Particle Mean Free Path Research Articles

On the second law of thermodynamics: A global flow spontaneously induced by a locally nonchaotic energy barrier

It is well known that, inherently, certain small-scale nonchaotic particle movements are not governed by thermodynamics. Usually, such phenomena are studied by kinetic theory and their energy properties are considered “trivial”. In current research, we show that, beyond the boundaries of the second law of thermodynamics where Boltzmann’s H-theorem does not apply, there may also be large-sized systems of nontrivial energy properties: when the system is isolated, entropy can decrease; from a single thermal reservoir, the system can absorb heat and produce useful work without any other effect. The key concept is local nonchaoticity, demonstrated by using a narrow energy barrier. The barrier width is much less than the nominal particle mean free path, so that inside the barrier, the particle-particle collisions are sparse and the particle trajectories tend to be locally nonchaotic. Across the barrier, the steady-state particle flux ratio is intrinsically in a non-Boltzmann form. With a step-ramp structure, the nonequilibrium effect spreads to the entire system, and a global flow is generated spontaneously from the random thermal motion. The deviation from thermodynamic equilibrium is steady and significant, and compatible with the basic principle of maximum entropy. These theoretical and numerical analyses may shed light on the fundamentals of thermodynamics and statistical mechanics.

Structural properties of different sphere packings with arbitrary porosities for planetary-science applications

Granular solids in planetary science are found in the regolith that covers planetary surfaces as well as in the bulk of rubble-pile asteroids, comets and planetesimals. To help understand the physics of these planetary bodies, we aim at deriving the structural properties of granular packings over a large range of porosities. Relevant to fluid flow and gas diffusion are the void spaces inside the granular packings so that we analyze the mean free path of point-like particles, their diffusion constant and their total traveled path lengths. For mechanical and heat-transport properties, the coordination number and the absolute chain length of the inter-connected particles are important. Generally, we also derive the homogeneity and isotropy of the granular solids. We compare granular packings generated by four algorithms for porosities in the range between 85 and 42%, which are the upper and lower limits for natural packings of equal-sized spheres. All produced sphere packings arrive at very similar quantities for the mean free path, the free path probability distribution function, the diffusion constant and the total traveled path length for the entire range of porosities. Hence, transport processes governed by the void-space properties are independent of the specific generation algorithm for the granular packing. In contrast, heat conduction or mechanical stresses almost exclusively depend on the existence and properties of particle contacts and particle chains in the network of spheres. In this case, the four algorithms deliver very different results.Graphic abstract

Multi-species kinetic-fluid coupling for high-energy density simulations

Many physics problems are subject to a mix of continuum and non-equilibrium flows or conditions where one transition into the other, as a function of time and/or space. Numerically, such flows could be described with a purely kinetic method which, in the limit of small particle mean free paths, reproduces a continuum regime. However, from a computational perspective, such approaches are usually very expensive or simply not feasible, even on modern computational architectures. A solution is the construction of hybrid methods which connect the kinetic and hydrodynamic system of evolution equations. Such coupling usually requires specific boundary conditions between non-equilibrium and continuum regimes which come with their own numerical challenges and can introduce unphysical effects. Here, we present a hybrid model which uses a buffer region to transition from a kinetic into a fluid description following the original work of Degond et al. [1]. In the buffer region, both, the kinetic and hydrodynamic equations are solved simultaneously while being coupled via a so-called transition function. The latter also ensures a smooth conversion from the coupled model to either the kinetic or continuum approach at the interfaces of the buffer region. With that, the method avoids the need to find direct interface boundary conditions and allows one to localize the use of a high dimensional kinetic model only where it is needed. In our work, we extend the original method of Degond to flows with multiple particle species in 3D. We derive the coupled equations for the multispecies Vlasov-Bhatnagar-Gross-Krook (VBGK) model and its limiting Euler or Navier-Stokes hydrodynamic equations. To validate our model numerically, we simulate a Sod shock problem and study the effect of kinetic multi-species mixing in the preheat phase of a high energy-density plasma physics experiment.

