Direct Simulation Monte Carlo Solutions Research Articles

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

The Direct Simulation Monte Carlo (DSMC) method was widely used to simulate low density gas flows with large Knudsen numbers. However, DSMC encounters limitations in the regime of lower Knudsen numbers (Kn<0.05). In such cases, approaches from classical computational fluid dynamics (CFD) relying on the continuum assumption are preferred, offering accurate solutions at acceptable computational costs. In experiments aimed at imaging aerosolized nanoparticles in vacuo a wide range of Knudsen numbers occur, which motivated the present study on the analysis of the advantages and drawbacks of DSMC and CFD simulations of rarefied flows in terms of accuracy and computational effort. Furthermore, the potential of hybrid methods is evaluated. For this purpose, DSMC and CFD simulations of the flow inside a convergent–divergent nozzle (internal expanding flow) and the flow around a conical body (external shock generating flow) were carried out. CFD simulations utilize the software OpenFOAM and the DSMC solution is obtained using the software SPARTA. The results of these simulation techniques are evaluated by comparing them with experimental data (1), evaluating the time-to-solution (2) and the energy consumption (3), and assessing the feasibility of hybrid CFD-DSMC approaches (4).

A stochastic Fokker–Planck–Master model for diatomic rarefied gas flows

The direct simulation Monte Carlo (DSMC) method is widely used for numerical solutions of the Boltzmann equation. However, the associated computational cost becomes prohibitive in the near-continuum regime. To address this limitation, the particle-based Fokker–Planck (FP) method has been extensively studied in the past decade. The FP equation, which describes Brownian motion, does not require resolution of the collisional time and length scales. While several monatomic FP models have been proposed, the modeling of diatomic gases within the FP framework has received limited attention. In this paper, we propose a new diatomic kinetic model, named the Fokker–Planck–Master (FPM) model, which can accurately describe energy exchanges between translational-rotational and translational-vibrational modes. The FPM model combines the FP equation to describe the evolution of translational and rotational modes, and the master equation to describe the evolution of the vibrational modes. The numerical test cases include relaxation problems, Couette flows, and hypersonic flows past a vertical flat plate. The results demonstrate that the FPM model shows good agreement with both analytical and DSMC solutions.

Inversion in binary gas mixtures in rarefied flow conditions: Direct simulation Monte Carlo solution and comparison with the analytical solutions at free molecular regime

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called “inversion” in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions.

Multiple temperature model of nonlinear coupled constitutive relations for hypersonic diatomic gas flows

The rotational energy of diatomic gases would be activated by the process of intermolecular collisions in high-temperature hypersonic flows. In this paper, a multi-temperature nonlinear coupled constitutive model has been proposed for simulating the transfer of energy between translational and rotational motions in hypersonic non-equilibrium flows. In this model, the nonlinear coupled constitutive equations are modified by introducing a rotational energy relaxation model and a changeable viscosity ratio related to local temperature. To confirm its accuracy, the new model is applied to investigate steady shock wave structures and high-speed gas flows around a cylinder and across a flat plate. The computational results are compared with the multi-temperature Navier–Stokes (NS) equations, the direct simulation Monte Carlo (DSMC) solutions, and the experiment data. The final results show the new model would reproduce the NS results at low Knudsen numbers but behave quite differently from the NS results as the non-equilibrium degree is enhanced. The new model is in better agreement with the DSMC solutions and the experimental data than the NS solutions in the far-from-equilibrium regions, which demonstrates the potential of the new relaxation model in the simulation of hypersonic non-equilibrium flows.

