Direct Simulation Monte Carlo Approach Research Articles

A hybrid Gaussian mixture/DSMC approach to study the Fourier thermal problem

In rarefied gas dynamics scattering kernels deserve special attention since they contain all the essential information about the effects of physical and chemical properties of the gas–solid surface interface on the gas scattering process. However, to study the impact of the gas–surface interactions on the large-scale behavior of fluid flows, these scattering kernels need to be integrated in larger-scale models like Direct Simulation Monte Carlo (DSMC). In this work, the Gaussian mixture (GM) model, an unsupervised machine learning approach, is utilized to establish a scattering kernel for monoatomic (Ar) and diatomic (H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {H}_{2}$$\\end{document}) gases directly from Molecular Dynamics (MD) simulations data. The GM scattering kernel is coupled to a pure DSMC solver to study isothermal and non-isothermal rarefied gas flows in a system with two parallel walls. To fully examine the coupling mechanism between the GM scattering kernel and the DSMC approach, a one-to-one correspondence between MD and DSMC particles is considered here. Benchmarked by MD results, the performance of the GM-DSMC is assessed against the Cercignani–Lampis–Lord (CLL) kernel incorporated into DSMC simulation (CLL-DSMC). The comparison of various physical and stochastic parameters shows the better performance of the GM-DSMC approach. Especially for the diatomic system, the GM-DSMC outperforms the CLL-DSMC approach. The fundamental superiority of the GM-DSMC approach confirms its potential as a multi-scale simulation approach for accurately measuring flow field properties in systems with highly nonequilibrium conditions.

A direct Monte Carlo approach for the modeling of neutrals at the plasma edge and its self-consistent coupling with the 2D fluid plasma edge turbulence model HESEL

This paper presents a novel coupling of a kinetic description of neutrals with a fluid description of a fusion plasma. The code, plasma interacting super-atoms and molecules (PISAM), employs a grid-free Cartesian geometry and a direct simulation Monte Carlo approach to solve the kinetic equations of deuterium atoms and molecules. The grid-free geometry and the parallel nature of the neutral dynamics, in the absence of neutral–neutral interactions, allow for an unlimited and work-efficient parallelization of PISAM that always ensures a balanced workload. The highly optimized Python implementation obtains good performance while securing easy accessibility to new users. The coupling of PISAM with the edge turbulence model HESEL is outlined with emphasis on the technical aspects of coupling Message Passing Interface-parallelized Python and C++ codes. Furthermore, the paper presents and analyzes simulation results from running the coupled HESEL-PISAM model. These results demonstrate the impact of radial neutral transport and plasma–neutral dynamics perpendicular to the magnetic field. Specifically, they illustrate how the inward flow of neutral kinetic energy and the inhibition of radial electric shear, resulting from poloidal momentum transfer between atoms and ions, can affect the energy containment time. By comparing the results of the HESEL-PISAM model with those obtained from coupling HESEL with a diffusive-fluid-neutral model, the capabilities of diffusion models in predicting neutral transport in the plasma edge and scrape-off layer are elucidated.

Liquid evaporation from nanochannel in the presence of non-condensable gas by direct simulation Monte Carlo

Liquid-vapor phase change in the form of evaporation and with the confinement of nanostructures is of vast importance in many engineering applications. The direct simulation Monte Carlo (DSMC) approach was employed to simulate the liquid evaporation from two-dimensional nanochannels in the presence of non-condensable (NC) gases. The thermodynamic properties of vapor and NC gas flow fields as well as the evaporation rate of liquid were calculated. Factors such as channel opening ratio, meniscus receding depth and the NC gas density were examined to reveal their impacts on the liquid evaporation rates. The results showed that the gas mixture flow field transforms to one-dimensional away from the channel exit. The overall vapor pressure drop can be divided into that across the liquid–vapor interface, that within the channel, outside the channel, as well as that caused by the binary diffusion. The effect of NC gas on the evaporation rate was found to be the most outstanding with pinned meniscus and with a wide channel opening, in which case the normalized evaporation rate drops to 0.43. Moreover, the Maxwell-Stefan equation could capture the evaporation data with modified diffusion lengths. For receded meniscus (250 nm deep) and small opening ratio (0.417), the effective diffusion length was as large as 1.6 times that in the one-dimensional case.

