Direct Simulation Monte Carlo Research Articles

Quantum-kinetic chemistry model with an anharmonic oscillator model: Model extension and validation

This work proposes an extended version of the quantum-kinetic chemistry models, aiming to accurately reproduce experimental measurements and high-fidelity calculations in both thermal equilibrium and non-equilibrium. The extension involves the development of new formulations, incorporating a set of tunable parameters obtained from a non-linear least squares fit on the dataset. The newly derived analytical expressions are implemented in a direct simulation Monte Carlo (DSMC) solver. These formulations are applied to the 19 most representative chemical reactions of an air mixture involving dissociation and exchange reactions. The DSMC reaction rates demonstrate excellent agreement with the newly derived analytical expressions, providing verification of the successful implementation in the DSMC solver. The study demonstrates excellent reproduction of the baseline dataset for both thermal equilibrium and non-equilibrium. Furthermore, the new formulations are applied to simulate the surface heat flux during the second space transport system (STS-II) mission at an altitude of 92.35 km.

Gas-Kinetic Unified Algorithm for Aerodynamics Covering Various Flow Regimes by Computable Modeling of Boltzmann Equation

The gas-kinetic unified algorithm (GKUA), to solve the modeling of the Boltzmann equation, has been developed to study the aerothermodynamics problems with the effects of wall activation energy covering various flow regimes. The unified velocity distribution function equation could be accordingly presented on the basis of the Boltzmann-Shakhov model. To remove the dependence of the distribution function on velocity space, the conservational discrete velocity ordinate method has been developed for hypervelocity flows. The gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by the operator splitting technique. The discrete velocity numerical integration method with the Gauss-type weight function has been developed to evaluate the macroscopic flow variables. Specially, to model the real physical process between gas molecules and the surface, the Maxwell-type gas-surface interaction model has been presented by the MD (molecular dynamic) simulation to obtain the energy adaptability coefficient. The multi-processing domain decomposition strategy and parallel implementation of high parallel efficiency and expansibility designed for the gas-kinetic numerical method is presented with good load balance and data communication efficiency. To validate the accuracy and feasibility of the present algorithm, the supersonic flows past two-dimensional circular cylinder are simulated covering various flow regimes. The results are in good agreement with the related theoretical, DSMC (Direct Simulation Monte Carlo), N-S (Navier-Stokes), and experimental data. The hypersonic reentry flows with the effects of wall activation energy around the Tianzhou-5 cargo spacecraft are simulated by the present GKUA and the massive parallel strategy. It has been confirmed that the present algorithm from the gas-kinetic point of view probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes during the re-entry and disintegration of the large-scale spacecraft.

Bерифікація алгоритму методу пробних частинок на задачі осьового обтікання конуса розрідженим газовим струменем

This paper is concerned with an aerodynamic calculation of supersonic gas plume flows and the determination of the forces they exert on obstacles. The paper presents a development of the test particle statistical method (TPSM) to numerically simulate supersonic gas plumes over a wide range of flow conditions. The work is based on the idea of a combined approach, i.e., the use of the gas-dynamic parameter distribution at the nozzle exit or on the conventional boundary of the dense core of the plume as input data for a TPSM algorithm adapted from homogeneous flows to plume ones. Combining methods of continual aerodynamics (inside or near the nozzle, where a continuum flow takes place) and the TPSM (where the motion is described on a molecular-kinetic level) allows one to solve supersonic plume efflux problems for arbitrarily rarefied plumes. The TPSM plume algorithm was tested to verify its reliability on the problem of axial flow past a cone. At the initial stage of the use of the combined approach, consideration was given to a rather rarefied gas flow, for which the gas-dynamic parameters at the nozzle exit can be used as TPSM input data. The force distribution over the cone surface and the static pressure upstream of the cone were calculated. The TPSM results were found to be in satisfactory agreement with the available direct simulation Monte-Carlo and experimental data. It was concluded that using the plume velocity and density distributions at the continual zone exit found from the Navier–Stokes equations as TPSM input data would significantly improve the expected results. This use of the TPSM in an aerodynamic calculation of gas plumes is the first in Ukraine. The TPM offers saving in computational resources: the TPSM running time depends on a variety of factors, but it is many times shorter than that of the direct simulation Monte Carlo method.

