Direct Simulation Monte Carlo Scheme Research Articles

Application of Weighting Direct Simulation Monte Carlo Schemes to Weakly Ionized Rarefied Gas Flows

Application of Weighting Direct Simulation Monte Carlo Schemes to Weakly Ionized Rarefied Gas Flows

Kinetic Modeling of Ionospheric Outflows in Pressure Cooker Environments

AbstractPlasma escape from the high‐latitude ionosphere (ion outflow) serves as a significant source of heavy plasma to the magnetospheric plasma sheet and ring current regions. Outflows alter mass density and reconnection rates, hence global responses of the magnetosphere. A new fully kinetic and semi‐kinetic model, KAOS (Kinetic model of Auroral ion OutflowS), is constructed from first principles which traces large numbers of individual O+ ion macro‐particles along curved magnetic field lines, using a guiding‐center approximation, in order to facilitate calculation of ion distribution functions and moments. Particle forces include mirror and parallel electric field forces, a self‐consistent ambipolar electric field, and a parameterized source of ion cyclotron resonance wave heating, thought to be central to the transverse energization of ions. The model is initiated with a steady‐state ion density altitude profile and Maxwellian velocity distribution and particle trajectories are advanced via a direct simulation Monte Carlo scheme. This outlines the implementation of the kinetic outflow model, demonstrates the model's ability to achieve near‐hydrostatic equilibrium necessary for simulation spin‐up, and investigates L‐shell dependent wave heating and pressure cookers scenarios. This paper illustrates the model initialization process and numerical investigations of L‐shell dependent outflows and pressure cooker environments and serves to advance our understanding of the drivers and particle dynamics in the auroral ionosphere.

Velocity distribution of metal ions in the target region of HiPIMS: the role of Coulomb collisions

High power impulse magnetron sputtering (HiPIMS) discharges have become an important tool for the deposition of thin, hard coatings. Such discharges are operated at a very low working gas pressure in the order of 1 Pa. Therefore, elastic collisions between ions and other heavy particles are often calculated to occur with low frequency, using the hard sphere approximation. However, inside the magnetic trap region of the discharge, a very dense plasma is created and Coulomb collisions become the dominant collision process for ions. In this article, we show that Coulomb collisions are a necessary part of a complete description of ion movement in the magnetic trap region of HiPIMS. To this end, the velocity distribution function (VDF) of chromium and titanium ions is measured using high-resolution optical emission spectroscopy. The VDF of those ions is then described using a simple simulation which employs a direct simulation Monte Carlo scheme. The simulation describes the self-relaxation of the VDF that is initially a Thompson distribution as being created during the sputtering process. Measurement positions inside the discharge are matched to the simulation results choosing an appropriate relaxation time. In this fashion, excellent agreement between simulation and measurement is obtained. We find, that the distribution quickly becomes mostly Maxwellian with a temperature of 9 eV for titanium ions and 4.5 eV in the case of chromium ions. Only the high energy tail of the VDF retains the shape of the initial Thompson distribution. The observed high temperature is explained with an energy redistribution from the highly energetic Thompson distribution into an partly-thermalized Maxwell-like distribution. Finally, the temperature resulting from this energy redistribution is calculated using a simple analytical model which shows good agreement with the measurements.

Interspecies fractional collisions

Relaxing the direct simulation Monte Carlo (DSMC) method to be compatible with variable particle weights has significant advantages when applied to simulations with a large density variation, as the statistical noise in the low density region can be controlled. The stochastic weighted particle method [S. Rjasanow and W. Wagner, J. Comput. Phys. 124, 243–253 (1996)] enables the control of particle weight but has never been applied to a multicomponent gas mixture or plasma. This paper focuses on charge-exchange collisions, but the method is applicable to any interspecies collision reactions including ionization and excitation collisions, providing the correct collision rate between different species with different weights in the context of no-time-counter scheme, which can easily be extended to other DSMC schemes. The method is first verified through two test cases observing the depletion of high-energy ions as impacting background neutral particles and looking at the spatial variation and temporal evolution of different species. Then, the method is applied to a simple electric propulsion plume simulation, demonstrating the controllability of species ratio and capability in reducing statistical noise for the charge-exchange ions.

Extension of the low diffusion particle method for near-continuum two-phase flow simulations

The low diffusion (LD) particle method, proposed by Burt and Boyd, is modified for the near-continuum two-phase flow simulations. The LD method has the advantages of easily coupling with the direct simulation Monte Carlo (DSMC) method for multi-scale flow simulations and dramatically reducing the numerical diffusion error and statistical scatter of the equilibrium particle methods. Liquid- or solid-phase particles are introduced in the LD method. Their velocity and temperature updating are respectively, calculated from the motion equation and the temperature equation according to the local gas properties. Coupling effects from condensed phase to gas phase are modeled as momentum and energy sources, which are respectively, equal to the negative values of the total momentum and energy increase in liquid or solid phase. The modified method is compared with theoretical results for unsteady flows, and good agreements are obtained to indicate the reliability of the one-way gas-to-particle coupling models. Hybrid LD–DSMC algorithm is implemented and performed for nozzle discharging gas–liquid flow to show the prospect of the LD–DSMC scheme for multi-scale two-phase flow simulations.