Plasma flow and instabilities in the magnetic mirror with ion recycling and neutral back-flow

Magnetic mirror configurations are observed in natural settings and have various applications in laboratory plasmas, such as a magnetic expander of the open mirror fusion devices. The axial plasma flow in open mirror systems is significantly influenced by atomic processes involving neutrals, such as ionization and charge-exchange collisions. A quasi-two-dimensional computational model was developed to study these effects on accelerated plasma flow in magnetic mirror configurations. This model includes an emitting wall, a quasineutral flow/acceleration region with a magnetic expander, and a recycling/absorbing wall. Implemented in a hybrid quasineutral code, the model incorporates drift-kinetic ions, fluid electrons, and fully kinetic neutral atoms with collision processes simulated using the direct simulation Monte Carlo approach. Ion recycling on the wall is accounted for using empirical methods. The model demonstrates that slow atoms with short mean free paths create a dense plasma layer near the wall, modifying the plasma potential which can lead to large-scale perturbations due to ion–ion streaming instabilities.

Study of Side-Jet Interactions over a Hypersonic Cone Flow Using Kinetic Methods

The interaction between the hypersonic flow over a cone and a side jet creates complex multiple recirculation regions as well as separation, reattachment, and bow shocks known as Edney type-IV shock/shock interactions. The interactions between these multiscale structures are shown to be well captured with the direct simulation Monte Carlo (DSMC) method, which is a particle-kinetic approach. This computational technique was used because even though the freestream conditions were continuumlike, sufficiently strong gradients in the shock and jet expansion were observed to cause continuum breakdown. The time-accurate DSMC approach has been shown in previous work to capture the multiscale structures and unsteadiness in the shock structure that occurs in such flows, as was also observed in this work. These sources of unsteadiness in the shock system are shown to locally affect surface parameters, such as heat transfer and skin friction, as well as predict a low–frequency unsteadiness consistent with earlier values in the literature.

Effects of the neutral dynamics model on the particle-in-cell simulations of a Hall thruster plasma discharge

The dynamics of the neutral atoms in Hall thrusters affects several plasma processes, from ionization to electrons' mobility. In the context of Hall thruster's particle-in-cell (PIC) modeling, the neutrals are often treated kinetically, similar to the plasma species, and their interactions with themselves and the ions are resolved using the direct-simulation Monte–Carlo (DSMC) algorithm. However, the DSMC approach is computationally resource demanding. Therefore, modeling the neutrals as a 1D fluid has been also pursued in simulations that do not involve the radial coordinate and, hence, do not resolve the neutrals' radial expansion. In this article, we present an extensive study on the sensitivity of the PIC simulations of Hall thruster discharge to the model used for the neutral dynamics. We carried out 1D axial PIC simulations with various fluid and kinetic models of the neutrals as well as self-consistent quasi-2D axial-azimuthal simulations with different neutrals’ fluid descriptions. Our results show that the predictions of the simulations in either 1D or 2D configurations are highly sensitive to the neutrals' model, and that different treatments of the neutrals change the spatiotemporal evolution of the discharge. Moreover, we observed that considering the ion-neutral collisions causes a significant variation in the neutral temperature, thus requiring that the neutrals' energy equation to be included as well in their fluid system of equations. Finally, the self-consistent axial-azimuthal simulations highlighted that a neutrals’ model based on the continuity conservation equation only is not an appropriate choice and leads to physically unexpected high-frequency global discharge oscillations.

Coupled Monte Carlo–Deterministic simulation of thermal behavior of graphene on substrate

The purpose of this paper is to study thermal effects in a graphene sheet on substrate. The temperature evolution of the phonon branches and of the crystal lattice is obtained when also the presence of the substrate is taken into account. The numerical strategy is based on the coupling of stochastic and deterministic numerical methods; Boltzmann equation for charge transport is treated by using the Direct Simulation Monte Carlo approach, whose results are treated as source terms for the phonon Boltzmann equation which is solved deterministically. The results are useful also for applications as correct design of graphene-based prototypes, when heating effects in the presence of a substrate are relevant.

Arbitrary Lagrangian-Eulerian discrete velocity method with application to laser-induced plume expansion

We propose a direct arbitrary Lagrangian-Eulerian (ALE) variant of the previously developed discrete velocity method to solve the kinetic equation with the Bhatnagar–Gross–Krook (BGK) model collision integral. The effectiveness and robustness of the new scheme are demonstrated by computing the axisymmetric plume expansion into a low pressure gas due to evaporation from the solid surface caused by a nanosecond laser pulse. The numerical studies show that the new numerical scheme provides a very significant reduction of the required computing time as compared to the discrete velocity method on a fixed mesh. Cross-comparisons with the direct simulation Monte Carlo (DSMC) approach further confirm the accuracy of the new scheme and model kinetic equations as applied to plume expansion studies. Finally, the good parallel scalability of the new ALE scheme as implemented into the in-house code Nesvetay is demonstrated. The use of two very different approaches to numerical prediction of the transient flow pattern allows us to obtain a reliable numerical solution, which can be regarded as a new transient benchmark test.