Exposure computational models with voxel phantoms coupled to EGSnrc Monte Carlo code

In computational dosimetry of ionizing radiation, the energy deposited in radiosensitive organs and tissues is evaluated when an anthropomorphic simulator (phantom) is irradiated using Exposure Computational Models (ECMs). An ECM is a virtual scene with a phantom positioned mathematically relative to a radioactive source. The initial state includes information like the type of primary particle, its energy, starting point coordinates, and direction. Subsequently, robust Monte Carlo (MC) codes are used to simulate the particle's mean free path, interaction with the medium's atoms, and energy deposition. These are common steps for simulations involving photons and/or primary electrons. The GDN (Research Group on Numerical Dosimetry and the Research Group on Computational Dosimetry and Embedded Systems) has published ECMs with voxel phantoms irradiated by photons using the MC code EGSnrc. This work has led to specific computational tools development for various numerical dosimetry stages, including input file preparation, ECM execution, and result analysis. Since 2004, the GDN developed in-house applications like FANTOMAS, CALDose_X, DIP, and MonteCarlo. Certain previously used phantoms are reintroduced to provide historical context in the ECMs' production timeline, emphasizing additive modifications inherent in systematic theme studies. The dosimetric evaluations used the binary version of the MASH (Male Adult mesh) phantom, converted to the SID (Dosimetric Information System) text file type. This format has been used by the group since 2021 to couple a voxel phantom to the EGSnrc user code. The ECM included an environmental dosimetry problem simulation. Most of these tools are accessible on the GDN page (http://dosimetrianumerica.org).

Adaptive wave-particle decomposition in UGKWP method for high-speed flow simulations

With wave-particle decomposition, a unified gas-kinetic wave-particle (UGKWP) method has been developed for multiscale flow simulations. With the variation of the cell Knudsen number, the UGKWP method captures the transport process in all flow regimes without the kinetic solver’s constraint on the numerical mesh size and time step being determined by the kinetic particle mean free path and particle collision time. In the current UGKWP method, the cell Knudsen number, which is defined as the ratio of particle collision time to numerical time step, is used to distribute the components in the wave-particle decomposition. The adaptation of particles in the UGKWP method is mainly for the capturing of the non-equilibrium transport. In this aspect, the cell Knudsen number alone is not enough to identify the non-equilibrium state. For example, in the equilibrium flow regime with a Maxwellian distribution function, even at a large cell Knudsen number, the flow evolution can be still modelled by the Navier-Stokes solver. More specifically, in the near space environment both the hypersonic flow around a space vehicle and the plume flow from a satellite nozzle will encounter a far field rarefied equilibrium flow in a large computational domain. In the background dilute equilibrium region, the large particle collision time and a uniform small numerical time step can result in a large local cell Knudsen number and make the UGKWP method track a huge number of particles for the far field background flow in the original approach. But, in this region the analytical wave representation can be legitimately used in the UGKWP method to capture the nearly equilibrium flow evolution. Therefore, to further improve the efficiency of the UGKWP method for multiscale flow simulations, an adaptive UGKWP (AUGKWP) method is developed with the introduction of an additional local flow variable gradient-dependent Knudsen number. As a result, the wave-particle decomposition in the UGKWP method is determined by both the cell and gradient Knudsen numbers, and the use of particles in the UGKWP method is solely to capture the non-equilibrium flow transport. The current AUGKWP method becomes much more efficient than the previous one with the cell Knudsen number only in the determination of wave-particle composition. Many numerical tests, including Sod shock tube, normal shock structure, hypersonic flow around cylinder, flow around reentry capsule, and an unsteady nozzle plume flow, have been conducted to validate the accuracy and efficiency of the AUGKWP method. Compared with the original UGKWP method, the AUGKWP method achieves the same accuracy, but has advantages in memory reduction and computational efficiency in the simulation for flows with the co-existing of multiple regimes.

Using Thermomechanical Properties to Reassess Particles' Dispersion in Nanostructured Polymers: Size vs. Content.

