Direct Simulation Monte Carlo Method Research Articles

First operation and validation of simulations for the divertor cryo-vacuum pump in Wendelstein 7-X

Ten cryo-vacuum pumps (CVPs) were installed in the subdivertor region of each island divertor in the stellarator Wendelstein 7-X (W7-X) and operated for the first time during the recently completed plasma campaign OP2.1. A pumping speed of 70 ± 1m3s was measured during dedicated tests with known hydrogen gas injection. Based on a conductance model, the estimated pumping speed ranges from 86-93m3s for different sticking coefficients between 0.6 and 0.8. After completion of the initial tests the CVPs were operated successfully throughout the campaign, with regeneration performed once a week. Neutral gas pressures in the subdivertor in the range of 10−4mbar are well within the molecular flow regime and limit the particle exhaust capabilities of the CVPs. Simulations of the neutral gas pressure in the three-dimensional complex geometry of the subdivertor were performed using the DIVGAS code based on the direct simulation Monte Carlo method and a model implemented in the steady-state thermal package in ANSYS, which are in agreement with the measured values during plasma operation.

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.

A discrete unified gas kinetic scheme with sparse velocity grid for rarefied gas flows

In this paper, a discrete unified gas kinetic scheme (DUGKS) with a sparse grid method applied in velocity space (DUGKS-SG) is proposed for simulating rarefied gas flows. The DUGKS-SG decomposes the computationally demanding problem into smaller and independent subproblems, thereby reducing the computational costs and exhibiting good parallelism. Several numerical tests, including the two-dimensional Riemann problem and the lid-driven microcavity flow, have been conducted to validate the performance of the DUGKS-SG. Comparisons with the original DUGKS and the Direct Simulation Monte Carlo (DSMC) method demonstrate that DUGKS-SG can provide satisfactory results with improved efficiency. Specifically, a maximum speedup of 9.486 for a 2D case with 7 CPU cores and 13.035 for a 3D case with 8 CPU cores can be achieved. These results suggest that the proposed DUGKS-SG can serve as an efficient numerical method for rarefied gas flow simulations.

Calculation of Radiation Characteristics of Shock-Heated Air by the Direct Simulation Monte Carlo Method

Calculation of Radiation Characteristics of Shock-Heated Air by the Direct Simulation Monte Carlo Method

Small cluster formation in a free argon jet

A free argon jet flow accompanied by small clusters formation is studied with the direct simulation Monte Carlo method. Some near-continuum flow regimes characterized by Knudsen numbers in the 2×10−4−2×10−3 range are considered. A model for the argon clusters' growth/decay is proposed, taking into account the phase state of the clusters. The model consists of a chain of reactions leading to the clusters' formation, including the clusters' growth via triple/pair collisions of particles, and the clusters decay according to the collisional/unimolecular mechanism. The cluster size distributions in the jet far field are obtained. The results are compared with two experimental datasets. Good agreement is shown for most of the considered range of parameters. The triple particle collisions' influence on the argon clusters growth process is studied, and their important role in small cluster formation is demonstrated. It has been established that the cluster formation process is limited to an enough small spatial zone near the source outlet, of the order of several exit orifice diameters. The simulation shows a significant influence of cluster formation on the temperature and Mach number distributions, and a weak influence on the flow velocity. The formed clusters' translational temperatures and their velocities are close to the argon atoms' corresponding parameters. A non-equilibrium state, featured by a significant difference between the clusters' internal temperatures and the flow temperature, develops with distance from the source outlet.

A second-order particle Fokker-Planck model for rarefied gas flows

The direct simulation Monte Carlo (DSMC) method has become a powerful tool for studying rarefied gas flows. However, for the DSMC method to be effective, the cell size must be smaller than the mean free path, and the time step smaller than the mean collision time. These constraints make it difficult to use the DSMC method in multiscale rarefied gas flows. Over the past decade, the particle Fokker-Planck (FP) method has been studied to address computational cost issues in the near-continuum regime. To capture the main features of the Boltzmann equation, various FP models have been proposed, such as the quadratic entropic FP (Quad-EFP) and the ellipsoidal statistical FP (ESFP). Nevertheless, few studies have clearly demonstrated that the FP method offers a computational advantage over the DSMC method without sacrificing accuracy. This is because conventional particle FP methods have employed first-order accuracy schemes. The present study proposes a unified stochastic particle ESFP (USP-ESFP) model. This model improves the accuracy of shear stress and heat flux predictions. Additionally, a spatial interpolation scheme is introduced to the particle FP method. The numerical test cases include relaxation problem, Couette flows, Poiseuille flows, velocity perturbation, and hypersonic flows around a cylinder. The results show that the USP-ESFP model agrees well with both analytical and DSMC results. Furthermore, the USP-ESFP model is found to be less sensitive to cell size and time step than the DSMC method, resulting in a factor of four speed-up for the considered hypersonic flow around a cylinder.

