Reduced Fluid Model Research Articles

A two‐way coupling approach for simulating bouncing droplets

AbstractThis article presents a two‐way coupling approach to simulate bouncing droplet phenomena by incorporating the lubricated thin aerodynamic gap between fluid volumes. At the heart of our framework lies a cut‐cell representation of the thin air film between colliding liquid fluid volumes. The air pressures within the thin film, modeled using a reduced fluid model based on the lubrication theory, are coupled with the volumetric liquid pressures by the gradient across the liquid–air interfaces and solved in a monolithic two‐way coupling system. Our method can accurately solve liquid–liquid interaction with air films without adaptive grid refinements, enabling accurate simulation of many novel surface‐tension‐driven phenomena such as droplet collisions, bouncing droplets, and promenading pairs.

Gyrofluid simulations of turbulence and reconnection in space plasmas

A Hamiltonian two-field gyrofluid model is used to investigate the dynamics of an electron-ion collisionless plasma subject to a strong ambient magnetic field, within a spectral range extending from the magnetohydrodynamic (MHD) scales to the electron skin depth. This model isolates Alfvén, Kinetic Alfvén and Inertial Kinetic Alfvén waves that play a central role in space plasmas, and extends standard reduced fluid models to broader ranges of the plasma parameters. Recent numerical results are reviewed, including (i) the reconnection-mediated MHD turbulence developing from the collision of counter-propagating Alfvén wave packets, (ii) the specific features of the cascade dynamics in strongly imbalanced turbulence, including a possible link between the existence of a spectral transition range and the presence of co-propagating wave interactions at sub-ion scales, for which new simulations are reported, (iii) the influence of the ion-to-electron temperature ratio in two-dimensional collisionless magnetic reconnection. The role of electron finite Larmor radius corrections is pointed out and the extension of the present model to a four-field gyrofluid model is discussed. Such an extended model accurately describes electron finite Larmor radius effects at small or moderate values of the electron beta parameter, and also retains the coupling to slow magnetosonic waves.

Radial drift of plasma blobs in a toroidal magnetic field with fully kinetic and reduced fluid models

In curved magnetic geometries, field-aligned regions of enhanced plasma pressure and density, termed ‘blobs,’ move as coherent filaments across the magnetic field lines. Coherent blobs account for a significant fraction of transport at the edges of magnetic fusion experiments and arise in naturally-occurring space plasmas. This work examines the dynamics of blobs with a fully kinetic electromagnetic particle-in-cell code and with a drift-reduced fluid code. In low-beta regimes with moderate blob speeds, good agreement is found in the maximum blob velocity between the two simulation schemes and simple analytical estimates. The fully kinetic code demonstrates that blob speeds saturate near the initial sound speed, which is a regime outside the validity of the reduced fluid model.

Generalized Hamiltonian drift-fluid and gyrofluid reductions

We provide a procedure for deriving Hamiltonian reduced fluid models for plasmas, starting from a Hamiltonian gyrokinetic system in the approximation. The procedure generalizes, to a considerable extent, previous results. In particular, the evolution of moments with respect to the magnetic moment coordinate is also taken into account, together with background density and magnetic inhomogeneities. In the limit of vanishing Finite Larmor Radius (FLR) effects, an infinite family of reduced electron drift-fluid equations is derived, evolving all the electron moments , with and , where N and M are arbitrary non-negative integers counting the maximum order of the moments taken with respect to the parallel velocity and to the magnetic moment coordinates, respectively. An analogous result is found in the gyrofluid case and applied to the ion species. The gyrofluid result holds for , finite FLR effects and a low ratio β e between electron internal pressure and magnetic guide field pressure. In both the drift and gyrofluid case, the key for the identification of the Hamiltonian structure resides in changes of variables based on orthogonal matrices that diagonalize the Jacobi matrices associated with Hermite and Laguerre polynomials. In terms of the transformed variables, the drift and gyrofluid equations are cast in a simple form, which reduces to advection equations for Lagrangian invariants in the two-dimensional case with homogeneous background. Because the procedure requires to evolve, for a particle species s, all and only the moments , with and , not every choice for the set of moments is admissible, for fixed N and M. This might also explain the scarcity of Hamiltonian reduced fluid models obtained so far which account for anisotropic temperature fluctuations.