An evaluation of the hybrid Fokker–Planck-DSMC approach for high-speed rarefied gas flows

The Direct Simulation Monte Carlo (DSMC) method has been widely used for simulations of rarefied gas flows. It has been proven that the numerical solution of the DSMC method converges to the Boltzmann equation, providing a solid physical foundation for high Knudsen numbers. However, many aerospace engineering problems include both low and high-density regions in a single domain, making the DSMC method computationally expensive. Over the past decade, the particle Fokker–Planck (FP) method has been studied as a means to reduce computational costs near the continuum regime. While enhancing efficiency, the FP method loses physical accuracy at high Knudsen numbers. Aiming for universal accuracy and efficiency across the whole range of Knudsen numbers, the hybrid FP-DSMC method has been studied. Nevertheless, consistent comparisons among the DSMC, FP, and hybrid FP-DSMC methods have received limited attention so far. This paper presents a consistent comparative study of the DSMC, FP, and hybrid FP-DSMC methods to assess the accuracy and efficiency of the hybrid FP-DSMC method. The hybrid FP-DSMC solver is developed based on the open-source SPARTA framework. The benchmark problems include supersonic flow in a planar nozzle, hypersonic flow around a cylinder, and hypersonic flow around a THAAD-like missile. The results demonstrate that the hybrid FP-DSMC method can be more efficient than the DSMC method while being more accurate than the FP method. The speed-up achieved by the FP-DSMC method ranged from 1.5 to 14 times compared to the converged DSMC simulation.

An improved analytical method for rarefied aerothermodynamics on a complex geometry in free-molecular flows

A numerical method for high altitude aerothermodynamic analysis code for Very Low Earth Orbit (VLEO) satellite and spacecraft, named as HACS, is presented in this article. HACS is a computational tool that addresses the challenges by integrating the free-molecular gas kinetic model with novel particle sampling method and robust particle tracing algorithm to predict the shadow region on surfaces. To ensure the reliability and accuracy of the HACS, a rigorous verification and validation process has been systematically conducted by comparing the aerothermodynamic properties of forces, moments, and heat flux with the direct simulation Monte-Carlo (DSMC) results. The HACS is applied to the realistic geometries of VLEO satellite, the Apollo re-entry module, and spacecraft including fin and wing shapes. Results demonstrate that HACS provides the aerothermodynamic properties within the time frame of 60 seconds for the complex geometries and the accuracy of the HACS is guaranteed above 120 km altitude. The maximum relative error of the aerothermodynamic properties is less than 5% compared to the DSMC results at the lowest altitude of about 120 km, and has smaller relative errors as the altitude increases.

An efficient single-loop strategy for time-variant reliability sensitivity analysis based on Bayes’ theorem

Time-variant reliability sensitivity analysis aims to measure the effect of input variables on the time-variant failure probability, which can then be used to guide structural design and optimization. The traditional methods for calculating the sensitivity index require a nested sampling procedure and their computational complexity depends on the number of input variables, making them computationally the most expensive. To address the above limitations, an efficient single-loop strategy is developed for time-variant reliability sensitivity (TRS) analysis. Based on Bayes' theorem, the TRS index is represented by the difference between the probability density function (PDF) and failure-conditional PDF of the variable. This derivation enables TRS analysis to be implemented using only a single set of samples, with computational costs that don’t depend on the number of input variables. Then, the sensitivity index is normalized to identify influential variables effectively. Several case studies involving numerical and engineering problems are employed to verify the feasibility and effectiveness of the proposed strategy. The direct Monte Carlo simulation is introduced to assess the accuracy of the obtained results. The results indicate that the proposed strategy provides acceptable sensitivity analysis for time-variant events, while greatly conserving computational resources and enhancing efficiency.