Improving computational efficiency in DSMC simulations of vacuum gas dynamics with a fixed number of particles per cell

The present study addresses the challenge of enhancing computational efficiency without compromising accuracy in numerical simulations of vacuum gas dynamics using the direct simulation Monte Carlo (DSMC) method. A technique termed ‘fixed particle per cell (FPPC)’ was employed, which enforces a fixed number of simulator particles across all computational cells. The proposed technique eliminates the need for real-time adjustment of particle weights during simulation, reducing calculation time. Using the SPARTA solver, simulations of rarefied gas flow in a micromixer and rarefied supersonic airflow around a cylinder were conducted to validate the proposed technique. Results demonstrate that applying the FPPC technique effectively reduces computational costs while yielding results comparable to conventional DSMC implementations. Additionally, the application of local grid refinement coupled with the FPPC technique was investigated. The results show that integrating local grid refinement with the FPPC technique enables accurate prediction of flow behaviour in regions with significant gradients. These findings highlight the efficacy of the proposed technique in improving the accuracy and efficiency of numerical simulations of complex vacuum gas dynamics at a reduced computational cost.

The effects of energy accommodation of reflected gas molecules on flow structures during expired spacecraft reentry

To study the influence of energy accommodation of scattering gas molecules on flow fields during large expired spacecraft reentry, a more elaborated gas-surface interaction model, compared with full Maxwellian diffuse model, is employed in implicit algorithms based on Boltzmann model equation. The characteristic distributions around cylinder at different fluid regimes are accordingly obtained by implicit algorithms, Navier-Stokes solver and DSMC ((Direct Simulation Monte Carlo) method. And the consistency of these results is verified. It is confirmed that present algorithms are capable of solving external flow problems covering various fluid regimes. Then the simulation results see that under current conditions set in the paper, pressure and temperature are proportional to wall activation (ω=Tw/T∞, Tw is surface temperature, T∞ denotes as free stream temperature), but their amplitudes alter with ω at different fluid regimes. As for the effects of energy accommodation coefficients (αe), both pressure and temperature profiles vary in a linear way with αe. However, the variation ranges of these parameters are diverse with regard to different fluid regimes. These observations are favor to the construction of efficient forecasting software, which could predict the flight path of large defunct spacecraft. In this forecasting software, the external ballistics computations and aerothermodynamic simulations are synchronously carried out.

Design and Evaluation of a Hypersonic Waverider Vehicle Using DSMC

This work investigates the aerodynamic performance of a hypersonic waverider designed to operate at Mach 7, focusing on optimizing its design through advanced computational methods. Utilizing the Direct Simulation Monte Carlo (DSMC) method, the three-dimensional flow field around the specifically designed waverider was simulated to understand the shock wave interactions and thermal dynamics at an altitude of 90 km. The computational approach included detailed meshing around the vehicle’s critical leading edges and the use of three-dimensional iso-surfaces of the Q-criterion to map out the shock and vortex structures accurately. Additional simulation results demonstrate that the waverider achieved a lift–drag ratio of 2.18, confirming efficient aerodynamic performance at a zero-degree angle of attack. The study’s findings contribute to the broader understanding of hypersonic flight dynamics, highlighting the importance of precise computational modeling in developing vehicles capable of operating effectively in near-space environments.

Development of convolutional neural network-based surrogate model for three-dimensional vacuum plume prediction via direct simulation Monte Carlo method

The vacuum plume phenomenon encountered during lunar exploration missions poses significant challenges, such as impingement forces, heat fluxes, and spacecraft contamination. Numerical simulation represents the predominant method for evaluating the impacts of vacuum plumes. However, the conventional direct simulation Monte Carlo (DSMC) method, despite being the standard, is notably time-consuming and impractical for real-time analysis. Addressing this limitation, our research explores deep learning, specifically convolutional neural networks (CNN), for the efficient prediction of vacuum plume dynamics. We introduce a novel CNN-based DSMC method (CNN-DSMC-3D), leveraging a dataset obtained from three-dimensional DSMC simulations. This approach translates the spacecraft's shape and boundary conditions into a signed distance function and an identifier matrix. The CNN-DSMC-3D method effectively predicts the vacuum plume field, aligning closely with DSMC results across various lunar surface conditions. Crucially, the CNN-DSMC-3D method achieves a speed increase in four to six orders of magnitude over the conventional DSMC method, demonstrating substantial potential for real-time aerospace applications and offering a paradigm shift in the simulation of lunar landing scenarios.