PolyStokes: A Polynomial Model Reduction Method for Viscous Fluid Simulation

Standard liquid simulators apply operator splitting to independently solve for pressure and viscous stresses, a decoupling that induces incorrect free surface boundary conditions. Such methods are unable to simulate fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. By contrast, unsteady Stokes solvers retain coupling between pressure and viscosity, thus resolving these phenomena, albeit using a much larger and thus more computationally expensive linear system compared to the decoupled approach. To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using a quadratic field of 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to correctly capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model is able to reproduce the same fluid phenomena at a smaller computational cost. Futhermore, we performed a crowdsourced user survey to verify that our method produces imperceptible differences compared to the full unsteady Stokes method.

On cubic non-linearities in fluid and kinetic models

The kinetic origin of the cubic terms found in electromagnetic reduced fluid models of magnetized plasmas is identified and analyzed. The transition from cartesian to guiding center variables in kinetic systems is responsible for the introduction of such cubic terms, and the derivation of a set of such terms is explicitly shown. The role of these terms in fluid simulations, and their importance for energy conservation of cubic terms in both frameworks is discussed

Four-field Hamiltonian fluid closures of the one-dimensional Vlasov–Poisson equation

We consider a reduced dynamics for the first four fluid moments of the one-dimensional Vlasov–Poisson equation, namely, fluid density, fluid velocity, pressure, and heat flux. This dynamics depends on an equation of state to close the system. This equation of state (closure) connects the fifth-order moment—related to the kurtosis in velocity of the Vlasov distribution—with the first four moments. By solving the Jacobi identity, we derive an equation of state, which ensures that the resulting reduced fluid model is Hamiltonian. We show that this Hamiltonian closure allows symmetric homogeneous equilibria of the reduced fluid model to be stable.

Poisson brackets and truncations in nonlinear reduced fluid models for plasmas

The Hamiltonian structure for an infinite class of nonlinear reduced fluid models, derived from a Hamiltonian drift-kinetic system, is explicitly provided in terms of the N+1 fluid moments evolving in each model of the class, with N an arbitrary positive integer. This improves previous results, in which the existence of the Hamiltonian structure was shown, but the complete explicit expression for the Poisson bracket of each model of the class was not provided. We also show that, whereas the Hamiltonian functional of the fluid models can be derived from that of the drift-kinetic system, by projecting the perturbation of the distribution function onto its truncated series in terms of Hermite polynomials, this is not the case for the Poisson bracket. Indeed, the antisymmetric bilinear form obtained by means of the aforementioned projection, although, interestingly, ”very similar” to the Poisson bracket of the fluid models, turns out to differ from it. The difference is found to reside in the coefficients W(N)lmn of the bilinear form, when the indices are such that l+m+n is even and l≥N+1,m≥N+1,n≥N+1. We show with a counterexample, related to the case N=2, that such bilinear form, in general, does not satisfy the Jacobi identity. We provide a physical interpretation of the set of variables G0,G1,…,GN, in terms of which the Poisson bracket of the fluid models exhibits a direct-sum structure, and point out an analogy between the present fluid reduction problem and the problem of the truncated quantum harmonic oscillator.

Numerical turbulence simulations of intermittent fluctuations in the scrape-off layer of magnetized plasmas

Intermittent fluctuations in the boundary of magnetically confined plasmas are investigated by numerical turbulence simulations of a reduced fluid model describing the evolution of the plasma density and electric drift vorticity in the two-dimensional plane perpendicular to the magnetic field. Two different cases are considered: one describing resistive drift waves in the edge region and another including only the interchange instability due to unfavorable magnetic field curvature in the scrape-off layer. Analysis of long data time series obtained by single-point recordings is compared to predictions of a stochastic model describing the plasma fluctuations as a superposition of uncorrelated pulses. For both cases investigated, the radial particle density profile in the scrape-off layer is exponential with a radially constant scale length. The probability density function for the particle density fluctuations in the far scrape-off layer has an exponential tail. Radial motion of blob-like structures leads to large-amplitude bursts with an exponential distribution of peak amplitudes and the waiting times between them. The average burst shape is well described by a two-sided exponential function. The frequency power spectral density of the particle density is simply that of the average burst shape and is the same for all radial positions in the scrape-off layer. The fluctuation statistics obtained from the numerical simulations are in excellent agreement with recent experimental measurements on magnetically confined plasmas. The statistical framework defines a new validation metric for boundary turbulence simulations.

