Evolution Of Distribution Function Research Articles

Predicting the aerosol capture characteristics of fibrous filters. I. Exact- and tractable (3-moment) approximate-methods to incorporate aerosol polydispersity effects with a multi-mechanism, semi-analytic single-fiber particle capture fraction

We exploit recent results for the multi-mechanism capture of aerosols by an isolated fiber in a steady low-Re crossflow, and analyze the consequences of aerosol polydispersity on the local particle capture rate per unit volume within a “deep” and spatially uniform fibrous filter medium. The linear first-order PDE governing evolution of the aerosol size distribution (ASD) function, n(v;z), within such an idealized medium (where v is particle volume and z the axial distance measured into the fibrous filter) is shown to admit exact solutions by a method of successive quadratures, free of restrictive assumptions about the functional form of either the inlet or internal ASD, or the local, multi-mechanism single fiber capture fraction, ηcap(v). This solution not only provides access to the usual measure of filter performance—i.e., fraction of the inlet mass (or volume) flow rate which is captured, but enables the accuracy of potentially attractive rational approximate methods to be studied. As examples, when the suspended local ASD is, and remains, single-mode log-normal-like, we formulate and examine the accuracy and efficiency of two tractable, mathematically closed, 3-moment methods for predicting the overall performance of ‘thick’ (LF/df≫1) non-woven filters comprised of fibers of a single diameter in a uniform ‘matte’ of small solid-fraction, ϕf. To fix ideas, we illustrate our present theoretical-computational methods, and some of their practical implications, for the particular case of a LF/df=300 filter comprised of 2.5 μm diam. glass fibers challenged by an aerosol population of spherical KCl(s) particles under conditions close to those experimentally studied (and computationally simulated using Brownian-dynamics + CFD methods) in the recent literature. We first consider a ‘coagulation-aged’ aerosol population having an entrance volume fraction of 0.1×10-6 with dp,g=200 nm, suspended in a U0=15 cm/s airstream (at p=1 atm, 300 K, Brownian diffusion dominated and interception influenced conditions). To demonstrate the expected improvement in filter efficiency near the previous maximum penetration particle size (MPPS) (ca. 500 nm diam) via the onset of sub-critical particle inertia effects, we also consider a carrier gas velocity of 200 cm/s, revealing a ca. 10-fold gain in local capture efficiency. Such improvements, while offset by the increased penetration of much smaller particles, should be attainable in many fibrous filter applications, provided the associated increase in gas pumping power remains manageable. We conclude with our current assessment of the relative advantages of each of the predictive methods developed and explored here. Tractable generalizations of our essentially deterministic, pseudo-continuum methods—e.g., to enable inclusion of the systematic effects of accessible fiber diameter and orientation distributions also appear to be feasible—as briefly outlined in Section 5.

A unified gas-kinetic scheme for micro flow simulation based on linearized kinetic equation

The flow regime of micro flow varies from collisionless regime to hydrodynamic regime according to the Knudsen number Kn, which is defined as the ratio of the mean free path over the local characteristic length. On the kinetic scale, the dynamics of a small-perturbed micro flow can be described by the linearized kinetic equation. In the continuum regime, according to the Chapman-Enskog theory, hydrodynamic equations such as linearized Euler equations and Navier-Stokes equations can be derived from the linearized kinetic equation. In this paper, we are going to propose a unified gas kinetic scheme (UGKS) based on the linearized kinetic equation. For the simulation of small-perturbed micro flow, the linearized scheme is more efficient than the nonlinear one. In the continuum regime, the cell size and time step of UGKS are not restricted to be less than the particle mean free path and collision time, and the UGKS becomes much more efficient than the traditional upwind-flux-based operator-splitting kinetic solvers. The important methodology of UGKS is the following. Firstly, the evolution of microscopic distribution function is coupled with the evolution of macroscopic flow quantities. Secondly, the numerical flux of UGKS is constructed based on the integral solution of kinetic equation, which provides a genuinely multiscale and multidimensional numerical flux. The UGKS recovers the solution of linear kinetic equation in the rarefied regime, and converges to the solution of the linear hydrodynamic equations in the continuum regime. An outstanding feature of UGKS is its capability of capturing the accurate viscous solution in bulk flow region once the hydrodynamic flow structure can be resolved by the cell size even when the cell size is much larger than the kinetic length scale, such as the capturing of the viscous boundary layer with a cell size being much larger than the particle mean free path. Such a multiscale property is called unified preserving (UP) which has been studied in (Guo, et al. arXiv preprint arXiv:1909.04923, 2019). In this paper, a mathematical proof of the UP property for UGKS will be presented and this property is applicable to UGKS for solving both linear and nonlinear kinetic equations.

