Ellipsoidal statistical Bhatnagar–Gross–Krook model with velocity-dependent collision frequency

The Bhatnagar-Gross-Krook (BGK) model and its extensions (ellipsoidal statistical BGK, Shakhov BGK, and unified BGK) are used in particle-based fluid dynamics and compared with the Direct Simulation Monte Carlo (DSMC) method. To this end, different methods are investigated that allow efficient sampling of the Shakhov and the unified target distribution functions. As a consequence, particle simulations based on the Shakhov BGK and unified BGK models are possible in an efficient way. Furthermore, different energy conservation schemes are tested for the BGK models. It is shown that the choice of the energy conservation scheme has a major impact on the quality of the results for each model. The models are verified with a Couette flow problem at different Knudsen numbers and wall velocities. Furthermore, the models are compared with the DSMC results of a hypersonic flow around a 70° blunted cone. It is shown that the unified BGK model is able to reproduce rarefied gas phenomena. Furthermore, it is shown that the difference in the reproduction of the shock structure is not significant between the ellipsoidal statistical BGK and Shakhov BGK models for the flow around a 70° blunted cone and significantly depends on the energy conservation method. The choice of the energy conservation method is especially crucial for the Shakhov model whereas the ellipsoidal statistical BGK model is much more robust concerning the energy conservation scheme. Additionally, a computational time study is performed to show the efficiency of BGK-based simulations for low Knudsen number flows compared with DSMC.

The Boltzmann equation is the governing equation for rarefied gas dynamics, which is relevant to the aerodynamic design of rarefied hypersonic vehicles. Direct solution of the Boltzmann equation for realistic gas flows using either analytic or computational techniques is almost impossible. However, the Direct Simulation Monte Carlo (DSMC) method has been developed to provide realistic solutions for rarefied gas flows. Bird formally introduced this method hi 1969, and ever since several arguments have developed as to whether the DSMC method provides solutions of the Boltzmann equation. This paper demonstrates that the Boltzmann equation and the DSMC method have a common origin which rests on the fundamental tenets of kinetic theory. These fundamental tenets are mathematically condensed into the Boltzmann equation. On the other hand, the DSMC method is a numerical implementation of these same fundamental tenets. Thus, the Boltzmann equation and the DSMC method express the same idea but use a different language. The Boltzmann equation uses the language of mathematics whereas the DSMC method uses the language of physics. It is explained how both approaches are related and it is concluded that the DSMC method does indeed provide solutions of the Boltzmann equation. It is also shown that both approaches can be easily extended to a wider range of problems than the literature has recognised (such as to high-density gases and liquids). One important issue is how to set the range of the molecular potential used in deflection angle formulae. It is shown that, contrary to traditional expectations, within a macroscopic volume the range of the potential is always finite. Finally, the thorny issue of molecular chaos is examined and it is concluded that it is not as limiting as commonly feared.