Parameter Optimization of Reduced Fluid Model via Sparse Point Measurements

Model order reduction is a modeling method that can facilitate the analysis and the control synthesis of distributed parameter systems, which are governed by partial differential equations. Many techniques have been proposed to further improve the property of the reduced order model based on full state information, which is usually not available in practical applications. In this paper, a reduced-order modeling method based on sparse point measurements is proposed for the benchmark system of flow past a cylinder. This method involves solving the dynamic optimization problems online, whose objective functions measure the differences between the predicted and observed variables on each time horizon. In this method, the sensor placement is also discussed, and we determine the sensor locations depending on a model-free optimization technique. Several numerical examples of different scenarios are simulated not only to illustrate the effectiveness of the proposed method but also to verify the discussion on the sensor placement strategy.

Linear stability of magnetic vortex chains in a plasma in the presence of equilibrium electron temperature anisotropy

The linear stability of chains of magnetic vortices in a plasma is investigated analytically in two dimensions by means of a reduced fluid model assuming a strong guide field and accounting for equilibrium electron temperature anisotropy. The chain of magnetic vortices is modeled by means of the classical ‘cat’s eyes’ solutions and the linear stability is studied by analysing the second variation of a conserved functional, according to the energy-Casimir method. The stability analysis is carried out on the domain bounded by the separatrices of the vortices. Two cases are considered, corresponding to a ratio between perpendicular equilibrium ion and electron temperature much greater or much less than unity, respectively. In the former case, equilibrium flows depend on an arbitrary function. Stability is attained if the equilibrium electron temperature anisotropy is bounded from above and from below, with the lower bound corresponding to the condition preventing the firehose instability. A further condition sets an upper limit to the amplitude of the vortices, for a given choice of the equilibrium flow. For cold ions, two sub-cases have to be considered. In the first one, equilibria correspond to those for which the velocity field is proportional to the local Alfvén velocity. Stability conditions imply: an upper limit on the amplitude of the flow, which automatically implies firehose stability, an upper bound on the electron temperature anisotropy and again an upper bound on the size of the vortices. The second sub-case refers to equilibrium electrostatic potentials which are not constant on magnetic flux surfaces and the resulting stability conditions correspond to those of the first sub-case in the absence of flow.

Numerical simulation of continuous separation of microparticles in two-stage acousto-microfluidic systems

Numerical modelling of acousto-microfluidic particle manipulation systems cannot only be used to explain the complex phenomena observed in experiments, but can also be applied to optimise their performances. In this work, we present numerical simulations of continuous-flow-based two-stage acoustic microparticle separations with a reduced-fluid model, which is consisted of three main parts: (1) an acoustic focusing zone; (2) a transition zone; and (3) an acoustic separation zone. The acoustophoresis of microparticles of various sizes in the fluid channel was modelled based on Newton's second law, where the acoustic radiation forces and the flow-induced drag forces, the main driving terms for particle motion, were solved from the Gorkov equation and the Navier-Stokes equations, respectively. It was found that an acoustic focusing process configured with appropriate force amplitudes can focus all particles to the same flow vector before entering the separation zone and thus can improve the separation efficiency, and that a sheath flow injected from the transition zone can push the sample flow onto the side boundaries, which can broaden the effective separation range for more robust separations. Based on the mechanism analyses, we here numerically demonstrated acoustofluidic separation of 5 different particle fractions simultaneously in a continuous microfluidic channel ending with 9 equally spaced outlets. We also predicted here that, with carefully designed acoustic and flow fields, it is capable to acoustically separate two different particle fractions with a diameter difference of 4% (difference in acoustic mobility of only ~1.08).

Asymmetry effects driving secondary instabilities in two-dimensional collisionless magnetic reconnection

In the framework of the studies on magnetic reconnection, much interest has been recently devoted to asymmetric magnetic configurations, which can naturally be found in solar and astrophysical environments and in laboratory plasmas. Several aspects of this problem have been investigated, mainly in a two-dimensional geometry and by means of particle-in-cell (PIC) simulations. Still, there are open questions concerning the onset and the effects of secondary instabilities in the nonlinear phase of an asymmetric reconnection process. In this work, we focus on the conditions that lead to the appearance of the Kelvin-Helmholtz instability following an asymmetric reconnection event in a collisionless plasma. This investigation is carried out by means of two-dimensional numerical simulations based on a reduced fluid model assuming a strong guide field. We show that, unlike the symmetric case, in the presence of asymmetry, a Kelvin-Helmholtz-like instability can develop also for a finite equilibrium electron temperature. In particular, simulations indicate the formation of steep velocity gradients, which drive the instability, when the resonant surface of the equilibrium magnetic field is located sufficiently far from the peak of the equilibrium current density. Moreover, a qualitative analysis of the vorticity dynamics shows that the turbulent behavior induced by the secondary instability not only is confined inside the island but can also affect the plasma outside the separatrices. The comparison between simulations carried out with an adiabatic closure and a Landau-fluid closure for the electron fluid indicates that the latter inhibits the secondary instability by smoothing velocity gradients.

