Two-fluid Simulations Research Articles

Two-fluid simulations of waves in the solar chromosphere

Waves and shocks traveling through the solar chromospheric plasma are influenced by its partial ionization and weak collisional coupling, and may become susceptible to multi-fluid effects, similar to interstellar shock waves. In this study, we consider fast magneto-acoustic shock wave formation and propagation in a stratified medium, that is permeated by a horizontal magnetic field, with properties similar to that of the solar chromosphere. The evolution of plasma and neutrals is modeled using a two-fluid code that evolves a set of coupled equations for two separate fluids. We observed that waves in neutrals and plasma, initially coupled at the upper photosphere, become uncoupled at higher heights in the chromosphere. This decoupling can be a consequence of either the characteristic spatial scale at the shock front, that becomes similar to the collisional scale, or the change in the relation between the wave frequency, ion cyclotron frequency, and the collisional frequency with height. The decoupling height is a sensitive function of the wave frequency, wave amplitude, and the magnetic field strength. We observed that decoupling causes damping of waves and an increase in the background temperature due to the frictional heating. The comparison between analytical and numerical results allows us to separate the role of the nonlinear effects from the linear ones on the decoupling and damping of waves.

Time-Dependent Two-Fluid Magnetohydrodynamic Model and Simulation of the Chromosphere

The sun's chromosphere is a highly dynamic, partially-ionized region where spicules (hot jets of plasma) form. Here we present a two-fluid MHD model to study the chromosphere, which includes ion-neutral interaction and frictional heating. Our simulation recovers a magnetic canopy shape that forms quickly, but is also quickly disrupted by the formation of a jet. Our simulation produces a shock self-consistently, where the jet is driven by the frictional heating, which is much greater than the ohmic heating. Thus, our simulation demonstrates that the jet could be driven purely by thermal effects due to ion-neutral collisions and not by magnetic reconnection. We plan to improve the model to include photo-chemical effects and radiation.

Dusty clumps in circumbinary discs

ABSTRACT Recent observations have revealed that protoplanetary discs often exhibit cavities and azimuthal asymmetries such as dust traps and clumps. The presence of a stellar binary system in the inner disc regions has been proposed to explain the formation of these structures. Here, we study the dust and gas dynamics in circumbinary discs around eccentric and inclined binaries. This is done through two-fluid simulations of circumbinary discs, considering different values of the binary eccentricity and inclination. We find that two kinds of dust structures can form in the disc: a single horseshoe-shaped clump, on top of a similar gaseous over-density; or numerous clumps, distributed along the inner disc rim. The latter features form through the complex interplay between the dust particles and the gaseous spirals caused by the binary. All these clumps survive between one and several tens of orbital periods at the feature location. We show that their evolution strongly depends on the gas–dust coupling and the binary parameters. Interestingly, these asymmetric features could in principle be used to infer or constrain the orbital parameters of a stellar companion – potentially unseen – inside the inner disc cavity. Finally, we apply our findings to the disc around AB Aurigae.

Ion-neutral decoupling in the nonlinear Kelvin–Helmholtz instability: Case of field-aligned flow

Nonlinear magnetic Kelvin-Helmholtz instability (KHI), and the turbulence it creates appear in many astrophysical systems. This includes those systems where the local plasma conditions are such that the plasma is not fully ionized, for example in the lower solar atmosphere and molecular clouds. In a partially ionized system, the fluids couple via collisions which occur at characteristic frequencies, therefore neutral and plasma species become decoupled for sufficiently high-frequency dynamics. Here, we present high-resolution 2D two-fluid simulations of the nonlinear KHI for a system that traverses the dynamic scales between decoupled fluids and coupled dynamics. We discover some interesting phenomena, including the presence of a density coupling that is independent of the velocity coupling. Using these simulations, we analyze the heating rate, and two regimes appear. The first is a regime where the neutral flow is decoupled from the magnetic field that is characterized by a constant heating rate, then at larger scales, the strong coupling approximation holds the heating rate with the KHI layer width to the power of –2. There is an energy cascade in the simulation, but the nature of the frictional heating means the heating rate is determined by the largest scale of turbulent motions, a fact that has consequences for understanding turbulent dissipation in multifluid systems.

