Maxwellian Velocity Distribution Function Research Articles

Gas Kinetic Scheme Coupled with High-Speed Modifications for Hypersonic Transition Flow Simulations.

The issue of hypersonic boundary layer transition prediction is a critical aerodynamic concern that must be addressed during the aerodynamic design process of high-speed vehicles. In this context, we propose an advanced mesoscopic method that couples the gas kinetic scheme (GKS) with the Langtry-Menter transition model, including its three high-speed modification methods, tailored for accurate predictions of high-speed transition flows. The new method incorporates the turbulent kinetic energy term into the Maxwellian velocity distribution function, and it couples the effects of high-speed modifications on turbulent kinetic energy within the computational framework of the GKS solver. This integration elevates both the transition model and its high-speed enhancements to the mesoscopic level, enhancing the method's predictive capability. The GKS-coupled mesoscopic method is validated through a series of test cases, including supersonic flat plate simulation, multiple hypersonic cone cases, the Hypersonic International Flight Research Experimentation (HIFiRE)-1 flight test, and the HIFiRE-5 case. The computational results obtained from these cases exhibit favorable agreement with experimental data. In comparison with the conventional Godunov method, the new approach encompasses a broader range of physical mechanisms, yielding computational results that closely align with the true physical phenomena and marking a notable elevation in computational fidelity and accuracy. This innovative method potentially satisfies the compelling demand for developing a precise and rapid method for predicting hypersonic boundary layer transition, which can be readily used in engineering applications.

Evaluation of stochastic particle Bhatnagar–Gross–Krook methods with a focus on velocity distribution function

For precise application of Bhatnagar–Gross–Krook (BGK) methods, assessing its accuracy in non-equilibrium flows is necessary. Generally, this assessment relies on macroscopic parameters, which are moments of the velocity distribution function (VDF). However, in non-equilibrium flows, the significance of each moment diminishes as the VDF deviates from the Maxwellian VDF. This study investigates the VDF in non-equilibrium flows. Two Prandtl-corrected BGK methods, the ellipsoidal statistical BGK and Shakhov BGK (SBGK), are compared with the direct simulation Monte Carlo method. To observe the VDF while excluding the effects of convection, the homogeneous relaxation of the initial non-equilibrium state is analyzed. The VDF in Couette flow and normal shock waves, where collision and convection coexist, is then examined. When comparing the accuracy of the BGK methods using higher-order moments, inconsistencies are observed. However, when comparing the VDFs, the SBGK method reproduces the non-equilibrium VDF more accurately. The results demonstrate the importance of the VDF in the evaluation of non-equilibrium flows.

Non-Maxwellian electron effects on the macroscopic response of a Hall thruster discharge from an axial–radial kinetic model

A 2D axial–radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

Three-dimensional coupling of electron cyclotron drift instability and ion–ion two stream instability

Electron cyclotron drift instability (ECDI) and ion–ion two stream instability (IITSI) are both kinetic instabilities that can be present in low-temperature, partially magnetized plasmas. The coupling of instabilities in a three-dimensional configuration leads to the existence of more than one unstable roots to the kinetic dispersion relation. In this paper, a generalized method has been developed for numerically evaluating solutions to the three-dimensional dispersion relation for coupled ECDI and IITSI, assuming cold singly and doubly charged ions and a Maxwellian velocity distribution function for the electrons. The present study demonstrates the coupling between ECDI and IITSI that affects the most unstable mode as a function of the wavenumbers in three dimensions and various plasma properties, including the applied electric field, magnetic field, electron temperature, ion velocities, and plasma density. One of the most notable results is that, while the most unstable mode with the largest growth rate is in the direction of the E×B drift in the two-dimensional cases, the most unstable mode for the three-dimensional configuration occurs in the oblique direction between the applied electric field and the E×B drift. This agrees with experimental observations in cross field plasma sources using coherent Thomson scattering.

