Direct Simulation Monte Carlo Method Research Articles

On the synchronisation of three-dimensional shock layer and laminar separation bubble instabilities in hypersonic flow over a double wedge

Linear three-dimensional instability is studied in the shock layer and the laminar separation bubble (LSB) induced by shock-wave/boundary-layer interactions in a Mach 7 flow of nitrogen over a double wedge with a$30^{\circ }\text {--}55^{\circ }$cross-sectional profile. At a free-stream unit Reynolds number$Re=5.2\times 10^{4}\,{\rm m}^{-1}$this flow exhibits rarefaction effects and has shock thicknesses comparable to the thickness of the boundary layer at separation. Flow features have been fully resolved using a high-fidelity massively parallel implementation of the direct simulation Monte Carlo method that captures the flow evolution from the inception of three-dimensionality, through linear growth of instabilities, to the early stages of nonlinear saturation. It is shown that the LSB sustains self-excited, small-amplitude perturbations that originate past the primary separation line and lead to spanwise-periodic wall striations inside the bubble and downstream of the primary reattachment line, as known from earlier experiments, simulations and instability analyses. A spanwise-periodic instability, synchronised with that in the separation zone, is identified herein for the first time, which exists in the internal structure of the separation and detached shock layers, and manifests itself as spanwise-periodic cats-eyes patterns in the global mode amplitude functions. The growth rate and the spanwise-periodicity length of linear disturbances in the shock layers and the LSB are found to be identical. Linear amplification of the most unstable three-dimensional flow perturbations leads to synchronised low-frequency unsteadiness of the triple point, with a Strouhal number of$St\approx 0.028$.

Multiscale simulation of hollow cathode assisted internal plasma treatment process

Hollow cathode discharges can be used for the plasma-assisted surface treatment and coating of the interior surface of blind hole geometries. Due to the geometry, it is difficult to find the setting parameters for a reproducible result. Therefore, a probabilistic simulation based on a Monte Carlo method has been performed to simulate a hollow cathode discharge in a blind hole with an aspect ratio ≫ 1. First, different neutral gas flows inside a blind hole geometry were investigated with a direct simulation Monte Carlo method and compared to impact pressure measurements. Then, the resulting neutral gas pressure distribution was used in a Particle-in-Cell plasma simulation as a background gas to determine the electrical potential distribution, the charge carrier distribution and mean energy inside the blind hole geometry. The plasma potential distribution is compared with a substrate coated with an amorphous diamond like carbon (a-C:H) thin film and shows good agreement.

Utility computable modeling of a Boltzmann model equation for bimolecular chemical reactions and numerical application

A Boltzmann model equation (kinetic model) involving the chemical reaction of a multicomponent gaseous mixture is derived based on Groppi's work [“A Bhatnagar–Gross–Krook-type approach for chemically reacting gas mixtures,” Phys. Fluids 16, 4273 (2004)], in which the relaxation parameters of elastic collision frequency for rigid elastic spheres are obtained based on the collision term, and the pivotal collision frequency of the chemical reaction is deduced from the chemical reaction rate that is determined by the direct simulation Monte Carlo (DSMC) method. This kinetic model is shown to be conservative, and the H theorem for an endothermic reaction is proven. In the framework of the gas-kinetic unified algorithm, the discrete velocity method, finite volume method, and implicit scheme are applied to solve the proposed kinetic model by introducing a suitable boundary condition at the wall surface. For hypersonic flows around a cylinder, the proposed kinetic model and the corresponding numerical methods are verified for both endothermic and exothermic reactions by comparison of the model's results with results from the DSMC method. The different influences of endothermic and exothermic reactions are also given. Finally, the proposed kinetic model is also used to simulate an exothermic reaction-driven flow in a square cavity.

Thermal transpiration flows induced by differences in accommodation coefficients

The behavior of a rarefied gas between two parallel plates, each having its uniform temperature but having a square wave-like distribution of the accommodation coefficients, is numerically analyzed using the Direct Simulation Monte Carlo method. The gas temperature near the plate becomes close to the plate temperature only if the accommodation coefficient of the wall is large enough. A lower accommodation coefficient leads to the difference between the gas temperature and the adjacent wall. This temperature distribution induces a gas flow similar to the well-known thermal transpiration flow along the plate. When only one plate has a wave-like distribution of the accommodation coefficients, an array of vortical flow is induced between the plates. The magnitude of the vortex is linear to the difference in the accommodation coefficients. When both plates have a wave-like distribution of the accommodation coefficients, a one-way flow through the channel between the plates is induced. This flow depends on the phase difference of the discontinuity point. The size of one-way flow is nonlinear to the difference in accommodation coefficients.

