Direct Simulation Monte Carlo Research Articles

Numerical and experimental study of supersonically expanding argon plasma using a micrometer hollow cathode discharge

Pulsed discharge nozzles (PDNs) have been successfully used for decades to produce rotationally cold (Trot ∼ 20 K) radicals and ions of astrophysical interest and to characterize these species spectroscopically. In this work, an evolution of the PDN, the piezostack pulsed discharge nozzle (P2DN), is used for the first time to investigate the characteristics of the still poorly understood supersonic plasma expansion. The P2DN allows for a better control of the reservoir pressure of which an accurate measurement is required to characterize the plasma expansion. This new source, thus, gives the opportunity to further optimize the plasma conditions and extend its use to new target species. The spatial distribution of an argon plasma and the effect of the supersonic flow for different pressures are studied by combining a two-dimensional extended fluid model (extFM) and a direct simulation Monte Carlo (DSMC) method. The combined simulation is validated with experimental results obtained through emission spectroscopy associated with a group-code collisional-radiative model to retrieve the plasma parameters. The validated numerical approach (DSMC-extFM) allows for an accurate characterization of the plasma structure in our typical experimental conditions (a reservoir pressure ranging from 90 to 905 mbar). Thus, this simulation will be used in future studies to improve the plasma conditions to favor the synthesis of (transient) hydrocarbon species as found in space, by seeding the argon gas with a suitable precursor, such as acetylene.

Investigation of local Nusselt number in a rarefied microchannel gas flow using Direct Simulation Monte Carlo method

Heat transfer in microscale and rarefied gaseous flows has been investigated for high Knudsen number (Kn) flows using the Direct Simulation Monte Carlo (DSMC) method. A wide range of Kn [0.05 – 5], pressure ratio [1.2 – 4] and wall temperature [323 K – 723 K] is studied. The heat flux and local Nusselt number (Nu) are found to exhibit three distinct sections along the length of the microchannel – heat transfer-dominated section, constant heat transfer section and exit section. The defining roles of different effects such as non-equilibrium conditions at the wall, viscous dissipation, axial conduction, pressure work, and shear work have been considered implicitly in the DSMC method. These parameters are seen to influence the heat transfer differently at lower and higher Kn. These results help to quantify the fraction of total heat exchange and reveals the complex interplay of various terms of the energy equation in gas flow in microchannels.

An improved continuum model for hypersonic thermal nonequilibrium flow in the near-continuum regime

In this work, the rarefied Couette flow of diatomic gases with thermal nonequilibrium effects is investigated by the direct simulation Monte Carlo (DSMC) method, and a macroscopic computational model is developed to consider the local rarefaction effects for diatomic gases in the near-continuum regime. The nonlinear transport properties of the diatomic gases are studied, indicating that effective viscosity and effective translational thermal conductivity in the shear nonequilibrium state are affected by translational nonequilibrium effects, which obey the same laws for both monatomic and diatomic gases. The transport coefficients of internal energy modes are affected by both translational nonequilibrium and internal energy relaxation, therefore, the effective rotational and vibrational thermal conductivities are related to the effective viscosity through a modified Eucken relation that accounts for internal energy relaxation. Conclusively, effective constitutive relations are newly established as a function of the shear nonequilibrium parameter and the modified Eucken factors for thermal nonequilibrium flows, and these are integrated into the macroscopic two-temperature model. Subsequently, it is assessed in the simulation of hypersonic flows over flat plates and cylinders at various Knudsen numbers. The results show that the surface shear stress and heat flux obtained by the proposed model agree well with the DSMC results, indicating significantly improved performance compared to the conventional Navier–Stokes two-temperature model for hypersonic flows in the near-continuum regime.