Wave–particle interaction in contact with a chaotic thermostat

This numerical investigation explores the properties of the wave–particle interaction in a situation in which the particle is simultaneously in contact with a chaotic thermostat [G. J. Morales, Phys. Rev. E 99, 062218 (2019)]. The role of the thermostat is to establish a Maxwellian velocity distribution function through deterministic chaotic orbits. The particle response is quantified by calculating the complex mobility, μk, ω, from the numerically obtained orbits for a wave of constant amplitude, with wave number k and frequency ω. It is found that in the limit of weak coupling to the thermostat, the behavior is that predicted by the plasma dispersion function, which implies collisionless Landau damping. As the coupling to the thermostat is increased (equivalent to increasing collisionality), the behavior follows the generalized collisional plasma dispersion function [Fried et al., Phys. Fluids 9, 292 (1966)]. For strong coupling, the response agrees with the Braginskii mobility. The nonlinear mobility associated with intermittent particle trapping is obtained for the various collisional regimes.

Isentropic plasma sheath model for improved fidelity

A model is developed for a collisionless plasma sheath assuming isentropic electrons in contrast to the standard isothermal electron assumption. This approach is enabled by the approximation of a Maxwellian electron velocity distribution function across the sheath, which is justified by near wall measurements. The conservation of entropy leads to a modified Boltzmann relation and a modified Bohm criterion. The predicted floating sheath potential is in excellent agreement with experimental data. Takamura's model for a space-charge limited plasma sheath near an emissive surface is also modified for isentropic electrons and with that modification agrees well with numerical results from a full fluid plasma model.

Comprehensive kinetic theory of inverse sheath for a strong electron-emitting electrode in a low-pressure isotropic plasma

The basic kinetic theory of an electron emitting inverse sheath was presented in T Gyergyek, J Kovačič, I Gomez, J P Gunn, S Costea and M Mozetič (2020 Phys. of Plasmas 27, 023 520). Here we extend this theory to find the potential profile and kinetic energy flux in inverse sheath for floating and current carrying electron emitting electrode/wall. The values of emitted-electron temperature, number and current densities are explored for the existence and nonexistence of inverse sheath for floating and current carrying electrode/wall. For this we consider half Maxwellian velocity distribution functions of species (emitted-electron, plasma-electron and ions) at their respective emerging boundaries. The species charge densities are calculated self-consistently from the prior assumed positive sheath structure. The Poisson’s equation is then solved numerically for floating and current carrying electrode/wall with varying normalized emitted-electron and ion temperatures. The resulting inverse sheath solution is valid for limited range of emitted-electron and ion temperatures in case of floating electrode/wall. The kinetic energy flux relations for each species are derived in inverse sheath. The numerical solutions of these relations for floating and current carrying electrode/wall are presented for valid range of parameters. These solutions shows that the total or kinetic flux received by floating electrode/wall surface decreases with increasing of emitted-electron temperature and even approaches to zero for equal values of emitted-electron and plasma-electron temperatures.

Development of a Plasma Chemistry Model for Helicon Plasma Thruster analysis

The present work is part of a wider project aimed at improving the description of the plasma dynamics during the production phase of a Helicon Plasma Thruster. In particular, the work was focused on the development of a chemical model for Argon- and Xenon-based plasma. The developed model consists of a collisional radiative model suitable to describe the dynamics of the 1s and 2p excited levels. The model is meant to be complementary to 3D-VIRTUS, a numerical tool which enforces a fluid description of plasma, developed by the University of Padova to analyse helicon discharges. Once identified, the significant reactions for both propellants, the reaction rate coefficients, have been integrated exploiting cross sections from literature and assuming a Maxwellian velocity distribution function for all the species. These coefficients have been validated against experimental measurements of an Argon Inductively Coupled Plasma and compared with a well-established code. For Argon, the selected reactions have been reduced through a proposed lumping methodology. In this way, it was possible to reduce the number of equations of the system to solve, and implement them into 3D-VIRTUS. A validation against an experimental case taken from literature was performed, showing good agreement of the results. Regarding the Xenon model, only a verification has been performed against the results of another collisional-radiative model in literature. Finally, a predictive analysis of the propulsive performances of a Helicon Plasma Thruster for both Argon and Xenon is presented.