Nanoparticle-filled polymers (i.e., nanocomposites) can exhibit characteristics unattainable by the unfilled polymer, making them attractive to engineer structural composites. However, the transition of particulate fillers from the micron to the nanoscale requires a comprehensive understanding of how particle downsizing influences molecular interactions and organization across multiple length scales, ranging from chemical bonding to microstructural evolution. This work outlines the advancements described in the literature that have become relevant and have shaped today's understanding of the processing-structure-property relationships in polymer nanocomposites. The main inorganic and organic particles that have been incorporated into polymers are examined first. The commonly practiced methods for nanoparticle incorporation are then highlighted. The development in mechanical properties-such as tensile strength, storage modulus and glass transition temperature-in the selected epoxy matrix nanocomposites described in the literature was specifically reviewed and discussed. The significant effect of particle content, dispersion, size, and mean free path on thermomechanical properties, commonly expressed as a function of weight percentage (wt.%) of added particles, was found to be better explained as a function of particle crowding (number of particles and distance among them). From this work, it was possible to conclude that the dramatic effect of particle size for the same tiny amount of very small and well-dispersed particles brings evidence that particle size and the particle weight content should be downscaled together.

Simulation and comparison between multi-fluid and kinetic models of impurity transport

Impurity transport is a highly significant research topic in international fusion plasma simulations, which are mainly simulated by numerical codes at present. Most of the numerical simulation codes for impurity transport adopt multi-fluid or kinetic model to treat impurity particles. Therefore, it is necessary to select a suitable transport model for the simulation process. For impurity particles, if the mean free path of particles λ is much smaller than the gradient scale length of particles λ g, it is sufficient to treat the particles by the multi-fluid model. However, under some conditions, λ will be much larger than λ g. The applicability of the fluid model is limited when λ is larger than or equal to λ g. A comparison with the simulations on impurity transport treated with multi-fluid and kinetic models is necessary, respectively. In this study, the simulation results of carbon (C) impurity transport in the EAST scrape-off layer with the 2D edge plasma fluid code SOLPS-ITER and the 2D Monte Carlo impurity transport code DIVIMP are compared. The comparison between the distributions of carbon impurities ( C 0 ∼ C + 6) in the different ionization states and the CIII emissivity predicted by SOLPS-ITER and DIVIMP shows that the density distributions of carbon atoms C 0 predicted by the SOLPS-ITER and DIVIMP codes are similar. However, for carbon ions in different ionization states, the variations between the density distributions simulated from the SOLPS-ITER and DIVIMP codes can become larger with the increase in ionization states. DIVIMP performs slightly better than SOLPS-ITER in reproducing the shape of the CIII profile when drifts are switched off in SOLPS-ITER, but the difference is extremely small in terms of the uncertainties involved in these calculations.

The role of resistivity in hot accretion flows with anisotropic pressure: Comparing magnetic field models

Abstract In hot accretion flows, such as the accretion flow in the Galactic center (Sgr A*) and in M 87, the collisional mean free path of the charged particles is significantly larger than the typical length-scale of the accretion flows. Under these conditions, the pressure perpendicular to the magnetic field and that parallel to the magnetic field are not the same; therefore, the pressure is anisotropic to magnetic field lines. On the other hand, the resistivity as a dissipative mechanism plays a key role in the structure and the heating of hot accretion flows. In the present paper, we study the dynamics of resistive hot accretion flows with anisotropic pressure when the magnetic fields have even z-symmetry about the midplane. By presenting a set of self-similar solutions, we find that if the magnetic fields have even z-symmetry or the viscosity form depends on the strength of magnetic field, the disc properties can be entirely different. In the presence of symmetric fields, the velocity components and the disc temperature increase considerably. Also, we show that the increase in infall velocity and temperature due to the anisotropic pressure can be more significant if the resistivity is taken into account. Our results indicate that the resistivity can be an effective mechanism for the heating of hot accretion flows in the high-limit of the magnetic diffusivity parameter. Moreover, the heating due to the anisotropic pressure is comparable to the resistive heating, only when the strength of anisotropic pressure is about unity. The increase of disc temperature can lead to the acceleration of the electrons in such flows. This helps us to explain the origin of phenomena such as the flares in Sgr A*. Our results predict that the presence of resistivity makes it easier for outflows to launch from hot accretion flows.