A flux-balanced fluid model for collisional plasma edge turbulence: Model derivation and basic physical features

We propose a new reduced fluid model for the study of the drift wave–zonal flow dynamics in magnetically confined plasmas. Our model can be viewed as an extension of the classic Hasegawa-Wakatani (HW) model and is based on an improved treatment of the electron dynamics parallel to the field lines, to guarantee a balanced electron flux on the magnetic surfaces. Our flux-balanced HW (bHW) model contains the same drift-wave instability as previous HW models, but unlike these models, it converges exactly to the modified Hasegawa-Mima model in the collisionless limit. We rely on direct numerical simulations to illustrate some of the key features of the bHW model, such as the enhanced variability in the turbulent fluctuations and the existence of stronger and more turbulent zonal jets than the jets observed in other HW models, especially for high plasma resistivity. Our simulations also highlight the crucial role of the feedback of the third-order statistical moments in achieving a statistical equilibrium with strong zonal structures. Finally, we investigate the changes in the observed dynamics when more general dissipation effects are included and, in particular, when we include the reduced model for ion Landau damping originally proposed by Wakatani and Hasegawa.

Drift driven cross-field transport and scrape-off layer width in the limit of low anomalous transport

The impact of the -drift in the cross-field transport and its effect on the density and power scrape-off layer (SOL) width in the limit of low anomalous transport is studied with the fluid code SolEdge2D. In the first part of the work, the simulations are run with an isothermal reduced fluid model. It is found that a -drift dominated regime is reached in all geometries studied (JET-like, ASDEX-like and circular analytic geometries), and that the transition toward this regime comes along with the apparition of supersonic shocks, and a complex parallel equilibrium. The parametric dependencies of the SOL width in this regime are investigated, and the temperature and the poloidal magnetic field are found to be the principal parameters governing the evolution of the SOL width. In the second part of this paper, the impact of additional physics is studied (inclusion of the centrifugal drift, self-consistent variation of temperature and the treatment of the neutral species). The addition of centrifugal drift and neutral species are shown to play a role in the establishment of the parallel equilibrium, impacting the SOL’s width, although the role of the centrifugal drift is limited to a low diffusion level. Finally, the numerical results are compared with the estimate of the Goldston’s heuristic drift based model (HD-model), the starting point of our study, and which has shown good agreement with experimental scaling laws. We find that the particles SOL widths in the -drift dominated regime are at least two times smaller than the estimate of the HD-model. Moreover, in the parametric dependencies proposed by the HD-model, the dependency with is retrieved, but not the one on T.

Analysis of equilibrium and turbulent fluxes across the separatrix in a gyrokinetic simulation

The SOL width is a parameter of paramount importance in modern tokamaks as it controls the power density deposited at the divertor plates, critical for plasma-facing material survivability. An understanding of the parameters controlling it has consequently long been sought [Connor et al. Nucl. Fusion 39(2), 169 (1999)]. Prior to Chang et al. [Nucl. Fusion 57(11), 116023 (2017)], studies of the tokamak edge have been mostly confined to reduced fluid models and simplified geometries, leaving out important pieces of physics. Here, we analyze the results of a DIII-D simulation performed with the full-f gyrokinetic code XGC1 which includes both turbulence and neoclassical effects in realistic divertor geometry. More specifically, we calculate the particle and heat E × B fluxes along the separatrix, discriminating between equilibrium and turbulent contributions. We find that the density SOL width is impacted almost exclusively by the turbulent electron flux. In this simulation, the level of edge turbulence is regulated by a mechanism that we are only beginning to understand: ∇B-drifts and ion X-point losses at the top and bottom of the machine, along with ion banana orbits at the low field side, result in a complex poloidal potential structure at the separatrix which is the cause of the E × B drift pattern that we observe. Turbulence is being suppressed by the shear flows that this potential generates. At the same time, turbulence, along with increased edge collisionality and electron inertia, can influence the shape of the potential structure by making the electrons non-adiabatic. Moreover, being the only means through which the electrons can lose confinement, it needs to be in a balance with the original direct ion orbit losses to maintain charge neutrality.