Self-Similarity of Continuous-Spectrum Radiative Transfer in Plasmas with Highly Reflecting Walls

Radiative Transfer (RT) in a continuous spectrum in plasmas is caused by the emission and absorption of electromagnetic waves (EM) by free electrons. For a wide class of problems, the deviation of the velocity distribution function (VDF) of free electrons from the thermodynamic equilibrium, the Maxwellian VDF, can be neglected. In this case, RT in the geometric optics approximation is reduced to a single transport equation for the intensity of EM waves with source and sink functions dependent on the macroscopic parameters of the plasma (temperature and density of electrons). Integration of this equation for RT of radio-frequency EM waves in laboratory plasmas with highly reflecting metallic walls is substantially complicated by the multiple reflections which make the waves with the long free path the dominant contributors to the power balance profile. This in turn makes the RT substantially nonlocal with the spatial–spectral profile of the power balance determined by the spatial integrals of the plasma parameters. The geometric symmetry of the bounding walls, especially when enhanced by the diffuse reflectivity, provides a semi-analytic description of the RT problem. Analysis of the accuracy of such an approach reveals an approximate self-similarity of the power balance profile and the radiation intensity spectrum in both approximate and ab initio modeling. This phenomenon is shown here for a wide range of plasma parameters and wall reflectivity, including data from various numeric codes. The relationship between the revealed self-similarity and the accuracy of numeric codes is discussed.

Charge-state dynamics of heavy-ion beams penetrating a hydrogen plasma

The dynamics of charge-state fractions of Au- and U-ion beams, penetrating hydrogen plasmas, is investigated for 1.4 and 3.6 MeV/u ion energies and plasma electron temperatures Te = (0.6 − 3.0) eV and densities Ne =(1016 − 1020) cm−3. Calculations of the fractions as a function of the plasma thickness are performed by solving the balance rate equations with calculated ionization and recombination rates including those for dielectronic recombination (DR). The DR rates are calculated by the recently created computer code DRIMP using the fully relativistic multi-configuration approach and the so-called shifted Maxwellian velocity distribution function. Calculated by this method DR rates of Au- and U-ion beams passing through a hydrogen plasma are presented for the first time. Using the results obtained, the temperatures Te and densities Ne are found at which the equilibrium mean charges q̄ of Au- and U-ion beams are maximal for hydrogen-plasma length L≤ 100 cm. Dependencies of the plasma equilibrium thicknesses Xeq on the initial charge q0 of incident ions are investigated, and it is found that Xeq values are minimal at q0≈q̄.

Exact solutions of plasma flow on a rigid oscillating plate under the effect of an external non-uniform electric field

The paper is concerned with the flow behavior of a helium-plasma bounded by a rigid, oscillating flat plate. The influences of an unsteady nonlinear external electric field (NEEF) on the plasma fluid were investigated. Attention has been focused on the Bhatnagar-Gross-Krook (BGK)–type of the Boltzmann kinetic equation (BKE) to study the gas dynamics with the Maxwellian velocity distribution function (VDF). An analytical solution of the model equations was given using the moment and the traveling wave methods. The role of the mean velocity, shear stress, viscosity coefficient, and electromagnetic fields are investigated. We found that the effect of the NEEF is powerful in plasma. It made it fluctuate and perturbed strongly compared to the effect of the nonlinear external magnetic field (NEMF). So, in the plasma controlling process, we should use NEMF instead of NEEF to save the equilibrium state (ES) for a plasma fluid. The thermodynamic properties were investigated, and qualitative agreements with previous related papers were demonstrated using 3-Dimensional graphics for the calculating variables. This study's significance was due to its vast applications in numerous fields, such as in physics, electrical engineering, micro-electro-mechanical systems (MEMS), or Nano-electro-mechanical systems (NEMS) devices, and various commercial-industrial applications.

On Analytical Solution of a Plasma Flow over a Moving Plate under the Effect of an Applied Magnetic Field

Our objective of this investigation is to mainly focus on the behavior of a plasma gas that is bounded by a moving rigid flat plate; its motion is damping with time. The effects of an external magnetic field on the electrons collected with each other, with positive ions, and with neutral atoms in the plasma fluid are studied. The BGK type of the Boltzmann kinetic equation is used to study the gas dynamics various regimes with Maxwellian velocity distribution functions. An analytical solution of the model equations for the unsteady flow was given using the moment and the traveling wave methods. The manner of the mean velocity of plasmas is illustrated, which is compatible with the variation of the shear stress, viscosity coefficient, and the initial and boundary conditions. Besides, the thermodynamic prediction is investigated by applying irreversible thermodynamic principles and extended Gibbs formula. Finally, qualitative agreements with previous related papers were demonstrated using 3-dimensional graphics for calculating the variables. The significance of this study is due to its vast applications in numerous fields such as in physics, engineering, commercial, and industrial applications.