Real-time vacuum plume flow field reconstruction during lunar landings based on deep learning

In space missions, the vacuum plume generated by rocket engines can negatively impact spacecraft. Therefore, researching the vacuum plume is crucial to guarantee the regular operation of spacecraft. The conventional numerical simulation methodology, the direct simulation Monte Carlo (DSMC) method, is time-consuming and lacks real-time calculation capabilities. Recently, deep learning (DL) methods have emerged in the field of fluid dynamics. In this study, a DL model trained by a convolutional neural network with multiple decoders is introduced to predict the vacuum plume flow field during lunar landings. The network processes shape topology information and boundary conditions as inputs, yielding flow field data including velocity and pressure fields as outputs. Meanwhile, the flow field prediction results under different conditions and training methods are discussed. The results show that the predicted flow field under different lunar surface conditions is in accord with the DSMC results. The maximum mean and standard deviation errors of the data distribution of each flow field do not exceed 9.72% and 9.07%, respectively. Different training methods with flat and inclined lunar surfaces also have an impact on the prediction results. Compared with the DSMC method, the DL method exhibits higher efficiency with a speedup of about four orders of magnitude, indicating that the DL-based flow field reconstruction method has strong application prospects in the real-time computation of vacuum plume flow fields.

Gaseous diffusion as a correlated random walk.

The mean-square displacement per collision of a molecule immersed in a gas at equilibrium is given by its mean-square displacement between two consecutive collisions (mean-square free path) corrected by a prefactor in the form of a series. The nth term of the series is proportional to the mean value of the scalar product r_{1}·r_{n}, where r_{i} is the displacement of the molecule between the (i-1)-th and ith collisions. Simple arguments are used to obtain approximate expressions for each term. The key finding is that the ratio of consecutive terms in the series closely approximates the so-called mean persistence ratio. Exact expressions for the terms in the series are considered and their ratios for several consecutive terms are calculated for the case of hard spheres, showing an excellent agreement with the mean persistence ratio. These theoretical results are confirmed by solving the Boltzmann equationby means of the direct simulation Monte Carlo method. By summing the series, the mean-square displacement and the diffusion coefficient can be determined using only two quantities: the mean-square free path and the mean persistence ratio. A simple and an improved expression for the diffusion coefficient D are considered and compared with the so-called first and second Sonine approximations to D as well as with computer simulations of the Boltzmann equation. It is found that the improved diffusion coefficient shows very good agreement with simulation results over all intruder and molecule mass ranges. When the intruder mass is smaller than that of the gas molecules, the improved formula even outperforms the first Sonine approximation.

Numerical simulation of plume–surface interaction and lunar dust dispersion during lunar landing using four engines

As the lander approaches the lunar surface, the engine plumes impinge on the lunar regolith and entrain lunar dust from the surface. This plume–surface interaction and the resulting dispersion of lunar dust form a multi-physics, multi-scale problem, which becomes even more complex under multi-engine conditions. This study employed the direct simulation Monte Carlo method to simulate the plume–surface interaction flow field of a four-engine lunar lander at various landing altitudes and lunar surface angles. Flow characteristics were analyzed, and the impact of the plume and backflow on the lander was assessed. Subsequently, lunar dust simulation was conducted using the plume field as a basis. The study determined the spatial distribution of particles with different diameters at various landing altitudes and surface angles, as well as their impact velocities on the lander. Furthermore, taking into account the variations in the lander's altitude and attitude, a dynamic simulation of lunar dust during the landing process was conducted. This process resulted in the dynamic distribution of lunar dust during landing, laying the groundwork for real-time simulation of lunar dust distribution and reliable visualization during landing simulations. These findings are valuable for assessing and mitigating the hazards posed by lunar dust.