Ten-moment fluid model with heat flux closure for gasdynamic flows

The gasdynamic 10-moment equations are revisited, applying a Chapman-Enskog type expansion to achieve the closure for heat flux. The resulting gradient-based closure performs favorably against previously employed closures in regions of shocks when compared to the kinetic results obtained from Direct Simulation Monte Carlo simulations. The 10-moment model can capture finite kinetic effects due to non-Maxwellian velocity distribution functions, as compared to the 5-moment Navier-Stokes equations, and remains hyperbolic-parabolic with real wave speeds under all initial conditions. Self-consistent kinetic boundary conditions are derived without consideration of a distinct Knudsen layer. The model is applied to a number of canonical gasdynamics problems in one and two dimensions such as steady normal and oblique shocks, as well as the Sod shock tube problem.

An improved DSMC method for acoustic agglomeration of solid particles assisted by spray droplets

The direct simulation Monte Carlo (DSMC) method was improved for describing the process of acoustic agglomeration assisted by spray droplets. The agglomeration and rebound as consequences of inter-particle collisions were modeled by considering all possible particle types, in particular, mixed-phase particles associated with the immersion and distribution mechanisms. Numerical predictions by the present method were validated by both analytical solutions and experimental data. The dynamic process of the acoustic agglomeration in terms of the size and type evolution as well as the evolution of number concentration of different particle types were examined. Results suggest that the strong interaction between the solid particles and the spray droplets causes the significant enhancement of acoustic agglomeration. Furthermore, the effect of frequency varying from audible to ultrasonic range on the acoustic agglomeration was evaluated. Overall, a better performance of acoustic agglomeration can be achieved at a lower frequency, regardless of addition of spray droplets. Agglomeration efficiency of 64.8 % can be achieve at acoustic frequency of 1000 Hz in case with spray droplets. This study provides not only a numerical model for describing the agglomeration of complex particles in particle-laden gas flows, but also important insights into the acoustic agglomeration assisted by spray droplets.

Thermal rarefied gas flow simulations with moving boundaries based on discrete unified gas kinetic scheme and immersed boundary method

Rarefied gas flows in complex geometries with heat transfer and moving boundaries have received remarkable attention due to the prevailing of micro/nano science and engineering applications. This study presents an immersed boundary (IB)-based discrete unified gas kinetic scheme (DUGKS) for the simulation of complex moving boundary thermal rarefied flows. The fluid and temperature fields are solved according to the DUGKS with Shakhov collision model (SDUGKS), which is capable of modeling non-unit Prandtl number problems in all Knudsen regimes. The IB approach, featured by its superior performance to incorporate complicated boundary constraints on Cartesian grids, is applied here to handling the interaction between the micro gas and curved boundaries. Specifically, for the IB adopted in this study, we use the least square method for the reconstruction of the distribution functions near the boundary. The accuracy of the IB-SDUGKS is explored comprehensively in six benchmark problems, including the micro lid-driven cavity flow, thermal equilibrium gas in a circular domain, thermal transpiration in an enclosure, moving shuttle in a rarefied gas, movement of a piston with pressure differences and particle free motion in a lid-driven cavity. The present results agree well with the corresponding analytical and direct simulation Monte Carlo (DSMC) solutions, illustrating the versatility of the present IB-SDUGKS complex thermal rarefied flows.

Localised sublimation under cometary dust mantles

Observations of cometary surfaces have indicated that the amount of exposed surficial water ice is insufficient to supply the measured water production rates within the inner solar system. The measured water production rates are also a small fraction of that which would arise from freely subliming pure water ice bodies of similar surface area. A plausible mechanism, given the now well-established, high porosity of comets is that water ice is subliming below the visible surface with the produced gas escaping through the pores in a non-subliming hot dust matrix. If this process leads to an additional heating of the evolving gas, this may provide an elegant explanation for the temperature of the gas required to match measured terminal velocities. We investigate this process using a series of numerical models based on the Direct Simulation Monte Carlo (DSMC) method for rarified gas dynamics. We show that inhomogeneities in the source are not evident in the flow field once the thickness of the porous layer reaches a specific size and that lateral flow through the porous layer (parallel to the surface) can broaden the apparent source size when the source is small and isolated. The temperatures of the gas flow at the visible surface can be significantly increased and will respond to the temperature distribution within the non-volatile porous layer. The influence of the structural properties of the porous layer (how the porous layer is mechanically constructed) is shown to have only limited influence on the final gas parameters at the surface.