Macroscopic and parametric study of a kinetic plasma expansion in a paraxial magnetic nozzle

A kinetic paraxial model of a collisionless plasma stationary expansion in a convergent-divergent magnetic nozzle (MN) is analyzed. Monoenergetic and Maxwellian velocity distribution functions of upstream ions are compared, leading to differences in the expansion only on second and higher-order velocity moments. Individual and collective magnetic mirror effects are analyzed. Collective ones are small on the electron population since only a weak temperature anisotropy develops, but they are significant on the ions all over the nozzle. Momentum and energy equations for ions and electrons are assessed based on the kinetic solution. The ion response is different in the hot and cold limits, with the anisotropic pressure tensor being relevant in the first case. Heat fluxes of parallel and perpendicular energies have a dominant role in the electron energy equations. They do not fulfill a Fourier-type law; they are large even when electrons are near isothermal. A crude electron fluid closure based on a constant diffusion-to-convective thermal energy ratio is shown equivalent to the much invoked polytropic law. Analytical dimensionless parameter laws are derived for the nozzle total electric potential fall and the downstream residual electron temperature. Electron confinement and related current control by a thin Debye sheath and a semi-infinite divergent MN are compared.

Temperature anisotropy relaxation processes in dense plasma

In this work the relaxation processes of dense plasmas were studied. The relaxation rate of a Maxwellian velocity distribution function that has an initially anisotropic temperature (T∥ ≠ T^) is an important physical process inertial confinement fusion plasmas. Relaxation characteristics in dense plasmas were studied on the basis of the effective potentials using the Coulomb logarithm. The effective potential is derived using the long wavelength expansion of the polarization function and quantum potential which takes into account the finite value of the interaction potential at close distance. In presented work temperature anisotropy relaxation processes in dense, non-isothermal plasma are considered. These interaction potential between particles take into account such collective effects as the ionization energy depression (reduction) and exchange-correlation effects. Therefore, this allowed us to examine the sensitivity of the computed relaxation time and the corresponding equilibrium plasma temperature on the quality of the description of the screening effect in dense plasmas.

The Characterization of Secondary Interstellar Neutral Oxygen beyond the Heliopause: A Detailed Analysis of the IBEX-Lo Oxygen Observations

Abstract In this study, we analyze the directional distribution of the secondary interstellar neutral (ISN) O population observed by the IBEX-Lo neutral atom camera on the Interstellar Boundary EXplorer (IBEX) via the comparison with simulated ISN O intensity maps produced by an analytical model. In the analytical model, we assume that there are primary and secondary ISN populations at the heliopause. We further assume that each population is represented by a Maxwellian velocity distribution function with its own flow parameters. For the viewing directions of IBEX-Lo, we compute the incoming atom speeds at the heliopause with a Keplerian equation of motion in the solar gravity field. Then, we calculate analytically the distribution function to obtain the ISN intensities at Earth’s orbit. We compare the simulated O intensity maps with the IBEX-Lo O sky map to determine the most likely flow parameters of the secondary ISN O population. Using this method, we find the most likely flow parameters of the secondary ISN O population: km s−1, , , and K. The results indicate that the secondary ISN O flow direction is deflected toward lower ecliptic longitude and higher negative ecliptic latitude from the ISN gas flow direction at the heliopause. The secondary ISN O flow direction is more deflected from the ISN gas flow direction than the secondary ISN He flow direction.

On the multifractality of plasma turbulence in the solar wind

AbstractIn this work we have analyzed turbulent plasma in the kinetic scale by the characterization of magnetic fluctuations time series. Considering numerical Particle-In-Cell (PIC) simulations we apply a method known as MultiFractal Detrended Fluctuation Analysis (MFDFA) to study the fluctuations of solar-wind-like plasmas in thermodynamic equilibrium (represented by Maxwellian velocity distribution functions), and out of equilibrium plasma represented by Tsallis velocity distribution functions, characterized by the kappa (κ) parameter, to stablish relations between the fractality of magnetic fluctuation and the kappa parameter.