Results of numerical simulations of high-altitude aerothermodynamics of the EXPERT reentry capsule along its descent trajectory are presented. Aerodynamic characteristics for different angles of attack and rolling of the capsule at altitude of 150 down to 20 km are studied. An engineering local bridging method is used in computations. The uncertainty of the engineering method in the transitional regime is determined by comparisons with results obtained by DSMC simulations. I. Introduction During disorbiting, the spacecraft experiences the influence of the atmosphere with significantly varying parameters. To estimate the aerodynamic loads and to predict the landing area, one should know, already at the stage of design, the coefficients of aerodynamic forces and moments of the spacecraft for varying temperature, density, flight velocity, and angles of attack and sideslip for numerous possible descent trajectories. Thousands of various variants have to be computed. At the beginning of descent, at high altitudes where the gas is strongly rarefied, numerical simulations do not involve many difficulties, because theoretical approaches have been developed for simple shapes and the Test Particle Monte Carlo method can be readily used for more complicated shapes. On the other hand, various engineering approaches have been developed for calculating aerodynamic characteristics of various bodies at the final segment of the descent trajectory, in the continuum regime, which are fairly effective. The main problem is the analysis of spacecraft aerodynamics in the transitional regimes between the free-molecular and continuum regimes. The Navier-Stokes equations yield, strictly speaking, incorrect results in the transitional regime and require special modifications for taking into account flow slipping. The Direct Simulation Monte Carlo (DSMC) method provides rather accurate values of aerodynamic characteristics with allowance for physical and chemical processes but requires large amounts of computer memory and performance. Application of the software based on the DSMC method is unreasonably expensive at the initial stage of spacecraft design and trajectory analysis. A possible solution of this problem is the approximate engineering methods. These methods allow calculating aerodynamic characteristics of bodies of arbitrary geometry for multiple variants of free-stream parameters within a reasonable time and then to refine the results at the most important segments of the flight trajectory by the DSMC method. The use of fast approximate engineering methods is the most acceptable approach to solving problems of spacecraft aerodynamics at the stage of conceptual design. The present paper describes the computations of aerodynamics of the EXPERT capsule by an engineering local bridging method at altitudes of 150 down to 20 km with flow parameters corresponding to the expected descent trajectory. The results in the transitional regime are refined by the DSMC method. The errors of the engineering method are studied in detail, and the contribution of various elements of the EXPERT capsule to aerodynamic characteristics is analyzed. II. EXPERT geometry The KHEOPS configuration (Fig. 1) is a blunted pyramidal shape featuring four flaps. The computational model was constructed by the RuSat software system. The total number of panels is approximately 36 thousand. The nose is made in the form of an ellipse with a misalignment of 0.9165. The local radius of the nose part is 0.6 m. The ellipse-cone junction is described by a clothoid. The conical body has a cone angle of 12.5°. The cone is truncated by planes at an angle of 8.35° to the axis of symmetry. Four control flaps are deflected by 20 °.

This paper describes the basic study of the simulation of chemical vapor deposition (CVD) step coverage that is used to estimate the deposition of thin films in LSI production. Given the trend toward extremely high density in the semiconductor manufacturing process, such as that of LSIs, the deposition process of silicon thin film on the circuit pattern needs to be accurate. To optimize the deposition of thin film around the trench of the order of less than a micron, computer simulations of the step coverage and of the film growth rate in the process are very important. In this simulation reported in this paper, the thin film deposition process, whose reaction is modeled as silicon from silane (SiH4) and silylene (SiH2), is solved using the direct simulation Monte Carlo (DSMC) method. The DSMC method is usually adopted in this case, but, since 1) this method solves the Boltzmann equation with a huge amount of molecular collisions and free motions and 2) sticking probability of molecules is of the order of 10-5 [1], even a high performance computer needs much time for this calculation. Although the performance of supercomputers and other types of computers is improving, the DSMC method as conventionally applied seems to be reaching its limit in terms of practicality.

The flow characteristics and heat transfer in micro couette flow were studied numerically using three methods: Burnett equations, direct simulation Monte Carlo (DSMC) method and information preservation (IP) method. The results of these three methods are compared to each other. Convergent solutions to the conventional Burnett equations with second order velocity-slip and temperature-jump boundary conditions were obtained on arbitrary fine numerical grids for all Knudsen numbers (Kn) up to the limit of the equations' validity. The non-dimensional wall shear stress and heat flux shows good agreement with DSMC results and are better than first order slip boundary condition. The velocity slip and temperature jump calculated from Burnett equations fit DSMC results well when Kn < 0.3 but departure at higher Kn number. The IP method shows an identical velocity distribution with DSMC at low speed, but costs much less time. Perfect agreement was achieved in the density, velocity, temperature and pressure distributions between the results of Burnett equations and IP method at Kn = 0.02 and 0.1. Burnett equations and IP method are better than Navier-Stokes equation and DSMC method, respectively, in the flow and heat transfer in MEMS or NEMS systems

This paper demonstrates the newly developed particle method in the Direct Simulation Monte Carlo (DSMC) framework, and we refer to it as the Maxwellian–DSMC (M–DSMC) method. The M–DSMC solver is utilized for various fluid flow problems with different length scales varied from continuum to transitional regime. The hypersonic flow of argon over a cylinder simulation is carried out using M–DSMC and the same simulation results are compared with the regular DSMC method. The present work shows that the M–DSMC results are quite a good match with the DSMC results. The computational cost is significantly reduced by using M–DSMC method and the same is compared with regular DSMC method.