Termination of Inward Migration for a Gap-opening Planet through Dust Feedback

Abstract Planetary migration due to disk–planet interaction is one of the most important processes determining the architecture of planetary systems. A sufficiently massive planet forms a density gap, and migrates together with the gap. By carrying out two-dimensional and two-fluid (gas and dust grains) hydrodynamic simulations, we investigated the effects of dust feedback on the migration of the gap-opening planet, which has not been considered in previous studies. We found that the gas surface density at the outer edge of the gap becomes smaller due to dust feedback, and thus the torque exerted from the outer disk decreases. This mechanism becomes more effective as the gap becomes wider and deeper. In particular, when the mass of the planet is Jupiter-sized and the turbulent viscosity is α = 3 × 10−4, the planet can migrate outward due to the reduction of the torque exerted from the outer disk. Even for a smaller planet, the migration slows down significantly. This termination of the inward migration triggered by dust feedback may explain why ring and gap structures can be frequently observed within protoplanetary disks.

Implementation of pseudo-turbulence closures in an Eulerian–Eulerian two-fluid model for non-isothermal gas–solid flow

The Eulerian–Eulerian two-fluid model is widely used for computational fluid dynamics simulations of gas–solid flows. For non-isothermal flows, the averaged conservation equations solved in the two-fluid model require closures for drag, gas–solid heat transfer, pseudo-turbulent velocity fluctuations and the pseudo-turbulent heat flux (PTHF). However, the pseudo-turbulence terms are usually neglected in two-fluid simulations due to the lack of accurate correlations. With the increase in computational power, closures for these terms are now available from particle-resolved direct-numerical simulation (PR-DNS). Here, the PTHF closure as well as the heat-transfer closure (i.e.,the Nusselt number) extracted from PR-DNS are implemented in the two-fluid thermal energy equation in OpenFOAM. The implementation is validated by comparing the simulation results with the PR-DNS data for the temperature profiles. Based on the analysis of the thermal energy budget, the PTHF can have a significant contribution and neglecting it can lead to large errors.

Vortex instabilities triggered by low-mass planets in pebble-rich, inviscid protoplanetary discs

Abstract In the innermost regions of protoplanerary discs, the solid-to-gas ratio can be increased considerably by a number of processes, including photoevaporative and particle drift. Magnetohydrodynamic disc models also suggest the existence of a dead zone at R ≲ 10 au, where the regions close to the mid-plane remain laminar. In this context, we use two-fluid hydrodynamical simulations to study the interaction between a low-mass planet (∼1.7 M⊕) on a fixed orbit and an inviscid pebble-rich disc with solid-to-gas ratio ϵ ≥ 0.5. For pebbles with Stokes numbers St = 0.1, 0.5, multiple dusty vortices are formed through the Rossby wave instability at the planet separatrix. Effects due to gas drag then lead to a strong enhancement in the solid-to-gas ratio, which can increase by a factor of ∼103 for marginally coupled particles with St = 0.5. As in streaming instabilities, pebble clumps reorganize into filaments that may plausibly collapse to form planetesimals. When the planet is allowed to migrate in an Minimum Mass Solar Nebula (MMSN) disc, the vortex instability is delayed due to migration but sets in once inward migration stops due a strong positive pebble torque. Again, particle filaments evolving in a gap are formed in the disc while the planet undergoes an episode of outward migration. Our results suggest that vortex instabilities triggered by low-mass planets could play an important role in forming planetesimals in pebble-rich, inviscid discs, and may significantly modify the migration of low-mass planets. They also imply that planetary dust gaps may not necessarily contain planets if these migrated away.

