was validated by comparing the simulated drop-size distributions and holdup with experimental measurements. After the validation of the model, the effects of the operating conditions (feed rates and pulsation) on the dispersed phase holdup and drop-size distributions were studied. The results of linked CFD-DPBM model and experiments revealed that the dispersed phase holdup was increased when the organic and aqueous flow rates increased and when the intensity of pulse was increased, the holdup increased. Increasing the dispersed and continuous feed rates caused the Sauter mean diameter of the drops decreased and when the intensity of pulse was increased, because of high droplets break up rate, the Sauter mean diameter decreased. Results of linked CFD-DPBM model show that the CFD-DPBM tool is able to predict hydrodynamic parameters in a pulsed packed column.

Hybrid parallelization of the XTOR-2F code for the simulation of two-fluid MHD instabilities in tokamaks

A hybrid MPI/OpenMP parallel version of the XTOR-2F code [Lütjens and Luciani, J. Comput. Phys. 229 (2010) 8130] solving the two-fluid MHD equations in full tokamak geometry by means of an iterative Newton–Krylov matrix-free method has been developed. The present work shows that the code has been parallelized significantly despite the numerical profile of the problem solved by XTOR-2F, i.e. a discretization with pseudo-spectral representations in all angular directions, the stiffness of the two-fluid stability problem in tokamaks, and the use of a direct LU decomposition to invert the physical pre-conditioner at every Krylov iteration of the solver. The execution time of the parallelized version is an order of magnitude smaller than the sequential one for low resolution cases, with an increasing speedup when the discretization mesh is refined. Moreover, it allows to perform simulations with higher resolutions, previously forbidden because of memory limitations.

Effects of relativistic electrons and spatial non-uniformities of density on periodic absolute parametric instability in a plasma waveguide

The effects of relativistic electrons and spatial plasma nonuniformity of density on periodic absolute parametric instability (API) of electrostatic waves in a relativistic 1−D nonuniform magnetoactive plasma are investigated in a cylindrical geometry. We use the separation method to solve the two-fluid plasma equations, which describe the system. The method used enables us to determine the frequencies and growth rates of unstable modes and the self-consistent electric field. Plasma electrons are considered to have a relativistic velocity. Increment has found in the buildup of the oscillations, and it is shown that the spatial nonuniformity of the plasma exerts a stabilizing effect on the parametric instability. It is shown that the growth rate of the API in relativistic plasma is reduced compared to nonrelativistic plasma. Independent of the geometry of the problem (plane and cylinder), the results of the API in a relativistic plasma waveguide are still valid.The proposal approach (Ȝseparation methodȝ) is significantly simpler that the method ordinarily employed in the theory of parametric resonance in nonuniform plasma. Therefore, it is of special interest to apply the separation method to solution of different problems involving parametric excitation of electrostatic waves in bounded nonuniform plasma.

On the interaction of Taylor length scale size droplets and isotropic turbulence

Droplets in turbulent flows behave differently from solid particles, e.g. droplets deform, break up, coalesce and have internal fluid circulation. Our objective is to gain a fundamental understanding of the physical mechanisms of droplet–turbulence interaction. We performed direct numerical simulations (DNS) of 3130 finite-size, non-evaporating droplets of diameter approximately equal to the Taylor length scale and with 5 % droplet volume fraction in decaying isotropic turbulence at initial Taylor-scale Reynolds number $\mathit{Re}_{\unicode[STIX]{x1D706}}=83$. In the droplet-laden cases, we varied one of the following three parameters: the droplet Weber number based on the r.m.s. velocity of turbulence ($0.1\leqslant \mathit{We}_{rms}\leqslant 5$), the droplet- to carrier-fluid density ratio ($1\leqslant \unicode[STIX]{x1D70C}_{d}/\unicode[STIX]{x1D70C}_{c}\leqslant 100$) or the droplet- to carrier-fluid viscosity ratio ($1\leqslant \unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x1D707}_{c}\leqslant 100$). In this work, we derive the turbulence kinetic energy (TKE) equations for the two-fluid, carrier-fluid and droplet-fluid flow. These equations allow us to explain the pathways for TKE exchange between the carrier turbulent flow and the flow inside the droplet. We also explain the role of the interfacial surface energy in the two-fluid TKE equation through the power of the surface tension. Furthermore, we derive the relationship between the power of surface tension and the rate of change of total droplet surface area. This link allows us to explain how droplet deformation, breakup and coalescence play roles in the temporal evolution of TKE. Our DNS results show that increasing $\mathit{We}_{rms}$, $\unicode[STIX]{x1D70C}_{d}/\unicode[STIX]{x1D70C}_{c}$ and $\unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x1D707}_{c}$ increases the decay rate of the two-fluid TKE. The droplets enhance the dissipation rate of TKE by enhancing the local velocity gradients near the droplet interface. The power of the surface tension is a source or sink of the two-fluid TKE depending on the sign of the rate of change of the total droplet surface area. Thus, we show that, through the power of the surface tension, droplet coalescence is a source of TKE and breakup is a sink of TKE. For short times, the power of the surface tension is less than $\pm 5\,\%$ of the dissipation rate. For later times, the power of the surface tension is always a source of TKE, and its magnitude can be up to 50 % of the dissipation rate.

Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels

Data generated by direct numerical simulations (DNS) of bubbly up-flow in a periodic vertical channel is used to generate closure relationships for a simplified two-fluid model for the average flow. Nearly spherical bubbles, initially placed in a fully developed parabolic flow, are driven relatively quickly to the walls, where they increase drag and slowly reduce the flow rate. Once the flow rate has been decreased enough, some of the bubbles move back into the channel interior and the void fraction there approaches the value needed to balance the weight of the mixture and the imposed pressure gradient. A database is generated by averaging the DNS results over planes parallel to the walls, and a Model Averaging Neural Network (MANN) is used to find the relationships between unknown closure terms in a simple model equations for the average flow and the resolved variables. The closure relations are then tested, by following the evolution of different initial conditions, and it is found that the model predictions are in reasonably good agreement with DNS results.

Relativistic Modeling Capabilities in PERSEUS Extended-MHD Simulation Code for HED Plasmas

A relativistic version of the PERSEUS extended magnetohydrodynamic simulation code for high-energy-density (HED) plasmas is developed and validated. Nonrelativistic PERSEUS solves the two-fluid equations, formulated in terms of a generalized Ohm’s law, so as to model about nine orders of magnitude in density variation using a local semi-implicit method. Relativistic PERSEUS preserves this structure and, therefore, retains these advantageous properties, enabling it to model a broader range of relativistic HED phenomena than the existing particle and fluid codes. The relativistic code is validated against a published particle-in-cell (PIC) simulation of the penetration of a laser into a supercritical hydrogen plasma with applications to fast ignition. PERSEUS recovers the penetration ahead of the laser of magnetized relativistic jets. As in the PIC simulation, multiple jets form and subsequently coalesce into larger centralized jets. The relativistic code also recovers expected nonrelativistic results for a problem in which the smallest electron length and time scales are under-resolved, and which is, therefore, inaccessible to PIC codes and conventional (explicit) fluid codes. The code can now be applied to relativistic phenomena with under-resolved electron dynamical scales, e.g., $X$ -pinches and electrode surface plasmas.

Prediction of mono-disperse gas–liquid turbulent flow in a vertical pipe

A two-fluid model in the Eulerian–Eulerian framework has been implemented for the prediction of gas volume fraction, mean phasic velocities, and the liquid phase turbulence properties for gas–liquid upward flow in a vertical pipe. The governing two-fluid transport equations are discretized using the finite volume method and a low Reynolds number k−ɛ model is used to predict the turbulence field for the continuous liquid phase. In the present analysis, a fully developed one-dimensional flow is considered where the gas volume fraction profile is predicted using the radial force balance for the bubble phase. The current study investigates: (1) the turbulence modulation terms which represent the effect of bubbles on the liquid phase turbulence in the k–ε transport equations; (2) the role of the bubble induced turbulent viscosity compared to turbulence generated by shear; and (3) the effect of bubble size on the radial forces which results in either a center-peak or a wall-peak in the gas volume fraction profiles. The results obtained from the current simulation are generally in good agreement with the experimental data, and somewhat improved over the predictions of some previous numerical studies.