A direct simulation Monte Carlo (DSMC) method is implemented on graphical processing unit (GPU) architecture to simulate the deposition phenomena that occur on a glass panel during modern organic light emitting diodes (OLED) manufacturing processes with a huge number of simulated particles because a fully three-dimensional (3D) particle simulation is required to accurately predict deposition thickness and uniformity on the substrate in a low-pressure environment. To examine the accuracy and efficiency of the newly developed DSMC code running on GPU architecture, verification and validation tests were performed by solving several benchmarking problems. Under three conditions which have been applied in previous OLED deposition process test experiments, the new DSMC model successfully simulates the transport of organic material evaporated from the heated nozzle array and its deposition onto a glass panel inside a low-pressure chamber. The uniformity and normalized thickness between the DSMC and experimental results for the three test cases are then compared. The normalized thickness of the organic material deposited to the large glass panel calculated from the DSMC is found to have an error of less than 4%, and the uniformity values calculated from the absolute deposition thickness show good agreement with those obtained from experimental data in all test cases.

Enceladus, a moon of Saturn, is of great astrobiological interest due to Cassini mission discoveries [1–5]. Its southern polar region features "Tiger Stripes"—fractures that vent water vapor, gases, and particles, including molecular hydrogen (H₂), which likely originates from hydrothermal activity [5]. The intermittent presence of H₂ raises questions about plume dynamics and the subsurface environment. Models suggest that the ejection of gases and particles is influenced by the geometry of the conduits beneath the icy shell [1]. Recent studies propose volatile-driven eruptions, constrained by plume observations, that shed light on cryovolcanic processes [6]. Yet, the mechanisms of hydrogen transport and the chemical properties of the subsurface ocean remain unresolved.This study investigates interactions between condensable (H₂O) and non-condensable (H₂) gases under varying subsurface conditions. Using the Direct Simulation Monte Carlo (DSMC) method, we model gas flow originating in a cavern, moving through a conduit, and expanding into vacuum.To capture the complex flow regimes of Enceladus’ plumes, we use the DSMC method [7], which accurately simulates particle interactions and non-equilibrium effects across transitions from near-continuum to free-molecular flows. DSMC represents real gas molecules through computational particles, allowing detailed tracking of pressure gradients, thermal nonequilibrium, and energy exchange. We use the PLANET DSMC code [8–13] to simulate the flows.The subsurface model assumes a "misty cavern" at the gas-liquid interface where H₂O vapor forms via sublimation and evaporation, while H₂ is added to reproduce the 99:1 H₂O:H₂ mass ratio seen by Cassini at the vent. The model geometry applies cylindrical symmetry and includes specified boundary conditions at the cavern and conduit walls. Simulation parameters such as cell size, time step, and convergence criteria are tightly controlled.Next, we analyze how steady-state plume dynamics respond to changes in several parameters: conduit radius, depth above a constriction, temperature profile along the walls, H₂ inflow flux, and reservoir size. Variations in vent radius and wall temperature, in particular, have a notable impact on the resulting gas composition and distribution at the surface vent. These findings help constrain the physical characteristics of Enceladus’ subsurface vents and support the hypothesis that even very small concentrations of H₂ plays a role in sustaining long-term vent activity. A key finding is that flow through high aspect ratio conduits can lead to very large build ups of hydrogen in the underground reservoir.