Two-fluid simulations of waves in the solar chromosphere

Solar chromosphere consists of a partially ionized plasma, which makes modeling the solar chromosphere a particularly challenging numerical task. Here we numerically model chromospheric waves using a two-fluid approach with a newly developed numerical code. The code solves two-fluid equations of conservation of mass, momentum, and energy, together with the induction equation for the case of the purely hydrogen plasma with collisional coupling between the charged and neutral fluid components. The implementation of a semi-implicit algorithm allows us to overcome the numerical stability constraints due to the stiff collisional terms. We test the code against analytical solutions of acoustic and Alfvén wave propagation in uniform medium in several regimes of collisional coupling. The results of our simulations are consistent with the analytical estimates, and with other results described in the literature. In the limit of a large collisional frequency, the waves propagate with a common speed of a single fluid. In the other limit of a vanishingly small collisional frequency, the Alfvén waves propagate with an Alfvén speed of the charged fluid only, while the perturbation in neutral fluid is very small. The acoustic waves in these limits propagate with the sound speed corresponding to either the charges or the neutrals, while the perturbation in the other fluid component is negligible. Otherwise, when the collision frequency is similar to the real part of the wave frequency, the interaction between charges and neutrals through momentum-transfer collisions cause alterations of the waves frequencies and damping of the wave amplitudes.

RANS-based predictions of dense solid–liquid suspensions in turbulent stirred tanks

This work is aimed at investigating the impact of different meso-scale models and constitutive equations for the RANS-based two-fluid model simulations of a turbulent solid–liquid stirred vessel with high solids loading. The model assessment is preceded with a grid convergence study, which confirms the variability of the discretization requirements depending on the observed variable. The simulation results demonstrate that for the investigated system the high solids loading contribution modelled by the kinetic theory of granular flows is negligible, both in incomplete and complete suspension conditions. Instead, the particle concentration fluctuations contribution included in the momentum equations dramatically affect the predictions, particularly in incomplete suspension conditions. The evaluation of the models is completed by the comparison of the predicted solids concentration profiles with experimental data measured by Electrical Resistance Tomography. The computational strategy for achieving realistic predictions of the solid distribution both in complete and incomplete suspension conditions is outlined.

A numerical study on concentration polarization in 3D cylindrical fluidized beds with vertically immersed membranes

Concentration polarization is a serious issue for systems in which high-flux, palladium-based, hydrogen perm-selective membranes are used to extract hydrogen, not only for packed-bed reactor systems, but also for fluidized bed membrane reactors. Two-Fluid Model simulations of 3D cylindrical lab-scale fluidized beds with single and multiple vertically immersed cylindrical membranes with state-of-the-art hydrogen permeability were performed to quantify concentration polarization, to study the interaction between concentration polarization zones of multiple vertically immersed membranes and to link these findings to the bed hydrodynamics. Simulations of a fluidized bed consisting of 500 μm particles using a H2/N2 mixture as fluidization gas with a single immersed membrane, show that reduced hydrogen concentrations prevail mostly within 1 cm from the membrane surface, and have disappeared beyond about 2 cm distance from the membrane surface. A 3D simulation was compared to a Cartesian 2D simulation that represents a slice of the 3D system, to assess the accuracy of 2D simulations. The 2D simulation did not fully capture the hydrodynamics and radial mass transfer effects of 3D cylindrical fluidized bed membrane reactors and overestimated the severity of concentration polarization and the formation of densified zones near the membrane surface. In particular, the hydrogen fluxes were determined at 0.217 mol/(m2 s Pa0.5) for the 3D case, and 0.133 mol/(m2 s Pa0.5) for the 2D case. The severity of densified zones is also affected by the particle size, because for smaller particles (250 μm) the emulsion phase density near the membrane surface is higher than for systems with larger particles (500 μm), which results in increased concentration polarization and a reduced extractive hydrogen flux. Therefore, employing relatively large particles of at least 500 μm in fluidized beds with modern high-flux membranes is advised. In fluidized beds with multiple membranes, interaction between concentration polarization zones of each membrane was observed. The interaction becomes more significant at smaller inter-membrane distances, especially below 2 cm, and decreasing the bed diameter decreases the system performance even more due to hydrogen depletion. Vertically immersed membranes also affect the fluidized bed hydrodynamics by reducing bubble size and increasing the number of small bubbles.