Regular reflection of shock waves in steady flows: viscous and non-equilibrium effects

Numerical analysis of a steady monatomic gas flow about the point of the regular reflection of a strong oblique shock wave from the symmetry plane is conducted with the Navier–Stokes–Fourier (NSF) equations, the regularized Grad 13-moment (R13) equations and the direct simulation Monte Carlo (DSMC) method. In contrast to the inviscid solution to this problem completely defined by the Rankine–Hugoniot (RH) relations, all three models predict a complicated flow structure with strong thermal non-equilibrium and a long wake with flow parameters not predicted by the RH relations. The temperature $T_y$ related to thermal motion of molecules in the direction normal to the symmetry plane has a maximum inside the reflection zone while in a planar shock wave the maximum is observed for the $T_x$ temperature. The R13 equations predict these features much better than the NSF equations and are in good agreement with the benchmark DSMC results. An analysis of the flow with the conservation equations was conducted in order to evaluate the effects of various processes on a fluid element moving along the symmetry plane. In contrast to the shock wave where effects of viscosity and heat conduction are one-dimensional with zeroth net contribution to the fluid-element energy across the shock, the flow across the zone of the shock reflection is dominated by two-dimensional effects with positive net contribution of viscosity and negative contribution of heat conduction to the fluid-element energy. These effects are believed to be the main source of the wake with parameters deviating from the RH values.

MCS-based quantile value approach for reliability-based design of tunnel face support pressure

This paper develops a new approach for reliability-based design (RBD) of tunnel face support pressure from a quantile value perspective. A surrogate model is constructed to calculate the collapse pressures of the random samples generated by a single run of Monte Carlo simulation (MCS). The cumulative distribution function (CDF) of the collapse pressure is then obtained and the support pressure aiming at a target failure probability is chosen as the upper quantile value of the collapse pressures. The proposed approach does not require repetitive reliability analyses compared to the existing methods. Moreover, a direct relationship between the target failure probability and the required support pressure is established. An illustrative example is used to demonstrate the implementation procedure. The accuracy of the reliability-based support pressures is verified by direct MCS incorporating with three-dimensional numerical simulations. Finally, the influencing factors, including the sample size of MCS, the correlation coefficient between random variables, the choice of experimental points, and the surrogate model, are investigated. This method can play a complementary role to available approaches due to its advantages of simplicity and efficiency.

Coexistence of distinct nonuniform nonequilibrium steady states in Ehrenfest multiurn model on a ring.

The recently proposed Ehrenfest M-urn model with interactions on a ring is considered as a paradigm model which can exhibit a variety of distinct nonequilibrium steady states. Unlike the previous three-urn model on a ring which consists of a uniform steady state and a nonuniform nonequilibrium steady state, it is found that for even M≥4, an additional nonequilibrium steady state can coexist with the original ones. Detailed analysis reveals that this additional nonequilibrium steady state emerged via a pitchfork bifurcation which cannot occur if M is odd. Properties of this nonequilibrium steady state, such as stability, and steady-state flux are derived analytically for the four-urn case. The full phase diagram with the phase boundaries is also derived explicitly. The associated thermodynamic stability is also analyzed, confirming its stability. These theoretical results are also explicitly verified by direct Monte Carlo simulations for the three-urn and four-urn ring models.