Kosterlitz-Thouless-type caging-uncaging transition in a quasi-one-dimensional hard disk system

Molecular dynamics computer simulations of collective excitations in a system of hard disks confined to a narrow channel of the specific width, that resembles 2D triangular lattice at disk close packing, are performed. We found that transverse excitations, which for hard-disk system are absent in the limit of 1D and are of acoustic nature in the limit of 2D, in the case of q1D hard-disk system emerge in the form of transverse optical excitations and could be considered as a tool to detect the structural transition to a zigzag ordering. By analyzing density evolution of longitudinal static structure factor and pair distribution function we have shown that driving force of zigzag ordering is caging phenomenon that in the case of hard-disk system is governed by excluded volume interaction with first and second neighbors and is of entropic origin.

Evolution of parton showers and parton distribution functions

Initial state evolution in parton shower event generators involves parton distribution functions. We examine the probability for the system to evolve from a higher scale to a lower scale without an initial state splitting. A simple argument suggests that this probability, when multiplied by the ratio of the parton distributions at the two scales, should be independent of the parton distribution functions. We call this the PDF property. We examine whether the PDF property actually holds using Pythia and Deductor. We also test a related property for the Deductor shower and discuss the physics behind the results.

Evolution of the electron energy distribution function in a dusty plasma irradiated by hard radiation.

Evolution of the electron energy distribution function in a dusty plasma irradiated by hard radiation.

Nine Outstanding Questions of Solar Wind Physics.

In situ measurements of the solar wind have been available for almost 60 years, and in that time plasma physics simulation capabilities have commenced and ground‐based solar observations have expanded into space‐based solar observations. These observations and simulations have yielded an increasingly improved knowledge of fundamental physics and have delivered a remarkable understanding of the solar wind and its complexity. Yet there are longstanding major unsolved questions. Synthesizing inputs from the solar wind research community, nine outstanding questions of solar wind physics are developed and discussed in this commentary. These involve questions about the formation of the solar wind, about the inherent properties of the solar wind (and what the properties say about its formation), and about the evolution of the solar wind. The questions focus on (1) origin locations on the Sun, (2) plasma release, (3) acceleration, (4) heavy‐ion abundances and charge states, (5) magnetic structure, (6) Alfven waves, (7) turbulence, (8) distribution‐function evolution, and (9) energetic‐particle transport. On these nine questions we offer suggestions for future progress, forward looking on what is likely to be accomplished in near future with data from Parker Solar Probe, from Solar Orbiter, from the Daniel K. Inouye Solar Telescope (DKIST), and from Polarimeter to Unify the Corona and Heliosphere (PUNCH). Calls are made for improved measurements, for higher‐resolution simulations, and for advances in plasma physics theory.

Discrete unified gas kinetic scheme for a reformulated BGK–Vlasov–Poisson system in all electrostatic plasma regimes

In this paper, the discrete unified gas kinetic scheme (DUGKS), which is a novel direct kinetic method, is developed for a reformulated BGK–Vlasov–Poisson system in all electrostatic plasma regimes characterized by a wide range of Knudsen number and normalized Debye length. The current scheme is constructed for multiscale plasma simulation, while the temporal and spatial step sizes of the method are not restricted by the Knudsen number and normalized Debye length. One key feature of this method is the un-splitting treatment of the particle transport, collision and acceleration in both the distribution function evolution and the numerical flux evaluation, which enables the method to economically and accurately provide a satisfactory solution for all Knudsen number regimes. With the coupling of the appropriate time discretization of the distribution function and a reformulated Poisson equation, the method further provides an easy-to-implement and efficient way for the investigation of electrical potential in all normalized Debye regimes. As a result, the proposed DUGKS becomes an asymptotic preserving scheme, which automatically degenerates to be consistent with the discretization of corresponding limiting models. Several numerical experiments in different electrostatic plasma regimes are presented to validate the proposed method.