Direct-simulation Monte Carlo modeling of reactor-scale gas-dynamic phenomena in a multiwafer atomic-layer deposition batch reactor

Atomic layer deposition (ALD) using multiwafer batch reactors has now emerged as the manufacturing process of choice for modern microelectronics at a massive scale. Stringent process requirements of thin film deposition uniformity within wafer (WiW) and wafer–wafer (WTW) in the batch, film conformity along submicrometer wafer features, thin film quality, and the utilization of expensive precursors in the reactor dictate ALD reactor design and process parameter optimization. This paper discusses a particle-based direct-simulation Monte Carlo (DSMC) of the full reactor scale simulation that overcomes the low Knudsen number limitation of typical continuum computational fluid dynamics approaches used for modeling low-pressure ALD reactors. A representative industrial multiwafer batch reactor used for the deposition of Si-based thin films with N2 and Si2Cl6 (hexachlorodisilane) as process feed gases with pressures in the range 43–130 Pa and a uniform reactor temperature of 600 °C is simulated. The model provides detailed insights into the flow physics associated with the transport of the precursor species from the inlets, through wafer feed nozzles, into the interwafer regions, and finally through the outlet. The reactor operating conditions are shown to be in the slip/transitional flow regime for much of the reactor volume and especially the feed gas nozzle and interwafer regions (where the Knudsen number approaches ∼0.2), justifying the need for a high-Knudsen number DSMC approach as in this work. For the simulated conditions, the nonuniformity of precursor species immediately above the wafer surface is predicted to be within <1% for a given wafer and <2% across the entire multiwafer stack. Results indicate that higher pressure degrades WiW and WTW uniformity. A reactor flow efficiency is defined and found to be ∼99%, irrespective of the chamber pressure.

Velocity-Space Hybridization of Direct Simulation Monte Carlo and a Quasi-Particle Boltzmann Solver

This paper presents a new method for modeling rarefied gas flows based on hybridization of direct simulation Monte Carlo (DSMC) and discrete velocity method (DVM)-based quasi-particle representations of the velocity distribution function. It is aimed at improving the resolution of the tails of the distribution function (compared with DSMC) and computational efficiency (compared with DVM). Details of the method, such as the collision algorithm and the particle merging scheme, are discussed. The hybrid approach is applied to the study of noise in a Maxwellian distribution, computation of electron-impact ionization rate coefficient, as well as numerical simulation of a supersonic Couette flow. The hybrid-based solver is compared with pure DSMC and DVM approaches in terms of accuracy, computational speed, and memory use. It is shown that such a hybrid approach can provide a lower computational cost than a pure DVM approach, while being able to retain accuracy in modeling high-velocity tails of the distribution function. For problems where trace species have a significant impact on the flow physics, the proposed method is shown to be capable of providing better computational efficiency and accuracy compared with standard fixed-weight DSMC.

Kinetic Modeling of Spacecraft Surfaces in a Plume Backflow Region

Plasma-surface interactions caused by electric propulsion devices are an important spacecraft aspect of the design that is difficult to measure in ground-based facilities. The negatively biased solar panel surfaces attract the slow-moving charge exchange (CEX) ions generated inside an ion core plume, which can cause surface sputtering on the protective coatings of the solar panels. We use a fully kinetic particle-in-cell direct simulation Monte Carlo (PIC-DSMC) approach that models both electron and ion trajectories to allow us to fully characterize the plasma sheath formed near these surfaces and to understand how the plasma sheath affects the trajectories of CEX ions, their incident energies and angles, and surface sputtering rates. We find that, outside the plasma core, the ion and electron distribution functions are highly non-Maxwellian, and the assumption of electron temperatures is questionable. We introduce a novel floating potential ground boundary condition that enables us to emulate the spacecraft ground for a high range of plasma number densities and surface charging conditions. Finally, we estimate the erosion of the surface using the kinetic results and surface yield empirical relations.

Thermophysical model for icy cometary dust particles

Context. Cometary dust particles are subjected to various forces after being lifted off the nucleus. These forces define the dynamics of dust, trajectories, alignment, and fragmentation, which, in turn, have a significant effect on the particle distribution in the coma. Aims. We develop a numerical thermophysical model that is applicable to icy cometary dust to study the forces attributed to the sublimation of ice. Methods. We extended the recently introduced synoptic model for ice-free dust particles to ice-containing dust. We introduced an additional source term to the energy balance equation accounting for the heat of sublimation and condensation. We use the direct simulation Monte Carlo approach with the dusty gas model to solve the mass balance equation and the energy balance equation simultaneously. Results. The numerical tests show that the proposed method can be applied for dust particles covering the size range from tens of microns to centimetres with a moderate computational cost. We predict that for an assumed ice volume fraction of 0.05, particles with a radius, r ≫ 1 mm, at 1.35 AU, may disintegrate into mm-sized fragments due to internal pressure build-up. Particles with r < 1 cm lose their ice content within minutes. Hence, we expect that only particles with r > 1 cm may demonstrate sustained sublimation and the resulting outgassing forces.