Continuous Approach in the Theory of Interaction of Charged Particles with Solids

In this paper, the main radiative processes, for the first time, are considered within the framework of continuum physics, where the motion of charged particles occurs in a continuous resisting medium.Within the analysis, new characteristics of the interaction of ions with matter are introduced. The proposed formalism also uses some results of modern corpuscular theory. It allows giving a physically clear space-time description of the processes and phenomena analyzed in this paper. Further in this paper, the authors have estimated, with a high degree of accuracy, the ranges of ions in various materials; constructed their energy loss profiles; determined the mean free path and mean free time of the charged particles in matter, and proposed the formalism for analyzing the radiative change in the physical properties of solids. For the first time, the physical nature of the main characteristic of the radiation resistance of materials, i.e., the threshold displacement energy, has been studied in detail, and the formula for its calculation has been derived by the authors.

A measurement of the effective mean free path of solar wind protons

Weakly collisional plasmas are subject to nonlinear relaxation processes, which can operate at rates much faster than the particle collision frequencies. This causes the plasma to respond like a magnetised fluid despite having long particle mean free paths. In this Letter the effective collisional mechanisms are modelled in the plasma kinetic equation to produce density, pressure and magnetic-field responses to compare with spacecraft measurements of the solar wind compressive fluctuations at 1 AU. This enables a measurement of the effective mean free path of the solar wind protons, found to be ${\approx }4 \times 10^{5}$ km, which is approximately $10^{3}$ times shorter than the collisional mean free path. These measurements are shown to support the effective fluid behaviour of the solar wind at scales above the proton gyroradius and demonstrate that effective collision processes alter the thermodynamics and transport of weakly collisional plasmas.

Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum

We consider the acceleration of charged particles in relativistic shearing flows, with Lorentz factor up to Γ0 ∼ 20. We present numerical solutions to the particle transport equation and compare these with results from analytical calculations. We show that in the highly relativistic limit the particle energy spectrum that results from acceleration approaches a power law, , with a universal value for the slope of this power law, where α parameterizes the power-law momentum dependence of the particle mean free path. At mildly relativistic flow speeds, the energy spectrum becomes softer and sensitive to the underlying flow profile. We explore different flow examples, including Gaussian and power-law-type velocity profiles, showing that the latter yield comparatively harder spectra, producing for Γ0 ≃ 3 and Kolmogorov turbulence. We provide a comparison with a simplified leaky-box approach and derive an approximate relation for estimating the spectral index as a function of the maximum shear flow speed. These results are of relevance for jetted, high-energy astrophysical sources such as active galactic nuclei, since shear acceleration is a promising mechanism for the acceleration of charged particles to relativistic energies and is likely to contribute to the high-energy radiation observed.

Cosmic-ray Transport in Magnetohydrodynamic Turbulence

Abstract This paper studies cosmic-ray (CR) transport in magnetohydrodynamic (MHD) turbulence. CR transport is strongly dependent on the properties of the magnetic turbulence. We perform test particle simulations to study the interactions of CR with both total MHD turbulence and decomposed MHD modes. The spatial diffusion coefficients and the pitch angle scattering diffusion coefficients are calculated from the test particle trajectories in turbulence. Our results confirm that the fast modes dominate the CR propagation, whereas Alfvén and slow modes are much less efficient and have shown similar pitch-angle scattering rates. We investigate the cross field transport on large and small scales. On large/global scales, normal diffusion is observed and the diffusion coefficient is suppressed by M A ζ compared to the parallel diffusion coefficients, with ζ closer to 4 in Alfvén modes than that in total turbulence, as theoretically expected. For the CR transport on scales smaller than the turbulence injection scale, both the local and global magnetic reference frames are adopted. Superdiffusion is observed on such small scales in all the cases. Particularly, CR transport in Alfvén modes show clear Richardson diffusion in the local reference frame. The diffusion transitions smoothly from the Richardson’s one with index 1.5 to normal diffusion as the particle mean free path decreases from λ ∥ ≫ L to λ ∥ ≪ L, where L is the injection/coherence length of turbulence. Our results have broad applications to CRs in various astrophysical environments.