In the first part of this paper presents a gas-kinetic scheme based on the Bhatnagar–Gross–Krook (BGK) model for the microflow simulations in the near continuum flow regime. The current method improves the previous gas-kinetic BGK Navier–Stokes (BGK–NS) solver by (i) implementing a general non-equilibrium state based on the Chapman–Enskog expansion of the BGK model up to the Knudsen number squared $(\hbox{\it Kn})^2$ in the gas distribution function, (ii) using the compatibility condition to evaluate all high-order time derivative terms in the Chapman–Enskog expansion, and (iii) implementing the kinetic boundary condition for the gas distribution function to obtain ‘slip’ boundary automatically. As a result, the gas-kinetic BGK–Burnett scheme improves the validity of the BGK–NS solver for the microchannel flow simulations even in the slip flow regime, where the Navier–Stokes equations with the slip boundary conditions are considered to be legitimately valid. Owing to the correction to the heat transport in the energy flux, the Prandtl number in the gas-kinetic BGK–Burnett method can take any value to capture both viscous and heat conduction effects. Since the current method is based on the direct evaluation of the gas distribution function and captures its time evolution, it is different from those methods that are based on the macroscopic Burnett or extended hydrodynamic equations. The second part of this paper is about the application of the newly developed gas-kinetic BGK–Burnett method in the microchannel flows. First, we verify the method in the pressure- and external-force-driven Poiseuille flows, where the reliable direct simulation Monte Carlo (DSMC) results are available. In the study of Poiseuille flow with $\hbox{\it Kn}=0.1$, the qualitative differences in the pressure distribution in the cross-stream direction between the Navier–Stokes and the DSMC results are resolved by the gas-kinetic BGK–Burnett scheme. It demonstrates that the BGK–Burnett method could give a more realistic description of flow motion than the Navier–Stokes method even in the slip flow regime. After that, the current method is used to simulate the microchannel flows, where the experimental data are available. In this study, the similarity in the pressure distribution along the straight microchannel is verified first. Then, the mass flow rates for different gases, such as argon, helium and nitrogen, in the long microchannel of submicron height are computed and compared with the experimental measurements.

Molecular models for simulation of rarefied gas flows using direct simulation Monte Carlo method

Adaptive domain decomposition for unstructured meshes applied to the direct simulation monte carlo method

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.

Abstract. Numerical simulations using the Direct Simulation Monte Carlo (DSMC) method are known to be useful for analyses of aerodynamic effects on in-situ rocket measurements in the lower thermosphere, but the DSMC analysis of a spin modulation caused by an asymmetric flow around the rocket spin axis has been restricted to the two-dimensional and axially symmetric simulations in actual sounding rocket experiments. This study provides a quantitative analysis of the spin modulation using a three-dimensional (3-D) simulation of the asymmetric flow with the DSMC method. Clear spin modulations in the lower thermospheric N2 density measurement by a rocket-borne instrument are simulated using the rocket attitude and velocity, the simplified payload structure, and the approximated atmospheric conditions. Comparison between the observed and simulated spin modulations show a very good agreement within 5% at around 100km. The results of the simulation are used to correct the spin modulations and derive the absolute densities in the background atmosphere.

The direct simulation Monte Carlo (DSMC) method is the primary numerical technique for analysis of rarefied gas flows. While recent progress in computational chemistry is beginning to provide vibrationally resolved transition and reaction cross sections that can be employed in DSMC calculations, the particle nature of the standard DSMC method makes it difficult to use this information in a statistically significant way. The current study introduces a new technique that makes it possible to resolve all of the vibrational energy levels by using a master equation approach along with temperature-dependent transition rates. The new method is compared to the standard DSMC technique for several heat bath and shock wave conditions and demonstrates the ability to resolve the full vibrational manifold at the expected overall rates of relaxation. The ability of the new master equation approach to the DSMC method for resolving, in particular, the high-energy states addresses a well-known, longstanding deficiency of the standard DSMC method.

The massively parallel computer Connection Machine is utilized to map an improved version of the direct simulation Monte Carlo (DSMC) method for solving flows with the Boltzmann equation. The kinetic theory is required for analyzing hypersonic aerospace applications, and the features and capabilities of the DSMC particle-simulation technique are discussed. The DSMC is shown to be inherently massively parallel and data parallel, and the algorithm is based on molecule movements, cross-referencing their locations, locating collisions within cells, and sampling macroscopic quantities in each cell. The serial DSMC code is compared to the present parallel DSMC code, and timing results show that the speedup of the parallel version is approximately linear. The correct physics can be resolved from the results of the complete DSMC method implemented on the connection machine using the data-parallel approach. 41 refs.