Laser-induced fluorescence velocimetry for a hypersonic leading-edge separation

Two-dimensional mapping of the velocity distribution for a hypersonic leading-edge separation flowfield generated by a “tick” shaped geometry is presented for the first time. Discrete measurements of two velocity components were acquired at a flow condition having a total specific enthalpy of 3.8 MJ/kg by imaging nitric oxide fluorescence over numerous runs of the hypersonic tunnel at the Australian Defence Force Academy (T-ADFA). The measured freestream velocity distribution exhibited some non-uniformity, which is hypothesized to originate from images acquired using a set of ultraviolet specific mirrors mounted on the shock tunnel deflecting under load during a run of the facility, slightly changing the laser sheet orientation. The flow separation point was measured to occur at 1.4 ± 0.2 mm from the model leading edge, based on the origin of the free shear layer emanating from the expansion surface. Reattachment of this free shear layer on the compression surface occurred at 59.0 ± 0.2 mm from the model vertex. Recirculating the flow bound by the separation and reattachment points contained supersonic reverse flow and areas of subsonic flow aligned with the location of three identified counter-rotating vortices. A comparison of the recirculation flow streamline plots with those computed using Navier–Stokes and direct simulation Monte Carlo (DSMC) codes showed differences in flow structures. At a flow time close to that produced by the facility, flow structures generated by the DSMC solution were seen to agree more favorably with the experiment than those generated by the Navier–Stokes solver due to its ability to better characterize separation by modeling the strong viscous interactions and rarefaction at the leading edge. The primary reason for this is that the no-slip condition used in the Navier–Stokes solution predicts a closer separation point to the leading edge and structures when compared to the DSMC solution, which affects surface shear stress and heat flux, leading to a difference in flow structures downstream of the separation.

Coupling of state-resolved rovibrational coarse-grain model for nitrogen to stochastic particle method for simulating internal energy excitation and dissociation.

We propose to couple a state-resolved rovibrational coarse-grain model to a stochastic particle method for simulating internal energy excitation and dissociation of a molecular gas. A coarse-grained model for a rovibrational reaction mechanism of an ab initio database developed at the NASA Ames Research Center for the N2-N system is modified based on variably spaced energy bins. The thermodynamic properties of the modified coarse-grained model allow us to closely match those obtained with the full set of rovibrational levels over a wide temperature range, while using a number of bins significantly smaller than the complete mechanism. The chemical-kinetic behavior of equally and variably spaced bin formulations is compared by simulating internal energy excitation and dissociation of nitrogen in an adiabatic, isochoric reactor. We find that the variably spaced formulation is better suited for reproducing the dynamics of the full database at conditions of interest in the Earth atmospheric entry. Also in this paper, we discuss the details of our particle method implementation for the uniform rovibrational collisional bin model and describe changes to the Direct Simulation Monte Carlo (DSMC) collision algorithm, which become necessary to accommodate our state-resolved reaction mechanism for excitation and dissociation reactions. The DSMC code is then verified against equivalent master equation calculations. In these simulations, state-resolved cross sections are used in analytical form. These cross sections verify micro-reversibility relations for the rovibrational bins and allow for fast execution of the DSMC code. In our verification calculations, we obtain very close agreement for the concentrations profiles of N and N2, as well as the translational and rovibrational mode temperatures obtained independently through both methods. In addition to macroscopic moments, we compare discrete internal energy populations predicted at selected time steps via DSMC and the master equations. We observe good agreement between the two sets of results within the limits imposed by statistical scatter, which is inherent to particle-based DSMC solutions. As future work, the rovibrational coarse-grain model coupled to the particle method will allow us to study 3D reentry flow configurations.

Cubic Fokker–Planck method for rarefied monatomic gas flow through a slit and an orifice

The flow through a thin slit and a thin orifice is studied with the Direct Simulation Monte Carlo (DSMC), cubic Fokker–Planck (FP), and a coupled FP-DSMC hybrid method. Pressure driven monatomic gas flows through a slit and an orifice with various values of degree of rarefaction and pressure ratio are computed. The DSMC method is physically accurate for all flow regimes; however it is computationally expensive in high density, near continuum regions. An alternative stochastic particle scheme, the cubic FP kinetic model has addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. The ability of the cubic FP method to reproduce breakdown of translational equilibrium is discussed. In addition, a coupled FP-DSMC hybrid scheme is employed aiming at an efficient and accurate solution. The FP-DSMC hybrid scheme employs DSMC in rarefied regions and FP method in near continuum flow regions. Numerical procedures of the cubic FP method are implemented within the framework of an existing DSMC-solver, SPARTA. The FP-DSMC hybrid solution reproduces pure DSMC solution with improved computational efficiency up to a factor of eight for vacuum flow through a thin slit.