A DSMC-CFD coupling method using surrogate modelling for low-speed rarefied gas flows

A new Micro-Macro-Surrogate (MMS) hybrid method is presented that couples the Direct Simulation Monte Carlo (DSMC) method with Computational Fluid Dynamics (CFD) to simulate low-speed rarefied gas flows. The proposed MMS method incorporates surrogate modelling instead of direct coupling of DSMC data with the CFD, addressing the limitations CFD has in accurately modelling rarefied gas flows, the computational cost of DSMC for low-speed and multiscale flows, as well as the pitfalls of noise in conventional direct coupling approaches. The surrogate models, trained on the DSMC data using Bayesian inference, provide noise-free and accurate corrections to the CFD simulation enabling it to capture the non-continuum physics. The MMS hybrid approach is validated by simulating low-speed, steady-state, force-driven rarefied gas flows in a canonical 1D parallel-plate system, where corrections to the boundary conditions and stress tensor are considered and shows excellent agreement with DSMC benchmark results. A comparison with the typical domain decomposition DSMC-CFD hybrid method is also presented, to demonstrate the advantages of noise-avoidance in the proposed approach. The method also inherently captures the uncertainty arising from micro-model fluctuations, allowing for the quantification of noise-related uncertainty in the predictions. The proposed MMS method demonstrates the potential to enable multiscale simulations where CFD is inaccurate and DSMC is prohibitively expensive.

Establishment of particle motion model and study of particle volume fraction distribution in the turbo air classifier based on DSMC

In order to study the particles' motion law and collision behavior in the flow field of the turbo air classifier, A particle motion simulation platform is developed using Compute Unified Device Architecture (CUDA) based on the Visual Studio. The particle-eddy interaction model, the particle-wall collision model, and the inter-particle collision Direct Simulation Monte Carlo (DSMC) solver are implemented. The results show that the deviation between calculated cut size d50 and experimental d50 using DSMC is smallest compared to DSMC, DDPM and DPM. DSMC has better acceleration performance compared to DDPM. The more particles need to be calculated, the more significant this advantage is. The influences of feeding rate on particle volume fractions and inter-particle collision number are analyzed. The increase of feeding rate leads to the increase of the particle volume fractions and inter-particle collision number.

Direct simulation Monte Carlo-driven optimization of vacuum intakes for air-breathing electric thrusters in very low earth orbits

The performance of an atmosphere-breathing electric propulsion (ABEP) intake has been investigated with a focus on the direct simulation Monte Carlo (DSMC) method. A numerical dataset was derived from extensive DSMC analysis of rarefied flow across various intake configurations. The intake geometry, based on a concept from the literature, comprises a cylindrical body with four annular coaxial channels and a conical convergent diffuser. By maintaining the aspect ratio of the coaxial channels, the DSMC simulations were performed by changing three key parameters: inlet area, convergent diffuser angle, and operating discharge voltage, at altitudes ranging from 140 to 200 km. The analysis of the ABEP system revealed that altitude has the most significant influence on the discharge power, while the effects of the diffuser angle and inlet area are comparatively smaller. Analysis at fixed altitudes reveals a strong influence of altitude on discharge power, while the diffuser angle and the inlet area play a minor role. The results also show that the sensitivity of the discharge power to the diffuser angle increases as the altitude approaches the highest level of 200 km. Furthermore, an evolutionary-based optimization methodology was applied, taking into account the requirements of a drag-to-thrust ratio of less than 1 and a discharge power of less than 12 kW. Optimization analysis in the full altitude range revealed that the optimal diffuser angle falls within the narrow range of 15°–20°, corresponding to an optimal operating altitude range of 170–178 km.