Revisiting low-fidelity two-fluid models for gas–solids transport

Two-phase gas–solids transport models are widely utilized for process design and automation in a broad range of industrial applications. Some of these applications include proppant transport in gaseous fracking fluids, air/gas drilling hydraulics, coal-gasification reactors and food processing units. Systems automation and real time process optimization stand to benefit a great deal from availability of efficient and accurate theoretical models for operations data processing. However, modeling two-phase pneumatic transport systems accurately requires a comprehensive understanding of gas–solids flow behavior. In this study we discuss the prevailing flow conditions and present a low-fidelity two-fluid model equation for particulate transport. The model equations are formulated in a manner that ensures the physical flux term remains conservative despite the inclusion of solids normal stress through the empirical formula for modulus of elasticity. A new set of Roe–Pike averages are presented for the resulting strictly hyperbolic flux term in the system of equations, which was used to develop a Roe-type approximate Riemann solver. The resulting scheme is stable regardless of the choice of flux-limiter. The model is evaluated by the prediction of experimental results from both pneumatic riser and air-drilling hydraulics systems. We demonstrate the effect and impact of numerical formulation and choice of numerical scheme on model predictions. We illustrate the capability of a low-fidelity one-dimensional two-fluid model in predicting relevant flow parameters in two-phase particulate systems accurately even under flow regimes involving counter-current flow.

Solitary fast magnetosonic waves propagating obliquely to the magnetic field in cold collisionless plasma

Solutions describing solitary fast magnetosonic (FMS) waves (FMS solitons) in cold magnetized plasma are obtained by numerically solving two-fluid hydrodynamic equations. The parameter domain within which steady-state solitary waves can propagate is determined. It is established that the Mach number for rarefaction FMS solitons is always less than unity. The restriction on the propagation velocity leads to the limitation on the amplitudes of the magnetic field components of rarefaction solitons. It is shown that, as the soliton propagates in plasma, the transverse component of its magnetic field rotates and makes a complete turn around the axis along which the soliton propagates.

Numerical Study of Compressor Fouling Mechanism Based on Eulerian–Eulerian Approach

The paper describes a fluid dynamic numerical model to study the fouling mechanism for turbomachines (such as large gas turbines and small turbochargers). For this purpose, the gas-particle behaviour over a compressor blade is modelled using an open source CFD tool, OpenFOAM®. Through this model, two-fluid implicit equations system, with Eulerian-Eulerian approach is considered. This approach uses RANS turbulent models and also takes into account the particle fluid interactions. Applying Johnson-Kendal-Robert (JKR) theory, the particle deposition over the blade surface is predicted. According to the simulation results, there are certain regions which are facing the huge amount of deposition. The surfaces of these regions are prone to form an almost thick layer of sticked particles and affect the boundary layers and blade aerodynamics significantly. In order to capture this effect, a model to predict the deformation of the boundary with respect to deposition rate is also introduced.

A horizontal stratified gas–liquid two-phase flow model for the two-fluid model in the WCOBRA/TRAC-TF2 PWR safety analysis code

For the two-phase flow in a horizontal pipe, individual phases may separate because of gravity. This horizontal stratification significantly impacts interfacial drag, interfacial heat transfer and wall drag of the two-phase flow. Due to the low interfacial drag and the low interfacial heat transfer, the horizontal stratification in reactor coolant systems is a highly important phenomenon during a postulated loss-of-coolant accident in PWR. The horizontal stratification also impacts other SBLOCA phenomena such as the offtake phenomenon and the cold leg direct contact condensation phenomenon.For the two-fluid model in the WCOBRA/TRAC-TF2 LOCA safety analysis computer code, a horizontal stratification criterion was developed following the inviscid Kelvin–Helmholtz neutral stability analysis. The horizontal stratification criterion combines the Taitel–Dukler model and the Wallis–Dobson model and captures the test data better. The stratification criterion was assessed against various experimental data with a wide range of pressures and pipe sizes. The stratification criterion also matches well with the transition boundaries predicted by the viscous Kelvin–Helmholtz model.The adequacy of the horizontal stratification model is confirmed by examining the predicted flow regime in a horizontal pipe with the measured data of the high pressure TPTF two-phase flow experiments. The void fractions for the horizontal flow are predicted with a good accuracy, which indicates that the interfacial drag and the wall drag in the two-fluid model are properly modeled and the two-fluid momentum equations with the hydraulic slope term reasonably capture the influence of inlet and outlet boundary conditions expected in the PWR LOCA simulation.