Particle simulation of nonequilibrium gas flows based on ellipsoidal statistical Fokker–Planck model

In order to improve the efficiency of particle simulation method for gas flows spanning the rarefied and continuum regimes, one promising approach is to employ the Fokker–Planck-type gas kinetic model. The ellipsoidal statistical Fokker–Planck (ES-FP) model treats the evolution of individual molecular velocity as a continuous stochastic process, which can be equivalently described by the Langevin dynamics where the forces acting on each gas molecule are a linear drag force and an anisotropic fluctuating force. By utilizing the exact stochastic integral solution of the corresponding Langevin equation, the ES-FP model is numerically implemented in a particle Monte Carlo manner. In a validation of the particle scheme, excellent agreement is obtained between the simulation results and the analytical solution for the same initial-value problem of ES-FP equation. The ES-FP particle method is used to simulate the relaxation process in a nonequilibrium gas, and agrees well with the direct simulation Monte Carlo (DSMC) method in the predictions of viscous stress and heat flux. Furthermore, extensive ES-FP particle simulations are performed for Couette flows at different Knudsen numbers ranging from 0.001 to 10, as well as supersonic flat-plate flows at different Knudsen numbers ranging from 0.001 to 0.1. Both flow fields and surface quantities are investigated and compared with the DSMC or theoretical solutions. In the ES-FP simulations, the Prandtl number of gas can be corrected, and the physical behaviors in different flow regimes can be properly captured. With reasonable agreement between the ES-FP and DSMC results, the ES-FP simulation is found to be more efficient than DSMC and can significantly save memory cost and CPU time, especially for multi-dimensional flows in the low-Knudsen-number regime.

A perturbation-based solution of Burnett equations for gaseous flow in a long microchannel

In this paper, an analytical investigation of two-dimensional conventional Burnett equations has been undertaken for gaseous flow through a long microchannel. The analytical solution is obtained by using perturbation analysis around the classical Navier–Stokes solution with appropriate boundary conditions. The perturbation expansion is employed with the smallness parameter $\unicode[STIX]{x1D716}$, taken as the ratio of height to length of the microchannel. The solution for pressure is obtained by solving the cross-stream momentum equation while the velocity distribution is obtained from the streamwise momentum equation. The resulting ordinary differential equations in pressure and velocity are third-order and second-order, respectively. The required boundary conditions for pressure are obtained from direct simulation Monte Carlo (DSMC) data. The obtained analytical solution matches the available DSMC solution well. This is perhaps the first analytical solution of the Burnett equations using the perturbation approach.

Assessment of the cubic Fokker–Planck–DSMC hybrid method for hypersonic rarefied flows past a cylinder

Hypersonic vehicles experience a wide range of Knudsen number regimes due to changes in atmospheric density. The Direct Simulation Monte Carlo (DSMC) method is physically accurate for all flow regimes, however it is relatively computationally expensive in high density, and low Knudsen number regions. Recent advances in the Fokker–Planck (FP) kinetic models have addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. Furthermore, a coupled FP–DSMC solution method has been devised aiming at a universally efficient yet accurate solution algorithm for rarefied gas flows. Well known Lofthouse case of a generic hypersonic flow about a cylinder (Mach 10, Kn 0.01, Argon) is selected to investigate the performance of a hybrid FP–DSMC implementation. The effect of molecular potential on the accuracy of the scheme is mainly analyzed. Furthermore, spatial resolution of cubic FP scheme is studied. Finally, detailed study of accuracy and efficiency of FP–DSMC hybrid scheme is discussed. It is found that the presented adaptive grid together with the FP–DSMC method results in a factor of six speed up for considered hypersonic flow about a cylinder.

On the vortical characteristics and cold-to-hot transfer of rarefied gas flow in a lid driven isosceles orthogonal triangular cavity with isothermal walls

Rarefied gas flows in nano-scale isosceles triangular cavities with a motion away from the square corner are considered over a wide range of flow rarefactions regimes, i.e., 0.01 ≤ Kn ≤ 10, using the direct simulation Monte Carlo (DSMC) method. We put emphasis to the rarefaction effects on the vortical behaviors and heat transfer patterns of rarefied monatomic and diatomic gases. We show that the primary vortices appearing in the triangular cavity are Moffat-type vortices, which disappear as velocity slip increases over the walls. However, an additional vortex is observed at Kn > 1 on the inclined wall attributable to the balance of the thermally driven flows from the cold to the hot region and flow induced by the primary vortex. We also report variations in the effective length scale of the fluid circulation with the Knudsen number. Our investigations specify that the competition between the Fourier term and shear stress gradient component of the heat flux constitutive relation results in the cold-to-hot transfer in triangular cavities. Also, heat flux patterns predicted by the asymptotic theory of the Boltzmann equation for the weakly non-linear flow and by the linearized form of the Regularized 13 moments (R13) equations are compared with the DSMC solution suitably at low Knudsen number regimes.