Sensitivity analysis and uncertainty quantification for a three-dimensional rarefied ionized hypersonic flow

The “communications blackout” phenomenon has bothered the aerospace industry for several decades. However, the unsatisfying numerical methods and the rapidly changing inflow conditions make the estimation of electrons’ density inaccurate. To lay the first step for model calibration and the anti-blackout design, the Artificial Neural Networks and the direct simulation Monte Carlo (DSMC) method with the quantum-kinetic (Q-K) model are brought together to perform the global sensitivity analysis and uncertainty quantification. The three-dimensional RAM-C II (the second flight of the Radio Attenuation Measurement experiments) head flows are simulated considering aleatory uncertainties (inflow uncertainties) and epistemic uncertainties (reaction parameters uncertainties). Under the inflow condition of an 81 km atmosphere and a velocity of 7.8 km/s, aleatory uncertainties are found to be the dominant type of uncertainty for the number density of electrons, especially the freestream velocity, which means the accurate measurement of the vehicle’s velocity is much more critical than the calibration of the model. The importance ranking is listed, and the “Three rules” for finding the essential reactions are proposed. The probability that the real value is smaller than the nominal value considering both uncertainties is 27%–68%.

Comprehensive Evaluation of the Massively Parallel Direct Simulation Monte Carlo Kernel “Stochastic Parallel Rarefied-Gas Time-Accurate Analyzer” in Rarefied Hypersonic Flows—Part A: Fundamentals

The Direct Simulation Monte Carlo (DSMC) method, introduced by Graeme Bird over five decades ago, has become a crucial statistical particle-based technique for simulating low-density gas flows. Its widespread acceptance stems from rigorous validation against experimental data. This study focuses on four validation test cases known for their complex shock–boundary and shock–shock interactions: (a) a flat plate in hypersonic flow, (b) a Mach 20.2 flow over a 70-degree interplanetary probe, (c) a hypersonic flow around a flared cylinder, and (d) a hypersonic flow around a biconic. Part A of this paper covers the first two cases, while Part B will discuss the remaining cases. These scenarios have been extensively used by researchers to validate prominent parallel DSMC solvers, due to the challenging nature of the flow features involved. The validation requires meticulous selection of simulation parameters, including particle count, grid density, and time steps. This work evaluates the SPARTA (Stochastic Parallel Rarefied-gas Time-Accurate Analyzer) kernel’s accuracy against these test cases, highlighting its parallel processing capability via domain decomposition and MPI communication. This method promises substantial improvements in computational efficiency and accuracy for complex hypersonic vehicle simulations.

Numerical investigation of micropropulsion systems for CubeSats: Gas species and geometrical effects on nozzle performance

In the present investigation, a numerical investigation is conducted to investigate cold-gas micropropulsion devices for CubeSat orbital control. Different micronozzle geometry configurations and gas species are analyzed to assess their influence on the flowfield structure, surface properties, and microthruster performance parameters. Due to the small length scales, the Direct Simulation Monte Carlo (DSMC) method is used to simulate rarefied argon and nitrogen in rectangular and curved micronozzles. The results indicate that nitrogen gas leads to a 23% increase in specific impulse at a relatively insignificant thrust cost. On the other hand, the use of a curved geometry increases the specific impulse and thrust generated by 23% and 35%, respectively. The results indicate that using lighter gases for cold gas microthrusters increases the micropropulsion efficiency. In addition, curved geometries significantly improve the overall performance of such devices, in contrast to rectangular geometries.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

Cathode-less RF plasma thruster design and optimisation for an atmosphere-breathing electric propulsion (ABEP) system