A novel hybrid DSMC-Fokker Planck algorithm implemented to rarefied gas flows

In this article, a new hybrid approach combining Fokker Planck (FP) and Direct Simulation Monte Carlo (DSMC) is introduced. The FP approach, which is an approximation of the Boltzmann equation, has been recently introduced for modeling rarefied gas flows. Compared to standard molecular methods like DSMC, it has a lower computational cost. However, the FP model may provide erroneous results in modeling some flow features, including vortex region and shock waves. The DSMC approach is accurate enough; yet, it has a high computational cost, especially at low Knudsen/low Mach number conditions. The purpose of this article is to suggest an optimized hybrid algorithm to benefit from high-speed modeling and sufficient accuracy simultaneously. We show a modified form of the gradient length Knudsen number, i.e., KnGL/Kn, must be considered to define the range of accuracy of the FP method. We used KnGL/Kn as a parameter for switching between DSMC and FP. The cavity flow and the nozzle flow are considered for the investigation of the proposed hybrid algorithm. The dependence of the accuracy and computational cost on KnGL/Kn is reported. The results demonstrate that the optimized KnGL/Kn is between 3 and 4, which gives suitable computational cost and acceptable accuracy.

Numerical study of heat transfer in Rayleigh-Bénard convection under rarefied gas conditions.

The focus of this research is to delineate the thermal behavior of a rarefied monatomic gas confined between horizontal hot and cold walls, physically known as rarefied Rayleigh-Bénard (RB) convection. Convection in a rarefied gas appears only for high temperature differences between the horizontal boundaries, where nonlinear distributions of temperature and density make it different from the classical RB problem. Numerical simulations adopting the direct simulation Monte Carlo approach are performed to study the rarefied RB problem for a cold to hot wall temperature ratio equal to r=0.1 and different rarefaction conditions. Rarefaction is quantified by the Knudsen number, Kn. To investigate the long-time thermal behavior of the system two ways are followed to measure the heat transfer: (i) measurements of macroscopic hydrodynamic variables in the bulk of the flow and (ii) measurements at the microscopic scale based on the molecular evaluation of the energy exchange between the isothermal wall and the fluid. The measurements based on the bulk and molecular scales agreed well. Hence, both approaches are considered in evaluations of the heat transfer in terms of the Nusselt number, Nu. To characterize the flow properly, a modified Rayleigh number (Ra_{m}) is defined to take into account the nonlinear temperature and density distributions at the pure conduction state. Then the limits of instability, indicating the transition of the conduction state into a convection state, at the low and large Froude asymptotes are determined based on Ra_{m}. At the large Froude asymptote, simulations following the onset of convection showed a relatively small range for the critical Rayleigh (Ra_{m}=1770±15) that flow instability occurs at each investigated rarefaction degree. Moreover, we measured the maximum Nusselt values Nu_{max} at each investigated Kn. It was observed that for Kn≥0.02,Nu_{max} decreases linearly until the transition to conduction at Kn≈0.03, known as the rarefaction limit for r=0.1, occurs. At the low Froude (parametric) asymptote, the emergence of a highly stratified flow is the prime suspect of the transition to conduction. The critical Ra_{m} in which this transition occurs is then determined at each Kn. The comparison of this critical Rayleigh versus Kn also shows a linear decrease from Ra_{m}≈7400 at Kn=0.02 to Ra_{m}≈1770 at Kn≈0.03.