A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows

Compressible flow has intrinsically multiple scale nature due to the large variations of gas density and characteristic scale of flow structure, especially in hypersonic and reentry problems. It is challenging to construct an accurate and efficient numerical algorithm to capture non-equilibrium flow physics across different regimes. In this paper, a unified gas kinetic scheme with adaptive velocity space (AUGKS) for multiscale flow transport will be developed. In near-equilibrium flow region, particle distribution function is close to the Chapman-Enskog expansion and can be formulated analytically, where only macroscopic conservative flow variables are updated. With the emergence of non-equilibrium effect, the AUGKS automatically switches to the original unified gas kinetic scheme (UGKS) with a discrete velocity space to follow the evolution of particle distribution function. A criterion is proposed to quantify the non-equilibrium and is used for the switch between continuous and discrete particle velocity space. Following the scale-dependent local evolving solution, the AUGKS presents the discretized gas dynamic equations directly on the cell size and time step scales, i.e., the so-called direct modeling method. As a result, the scheme is able to capture the cross-scale flow physics from particle transport to hydrodynamic wave propagation, and provides a continuous variation of solutions from the Boltzmann to the Euler equations. Different from conventional DSMC-Fluid hybrid method, the AUGKS does not need a buffer zone to match up kinetic and hydrodynamic solutions. Instead, continuous and discrete particle velocity spaces are automatically and robustly switched, such as translating the continuous Chapman-Enskog distribution function to the discrete grid points in the velocity space. Therefore, the AUGKS is feasible for the numerical simulations with unsteadiness and complex geometries. Compared with the asymptotic preserving (AP) method which solves kinetic equation uniformly over entire computational domain with discretized velocity space, the current velocity-space adaptive unified scheme speeds up the computation, reduces the memory requirement significantly, and maintains the equivalent accuracy for multiscale flow simulations. Many test cases validate the current approach. The AUGKS provides an effective tool for non-equilibrium flow studies.

Soot size distribution in lightly sooting premixed flames of benzene and toluene

The evolution of particle size distribution function (PSDF) of soot in premixed flames of benzene and toluene was studied on a burner stabilized stagnation (BSS) flame platform. The cold gas velocities were changed to hold the maximum flame temperatures of different flames approximately constant. The PSDFs of all the test flames exhibited a bimodal distribution, i.e., a small-size nucleation mode and a large-size accumulation mode. It was observed that soot nucleation and particle growth in the benzene flame were stronger than those in the toluene flame at short residence times. At longer residence times, the PSDFs of the two flames were similar, and the toluene flame showed a larger particle size distribution range and a higher particle volume fraction than the benzene flame.

Evolution of the Electron Distribution Function in the Presence of Inverse Bremsstrahlung Heating and Collisional Ionization.

The picosecond evolution of non-Maxwellian electron distribution functions was measured in a laser-produced plasma using collective electron plasma wave Thomson scattering. During the laser heating, the distribution was measured to be approximately super-Gaussian due to inverse bremsstrahlung heating. After the heating laser turned off, collisional ionization caused further modification to the distribution function while increasing electron density and decreasing temperature. Electron distribution functions were determined using Vlasov-Fokker-Planck simulations including atomic kinetics.

Extensive analysis of the lattice Boltzmann method on shifted stencils.

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S.S. Chikatamarla, and I.V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)].PRLTAO0031-900710.1103/PhysRevLett.117.010604 From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity. Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.