W fuzz layers: very high resistance to sputtering under fusion-relevant He + irradiations

In this study, we have modeled the sputtering process of energetic He+ ions colliding with W nano-fuzz materials, based on the physical processes, such as the collision and diffusion of energetic particles, sputtering and redeposition. Our modeling shows that the fuzzy nanomaterials with a large surface-to-volume ratio exhibit very high resistance to sputtering under fusion-relevant He+ irradiations, and their sputtering yields are mainly determined by the thickness of fuzzy nano-materials, the reflection coefficients and mean free paths of energetic particles, surface sputtering yields of a flat base material, and the geometry of nano-fuzz. Our measurements have confirmed that the surface sputtering yield of a W nano-fuzz layer with the columnar geometry of nano-fuzz in cross-section is about one magnitude of order lower than the one of smooth W substrates. This work provides a complete model for energetic particles colliding with the nano-fuzz layer and clarifies the fundamental sputtering process occurring in the nano-fuzz layer.

Resistive hot accretion flows with anisotropic pressure

Since the collisional mean free path of charged particles in hot accretion flows can be significantly larger than the typical length-scale of the accretion flows, the gas pressure is anisotropic to magnetic field lines. For such a large collisional mean free path, the resistive dissipation can also play a key role in hot accretion flows. In this paper, we study the dynamics of resistive hot accretion flows in the presence of anisotropic pressure. We present a set of self-similar solutions where the flow variables are assumed to be a function only of radius. Our results show that the radial and rotational velocities and the sound speed increase considerably with the strength of anisotropic pressure. The increase in infall velocity and in sound speed are more significant if the resistive dissipation is taken into account. We find that such changes depend on the field strength. Our results indicate that the resistive heating is $10\%$ of the heating by the work done by anisotropic pressure when the strength of anisotropic pressure is 0.1. This value becomes higher when the strength of anisotropic pressure reduces. The increase in disk temperature can lead to heating and acceleration of the electrons in such flows. It helps us to explain the origin of phenomena such as the flares in Galactic Center Sgr A*.

Unified gas-kinetic wave-particle methods V: Diatomic molecular flow

In this paper, the unified gas-kinetic wave-particle (UGKWP) method is further developed for diatomic gas with the energy exchange between translational and rotational modes for flow study in all regimes. The multiscale transport mechanism in UGKWP is coming from the direct modelling in a discretized space, where the cell's Knudsen number, defined by the ratio of particle mean free path over the numerical cell size, determines the flow physics simulated by the wave particle method. The non-equilibrium distribution function in UGKWP is tracked by the discrete particle and analytical wave. The weights of distributed particle and wave in different regimes depend on cell's Knudsen number, where distinguishable macroscopic flow variables of particle and wave are updated inside each control volume. The UGKWP becomes a particle method in the highly rarefied flow regime and converges to the gas-kinetic scheme (GKS) for the Navier-Stokes solution in the continuum flow regime without particles. In comparison with discrete velocity method (DVM)-based unified gas-kinetic scheme (UGKS), the computational cost and memory requirement in UGKWP have been reduced by several orders of magnitude for the high speed and high temperature flow simulation, where the translational and rotational non-equilibrium can be captured accurately in the transition and rarefied regime. As a result, for the hypersonic flow around a flying vehicle, the computation can be conducted using a personal computer for the studies in all regimes. The UGKWP method for diatomic gas will be validated in various cases from one dimensional shock structure to three dimensional flow over a sphere, and the numerical solutions will be compared with the reference DSMC results and experimental measurements.

Unified gas-kinetic wave-particle methods IV: multi-species gas mixture and plasma transport

In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) methods to the multi-species gas mixture and multiscale plasma transport. The construction of the scheme is based on the direct modeling on the mesh size and time step scales, and the local cell’s Knudsen number determines the flow physics. The proposed scheme has the multiscale and asymptotic complexity diminishing properties. The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale, and preserve the asymptotic Euler, Navier-Stokes, and magnetohydrodynamics (MHD) when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path. The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number. In the continuum regime, the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver. In the UGKWP, the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables, and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale. For plasma transport, the current scheme provides a smooth transition from particle-in-cell (PIC) method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime. In the continuum limit, the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time. In the highly magnetized regime, the cell size and time step are not restricted by the Debye length and plasma cyclotron period. The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.

