Accuracy and performance evaluation of low density internal and external flow predictions using CFD and DSMC

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 °.

The structure of a normal shock waves in dilute gas mixtures of various compositions (Xenon-Helium and Argon-Helium) are studied using Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC). The MD solutions reproduce well the experimental data, with the exception of the parallel temperature profile in the 24.7% Ar-He mixture, despite the satisfactory agreement between the parallel velocity profiles. The Generalized Hard Sphere (GHS) DSMC solutions match almost perfectly the MD data, for all the cases considered. Using textbook parameters for like particles and their arithmetic averages for unlike particles, the Variable Hard Sphere (VHS) model fails to describe the shock wave structure in the Xe-He mixture. For the Ar-He cases, the VHS diffusion cross-section better approximates that obtained with the Lennard-Jones potential. Therefore, the VHS DSMC solutions exhibit less significant differences with those obtained with MD. If the VHS model is properly parametrized, the agreement between DSMC and MD solutions increases significantly. These cases exemplify situations where MD simulations based on a reliable interatomic potential can be used to inform simplified collision schemes used in 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.

Two particle-based methods, the established Direct Simulation Monte Carlo (DSMC) method and the novel Low Diffusion (LD) method, are compared in terms of accuracy and performance. While DSMC enables the physically accurate simulation of rarefied gas flows, LD enables the efficient simulation of dense flows under the continuum assumption. Both methods have been utilized to simulate the gas flow around a 70° blunted cone in a wide range of Knudsen numbers from 0.001 to 0.1, and are validated against existing experimental heat flux measurements. Additionally, flow field variables in front of the heat shield as well as surface properties are compared. Finally, the research towards a coupled application of both methods to enable the efficient and physically accurate simulation of flows with large density gradients is addressed.

The direct simulation Monte Carlo (DSMC) method is used to solve the three-dimensional nano-scale gas film lubrication problem between a gas bearing slider and a rotating disk, and this solution is compared to the numerical solution of the compressible Reynolds equations with the slip flow correction based on the linearized Boltzmann equation as presented by Fukui and Kaneko [molecular gas film lubrication (MGL) method] [ASME J. Tribol. 110, 253 (1988)]. In the DSMC method, hundreds of thousands of simulated particles are used and their three velocity components and three spatial coordinates are calculated and recorded by using a hard-sphere collision model. Two-dimensional pressure profiles are obtained across the film thickness direction. The results obtained from the two methods agree well with each other for Knudsen numbers as large as 35 which corresponds to a minimum spacing of 2 nm. The result for contact slider is also obtained by the DSMC simulation and presented in this paper.

A hybrid particle scheme, which is based on the direct simulation Monte Carlo (DSMC) method and incorporates a low diffusion particle method for continuum flow simulation, is extended for improved efficiency and wider applicability. The hybrid scheme is intended for simulation of gas flows involving a wide range of Knudsen number regimes, where strong coupling is desired between calculations in rarefied and continuum flowfield regions. A series of modifications are presented for cell-based numerical weight and time step adaptation, diffusive transport of nonequilibrium internal energy modes, and decreased sensitivity to DSMC cell size. Proposed modifications are employed in a hybrid simulation of a Mach 20 flow around a cylinder with a global Knudsen number of 0.002, for which a factor of 2.6 efficiency increase relative to full DSMC computation is demonstrated. I. Introduction n a variety of gas flow problems, portions of the flowfield exhibit large translational nonequilibrium effects and require relatively expensive simulation techniques based on the Boltzmann equation, while other regions are within low Knudsen number (Kn) continuum flow regimes and can be prohibitively expensive to simulate using such Boltzmann equation techniques. These “multiscale” flow problems include flows around atmospheric entry vehicles and hypersonic aircraft, where local Kn values based on gradient length scales may be very small through much of the flowfield but become large within shock, forebody boundary layer and wake regions. Other examples include rocket exhaust plumes and fuel venting flows at high altitude, where the mean free path may differ by several orders of magnitude between nearfield and farfield regions. A. Simulation Methods for Multiscale Gas Flows Accurate simulations may be performed for some of these flows using a two step, one-way coupled approach. 1-3 In this type of approach, continuum and rarefied regions are considered independently and different simulation techniques are applied in each region. In general, however, effects of two-way coupling between rarefied and continuum regions must be considered, and a single simulation employing a coupled hybrid algorithm is required. In one popular type of hybrid algorithm, a computational fluid dynamics (CFD) technique based on the Navier-Stokes equations is used in low-Kn continuum regions, the direct simulation Monte Carlo (DSMC) method is employed in rarefied regions, and additional procedures are used for evaluation of continuum breakdown and two-way information transfer between CFD and DSMC calculations.