Obtaining of nanoparticles of Sc, Ti, V, Cr, Mn, Fe, Co, Ni with controlled sizes and properties using laser ablation

Using laser ablation, nanoparticles of Sc, Ti, V, Cr, Mn, Fe, Co, Ni with controlled sizes and properties were obtained. To obtain nanoparticles of a given size, not only ablation of massive targets was used, but fragmentation (exposure to a mixture of micro- and nanoparticles by laser radiation). The evolution of the particle size distribution function in the process of their laser fragmentation is investigated and its key parameters determining it are established, such as the energy density of a laser beam in a medium and the peak power of laser radiation. Different schemes for introducing laser radiation into colloidal systems have been used. To obtain nanoparticles that do not contain oxide films, organic alcohols (ethanol and propanol-2) were used as the working fluid. The obtained nanoparticles met the following criteria: 1). at least 95% of the particles in the preparation have the specified diameter; 2). The shape of the nanoparticles is close to spherical; 3). The composition of the nanoparticles does not include a significant amount of impurities and oxides; 4). In most of the nanoparticles, a metal crystal lattice is observed, although under certain conditions it is possible to obtain nanoparticles, both with the outer oxide layer and entirely consisting of oxides. The stability of the obtained colloid nanoparticles was investigated. It is assumed that the solvent is saturated with molecular hydrogen to prevent oxidation of nanoparticles during storage.

Entropic Divergence and Entropy Related to Nonlinear Master Equations

We reverse engineer entropy formulas from entropic divergence, optimized to given classes of probability distribution function (PDF) evolution dynamical equation. For linear dynamics of the distribution function, the traditional Kullback–Leibler formula follows from using the logarithm function in the Csiszár’s f-divergence construction, while for nonlinear master equations more general formulas emerge. As applications, we review a local growth and global reset (LGGR) model for citation distributions, income distribution models and hadron number fluctuations in high energy collisions.

Q2 evolution of gluon distribution function inside the hadrons with both shadowing and antishadowing effects

We deliberately attempt to solve the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu,Zhu-Ruan-Shen (GLR-MQ-ZRS) equation in the leading order of QCD to study the Q2 evolution of gluon distribution function G(x,Q2), Q2 is the resolution scale or the four momentum transfered squared and x is the Bjorken variable. Both the shadowing and antishadowing corrections due to gluon recombination processes are incorporated in GLR-MQ-ZRS equation and while solving this equation we have used Regge like behavior in gluon distribution function. We present a phenomenological study on the collective contribution due to both the shadowing and antishadowing effects on the gluon distribution functions at small-x. Our solution of GLR-MQ-ZRS equation is valid in the vicinity of saturation at small-x, where gluon recombination effects cannot be ignored. We have shown comparison of our computed results for different values of strong coupling constant αs with the PDFs of NNPDF3.0, PDF4LHC and CT14 groups.

Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow

The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by the evolution of macroscopic variables, and guarantees the conservation of the scheme in the final wave-particle formulation. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced. Multiscale preserving means UGKWP method preserves the flow regime from collisionless regime to hydrodynamic regime without requiring the cell size and time step to be less than the mean free path and collision time. Multi-efficiency preserving means the computational cost of the scheme including the computational time and memory cost is on the same level as the most efficient method in the corresponding regime, such as the particle methods in the rarefied regime and hydrodynamic solvers in continuum regime. The UGKWP method is shown to satisfy the MMP requirement. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include the hypersonic flow passing a circular cylinder at Mach numbers 20 and 30 and Knudsen numbers 1 and 10−4, low speed lid-driven cavity flow, laminar boundary layer, shock structure, and shock tube problems. These results validate the accuracy, efficiency, and multiscale and multi-efficiency property of UGKWP method.

Nonlinear inverse Landau damping of ion Bernstein waves on alpha particles

A novel scheme to realize the alpha-channeling concept in tokamak plasmas is proposed. It is based on (i) the free energy available due to the inversion of the population of alpha particles in velocity space and (ii) the nonlinear inverse Landau damping of ion Bernstein waves (IBW). The latter occurs at Doppler shifted half-integer harmonics of the ion-cyclotron resonance Based on single-particle, single-wave analysis, we derive the diffusion tensors in the phase space associated to the nonlinear IBW- interaction. Numerical solutions of a Fokker–Planck equation, including the collisional operator as well as the source and sink terms show that alpha power can be channeled into IBW power during the time evolution of the alpha particle distribution function towards a steady state. For sufficient large loss rate compared to the slowing down rate, inversion of population in velocity space and alpha channeling can occur also at the steady state.