General self-similar solution for expansion of non-Maxwellian plasmas

In this work, self-similar expansion of one-dimensional collisionless plasma into a vacuum has been calculated for non-Maxwellian particles. It is known that, during the expansion, the initial Maxwellian electron velocity distribution function (DF) is deformed. In this paper, this deformation is illustrated by the Vlasov simulation of plasma expansion. A part of deformation is associated with the appearance of high energy (super thermal) electrons. Therefore, a more complete and realistic plasma model has to include the effect of the super thermal electrons. It is found that the resulting DF of simulation has good consistency with the generalized Lorentzian (kappa) DF. Then the self-similar calculations were carried out for cold ions while the electrons having Lorentzian DF. Finally, it was found that, by increasing the population of energetic electrons, the expansion takes place faster and the ions are accelerated to higher energies.

Effect of non‐extensivity parameter q on the damping rate of dust ion acoustic waves in non‐extensive dusty plasma

Kinetic theory has been applied to study the damping characteristics of dust ion acoustic waves (DIAWs) in a dusty plasma comprising q‐non‐extensive distributed electrons and ions, while the dust particles are considered extensive following the Maxwellian velocity distribution function. It is found that the results of the three‐dimensional velocity distribution function are more accurate compared to the results of the one‐dimensional velocity distribution function. The numerical solution of the dispersion relation is carried out to study the effect of the non‐extensivity parameter q on the dispersion, the damping rate, and the range of the values of the normalized wavenumber ( k λD) for which the DIAWs are weakly damped. It is found that the change in the value of the electron non‐extensivity parameter qe has a minor effect on the dispersion, the damping rate, and the range of the values of the normalized wavenumber ( k λD) for which the DIAWs are weakly damped, while on the other hand, ion non‐extensivity parameter qi has a strong effect on these arguments. The effect of other parameters, such as the ratio of electron to ion number density and ratio of electron to ion temperature, on the damping characteristics of DIAWs is also highlighted.

Global kinetic hybrid simulation for radially expanding solar wind

AbstractWe present the results of a 1‐D global kinetic simulation of the solar wind in spherical coordinates without a magnetic field in the region from the Sun to the Earth's orbit. Protons are considered as particles while electrons are considered as a massless fluid, with a constant temperature, in order to study the relation between the hybrid and hydrodynamic solutions. It is shown that the strong electric field in the hybrid model accelerates the protons. Since the electric field in the model is related to electron pressure, each proton in the initial Maxwellian velocity distribution function moves under the same forces as in the classical Parker Solar wind model. The study shows that the hybrid model results in very similar velocity and number density distributions along the radial distance as in the Parker model. In the hybrid simulations, the proton temperature is decreased with distance in 1 order of magnitude. The effective polytropic index of the proton population slightly exceeds 1 at larger distances with the maximum value ∼1.15 in the region near the Sun. A highly non‐Maxwellian type of distribution function is initially formed. Further from the Sun, a narrow beam of the escaping protons is created which does not change much in later expansion. The results of our study indicates that already a nonmagnetized global hybrid model is capable of reproducing some fundamental features of the expanding solar wind shown in the Parker model and additional kinetic effects in the solar wind.

Temperature anisotropy relaxation of the one‐component plasma

The relaxation rate of a Maxwellian velocity distribution function that has an initially anisotropic temperature (T‖≠T⊥) is an important physical process in space and laboratory plasmas. It is also a canonical example of an energy transport process that can be used to test theory. Here, this rate is evaluated using molecular dynamics simulations of the one‐component plasma. Results are compared with the predictions of four kinetic theories; two treating the weakly coupled regime, namely (a) the Landau equation, and (b) the Lenard–Balescu equation, and two that attempt to extend the theory into the strongly coupled regime, namely (c) the effective potential theory and (d) the generalized Lenard–Balescu theory. The role of dynamic screening is studied, and is found to have a negligible influence on this transport rate. Oscillations and a delayed relaxation onset in the temperature profiles are observed at strong coupling, which are not described by the kinetic theories.