DSMC Data Research Articles

Unified stochastic particle simulation of polyatomic gas flows using SPARTACUS

Numerically simulating multiscale nonequilibrium gas flows that involve internal energy exchange is a challenging problem in engineering applications due to the complexity of the physical processes. The unified stochastic particle (USP) method, based on the Bhatnagar-Gross-Krook (BGK) model, is a promising approach for efficiently simulating multiscale flows. In this paper, we develop a USP method for polyatomic gases that can simulate hypersonic gas flows involving internal energy exchange across all flow regimes. This method is integrated into the open-source SPARTACUS solver, which was previously developed for monatomic gases. The newly developed USP method reassigns the velocity and internal energies of a subset of computational particles based on a precisely designed local target distribution function. A conservation algorithm is applied to ensure that the momentum and energy are conserved. To validate the accuracy and efficiency of the solver, we perform three benchmark cases, including flow past a two-dimensional cylinder, a three-dimensional 70° blunted cone, and an axisymmetric double cone configuration. The results demonstrate that SPARTACUS achieves excellent agreement with the reference results of DSMC and experimental data. Additionally, SPARTACUS provides significant computational efficiency advantages over the DSMC solver SPARTA, particularly in simulating three-dimensional gas flows.

Nonlinear coupled constitutive relation model for monatomic gas in the OpenFOAM framework

In high-speed rarefied gas flow simulations, the linear constitutive relations of the classical Navier-Stokes-Fourier (NSF) equations are no longer successfully applied. In the present work, the two-dimensional nonlinear coupled constitutive relations (NCCR) model of the monatomic gas in an algebraic-type form is successfully implemented in the rhoCentralFoam in OpenFOAM to simulate the high-speed rarefied gas flows. Methods: This solver initially resolves the NSF equations using high-resolution central schemes in the finite volume method. The implicit NCCR model is solved by the iterative process with the initial values from the linear constitutive relations of the NSF model. After the convergence, these nonlinear constitutive relations are embedded in the NSF equations to continue the solution. Results: This new modified solver would be validated for the argon gas flows past a circular cylinder in cross-flow at Mach of 5 and Kn = 0.05 and the sharp leading edge flat plate at Mach of 4 and Kn = 0.0042. Conclusion: The simulation results show that the NCCR model agrees well with the DSMC data for the temperature contours for both cases and the temperature and velocity along the shock stand-off distance and is better than the NSF model in the high-speed rarefied gas simulations for the circular cylinder case.

High-speed rarefied gas flow simulations using Quasi-Gas Dynamic equations with slip and jump boundary conditions

The high-speed rarefied gas flow simulation is essential for the aerothermodynamic design of high-altitude vehicles. Recently, the Quasi-Gas Dynamic (QGD) equations have been implemented in OpenFOAM as a CFD solver named QGDFoam. In computational fluid dynamics (CFD), the accuracy of the prediction of the surface quantities depends on the slip and jump boundary conditions applied to the surfaces. In the present work, various first-order and second-order slip and jump boundary conditions have been numerically implemented into the solver QGDFoam in OpenFOAM to obtain a completed solver for simulating the rarefied gas flows. It then captures the surface quantities of the gas flows, such as the surface gas pressure, the slip velocity, and surface gas temperature. This completed solver is validated for the sharp-leading-edge wedge, the compression ramp, and the NACA 0012 micro-airfoil cases. Using the QGD model with the slip and jump conditions extends its application and enhances it to simulate high-speed rarefied gas flows. The simulation results show that the slip and jump conditions have been successfully employed with the QGD equations and give good results for predicting the surface quantities compared with the DSMC data.

Second-order slip condition considering Langmuir isothermal adsorption for rarefied gas microflows

Effect of the slip boundary condition on rarefied gas flow simulations plays an important role to understand the behaviour of gas microflows in MEMS. Several second-order slip conditions were proposed by the models of the kinetic theory of gases to simulate the rarefied gas microflows, in which the so-called classical second-order slip condition was derived from the Karniadakis et al. model. In this paper, a new second-order slip condition is proposed to employ with the Navier-Stokes-Fourier equations for simulating the rarefied gas flows in microchannels. It is derived by combining the Langmuir isothermal adsorption and the Karniadakis et al. model, with the aim of achieving a more realistic physical model. The pressure-driven back-forward-step, the Couette and pressure-driven Poiseulle rarefied gas flows in microchannels are investigated to validate our new second-order slip condition. Slip velocities using our new second-order slip condition are better than those using the conventional Maxwell and the so-called classical second-order slip conditions, and are in very good agreement with the DSMC data for all cases considered.