Numerical Simulation of Axisymmetric Plasma Plume Based on Gas–Surface Interaction/Electric Field Coupling

Simulation of the transport of ions and neutral particles is important for studies of the flow field and working characteristics of jet flow plasma devices. The spatiotemporal evolution of plasma plumes in the small space within such devices is complex. To realize numerical simulations of particle transport in this situation, a variety of complex effects need to be considered, such as collisions between particles, chemical reactions, gas–surface interaction, and the electric field. In this study, five types of particles (H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , H, metal atoms, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{H}_{2}^{+}$ </tex-math></inline-formula> , and H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> ) are injected into a double-cylinder calculational domain. The direct simulation Monte Carlo (DSMC) method is used to simulate the spatiotemporal evolution of particles in the Maxwell and Cercignani–Lampis–Lord model with different accommodation coefficients. The particle accumulation phenomenon is explained, and the influence of the accommodation coefficient is elucidated. Wall chemical reactions are then introduced to make the gas–surface coupling more realistic. These reactions are found to have a great influence on the particle number density of H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , and H, while the flow field structure has little influence, which is due to the small differences in the particle velocity distributions of each component. Finally, a DSMC/particle-in-cell method is used to simulate a simple axisymmetric example, and the influence of gas–surface interaction, wall chemical reactions, and the electric field on the spatiotemporal evolution of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , and H in the flow field is demonstrated.

Collisional Radiative Modeling of Electronically Excited States in a Hypersonic Flow

The hypersonic, nonequilibrium thermochemistry flow over a cylinder was modeled to study nitric oxide (NO) ultraviolet (UV) emissions. The main focus of this work is to model the populations of the electronically excited states of and NO because the transitions from those states to ground states are responsible for UV emissions. Collisional radiative models were constructed for the purpose of predicting the electronically excited state populations using two approaches, the quasi-steady-state (QSS) method and a MassTR method. These methods were applied to flows generated by the direct simulation Monte Carlo method of ground state species for two types of collisional radiative models. The results showed that when the coupling between the formation of electronically excited states and flow transport was modeled, this resulted in higher populations of the electronically excited states of () and NO in the expansion and wake regions of the flow compared to the QSS result where flow transport is not included. In addition, it was shown that the () state is responsible for the most of the NO(A) state population downstream of the stagnation region.

An open source code for two-phase rarefied flows: rarefiedMultiphaseFoam

This paper presents the development and benchmarking tests of an open source code for providing solutions of two-phase rarefied flows. The solver is named in rarefiedMultiphaseFoam and it is developed in OpenFOAM, based on dsmcFoamPlus, which is used for the rarefied gas phase. The solver can produce both steady and transient results for arbitrary 2D/3D two-phase rarefied flows and includes a phase change model for materials that experience melting and solidification processes. The benchmarking tests include momentum and heat exchange between gas phase and solid phases, particle phase change, a converging nozzle that creates a solid particle beam, and transport of solid particles. The tests yield good agreement with analytical and experimental data where available, and are compared to previous numerical results from the literature. Program summaryProgram Title: rarefiedMulitphaseFoamCPC Library link to program files:https://doi.org/10.17632/8vxrc5gsb8.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: rarefiedMultiphaseFoam has been developed to help investigate multiphase problems with rarefied gas and solid phases. The rarefied gas phase is simulated with the direct simulation Monte Carlo (DSMC) method, with both one and two way coupling models available to simulate the momentum and heat transfer between the phases. It provides an easily extended, parallelised, environment.Solution method: rarefiedMultiphaseFoam implements an explicit time-stepping solver with stochastic molecular collisions appropriate for studying rarefied gas flow problems and uses a Green's function model to calculate the energy transfers between phases.