On Geometrical Invariants of the Magnetic Field Gradient Tensor in Turbulent Space Plasmas: Scale Variability in the Inertial Range

Abstract In a recent paper, Consolini et al. investigated the statistics of geometrical invariants of the coarse-grained gradient tensor of plasma velocity for a case study of space plasma turbulence. They showed how, at spatial scales near the proton inertial length, there is evidence for the occurrence of dissipation structures along the Vieillefosse’s tail. Here, we extend the previous analysis to the statistics of the geometrical invariants of the magnetic field coarse-grained gradient tensor, computed using magnetic field measurements by the ESA-Cluster mission in the solar wind region. In detail, we investigate the evolution of the joint probability distribution functions of the magnetic geometrical invariants at different scales in the inertial range of turbulent solar wind. The results show a clear dependence of the joint statistics of geometrical invariants on the distance from the proton inertial length scale in the inertial range, which seems to be compatible with a variation of the dimensionality of the fluctuation field from two dimensions to three dimensions at the smallest scales. Evidence of an increasing role of the ingoing spiral saddle and current-associated dissipation structures is found at the smallest investigated scales, where dissipation can occur.

Measurements of ion velocity and wave propagation in a hollow cathode plume

The mechanism responsible for the production of energetic ions in the plume of hollow cathodes for electric propulsion is still an open issue. These ions are of concern to cathode and thruster lifetime, particularly for cathodes operating at high (>20 A) discharge currents. Recent theoretical and experimental investigations suggest that there is a correlation between ion energy gain and ion acoustic turbulence. In this paper we present measurements of the evolution of the ion velocity distribution function in the near plume of a 100 A-class hollow cathode, operated in a regime in which the dominant mode is ion acoustic turbulence. Ion flow and thermal properties were related to measurements of the background plasma, fluctuation spectra, and dispersion relations obtained from an array of Langmuir probes. We found ions to flow outward from the cathode and accelerate downstream, to supersonic speeds, approximately aligned with the acoustic wave group velocity vectors. The directions of the ion flow and wave propagation were similar for a range of discharge currents and mass flow rates in the jet region of the plume. One operating condition showed a significant temperature increase, also in the direction of acoustic wave propagation, corresponding to the highest wave energy condition. These results are interpreted in the context of ion acoustic turbulence as a contributing mechanism for ion energy gain.

Generalized lattice Boltzmann model for frosting.

Frosting is a multiscale and multiphysics problem, which presents a significant challenge for numerical methods. In this study, a generalized lattice Boltzmann (LB) model is developed to simulate the frosting of humid air at representative elementary volume scale. In this model, three LB equations are introduced to describe the evolution of distribution functions for velocity, temperature, and humidity (i.e., mass fraction of water vapor in the humid air) fields, respectively. The frost layer is regarded as a porous medium, while the humid air is treated as a plain one. This unified LB model can be applied to describe the phase change and transport processes in these two subdomains seamlessly. Through the Chapman-Enskog analysis, the macroscopic equations for the frosting process can be recovered from the present LB model. Benchmark problems in conduction solidification, convection melting and frosting are simulated, and the numerical results match well with analytical or experimental solutions. Finally, this model is applied to simulate frost formation between two parallel plates, and the influences of air velocity, humidity, temperature, and cold wall temperature are evaluated.

Size Distribution of Nascent Soot in Premixed n-Hexane, Cyclohexane, and Methylcyclohexane Flames

The evolution of particle size distribution function (PSDF) of nascent soot in premixed flames of n-hexane, cyclohexane, and methylcyclohexane was studied on the burner-stabilized stagnation flame platform. The cold gas velocities were changed to keep the maximum flame temperatures of different flames approximately constant. The PSDFs of all fuels tested transited from the unimodal (nucleation mode only) to the bimodal (both nucleation and accumulation modes) distribution with increased particle residence time. Soot nucleation and growth in the n-hexane flame were found to appear later than those in the cyclohexane and methylcyclohexane flames. In addition, the methylcyclohexane flame exhibited the highest nucleation and accumulation rate at the early stage of soot formation. However, the particle nucleation strength and the soot mass growth rate of the cyclohexane flame exceeded those of the methylcyclohexane flame at long particle residence times.