A material-property-dependent sub-grid drag model for coarse-grained simulation of 3D large-scale CFB risers

Developing, verifying and validating sub-grid methods is of crucial significance for enabling accurate coarse-grained two-fluid modeling of rapid gas-fluidized flows. However, very few studies in the literature so far have been focused on how the sub-grid modification depends on material properties. As an extension of previous sub-grid efforts, this fundamental investigation attempts to derive a material-property-dependent drag modification based on generated data from an initially homogeneous state of periodic gas-particle suspensions. The newly constituted model is then verified by highly-resolved two-fluid simulation results. We further validate the accuracy of model predictions via systematic assessments to experimental results that encompass a wide variety of material properties in five three-dimensional large-scale circulating fluidized bed (CFB) risers. Besides, we introduce a deviation index (DI) to quantify the predictive capability of the extended model. Computational results demonstrate an essential dependence of drag correction on material properties as an additional factor. Hydrodynamic validation predictions achieve satisfactory accordance with experiments. The current model is potential to serve as a more general tool for efficiently reducing the number of pilot-scale tests and effectively designing and controlling industrial process devices.

Turbulent Dynamo in a Weakly Ionized Medium

Abstract The small-scale turbulent dynamo is an important process contributing to the cosmic magnetization. In partially ionized astrophysical plasmas, the dynamo growth of magnetic energy strongly depends on the coupling state between ions and neutrals and the ion-neutral collisional damping effect. A new damping stage of turbulent dynamo in a weakly ionized medium was theoretically predicted by Xu & Lazarian. By carrying out a 3D two-fluid dynamo simulation, we have for the first time numerically confirmed the physical conditions and the linear-in-time growth of the magnetic field strength of the damping stage of a dynamo. The dynamo-amplified magnetic field has a characteristic length as the damping scale, which increases with time and can reach the injection scale of turbulence after around eight turnover times of the largest eddy, given sufficiently low ionization fraction and a weak initial magnetic field. Due to the weak coupling between ions and neutrals, most turbulent energy carried by neutrals cannot be converted to magnetic energy, resulting in a relatively weak magnetic field at the end of the dynamo. This result has important implications for the growth of magnetic fields in the partially ionized interstellar medium and shock acceleration of Galactic cosmic rays.

Turbulent electromagnetic fields at sub-proton scales: Two-fluid and full-kinetic plasma simulations

Plasma dynamics is a multi-scale problem that involves many spatial and temporal scales. Turbulence connects the disparate scales in this system through a cascade that is established by nonlinear interactions. Most astrophysical plasma systems are weakly collisional, making a fully kinetic Vlasov description of the system essential. The use of reduced models to study such systems is computationally desirable, but careful benchmarking of physics in different models is needed. We perform one such comparison here between the fully kinetic Particle-In-Cell model and a two-fluid model that includes Hall physics and electron inertia, with a particular focus on the sub-proton scale electric field. We show that in general, the two fluid model captures large scale dynamics reasonably well. At smaller scales, the Hall physics is also captured reasonably well by the fluid code, but electron features show departures from the fully kinetic model. Implications for the use of such fluid models are discussed.

Extension of Landau-fluid closure to weakly collisional plasma regime

Both collisionless and collisional Landau fluid (LF) closure have been developed and implemented in two-fluid plasma simulation framework BOUT++, taking full advantage of the fast non-Fourier method (Dimits, 2014). The spectral range of good fit can be conveniently extended to larger domain by adding more Lorentzians. The scaling for upper limit of wavenumber resolved in collisionless LF operator and the fitting coefficients for the collisional LF operator, the later is extended to weakly collisional regime (νe∗∼0.01), are presented in this paper. The flux limited expression q∥eFL is reduced to free streaming expression q∥eFS in collisionless limit and gives classical Spitzer–Härm result q∥eSH in collisional limit as we expected. The collisional LF closure recovers the classical Spitzer–Härm result q∥eSH in collisional limit and is same as the collisionless LF closure in collisionless limit. The test cases also obviously demonstrate the nonlocal effects from LF closures and responding to different boundary conditions in open surface region. This work will make it more applicable and possible to include the Landau damping kinetic effect in the fluid simulation of tokamak plasma.