Atmosphere-breathing electric propulsion (ABEP) is a concept that ingests residual atmospheric gases as a source of propellant for an electric thruster, removing the need for onboard propellant storage. This would enable continuous low-thrust drag compensation, extending the lifetime of spacecraft in Very-Low Earth Orbit (VLEO); <250 km. VLEO is an appealing region for spacecraft operations, enabling new remote sensing missions with improved radiometric performance and spatial resolution, whilst reducing size, mass and power requirements, as well as mission cost. A preliminary design review and optimisation is therefore conducted for an ABEP system that uses the cathode-less radio frequency (RF) plasma thruster from Technology for Innovation & Propulsion (T4i) S.p.A. This removes the issue of thruster erosion by means of magnetic confinement and offers reduced susceptibility to varying atmospheric composition. A semi-empirical oxygen-nitrogen global source model (GSM) has been developed which considers the volume-averaged flux, momentum, and energy balance of the RF discharge. This includes a detailed chemistry model for the complex electron-molecular reactions and energy-loss channels of air plasma in the ionisation chamber. The GSM is coupled to an analytical model of flux balance for an air intake, verified by Direct Simulation Monte-Carlo (DSMC) simulation, to consider its design for maximum collection efficiency. This is then utilised in a robust multi-objective optimisation of the ABEP system, accounting also for spacecraft aerodynamics and power requirements.

Efficient optimization-based method for simultaneous calibration of load and resistance factors considering multiple target reliability indices

This study introduces an innovative optimization process for calibrating probabilistic load and resistance factors (LRFs) in limit state designs, effectively accommodating multiple target reliability indices. Given the impracticality of direct Monte Carlo simulations (MCS) for this task, a response surface method (RSM) is proposed to approximate load and resistance components separately rather than fitting conventional safety factors. This approach eliminates the need for additional implicit evaluations, thereby improving the efficiency of LRF calibration across multiple targets. The process is further enhanced by an adaptive boundary algorithm that updates search domains in real-time, streamlining the optimization. Validation through three examples—including one explicit and two implicit performance functions (a structural and a geotechnical example)—demonstrates that the method achieves accurate results with fewer iterations by dynamically narrowing search domains. Specifically, the accuracy of the proposed method is confirmed by comparing results with those from the literature for the explicit example and with basic MCS results applied to the initial implicit problems. Performance on the illustrative examples shows that the structural example achieves calibration for three targets within ten iterations. Additionally, this method eliminates the need for approximately ten thousand implicit evaluations when calculating limit state points for the geotechnical example.

Effects of the chirp rate on single-shot coherent Rayleigh-Brillouin scattering

Single-shot coherent Rayleigh-Brillouin scattering (CRBS) is modeled using direct simulation Monte Carlo (DSMC). In CRBS, an optical lattice generated by the interference of two pump lasers traps the neutral particles via the dipole force. The gradient of refractive index due to the trapped particles of the gaseous media leads to coherent scattering of a probe beam, resulting in the CRBS signal. Additionally, using a chirped laser beam, CRBS signals can be obtained in a single laser shot, shortening the measurement time of the gas flow (on the order of 100 ns). The DSMC results of single-shot CRBS assuming Maxwellian velocity distribution functions (VDFs) show good agreement with experimental data, including argon, carbon dioxide, and xenon, for different gas pressures. One of the key observations obtained from the present simulations is that the CRBS signals become asymmetric when using a fast chirp rate due to finite time for collisionless trapping and collisional thermalization. Published by the American Physical Society 2024

Multiscale simulation of rarefied gas dynamics via direct intermittent GSIS-DSMC coupling

The general synthetic iterative scheme (GSIS) has proven its efficacy in modeling rarefied gas dynamics, where the steady-state solutions are obtained after dozens of iterations of the Boltzmann equation, with minimal numerical dissipation even using large spatial cells. In this paper, the fast convergence and asymptotic-preserving properties of the GSIS are harnessed to remove the limitations of the direct simulation Monte Carlo (DSMC) method. The GSIS, which leverages high-order constitutive relations derived from DSMC, is applied intermittently, which not only rapidly steers the DSMC towards steady state, but also eliminates the requirement that the cell size must be smaller than the molecular mean free path. Several numerical tests have been conducted to validate the accuracy and efficiency of this hybrid GSIS-DSMC approach.