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.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

A stochastic Fokker–Planck–Master model for diatomic rarefied gas flows

The direct simulation Monte Carlo (DSMC) method is widely used for numerical solutions of the Boltzmann equation. However, the associated computational cost becomes prohibitive in the near-continuum regime. To address this limitation, the particle-based Fokker–Planck (FP) method has been extensively studied in the past decade. The FP equation, which describes Brownian motion, does not require resolution of the collisional time and length scales. While several monatomic FP models have been proposed, the modeling of diatomic gases within the FP framework has received limited attention. In this paper, we propose a new diatomic kinetic model, named the Fokker–Planck–Master (FPM) model, which can accurately describe energy exchanges between translational-rotational and translational-vibrational modes. The FPM model combines the FP equation to describe the evolution of translational and rotational modes, and the master equation to describe the evolution of the vibrational modes. The numerical test cases include relaxation problems, Couette flows, and hypersonic flows past a vertical flat plate. The results demonstrate that the FPM model shows good agreement with both analytical and DSMC solutions.

Efficient metamodel-based importance sampling coupled with single-loop estimation method for parameter global reliability sensitivity analysis

To efficiently estimate the main effects and total effects of uncertain distribution parameters on the uncertainty of failure probability, we construct single-loop estimation formulas by introducing auxiliary variables through the equal probability transformation. This approach circumvents the original nested triple-loop process. For generating samples used in the derived single-loop estimation formulas, direct Monte Carlo simulation can be employed. To reduce the number of samples in Monte Carlo simulation, the important sampling technique can be integrated into the proposed single-loop estimation formulas. Additionally, to enhance the efficiency of identifying the states (failure or safety) of all used samples, an adaptive Kriging model can be introduced. Subsequently, the adaptive Kriging model coupled with Monte Carlo simulation, and the adaptive Kriging model coupled with the importance sampling technique, are integrated into the derived single-loop formulas to concurrently and efficiently estimate the main effects and total effects of uncertain distribution parameters. The results of three case studies validate the accuracy and efficiency of the proposed method.

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.

Non-equilibrium and anisotropy in Titan atmosphere entry

Titan is the largest moon of Saturn. It is the only moon in our solar system provided with atmosphere. Therefore, exploration of Titan demands aerodynamic computations aimed at a proper design of space vehicles that will bring instruments to its surface. To this purpose, the aerodynamic computing methodologies are Computational Fluid Dynamics (CFD) for the solution of continuum flow fields and Direct Simulation Monte Carlo (DSMC) method for the solution of rarefied flow fields. The failure of the Navier-Stokes equations at high altitude is due to non-equilibrium (different translational, rotational and vibrational temperatures) and to anisotropy (different components in the x, y and z direction of the translational temperature). The limitations in using DSMC at low altitudes are due to technical limitations of the computer, i.e. memory capacity and processing speed. In the present work, non-equilibrium has been quantified by the relative differences between translational and rotational (θt-r), translational and vibrational (θt-v) temperatures. Similarly, the relative differences between the translational temperature component along x and those along y (θx-y) and z (θx-z) quantified anisotropy. For each test condition, the maximum values of θ are representative of non-equilibrium and anisotropy. Computations have been carried out considering the Huygens capsule in axial-symmetric flow, along Titan atmospheric entry in the altitude interval 100–470 km by means of the commercial codes: 1) Fluent v18.1 in the interval 100–295 km, 2) DS2V-4.5 64 bits in the interval 295–470 km. The present computations stated that in the Titan entry of a capsule of dimension comparable with that of Huygens, the altitude of about 182 km should be the limit altitude for a proper use of a CFD code.

A Coaxial Nozzle Attachment Improving the Homogeneity of the Gas Flow Sputtering

The Hollow Cathode Gas Flow Sputtering (GFS) provides special plasma conditions and is of extensive interest as a more affordable alternative to the high vacuum sputtering techniques. In the case of the tubular cathode a circular outlet symmetry stipulates homogeneity issues for both metallic and reactive deposition regimes. Using the results of Direct Simulation Monte Carlo (DSMC), we propose an external coaxial attachment which is manufactured and examined in a nozzle and a diffuser positioning. The impact on the homogeneity of Ti and TiO2 films is examined using profilometry and spectral ellipsometry. Our results demonstrate that the use of the nozzle attachment significantly enhances film homogeneity from about 3 cm2 to more than 12 cm2. It also secures better process control in terms of oxygen stoichiometry and film thickness. Some crucial general issues of the reactive GFS process are discussed.