Effect of internal degrees of freedom in rarefied gas problems: Plane Couette flow

Present paper studies the influence of internal degrees of freedom (IDOF) on rarefied gas flow in a classic plane Couette problem. The numerical study is performed using Direct Simulation Monte Carlo method in a wide range of gas rarefaction, plates relative velocities and for different values of accommodation coefficients (ACs) on the walls. It is shown that: (1) in presence of IDOF a noticeable thermal non-equilibrium between translational and internal energy modes is observed, which increases with gas rarefaction and with plates relative velocity; (2) presence of IDOF can noticeably decrease shear stress and heat flux to the walls, which is mainly explained by the drain of energy from over-excited translational mode into internal ones, leading to lower values of translational temperature and, hence, viscosity; (3) the magnitude of IDOF influence is a non-monotonic function of gas rarefaction, taking maximal values in transitional regimes Kn≈1; (4) values of ACs can significantly change the influence of IDOF, both qualitatively and quantitatively; (5) the rate of energy exchange between translational and internal modes has a noticeable influence on the results, but the energy exchange model itself is not principal as long as this rate is reproduced correctly; (6) increasing number of IDOF leads to more noticeable influence on shear stress and heat flux. Therefore the influence of IDOF will be more significant for polyatomic gases.

Effects of chemical energy accommodation on nonequilibrium flow and heat transfer to a catalytic wall

The accurate prediction of the aeroheating performance of hypersonic vehicles requires more detailed modeling of the catalysis process, rather than merely employing a catalytic coefficient. In this paper, the theoretical modeling, as well as the direct simulation Monte Carlo method, is used to preliminarily study the incomplete chemical energy accommodation effects, that is, only a part of the potential energy released in the heterogenous recombination reaction is transferred to the surface, while the remaining is retained as the vibrational energy of the desorbed molecule. An integrated model is proposed to describe the contribution of each energy mode in the rarefied nonequilibrium heat and mass transfer process. Based on the model and several Damköhler numbers, an analytical formula is derived, and is also shown to compare well with the numerical results. On account of the incomplete accommodations of the chemical and vibrational energy on the wall, a variation up to 20% is observed in predicting the stagnation point heat flux under typical nonequilibrium flow conditions. This study could enrich our understanding of the nonequilibrium heat transfer phenomenon and also shows a potential practical value.

A universal method of redistributing relaxation energies in inelastic molecular collisions

The Larsen–Borgnakke (L–B) model is most widely used to redistribute relaxation energies in direct simulation Monte Carlo (DSMC) method; the relaxation energies in an inelastic collision are determined beforehand and redistributed by sampling from the equilibrium distributions (not locally equilibrium distributions) of these energies that are appropriate to this collision. The present concern focuses on how to redistribute the selected energies when they have been determined to require relaxation. When certain energies are determined to require relaxation and redistribution in an inelastic collision, the hierarchical method and the serial or sequential method are used to complete the L–B procedure. Although the hierarchical method is theoretically accurate, it needs to complete the integration of the joint energy distribution function, so it is not applicable when the energy distribution function has a complex form or contains more than one discontinuous energy. Hence, researchers turn to the serial or sequential method, but the rationality proof of this method implies an assumption of locally thermal equilibrium, so it is obviously not appropriate in the DSMC method usually applied to hypersonic nonequilibrium flows. Then, a new method of redistributing relaxation energies to complete the L–B procedure is proposed, which does not include any integration process; the original form of the energy distribution function is used directly, so it can be expected to have excellent adaptability, especially for complex cases such as energy redistribution of reaction products. The effectiveness, efficiency, and extensive applicability of the method are proved by multiple examples.

Couette flow in a rectangular channel in the whole range of the gas rarefaction

The mass flow rate of a Couette flow in a long rectangular channel is calculated for constant or linear wall velocities in the whole range of the gas rarefaction and in a wide range of the width-to-height ratio. Analytical solutions for arbitrary width-to-height ratios are given for the hydrodynamic regime, the slip regime, and the free molecular regime. Therefore, both the velocity field and the mass flow rate can be calculated. In the transitional regime, simulations via direct simulation Monte Carlo method are performed. The results are provided as reduced flow rates in tabulated data, which can be used for any constant or linear increasing wall velocity (e.g., bounding walls of working chambers in positive displacement vacuum pumps).

One-way flow over uniformly heated U-shaped bodies driven by thermal edge effects

The thermal edge flow is a gas flow typically induced near a sharp edge (or a tip) of a uniformly heated (or cooled) flat plate. This flow has potential applicability as a nonmechanical pump or flow controller in microelectromechanical systems (MEMS). However, it has a shortcoming: the thermal edge flows from each edge cancel out, resulting in no net flow. In this study, to circumvent this difficulty, the use of a U-shaped body is proposed and is examined numerically. More specifically, a rarefied gas flow over an array of U-shaped bodies, periodically arranged in a straight channel, is investigated using the direct simulation Monte-Carlo (DSMC) method. The U-shaped bodies are kept at a uniform temperature different from that of the channel wall. Two types of U-shaped bodies are considered, namely, a square-U shape and a round-U shape. It is demonstrated that a steady one-way flow is induced in the channel for both types. The mass flow rate is obtained for a wide range of the Knudsen numbers, i.e., the ratio of the molecular mean free path to the characteristic size of the U-shape body. For the square-U type, the direction of the overall mass flow is in the same direction for the entire range of the Knudsen numbers investigated. For the round-U type, the direction of the total mass flux is reversed when the Knudsen number is moderate or larger. This reversal of the mass flow rate is attributed to a kind of thermal edge flow induced over the curved part of the round-U-shaped body, which overwhelms the thermal edge flow induced near the tip. The force acting on each of the bodies is also investigated.