A stochastic kinetic scheme for multi-scale plasma transport with uncertainty quantification

Plasmas present a diverse set of behaviors in different regimes. Given the intrinsic multi-scale nature of plasma dynamics, classical theoretical and numerical methods are often employed at separate scales with corresponding assumptions and approximations. Clearly, the coarse-grained modeling may introduce considerable uncertainties between the field solutions of flow and electromagnetic variables, and the real plasma physics. To study the emergence, propagation and evolution of randomness from gyrations of charged particles to magnetohydrodynamics poses great opportunities and challenges to develop both sound theories and reliable numerical algorithms. In this paper, a physics-oriented stochastic kinetic scheme will be developed that includes random inputs from both flow and electromagnetic fields via a hybridization of stochastic Galerkin and collocation methods. Based on the BGK-type relaxation model of the multi-component Boltzmann equation, a scale-dependent kinetic central-upwind flux function is designed in both physical and particle velocity space, and the governing equations in the discrete temporal-spatial-random domain are constructed. By solving Maxwell's equations with the wave-propagation method, the evolutions of ions, electrons and electromagnetic field are coupled throughout the simulation. We prove that the scheme is formally asymptotic-preserving in the Vlasov, magnetohydrodynamical, and neutral Euler regimes with the inclusion of random variables. Therefore, it can be used for the study of multi-scale and multi-physics plasma system under the effects of uncertainties, and provide scale-adaptive physical solutions under different ratios among numerical cell size, particle mean free path and gyroradius (or time step, local particle collision time and plasma period). Numerical experiments including one-dimensional Landau Damping, the two-stream instability and the Brio-Wu shock tube problem with one- to three-dimensional velocity settings, and each under stochastic initial conditions with one-dimensional uncertainty, will be presented to validate the scheme.

Mass-limited plasmas heated by laser-driven fast electrons as a powerful source of neutron and hard x-ray radiation

To create a plasma with extreme thermodynamic parameters, we propose to heat with a laser-accelerated fast electron beam a target of a size less than the mean free path of the heating particles. The effect of capture of fast electrons generated in an electrically neutral target due to the action of a self-consistent electrostatic field at its boundaries allows us to volumetrically heat a target over multiple flights of fast electrons through it. Using such a heating mode enables control of the target mass to be significantly less than the mass stopping range of the heating particles. Heating a mass-limited target by laser-driven relativistic electrons can produce a plasma with a temperature of ∼10’s keV and a density close to its initial solid-state density. Such plasma objects are expected to serve as powerful sources of neutron and hard x-ray radiation.

Numerical investigation of optimal divertor gas baffle closure on TCV

A first set of divertor gas baffles has recently been installed in the TCV tokamak. In order to explore the physics determining the benefits and limitations of divertor baffling and to guide the design of a possible second generation of baffles, the effect of baffle closure is investigated using the 2D transport code SolEdge2D-EIRENE with realistic wall geometries. The baffle extension is scanned, first imposing the same upstream conditions as in previous SOLPS-ITER studies, then extending the parameter space to access detached plasma conditions. In attached plasma cases, divertor neutral compression is maximised by a Low-Field Side baffle length with an opening between the separatrix and the baffle tip of approximately 5 λ q , resulting in an increase in neutral compression by a factor 4 with respect to the unbaffled case. In detached cases this ratio can be improved by up to a factor 25 using higher baffle closures. This difference in behaviour between attached and detached conditions is explained by a model based on the ionisation mean free path of neutral particles recycled from the target. In some conditions, the optimal baffle extension in terms of neutral compression is found to be subject to high levels of intercepted upstream heat flux, which results in a peak heat flux on the baffles comparable to the one impinging on the outer target. The individual roles of the High-Field Side and Low-Field Side baffles are disentangled by means of dedicated simulations, which show a lower global impact of the inner baffle. This study suggests that an outer baffle with a gap of approximately 3 λ q , slightly more closed than the one presently installed, could further enhance the neutral compression ratio in cases where the ionisation front is detached. The biggest unknown in these simulations is related to far SOL particle transport, which could result in higher levels of baffle recycling and thus limit baffle performance.

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.