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

The role of attractive-repulsive interactions in direct simulation Monte Carlo (DSMC) simulations is studied by comparing with traditional purely repulsive interactions. The larger collision cross section of the long-range LJ potential is shown to result in a higher collision frequency and hence a lower mean free path, by at least a factor of two, for given conditions. This results in a faster relaxation to equilibrium as is shown by comparing the fourth and sixth moments of the molecular velocity distribution obtained using 0-D DSMC simulations. A 1-D Fourier-Couette flow with a large temperature and velocity difference between the walls is used to show that matching transport properties will result in identical solutions using both LJPA and VSS models in the near-continuum regime. However, flows in the transitional regime with Knudsen number, Kn ∼ 0.5 show a dependence on the intermolecular potential in spite of matching the viscosity coefficient due to differences in the collision frequency. Attractive-repulsive potentials should be used when both transport coefficients and collision frequencies should be matched.

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 hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the “augmented Burnett equations” and the “BGK–Burnett equations.” This paper traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and the Grad’s 13 moment equations is elucidated by employing the Maxwell–Truesdell–Green iteration. Numerical solutions are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum–transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier–Stokes calculations, Burnett solutions of other investigators, and DSMC solutions as much as possible.

Following seminal studies such as Muñoz’s 2007 work on HD 209458b, which simulated heavy element escape beyond the Roche lobe, one-dimensional hydrocodes have flourished, routinely solving the Euler equations to model transonic outflows across an increasingly diverse population of exoplanets. However, the modelling frontier of escape is often shaped by the hand-off from continuum to rarefied flow (Kn ≳ 0.1) and non-equilibrium processes. Molecular-kinetic techniques, long the workhorse of Solar-System aeronomy, naturally bridge this gap, providing a self-consistent description of collisional, transitional and free-molecular regimes in a single framework. Here we make the case for a concerted push toward large-scale molecular-kinetic simulations of exoplanet outflows, highlighting two end-member scenarios along the escape spectrum where forthcoming observations may allow the theory to be tested and refined.Cosmic Shoreline. Characterising the transition from Jeans (particle-by-particle) escape to subsonic and ultimately transonic bulk outflow remains an open problem in escape theory. The onset of rapid escape (~1 bar Myr⁻¹) as ionising irradiation increases is a key parameter defining the phase boundary between airless and airy rocky worlds—the “Cosmic Shoreline” (Zahnle &amp; Catling 2017; Ji et al. 2025). Johnson et al. (2013) combined an analytic treatment with Direct Simulation Monte Carlo (DSMC; Bird 1994) to derive a critical heating rate for triggering transonic flow, working with the ansatz that the scaling of this transition extends smoothly from Pluto- to Earth-sized bodies. We will present new DSMC simulations that probe this transition for high-molecular-weight atmospheres on Earth-mass and super-Earth planets, refining the dynamics of rapid escape across this regime.Helium triplet and fractionation. Fractionation may help explain some of the non-detections of the neutral-helium triplet (1083 nm) in giant-planet outflows (Schulik &amp; Owen 2024). Multi-fluid hydrodynamics simulations have found that the neutral helium can actually be accelerated by gravity to accrete out of the flow at a downward velocity of ~1 km s⁻¹ (Xing et al. 2023; Schulik &amp; Owen 2024). We note that the ratio of the slip velocity to the thermal speed of the outflow scales with the Knudsen number for collisionality, ΔU/ Vth~ KnHe . Thus, we will discuss how a significant slip velocity may require Kn ≳ 0.1, a regime in which the fractionation process may be better described with molecular-kinetics, possibly with implications for predictions of the transit depth of the helium triplet.Moreover, the Direct Simulation Monte Carlo (DSMC) method offers some desirable properties over hydrocodes: it scales naturally to fully three-dimensional geometries, albeit at significant computational cost, and naturally treats non-equilibrium phenomena such as photoelectron heating and excited-state populations.

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.

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.

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 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.