Linear stability analysis of hypersonic boundary layers computed by a kinetic approach: a semi-infinite flat plate at $$\varvec{4.5\le \mathrm{M}_\infty \le 9}$$

Linear stability analysis is performed using a combination of two-dimensional Direct Simulation Monte Carlo (DSMC) method for the computation of the basic state and solution of the pertinent eigenvalue problem, as applied to the canonical boundary layer on a semi-infinite flat plate. Three different gases are monitored, namely nitrogen, argon and air, the latter as a mixture of 79\% Nitrogen and 21\% Oxygen at a range of free-stream Mach numbers corresponding to flight at an altitude of 55km. A neural network has been utilised to predict and smooth the raw DSMC data; the steady laminar profiles obtained are in very good agreement with those computed by (self-similar) boundary layer theory, under isothermal or adiabatic wall conditions, subject to the appropriate slip corrections computed in the DSMC method. The leading eigenmode results pertaining to the unsmoothed DSMC profiles are compared against those of the classic boundary layer theory. Small quantitative, but no significant qualitative differences between the results of the two classes of steady base flows have been found at all parameters examined. The frequencies of the leading eigenmodes at all conditions examined are practically identical, while perturbations corresponding to the DSMC profiles are found to be systematically more damped than their counterparts arising in the boundary layer at the conditions examined, when the correct velocity slip and temperature jump boundary conditions are imposed in the base flow profiles; by contrast, when the classic no-slip boundary conditions are used, less damped/more unstable profiles are obtained, which would lead the flow to earlier transition. On the other hand, the DSMC profiles smoothed by the neural network are marginally more stable than their unsmoothed counterparts.

Moving impingement heat transfer in a three-dimensional rarefied hydrogen gas jet based on the direct simulation Monte Carlo method coupled with the finite difference method

Temperature control systems in low-pressure environments are an important component of precision instruments. In this study, a comprehensive model coupling the direct simulation Monte Carlo (DSMC) method with the finite difference method is proposed to investigate the heat transfer due to the impact of a moving rarefied hydrogen jet. The cooling heat flow calculated using the DSMC method and radiant heat flow calculated using the boundary integral method are considered as the boundary conditions of the substrate, and the temperature distribution of the substrate is solved using the finite difference method. The influences of different inlet pressures, jet apertures, and impact distances on the jet heat transfer intensity are analyzed. The results indicate that the average convective heat transfer coefficient of jet impingement is positively correlated with the inlet pressure and jet aperture, whereas it is inversely correlated with the impingement distance. The structural parameters that meet the temperature control requirements in the substrate are determined using the proposed comprehensive model. These results can be used as a guide for the design of high-precision temperature control devices.

Machine learning product state distributions from initial reactant states for a reactive atom-diatom collision system.

A machine-learned model for predicting product state distributions from specific initial states (state-to-distribution or STD) for reactive atom-diatom collisions is presented and quantitatively tested for the N(4S) + O2(X3Σg -) → NO(X2Π) + O(3P) reaction. The reference dataset for training the neural network consists of final state distributions determined from quasi-classical trajectory (QCT) simulations for ∼2000 initial conditions. Overall, the prediction accuracy as quantified by the root-mean-squared difference (∼0.003) and the R2 (∼0.99) between the reference QCT and predictions of the STD model is high for the test set, for off-grid state-specific initial conditions, and for initial conditions drawn from reactant state distributions characterized by translational, rotational, and vibrational temperatures. Compared with a more coarse grained distribution-to-distribution (DTD) model evaluated on the same initial state distributions, the STD model shows comparable performance with the additional benefit of the state resolution in the reactant preparation. Starting from specific initial states also leads to a more diverse range of final state distributions, which requires a more expressive neural network compared with DTD. A direct comparison between QCT simulations, the STD model, and the widely used Larsen-Borgnakke (LB) model shows that the STD model is quantitative, whereas the LB model is qualitative at best for rotational distributions P(j') and fails for vibrational distributions P(v'). As such, the STD model can be well-suited for simulating nonequilibrium high-speed flows, e.g., using the direct simulation Monte Carlo method.