Observational diagnostics of elongated planet-induced vortices with realistic planet formation time-scales

Gap-opening planets can generate dust-trapping vortices that may explain some of the latest discoveries of high-contrast crescent-shaped dust asymmetries in transition discs. While planet-induced vortices were previously thought to have concentrated shapes, recent computational work has shown that these features naturally become much more elongated in the gas when simulations account for the relatively long timescale over which planets accrete their mass. In this work, we conduct two-fluid hydrodynamical simulations of vortices induced by slowly-growing Jupiter-mass planets in discs with very low viscosity ($\alpha = 3 \times 10^{-5}$). We simulate the dust dynamics for four particle sizes spanning 0.3 mm to 1 cm in order to produce synthetic ALMA images. In our simulations, we find that an elongated vortex still traps dust, but not directly at its center. With a flatter pressure bump and disruptions from the planet's overlapping spiral density waves, the dust instead circulates around the vortex. This motion (1) typically carries the peak off-center, (2) spreads the dust out over a wider azimuthal extent $\geq 180^{\circ}$, (3) skews the azimuthal profile towards the front of the vortex, and (4) can also create double peaks in newly-formed vortices. In particular, we expect that the most defining observational signature, a peak offset of more than $30^{\circ}$, should be detectable $>30\%$ of the time in observations with a beam diameter of at most the planet's separation from its star.

Candidate explanation for the mild core oscillations in dominant electron heating scenario on experimental advanced superconducting tokamak

The interchange-like transport is observed between two resonant surfaces (q = 1 and q = 4/3, where q is the safety factor) in a finite small positive magnetic shear regime with mild core oscillations in the experimental advanced superconducting tokamak strong on-axis electron heating H-mode plasmas. It is synchronized with the increasing gradient of the soft X-ray profile and the intensifying electron density fluctuations in the core. The analysis of two-fluid simulations combined with experimental measurements indicates the destabilization of collective resistive interchange modes with several toroidal mode numbers. The overall effect of modes leads to strong perturbations at the two resonant surfaces in contrast to that between them where the anomalous electron flux is low. Their radial displacement is beyond the resistive layer width which satisfies the condition for the nonlinear destabilization of tearing modes [L. Comisso et al., Phys. Plasmas 23, 100702 (2016)]. Evidence and analysis shown in this paper tend to understand the mechanism of mild oscillations in the core.

Confined pseudo-shocks as an energy source for the active solar corona

The Sun’s active corona requires an energy flux of ~103 W m−2 to compensate for radiative losses and to maintain its high temperature1. Plasma moves in the corona through magnetic loops2,3, which may be connected with the flows in and around sunspots4–6. Global energizing processes (for example, reconnection) play an important part in heating the corona7–9; however, energy and mass transport may also occur via shocks, waves or flows5,10,11. A full picture and the influence of such localized events, which significantly couple with various layers of the solar upper atmosphere, is still not clear. Using the Interface Region Imaging Spectrograph temporal image data of C ii 1,330 A, we observed the presence of pseudo-shocks around a sunspot. Unlike shocks12, pseudo-shocks exhibit discontinuities only in the mass density. A two-fluid numerical simulation reproduces such confined pseudo-shocks with rarefied plasma regions lagging behind them. We find that these pseudo-shocks carry an energy of ~103 W m−2, which is enough to locally power the inner corona and also generate bulk flows (~10−5 kg m−2 s−1), contributing to the localized mass transport. If they are ubiquitous, such energized and bulky pseudo-shocks above active regions could provide an important contribution to the heating and mass transport in the overlying solar corona. The IRIS spectrograph observed upward ‘pseudo-shocks’ (shocks exhibiting a discontinuity only in plasma density) over an active region of the Sun. Modelling shows that such pseudo-shocks contribute significantly to coronal heating and mass transport.