On the thermally-driven gas flow through divergent micro/nanochannels

A detailed study on thermally driven flows through divergent micro/nanochannels is presented. Rarefied gas flow behavior and thermal mass flow rate were investigated with different divergence angles ranging between 0[Formula: see text] and 7[Formula: see text] at two aspect ratios ([Formula: see text]) using particle-based direct simulation Monte-Carlo (DSMC) method. We compare our DSMC solutions for normalized thermal mass flow rate with the numerical solution of the Boltzmann–Krook–Walender (BKW) model and Bhatnagar–Gross–Krook (BGK) model and asymptotic theory over a wide range of Knudsen number in the transition regime. The flow field properties including Mach number, pressure, overall temperature and magnitude of shear stress are examined in detail. Based on our analysis, we observed an approximately constant velocity and pressure distribution at a microchannel with a small opening angle. Our results also demonstrate that the heat lines from weakly nonlinear form of Sone constitutive law and DSMC show good agreement at low Knudsen numbers. Moreover, we show that the effect of divergence angle is influential in increasing normalized thermal mass flow rate at early transition regime.

On the consequences of successively repeated collisions in no-time-counter collision scheme in DSMC

The current paper investigates the appropriateness and consequences of avoiding successively repeated collisions usually recommended in the Direct Simulation Monte Carlo (DSMC) solvers, which use the No Time Counter (NTC) collision technique. A sensitive test case, i.e., Fourier heat transfer problem at the early slip regime, is considered in the presence and absence of the successively repeated collisions. The effects of repeated collisions and avoiding them on different parameters such as heat flux, collision frequency, collision separation distance, accepted-to-selected collisions ratio, and probability distribution functions for the number of collisions and relative velocities of collision pairs are evaluated and discussed. The investigations are performed considering contributions of different computational parameters, i.e., cell sizes, time step, and number of particles. In addition to collision frequency and collision separation distance, which were reported previously as crucial parameters affecting the accuracy of the DSMC solution, we found that the level of repeated collisions also plays a pivotal role in the accuracy of the heat flux prediction in the Fourier problem. We show that direct avoiding of repeated collisions can lead to a distortion of collision probability distribution and consequently, to introduction of a systematic error in collision frequency and predicted heat flux. This error is negligible if one uses a large number of particles per cell, but it is of considerable importance if there are a few particles per cell.

Regulation of anti-Fourier heat transfer for non-equilibrium gas flows through micro/nanochannels

In this study we use the direct simulation Monte Carlo (DSMC) to elaborate on the heat transfer patterns in the pressure-driven rarefied flow through micro/nanochannels. Finite length planar micro/nanochannels are considered using with symmetrical wall heat flux boundary conditions, and the gas flow is considered to be in slip and transition regimes. When considering zero-conductive or cooled walls, the DSMC solution predicts a possibility of the anti-Fourier heat transfer, i.e., the transfer of heat from cold-to-hot regions of the flow field. It turns out that the competition between the contributions of temperature gradient and pressure gradient (shear stress) in the heat flux results in three different heat transfer regimes. The regimes consist of complete hot-to-cold heat transfer regime, the entire anti-Fourier regime, and localized anti-Fourier regime. While the heat flux due to the shear stress is directed from the outlet towards the inlet, the Fourier term is strongly influenced by viscous slip heating, which then acts as a heat source, and contributes to patterns of heat flux on the fluid layer adjacent to the walls. Furthermore, the heat flux regimes for complete or localized cold-to-hot transfer are classified according to the magnitude of the normalized heat flux and the Knudsen number. Additionally, effects of heat flux condition on the mass flow rate are discussed.

Particle Ellipsoidal Statistical Bhatnagar–Gross–Krook Approach for Simulation of Hypersonic Shocks

To reduce the computational costs related to direct simulation Monte Carlo computations for low Knudsen number flows, a new particle method, based on the ellipsoidal statistical Bhatnagar–Gross–Krook model of the Boltzmann equation, is investigated. In this method, a fraction of the computational particles in a cell is selected for velocity reassignment from the local Maxwellian distribution based the local collision frequency and an even smaller fraction is selected for internal mode energy reassignment based on the rotational and vibrational relaxation rates. A scaling algorithm is used to decrease the required number of particles per cell and to guarantee that the energy and momentum are conserved in each time step. The accuracy and efficiency of the ellipsoidal statistical Bhatnagar–Gross–Krook method are tested in three different test cases and the results are compared with the direct simulation Monte Carlo solution and experiment as well. The ellipsoidal statistical Bhatnagar–Gross–Krook approach is found to provide good agreement with direct simulation Monte Carlo solutions, however, a factor of 2 speed up compared with direct simulation Monte Carlo is only achieved when the collision frequency is sufficiently high.