A reduced Landau-gyrofluid model for magnetic reconnection driven by electron inertia

An electromagnetic reduced gyrofluid model for collisionless plasmas, accounting for electron inertia, finite ion Larmor radius effects and Landau-fluid closures for the electron fluid is derived by means of an asymptotic expansion from a parent gyrofluid model. In the absence of terms accounting for Landau damping, the model is shown to possess a non-canonical Hamiltonian structure. The corresponding Casimir invariants are derived and use is made thereof, in order to obtain a set of normal field variables, in terms of which the Poisson bracket and the model equations take a remarkably simple form. The inclusion of perpendicular temperature fluctuations generalizes previous Hamiltonian reduced fluid models and, in particular, the presence of ion perpendicular gyrofluid temperature fluctuations reflects into the presence of two new Lagrangian invariants governing the ion dynamics. The model is applied, in the cold-ion limit, to investigate numerically a magnetic reconnection problem. The Landau damping terms are shown to reduce, by decreasing the electron temperature fluctuations, the linear reconnection rate and to delay the nonlinear island growth. The saturated island width, on the other hand, is independent of Landau damping. The fraction of magnetic energy converted into perpendicular kinetic energy also appears to be unaffected by the Landau damping terms, which, on the other hand, dissipate parallel kinetic energy as well as free energy due to density and electron temperature fluctuations.

Hamiltonian closures in fluid models for plasmas

This article reviews recent activity on the Hamiltonian formulation of fluid models for plasmas in the non-dissipative limit, with emphasis on the relations between the fluid closures adopted for the different models and the Hamiltonian structures. The review focuses on results obtained during the last decade, but a few classical results are also described, in order to illustrate connections with the most recent developments. With the hope of making the review accessible not only to specialists in the field, an introduction to the mathematical tools applied in the Hamiltonian formalism for continuum models is provided. Subsequently, we review the Hamiltonian formulation of models based on the magnetohydrodynamics description, including those based on the adiabatic and double adiabatic closure. It is shown how Dirac’s theory of constrained Hamiltonian systems can be applied to impose the incompressibility closure on a magnetohydrodynamic model and how an extended version of barotropic magnetohydrodynamics, accounting for two-fluid effects, is amenable to a Hamiltonian formulation. Hamiltonian reduced fluid models, valid in the presence of a strong magnetic field, are also reviewed. In particular, reduced magnetohydrodynamics and models assuming cold ions and different closures for the electron fluid are discussed. Hamiltonian models relaxing the cold-ion assumption are then introduced. These include models where finite Larmor radius effects are added by means of the gyromap technique, and gyrofluid models. Numerical simulations of Hamiltonian reduced fluid models investigating the phenomenon of magnetic reconnection are illustrated. The last part of the review concerns recent results based on the derivation of closures preserving a Hamiltonian structure, based on the Hamiltonian structure of parent kinetic models. Identification of such closures for fluid models derived from kinetic systems based on the Vlasov and drift-kinetic equations are presented, and connections with previously discussed fluid models are pointed out.

Electron-scale reduced fluid models with gyroviscous effects

Reduced fluid models for collisionless plasmas including electron inertia and finite Larmor radius corrections are derived for scales ranging from the ion to the electron gyroradii. Based either on pressure balance or on the incompressibility of the electron fluid, they respectively capture kinetic Alfvén waves (KAWs) or whistler waves (WWs), and can provide suitable tools for reconnection and turbulence studies. Both isothermal regimes and Landau fluid closures permitting anisotropic pressure fluctuations are considered. For small values of the electron beta parameter $\unicode[STIX]{x1D6FD}_{e}$, a perturbative computation of the gyroviscous force valid at scales comparable to the electron inertial length is performed at order $O(\unicode[STIX]{x1D6FD}_{e})$, which requires second-order contributions in a scale expansion. Comparisons with kinetic theory are performed in the linear regime. The spectrum of transverse magnetic fluctuations for strong and weak turbulence energy cascades is also phenomenologically predicted for both types of waves. In the case of moderate ion to electron temperature ratio, a new regime of KAW turbulence at scales smaller than the electron inertial length is obtained, where the magnetic energy spectrum decays like $k_{\bot }^{-13/3}$, thus faster than the $k_{\bot }^{-11/3}$ spectrum of WW turbulence.

Reduced fluid models for self-propelled particles interacting through alignment

The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self-propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.