Simulation of bipolar charge transport in graphene on h-BN

Purpose The purpose of this paper is to simulate charge transport in monolayer graphene on a substrate made of hexagonal boron nitride (h-BN). This choice is motivated by the fact that h-BN is one of the most promising substrates on account of the reduced degradation of the velocity due to the remote impurities. Design/methodology/approach The semiclassical Boltzmann equations for electrons in the monolayer graphene are numerically solved by an approach based on a discontinuous Galerkin (DG) method. Both the conduction and valence bands are included, and the inter-band scatterings are taken into account as well. Findings The importance of the inter-band scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the inter-band scatterings produces sizable variations in the average values, like the current density, at zero Fermi energy, whereas, as expected, the effect of the inter-band scattering becomes negligible by increasing the absolute value of the Fermi energy. Research limitations/implications The correct evaluation of the influence of the inter-band scatterings on the electronic performances is deeply important not only from a theoretical point of view but also for the applications. In particular, it will be shown that the time necessary to reach the steady state is greatly affected by the inter-band scatterings, with not negligible consequences on the switching on/off processes of realistic devices. As a limitation of the present work, the proposed approach refers to the spatially homogeneous case. For the simulation of electron devices, non-homogenous numerical solutions are required. This last case will be tackled in a forthcoming paper. Originality/value As observed in Majorana et al. (2019), the use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the inter-band scatterings, is computationally very expensive because the valence band is highly populated and a huge number of particles is needed. Even by simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electron-hole pairs. The DG approach, used in this paper, does not suffer from the previous drawbacks and requires a reasonable computing effort.

Systematic direct simulation Monte Carlo approach to characterize the effects of surface roughness on accommodation coefficients

The effects of surface roughness on accommodation coefficients are studied using the direct simulation Monte-Carlo (DSMC) approach. DSMC simulations are carried out by varying the roughness parameters. An empirical relationship between surface roughness and accommodation coefficients is established.

Molecular dynamics simulation of particle trajectory for the evaluation of surface accommodation coefficients

Molecular dynamics approach has been employed in this work to model experiments for the interaction of a single particle with a surface. Some important modifications are implemented in the present approach as compared to the previous works. Investigations have been performed to compute the reflected distribution of molecules and calculate accommodation coefficients for interaction of noble gases with platinum and graphite surfaces. Through a few numerical tests, it is verified that the results obtained using the present approach are in good agreement with some established data in the literature. The method is then utilized to predict the surface accommodation coefficients, which are required as boundary conditions in the gas–surface interaction models, such as the Cercignani–Lampis–Lord (CLL) model, for mesoscopic simulations using the direct simulation Monte Carlo approach. Interestingly, it is shown that the CLL model shows some disagreement with the molecular dynamics simulation data for the distribution of reflected molecules. However, the CLL model can still be utilized in practical applications, as it is shown to predict the average macroscopic properties to a very high degree of accuracy. The behavior of accommodation coefficients with a change in the incident flow properties, such as bulk velocity, gas temperature, and gas-to-surface molecule mass ratio, is also investigated in detail.

Nonequilibrium air flow predictions with a high-fidelity direct simulation Monte Carlo approach

A conventional and high-fidelity direct simulation Monte Carlo approach is applied to a number of hypersonic cases where validation data is available. Changes in the collision model are found to have a small effect on surface properties, with nitrogen recombination being the only notable exception.

Validation of High-Temperature Air Reaction and Relaxation Models Using Emission Data

The analysis of infrared (IR) and ultraviolet (UV) emissions in hypersonic air flows is performed using conventional and high-fidelity direct simulation Monte Carlo approaches, with the objective of distinguishing between models of various high-temperature processes. A comparison between the computational results and the available shock-tube data is conducted for IR intensities in the NO band at relatively low freestream velocities from 2.97 to , and UV spectra of NO at a high velocity of . For all conditions, the high-fidelity model provides good agreement with the data. Little impact of the model refinement is observed at low velocities. At the high velocity, the equilibrium constant and the energy disposal for the exchange reaction, as well as the vibrational coupling of nitrogen dissociation, play a significant role and warrant further study.

Development of Knudsen thermal force for mass analysis of CH4/He gas mixture

Natural gas is known as the main source of energy and also contains significant and noble gases. Numerous researches have been performed to present novel methods for the detection and analysis of natural gas. In this study, Direct Simulation Monte Carlo (DSMC) method is used to evaluate the performance of a new micro gas sensor (MIKRA) for detection of helium in CH4/He gas mixture. In this sensor, the temperature difference of two arms inside a rectangular domain at low-pressure condition induces a Knudsen force which is proportional to physical properties of the gas. In order to define flow feature of a low-pressure gas inside the micro gas actuator, high order equation of Boltzmann is used to attain high precision results. To solve these equations, DSMC approach is used as a robust method for the non-equilibrium flow field. The effects of main factors such as length and gap of arms are comprehensively investigated in different ambient pressures. Furthermore, the effect of various concentrations of the CH4/He gas mixture on force generation is comprehensively studied. Our findings show that value of generated Knudsen force significantly different when the fraction of He in CH4/He gas mixture is varied. This indicates that this micro gas sensor could precisely detect the concentration of Helium gas inside a low-pressure CH4/He gas mixture.