Drag and lift coefficients of ellipsoidal particles under rarefied flow conditions.

The capability to simulate a two-way coupled interaction between a rarefied gas and an arbitrary-shaped colloidal particle is important for many practical applications, such as aerospace engineering, lung drug delivery, and semiconductor manufacturing. By means of numerical simulations based on the direct-simulation Monte Carlo (DSMC) method, we investigate the influence of the orientation of the particle and rarefaction on the drag and lift coefficients, in the case of prolate and oblate ellipsoidal particles immersed in a uniform ambient flow. This is done by modeling the solid particles using a cut-cell algorithm embedded within our DSMC solver. In this approach, the surface of the particle is described by its analytical expression and the microscopic gas-solid interactions are computed exactly using a ray-tracing technique. The measured drag and lift coefficients are used to extend the correlations, based on the sine-squared drag law, available in the continuum regime to the rarefied regime, focusing on the transitional and free-molecular regimes. The functional forms of the correlations for the ellipsoidal particles are chosen as a generalization from the spherical case. We show that the fits over the data from numerical simulations can be extended to regimes outside the simulated range of Kn. Our approach allows to achieve a higher precision when compared with existing predictive models from the literature. Finally, we underline the importance of this work in providing correlations for nonspherical particles that can be used for point-particle Euler-Lagrangian simulations to address the problem of contamination from finite-size particles in high-tech mechanical systems.

Novel direct simulation Monte Carlo method for spherocylinders

Traditional discrete element models of granular flows for non-spherical particles can be limited by costly contact detection. The aim of the current work is develop the foundation towards a more computationally efficient method to solve flows with non-spherical particles. This study investigates extending the direct simulation Monte Carlo, or DSMC, method to the rotational relaxation of spherocylinders. While rotational energies have been accounted for in gases within DSMC, similar models for non-spherical particles are unavailable. We propose a modified DSMC method that samples from post-collisional energies and scattering distributions derived from deterministic discrete element models. We demonstrate how to integrate derived distributions into an adapted DSMC method for spherocylinders. To assess the fidelity of the model, we consider the case of the rotational relaxing dilute gas. While the study considers spherocylinders, the methodology itself may be generalized to more complex particle shapes.

Ab initio simulation of rarefied flows of gaseous mixtures in the system of microbeams with different temperatures

The issues of gas flows induced by inhomogeneous temperatures in a rarefied gas and exerting Knudsen force on the immersed structures are classic in the field of rarefied gas dynamics. The rarefied flows of HeNe gaseous mixtures in the system of microbeams with different temperatures for different combinations of equilibrium temperatures, rarefaction parameters, and mole fractions are studied using direct simulation Monte Carlo (DSMC) method in combination with ab initio (AI) potential with quantum scattering. Moreover, an extra simulation for VSS model is conducted by the same method to investigate the influence of inter-atomic potentials. The research results demonstrate that the rarefaction parameter for the maximum value of Knudsen force on the beams is related to the beam temperature. By comparing the results of AI and VSS potentials, it is found that the difference between the results predicted by the two potentials in the free-molecular regime is almost negligible. However, in transition regime or in an extremely low (or extremely high) temperature environment, the difference between the results is significant. In particular, in the transition regime with equilibrium temperature of 10 K, the maximum relative deviations of heat transfer coefficient and Knudsen force are 19.43% and 194.29%, respectively.

A Variance-Reduced Direct Monte Carlo Simulation Method for Solving the Boltzmann Equation Over a Wide Range of Rarefaction

A Variance-Reduced Direct Monte Carlo Simulation Method for Solving the Boltzmann Equation Over a Wide Range of Rarefaction

A Numerical Study on a Pincette-like Device using Thermally Induced Knudsen Forces

In this study, we demonstrate that thermally induced Knudsen forces in a rarefied gas can be exploited to achieve a tweezer-like mechanism to trap and grasp a micro-object without physical contact. Using the direct simulation Monte Carlo (DSMC) method, we showed that a heated thin plate, mounted perpendicularly on a flat substrate, can act as the tweezer-like mechanism. Specifically, it is found that the pressure differences induced by the thermal edge flows around the corners of the beam and the tip of the thin plate, generate a force that attracts the beam to the plate horizontally and vertically.