Nonlocal heat flux effects on temperature evolution of the solar atmosphere

Context. Heat flux is one of the main energy transport mechanisms in the weakly collisional plasma of the solar corona. There, rare binary collisions let hot electrons travel over long distances and influence other regions along magnetic field lines. Thus, the fully collisional heat flux models might not describe transport well enough since they consider only the local contribution of electrons. The heat flux in weakly collisional plasmas at high temperatures with large mean free paths has to consider the nonlocality of the energy transport in the frame of nonlocal models in order to treat energy balance in the solar atmosphere properly. Aims. We investigate the impact of nonlocal heat flux on the thermal evolution and dynamics of the solar atmosphere by implementing a nonlocal heat flux model in a 3D magnetohydrodynamic simulation of the solar corona. Methods. We simulate the evolution of solar coronal plasma and magnetic fields considering both a local collision dominated and a nonlocal heat flux model. The initial magnetic field is obtained by a potential extrapolation of the observed line-of-sight magnetic field of AR11226. The system is perturbed by moving the plasma at the photosphere. We compared the simulated evolution of the solar atmosphere in its dependence on the heat flux model. Results. The main differences for the average temperature profiles were found in the upper chromosphere/transition region. In the nonlocal heat transport model case, thermal energy is transported more efficiently to the upper chromosphere and lower transition region and leads to an earlier heating of the lower atmosphere. As a consequence, the structure of the solar atmosphere is affected with the nonlocal simulations producing on average a smoother temperature profile and the transition region placed about 500 km higher. Using a nonlocal heat flux also leads to two times higher temperatures in some of the regions in the lower corona. Conclusions. The results of our 3D MHD simulations considering nonlocal heat transport supports the previous results of simpler 1D two-fluid simulations. They demonstrated that it is important to consider a nonlocal formulation for the heat flux when there is a strong energy deposit, like the one observed during flares, in the solar corona.

Development and validation of an enhanced filtered drag model for simulating gas-solid fluidization of Geldart A particles in all flow regimes

Coarse-grid two-fluid simulation of gas-solid fluidized bed reactors based on the kinetic theory of granular flow exhibits a significant dependence on drag models, especially for Geldart A particles. Many drag models are available in the literature, which have been reported to work for different systems. This study focused on the evaluation of an enhanced filtered drag model along with other different drag models derived from different methods for three-dimensional two-fluid model simulations of gas-solid fluidized beds of Geldart A particles covering a broad range of fluidization regimes, including bubbling fluidization, turbulent fluidization, fast fluidization, and dilute phase transport regimes. Eight drag models were selected, which included five heterogeneous drag models and three homogeneous drag models. Comparison with the available experimental data demonstrates the need for modification of homogeneous drag models to account for the effect of mesoscale structures (i.e., bubbles and clusters). The enhanced filtered drag model and energy-minimization multi-scale (EMMS) drag models were found to achieve superior predictions in all fluidization regimes, while the other drag models were only capable of predicting certain fluidization regimes. The results of this work provide a guideline for choosing appropriate drag models for simulating Geldart A particles and suggestions on developing more reliable and general drag models applicable in all flow regimes.

EMMS mixture model with size distribution for two-fluid simulation of riser flows

Further development of an energy-minimization multiscale modeling approach to simulating two-phase flow under turbulent conditions that considers the size distribution of mesoscale structures, i.e. bubbles and clusters, is presented. User-defined values of minimum and maximum cluster or bubble diameters were specified. A uniform size distribution was first considered as a test case, in which the drag force comprised contributions from each size group. The mathematical form of the objective function describing the energy for suspension and transport was not altered. The heterogeneity index of this new drag modification was then used to simulate pilot-scale circulating fluidized-bed risers involving Geldart group A particles. The results were validated against available experimental data. The model is capable of capturing both axial and radial profiles of flow-field variables.