Kinetic investigation of discharge performance for Xe, Kr, and Ar in a miniature ion thruster using a fast converging PIC-MCC-DSMC model

Abstract The present work develops a full particle-based model that couples the particle-in-cell plus Monte Carlo collision (PIC-MCC) simulation for plasma dynamics and the direct simulation Monte Carlo (DSMC) method for neutral dynamics in a synergistic iterative manner. This new model overcomes the slow convergence issue in the conventional direct coupling approach caused by the disparity of the time scales between the plasma and neutral dynamics. This model is applied to simulate the behavior of xenon (Xe) and its potential alternatives, krypton (Kr) and argon (Ar), in the discharge chamber of a miniature direct current (DC) ion thruster. The results show that a stable discharge is difficult to achieve for Kr and Ar under the operating conditions optimal for Xe. While increasing the discharge voltage can effectively improve the stability of discharge for Kr and Ar, other common strategies such as changing the magnetic field strength, propellant flow rate, and cathode current are not successful. The propellant utilization efficiency and discharge efficiency are affected by both discharge voltage and propellant flow rate. A maximum utilization efficiency and an optimal discharge efficiency are observed for all three propellants, with the values decreasing in the order of Xe, Kr, and Ar. Moreover, the discharge voltage corresponding to the optimal efficiency is inversely proportional to the square root of the propellant mass, indicating that the ion diffusional loss to the wall, rather than the ionization energy, is the dominant factor affecting the discharge performance for alternative propellants in a miniature DC thruster.

Numerical investigations of spatiotemporal dynamics of space-charge limited collisional sheaths

Electrostatic particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC) methods are used to compare the plasma dynamics of collisionless with collisional emissive sheaths in partially ionized environments. Space-charge limited emissive sheaths submersed in a plasma with a density of ∼1017 m−3 are examined using a PIC-DSMC solver, CHAOS. Collisionless emissive sheaths with plasma domains sufficiently long (30 and 60 Debye lengths, λD) are subject to strong oscillations due to two-stream electron instability, whereas emissive sheaths in weakly collisional conditions with a short domain (15 λD) exhibit self-spike (sawtooth) oscillations in the plasma field due to the trapped charge-exchange (CEX) ion population within the virtual cathode (VC) region. The two-stream electron instability leads to strong temporal fluctuations in the total emission current, with maximum deviations of 60% and 100% from the time-averaged current for the long plasma domains, whereas CEX collisions cause strong spikes in the emission current if the domain size is short. Our PIC-DSMC simulations show for the first time that the interaction of the two types of instabilities causes the strength of the self-spike to be weakened due to the strong fluctuations caused by the two-stream instability when a sufficiently long computational domain with ion-neutral collisions is employed. By conducting a two-dimensional Fast Fourier Transform (FFT) on the collisional and collisionless sheaths with long domains, we show that the transient evolution of CEX entrapment in the VC increases frequency of sheath oscillations up to two times the ion-acoustic frequencies observed in the collisionless sheath. CEX collisions weaken the VC region and result in a total emission current more than that obtained from the collisionless case for the same domain length. With a more rarefied neutral environment of 1019 m−3 in the plasma sheath, the total emission current increases only 4% in comparison with 14% for one order of magnitude denser environment, within 20 μs. In addition, the spike period is tested with different neutral temperatures and densities. While we do not observe any self-spike in the more rarefied environment, the spike period increased from 5 to 7.5 μs when the neutral temperature is increased from 300 to 2000 K in the denser environment with the simulation time of 20 μs.

An improved stochastic weighted particle method for boundary driven flows

The stochastic weighted particle method (SWPM) is a generalization of the Direct Simulation Monte Carlo (DSMC) method where particle weights are variable and dynamic. SWPM is backed by a strong theoretical foundation but has not been critically evaluated for problems of practical interest. A thorough assessment of SWPM for boundary-driven flows reveals significant numerical artifacts near the boundary, notably a diverging heat flux. To correct the boundary heat flux, two modifications to SWPM are proposed: separated grouping and a spatially-dependent weight transfer function. To gauge the relative efficiency of SWPM in comparison to DSMC, a high-Mach-number wheel flow which forms a strong density gradient is also simulated.