Rarefied cylindrical Couette problem: Effect of a very sharp transition from weakly rarefied flow near the inner cylinder to free molecular flow at very low average Knudsen numbers

We consider the rarefied cylindrical Couette flow. The outer cylinder rotates at a constant angular velocity, and the inner one is at rest. The Knudsen number, calculated from the average gas density, is much less than one. To find the fluid dynamic variables and fluxes, the direct simulation Monte Carlo method (DSMC) is used. We find the effect of a very sharp decrease in shear stress and energy fluxes transferred to the cylinders if the Knudsen number is sufficiently low and the rotation speed of the outer cylinder is sufficiently high. We show that when the average Knudsen number is sufficiently low, if the velocity is initially high and increases slightly further, the weakly rarefied flow regime near the inner cylinder transitions abruptly to a free molecular flow regime. Thus, the flow abruptly separates from the inner cylinder. We compare the fluxes obtained by solving the Navier–Stokes equations and by the DSMC method. It turns out that the solutions of the Navier–Stokes equations with slip boundary conditions also change abruptly at certain critical values, and these critical values are close to the appropriate DSMC values.

Multi-species modeling in the particle-based ellipsoidal statistical Bhatnagar-Gross-Krook method including internal degrees of freedom

The implementation of the ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) method in the open-source particle code PICLas is extended for multi-species modeling of polyatomic molecules, including internal energies with multiple vibrational degrees of freedom. For this, the models of Mathiaud et al. [1], Pfeiffer [2] and Brull [3,4] are combined. In order to determine the transport coefficients of the gas mixture, Wilke's mixing rules and collision integrals are compared. The implementation is verified with simulation test cases of a supersonic Couette flow as well as a hypersonic flow around a 70° blunted cone. The solutions of the ESBGK method are compared to the Direct Simulation Monte Carlo (DSMC) method to assess the accuracy, where overall good agreement is achieved. In general, collision integrals should be preferred for the determination of the transport coefficients, since the results using Wilke's mixing rules show larger deviations in particular for large mass ratios. Considering the computational efficiency of the methods, a considerable reduction in computational time is possible with the ESBGK model.

A general drag coefficient model for a spherical particle incorporating rarefaction and particle-to-gas temperature ratio effects

A concept and model of two critical Reynolds numbers Rep,cr1 and Rep,cr2 corresponding respectively to onsets of drag crisis and recovery are proposed. A drag model at limits of zero particle Mach and Knudsen numbers is constructed. On this basis, we develop a general drag coefficient model applicable for a spherical particle traveling in a gas by using a large number of available data derived from the previous experiments, direct numerical simulations and direct simulation Monte-Carlo methods. The scope of applicability of the proposed drag model covers all flow regimes relative to the particle characterized by particle Reynolds and Mach (or Knudsen) numbers and different particle-to-gas temperature ratios. Its comparison with two latest general drag models shows the significantly smaller relative error. Furthermore, quasi-one dimensional simulations against two supersonic nozzle gas-particle flow experiments are conducted with an in-house code to validate its accuracy in comparison with the two drag models.

A statistical boundary for 3D rarefied flows through meshes: implementation to a new version of dsmcFoam+ and wind tunnel validation

AbstractThe Direct Simulation Monte Carlo (DSMC) method has become a standard tool for rarefied aerodynamics and microchannel flows. However, the performance benefits of DSMC, such as adaptive grid sizes and number of particles, are constrained by the need to resolve small geometric details of mesh applications within relatively large simulation volumes. The requirement for a sufficient number of particles in even the smallest cells imposes a significant computational burden. A novel set of cyclic statistical boundary conditions is proposed to address the computational bottleneck associated with simulating micrometre-scale structures prevalent in atmospheric and space research under rarefied flow conditions. These conditions account for the geometric parameters of a geometric mesh and the angular dependency of impacting particles, aiming to alleviate the computational challenges posed by conventional approaches. Validation against wind tunnel measurements demonstrates excellent agreement for one of the implemented boundaries, able to simulate fine meshes for conditions of rocket soundings in the Mesosphere. The newly developed boundary conditions are implemented within the advanced DSMC solver, dsmcFoam+ framework. For this study, the solver is ported from OpenFOAM® version 2.4.0 to the OpenFOAM® version v2306 to leverage recent code developments, particularly in dynamic meshes, load balancing, and barycentric particle tracking. This advancement enhances the capabilities of DSMC simulations, offering improved fidelity and accuracy in capturing rarefied flow phenomena.