Modeling of Nitrogen Monoxide Formation and Radiation in Nonequilibrium Hypersonic Flows

The interaction between nitrogen vibrational-translational relaxation with different chemistry models in a hypersonic shock is studied. The vibrational-translational transition models considered include the Larsen–Borgnakke, Schwartz-Slawsky-Herzfeld, and forced harmonic oscillator models; and the nitrogen monoxide formation reaction was studied using the total collision energy chemistry model and reaction cross sections obtained from quasi-classical trajectory simulations. Using direct simulation Monte Carlo solutions that incorporate these models, improved nitrogen monoxide spectral radiation modeling was performed by employing quenching rates from recent experiments. Comparisons of electronic state populations and the corresponding radiative spectra for the different vibrational-translational and chemistry models are presented, and it is shown that the vibronic spectral structure of the nitrogen monoxide radiation could provide a way to test the fidelity of the internal energy transfer and chemistry models.

Investigation of convective heat transfer through constant wall heat flux micro/nano channels using DSMC

In this research, convective heat transfer of the argon gas flow through a micro/nano channel with uniform heat flux wall boundary condition is investigated using the direct simulation Monte Carlo (DSMC) method. Both of the hot wall (qw>0) and the cold wall (qw<0) cases are considered. Implementation of wall heat flux in the DSMC method is performed using the recently developed “iterative” technique. Our investigation considers heat transfer behavior in both of slip and transition flow regimes. We investigate the influence of rarefaction, i.e., Knudsen number, and viscous dissipation, i.e., Brinkman number, on the Nusselt number behavior. We use the generalized hard sphere (GHS) collision model to consider accurate variation of the heat conductivity with the temperature. The DSMC solutions for the Nusselt number are compared with different analytical expressions reported in the literature with suitable accuracy through the slip regime. We observe that the dependency of the Nusselt number on the Knudsen number decreases in nanochannels as Knudsen number increases into the transition regime, i.e., Nusselt number approaches nearly a constant value as Knudsen number goes beyond 1. Additionally, it is shown that the Nusselt number is a weak function of the Brinkman number.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III: Microflow Simulations

AbstractDue to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But, the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

Molecular Dynamics Simulations of Shock Waves in Mixtures of Noble Gases

We study the structure of a normal shock wave in noble gas mixtures (Xe-He and Ar-He) of various compositions using molecular dynamics and direct simulation Monte Carlo. The molecular dynamics simulations are first validated against experimental data. Good agreement is found between the molecular dynamics solutions and the experimental measurements, with the exception of the parallel temperature profile in the 24.7% Ar-He mixture, despite the satisfactory agreement between the parallel velocity profiles. Secondly, a validation against direct simulation Monte Carlo solutions obtained with the accurate generalized hard sphere model is presented. As expected, the generalized hard sphere direct simulation Monte Carlo and molecular dynamics solutions are in near-perfect agreement. Finally, molecular dynamics results are compared to those obtained with the lower fidelity variable hard sphere model, which, if inappropriately parametrized, fails to describe the shock wave structure. This work exemplifies how full molecular dynamics solutions could be used to precisely discern differences between phenomenological models of various accuracy.

A Numerical Method for Simulation of Microflows by Solving Directly Kinetic Equations with WENO Schemes

A numerical method for simulation of transitional-regime gas flows in microdevices is presented. The method is based on solving relaxation-type kinetic equations using high-order shock capturing weighted essentially non-oscillatory (WENO) schemes in the coordinate space and the discrete ordinate techniques in the velocity space. In contrast to the direct simulation Monte Carlo (DSMC) method, this approach is not subject to statistical scattering and is equally efficient when simulating both steady and unsteady flows. The presented numerical method is used to simulate some classical problems of rarefied gas dynamics as well as some microflows of practical interest, namely shock wave propagation in a microchannel and steady and unsteady flows in a supersonic micronozzle. Computational results are compared with Navier---Stokes and DSMC solutions.