Estimating external exposure from different source geometries in soil contaminated by gamma-ray emitting radionuclides: A computational model combining Monte Carlo simulation and mathematical transformation

A computational model is developed to estimate the external exposure to an arbitrary scenario describing the spatial distribution of the source in a contaminated soil by gamma-ray emitting radionuclides released after nuclear activities. This model welds together a Monte Carlo simulation approach and a mathematical transformation based on Tait–Bryan angles formalism. For Monte Carlo simulation, a source depth dependent optimized geometry describing a semi-infinite soil-cover-air medium of propagation is mathematically derived based on the physical attenuation of the gamma ray in matter. For each primary energy of interest, the proposed mathematical transformation is applied to the output of the Monte Carlo simulation to define the overlapping region between the contaminated soil volume and the proposed optimized volume. This approach made it possible to estimate the human external exposure associated with the scenario under investigation. Several scenarios were studied. The flux energy distribution of the detected radiation and the corresponding dose quantities are calculated. A benchmarking test with a direct Monte Carlo simulation and a comparison with published results in the literature show good agreement validating therefore the presented model.

Particle-in-Cell simulation of radio-frequency ion thruster RIT-1.0

Ion propulsion systems with a small thrust can be used in space missions for highly accurate controllability while possessing low power and propellant consumption. At the University of Giessen, radio-frequency ion thrusters are miniaturized with thrusts in the order of micro-Newtons (μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document}N-RIT). As the size of the thruster becomes smaller the plasma properties can not be measured inside a closed plasma chamber. The simulations are needed to investigate plasma properties like temperature, density profile which would aid further opimization. In this work, we discuss kinetic simulations of plasma inside the RIT-1.0 thruster using in-house developed PlasmaPIC code that incorporates Monte-Carlo collisions (MCC) and Direct Simulation Monte Carlo (DSMC) for neutral gas distribution. We briefly present the features of PlasmaPIC and simulation of stable plasma generation. We also show the influence of neutral gas and extraction properties for the RIT-1.0 thruster under consideration.

Direct numerical simulation of supersonic nanoparticles flow in free-molecule regime using the angular coefficient method

We employ molecular flow methods to numerically simulate the supersonic nanoparticles flow in free-molecule regime. To streamline the computational complexity, interaction forces between the gas and solid particles are disregarded. We first develop a discrete phase model (DPM) method that integrates the non-rigid body collision model, enabling an accurate simulation of nanoparticle diffusion under the influence of the drag force and Brownian motion force. The nanoparticles considered in this study have sizes below 10 nm, and the accuracy of the DPM method is verified by comparing its results with experimental data. Subsequently, we theoretically and numerically investigate the transmission probability and number density of N2 molecules flowing through two-dimensional (2D) channels and three-dimensional (3D) tubes by using the angular coefficient (AC) method and the direct simulation Monte Carlo (DSMC) method. The findings indicate that as the diameter of the nanoparticle (dp) decreases to 1 nm, the diffusion coefficient (D) and the root mean square displacement (x) of nanoparticles approach the N2 molecules. The microscopic velocity of most N2 molecules falls within the range of 62–1400 m/s, and the macroscopic velocity of N2 flow falls within the range of Ma = 1.28–1.35. In contrast to the DSMC method, the AC method exhibits enhanced accuracy even with a reduced number of grids and obviates the process for large-scale sampling. Additionally, the solution time required by the AC method is approximately 1/10 and 1/13–1/32 of the DSMC method in 3D cylindrical tubes and 2D channels, respectively. Moreover, the AC method demonstrates superior adaptability when dealing with complex geometries.