Oscillatory superfluid Ekman pumping in helium II and neutron stars

The linear response of a superfluid, rotating uniformly in a cylindrical container and threaded with a large number of vortex lines, to an impulsive increase in the angular velocity of the container is investigated. At zero temperature and with perfect pinning of vortices to the top and bottom of the container, we demonstrate that the system oscillates persistently with a frequency proportional to the vortex line tension parameter to the quarter power. This low-frequency mode is generated by a secondary flow analogous to classical Ekman pumping that is periodically reversed by the vortex tension in the boundary layers. We compare analytic solutions to the two-fluid equations by Chandler & Baym (J. Low Temp. Phys., vol. 62, 1986, pp. 119–142) with the spin-up experiments by Tsakadze & Tsakadze (J. Low Temp. Phys., vol. 39, 1980, pp. 649–688) in helium II and find that the frequency agrees within a factor of four, although the experiment is not perfectly suited to the application of linear theory. We argue that this oscillatory Ekman pumping mode, and not Tkachenko modes, provides a natural explanation for the observed oscillation. In neutron stars, the oscillation period depends on the pinning interaction between neutron vortices and flux tubes in the outer core. Using a simplified pinning model, we demonstrate that strong pinning can accommodate modes with periods of days to years, which are only weakly damped by mutual friction over longer time scales.

Mesh generation for confined fusion plasma simulation

XGC1 and M3D-C1 are two fusion plasma simulation codes being developed at Princeton Plasma Physics Laboratory. XGC1 uses the particle-in-cell method to simulate gyrokinetic neoclassical physics and turbulence (Chang et al. Phys Plasmas 16(5):056108, 2009; Ku et al. Nucl Fusion 49:115021, 2009; Admas et al. J Phys 180(1):012036, 2009). M3D-$$C^1$$C1 solves the two-fluid resistive magnetohydrodynamic equations with the $$C^1$$C1 finite elements (Jardin J comput phys 200(1):133---152, 2004; Jardin et al. J comput Phys 226(2):2146---2174, 2007; Ferraro and Jardin J comput Phys 228(20):7742---7770, 2009; Jardin J comput Phys 231(3):832---838, 2012; Jardin et al. Comput Sci Discov 5(1):014002, 2012; Ferraro et al. Sci Discov Adv Comput, 2012; Ferraro et al. International sherwood fusion theory conference, 2014). This paper presents the software tools and libraries that were combined to form the geometry and automatic meshing procedures for these codes. Specific consideration has been given to satisfy the mesh configuration and element shape quality constraints of XGC1 and M3D-$$C^1$$C1.

Numerical modelling of sawtooth crash using two-fluid equations

The nonlinear growth of internal kink modes is studied numerically using two-fluid equations. As already reported earlier (Günter et al 2015 Plasma Phys. Control. Fusion 57 014017), short sawtooth crash times (<100 μs) are found for typical ASDEX Upgrade parameters, in agreement with experimental observations. These fast sawtooth crashes are associated with large parallel electric field perturbations, giving rise to the generation of supra-thermal electrons. Slow sawtooth crashes (~ms) are obtained only for a sufficiently small () and a large local electron diamagnetic drift frequency at the surface, where q0 is the value of the safety factor q at the magnetic axis. Kink modes are shown to drive sheared plasma rotation, propagating from the surface towards the magnetic axis during the nonlinear phase. After the sawtooth crash, the driven plasma rotation is in the co- (counter-) current direction inside (outside) the surface, as observed in experiments.

Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system

Direct numerical simulations of bubbly multiphase flows are used to find closure terms for a simple model of the average flow, using Neural Networks (NNs). The flow considered consists of several nearly spherical bubbles rising in a periodic domain where the initial vertical velocity and the average bubble density are homogeneous in two directions but non-uniform in one of the horizontal directions. After an initial transient motion the average void fraction and vertical velocity become approximately uniform. The NN is trained on a dataset from one simulation and then used to simulate the evolution of other initial conditions. Overall, the resulting model predicts the evolution of the various initial conditions reasonably well.