DSMC Scheme for Phonon Transport in Solid Thin Films

Analysis of phonon dynamics based on a linearized Boltzmann transport equation is widely used for thermal analysis of solid thin films, but couplings among various phonon modes appear in some situations. We propose a direct simulation Monte Carlo (DSMC) scheme to simulate the phonon gas starting without the conventional linearization approximation. This requires no relaxation time as an input parameter, and we can investigate the couplings among phonons with different modes. A prototype code based on a simple phonon model was developed, and energy flux was evaluated for thin films of various thickness as a test calculation.

DSMC scheme to study phonon dynamics

In order to solve the Boltzmann transport equation (BTE) of phonons for investigating heat conduction in non-metallic solids, we propose employing a DSMC (direct simulation Monte Carlo) scheme to simulate the dynamics of phonons analogous to rarefied gas. In contrast to treating the BTE with conventional linear approximation, this scheme requires no relaxation times as input parameters. We can directly investigate couplings among phonons with different modes, although we have to assume an appropriate scattering model for phonon-phonon interactions. In this paper, we describe the DSMC scheme for phonon dynamics and present some results with our prototype codes for a simple solid model. In the first case of single-branch four-phonon processes, we carried out simulations of non-equilibrium thin films with a temperature gradient. We found that the temperature jump at the boundaries can be successfully achieved. For the second case of three phonon processes, we developed a simulation code that takes into consideration the different acoustic branches to evaluate the mode-dependent relaxation time and mean free path. This type of DSMC scheme for phonons enables us to include other relevant factors, such as optical branches and phonon-electron interactions.

A new scheme developed for the numerical simulation of the Boltzmann equation using the direct simulation monte-carlo scheme for the flow about a sphere

A new scheme, the modified direct simulation Monte-Carlo (MDSMC), for the numerical simulation of the Boltzmann equation for rarefied gas flow about a sphere is developed. The Taylor series expansion is used to obtain the modified equation of the first-order time discretization of the collision equation and the new scheme, MDSMC, is implemented to simulate the collision equation in the Boltzmann equation. In the new scheme (MDSMC) there exists a new extra term which takes into account the effect of the second-order collision. In the new scheme (MDSMC) there also exists a second-order term in time step in the probabilistic coefficients which has the effect of simulation with higher accuracy than the previous DSMC scheme. The results of the drag coefficient of the sphere using the MDSMC scheme show better agreement in comparison with the experimental data of Wegener (1961) than the results of the drag coefficient of the sphere using the DSMC scheme.

Time Splitting Error in DSMC Schemes for the Spatially Homogeneous Inelastic Boltzmann Equation

The paper is concerned with the numerical treatment of the uniformly heated inelastic Boltzmann equation by the direct simulation Monte Carlo (DSMC) method. This technique is presently the most widely used numerical method in kinetic theory. We consider three modifications of the DSMC method and study them with respect to their efficiency and convergence properties. Convergence is investigated with respect to both the number of particles and the time step. The main issue of interest is the time step discretization error due to various splitting strategies. A scheme based on the Strang-splitting strategy is shown to be of second order with respect to time step, while there is only first order for the commonly used Euler-splitting scheme. On the other hand, a no-splitting scheme based on appropriate Markov jump processes does not produce any time step error. It is established in numerical examples that the no-splitting scheme is about two orders of magnitude more efficient than the Euler-splitting scheme. The Strang-splitting scheme reaches almost the same level of efficiency as that of the no-splitting scheme, since the deterministic time step error vanishes sufficiently fast.

Surface Tension Effects on Post-Nucleation Growth of Water Microdroplets in Supersaturated Gas Mixtures

Immediately after nucleation in a superheated gas mixture, the very small size of liquid droplets affects the condensation growth of the droplets in two ways: (1) The droplet size may be comparable to the mean free path of the gas molecules, resulting in noncontinuum transport effects, and (2) surface tension effects may strongly alter the conditions at the interface of the droplet. In the study reported here, a direct simulation Monte Carlo scheme was used to model the molecular transport during quasi-equilibrium condensation growth of water microdroplets in a supersaturated mixture of water vapor and a noncondensable gas. In the simulation, the boundary condition at the droplet surface is treated in a manner that allows us to model surface tension effects on transport. Results of calculations are presented for water vapor and argon mixtures for which some experimental data on droplet growth rates exists. The simulation results indicate that surface tension effects play a significant role in the determination of droplet growth rates during early stages of droplet growth. In particular, the results indicate that the droplet growth rate, droplet temperature, and the temperature slip at the interface pass through maxima as the droplet grows. These trends are a consequence of the shift in equilibrium vapor pressure due to surface tension effects at small droplet radii. [S0022-1481(00)02702-X]

A direct simulation Monte Carlo scheme and uniformly distributed sequences for solving the Boltzmann equation

The Direct Simulation Monte Carlo (DSMC) scheme of Nanbu is considered for the solution of the Boltzmann equation in a simplified case. It is interpreted as a one-step method using particles combined with numerical quadratures after each step. A modified scheme in which the particles are ordered after each step is proposed. It is called the Low Discrepancy (LD) method. The error of the LD method is defined as the discrepancy of the set of particles relative to the exact solution. This error is estimated by means of other discrepancies, namely those of the sequences which perform the quadratures. The replacement of pseudo-random numbers used in the quadratures by uniformly distributed sequences is consequently suggested. Numerical comparisons are given between the DSMC scheme and the LD method that repeatedly uses the Hammersley sequence in the quadratures (LDH method).