Effect of viscosity on slip boundary conditions in rarefied gas flows

The viscosity of gases plays an important role in the kinetic theory of gases and in the continuum-fluid modeling of the rarefied gas flows. In this paper we investigate the effect of the gas viscosity on the surface properties as surface gas temperature and slip velocity in rarefied gas simulations. Three various viscosity models in the literature such as the Maxwell, Power Law and Sutherland models are evaluated. They are implemented into OpenFOAM to work with the solver “rhoCentralFoam” that solves the Navier-Stokes-Fourier equations. Four test cases such as the pressure driven backward facing step nanochannel, lid-driven micro-cavity, hypersonic gas flows past the sharp 25-55-deg. biconic and the circular cylinder in cross-flow cases are considered for evaluating three viscosity models. The simulation results show that, whichever the first-order or second-order slip and jump conditions are adopted, the simulation results of the surface temperature and slip velocity using the Maxwell viscosity model give good agreement with DSMC data for all cases studied.

Comprehensive assessment of newly-developed slip-jump boundary conditions in high-speed rarefied gas flow simulations

In this paper we numerically evaluate the recently developed Aoki et al. slip and jump conditions in high-speed rarefied gas flows for the first time. These slip and jump conditions are developed to be employed with the Navier–Stokes–Fourier equations. They were derived based on the Boltzmann equation with the first order Chapman–Enskog solution, and the analysis of the Knudsen layer. Four aerodynamic configurations are selected for a comprehensive evaluation of these conditions such as sharp-leading-edge flat plate, vertical plate, wedge and circular cylinder in cross-flow with the Knudsen number varying from 0.004 to 0.07, and argon as the working gas. The simulation results using the Aoki et al. boundary conditions show suitable agreement with the DSMC data for slip velocity and surface gas temperature. The accuracy of these boundary conditions is superior to the conventional Maxwell, Smoluchowski and Le boundary conditions.

Effect of Knudsen Layer on the heat transfer in hypersonic rarefied gas flows

Effect of the Knudsen layer on the surface heat transfer is investigated for the hypersonic flow applications in the slip and early transition flow regime. Knudsen Layer formulation is incorporated within an open source computational fluid dynamics framework, OpenFOAM. Navier-Stokes-Fourier constitutive relations, and first order slip and jump boundary conditions are modified based on the effective mean free path, which is dependent on the geometry of an obstacle and local flow gradients. The modified solver is validated against benchmark cases of low-speed Couette gas flow and hypersonic flow over a flat plate. Investigation is further extended to the high speed flow over a wedge and a circular cylinder covering a wide range of Knudsen numbers within the slip and early transition flow regime. Effect of Knudsen layer on the surface heat transfer is examined, and maximum improvement of 27% is achieved for a wedge case. Moreover, results show excellent agreement with the DSMC data, especially for the cylinder cases with high Knudsen numbers, Kn = 0.05 and 0.25. The simulation results convey that our approach greatly improves the predictive capabilities of the compressible N-S-F solver up to early transition flow regime (Kn ∼ 1). The current work is significant from the perspective of accurate thermal design of hypersonic vehicles.

Predicting the rarefied gas flow through circular nano/micro short tubes: A semi-empirical model

A new explicit semi-empirical model was proposed for predicting the steady state gas flow rate through short tubes under rarefied condition. The experimental data produced by Fujimoto and Usami and the DSMC data provided by Varoutis et al. were utilized to obtain the model coefficients. The model was further validated by the published experimental data. The proposed model was capable to make predictions while meeting all the criteria of the rarefied gas flow in the following ranges of the variables: 0<Pr < 0.7, 0<δ < 4000 and 0<ω < 13. The model was capable to predict the reduced flow rate of rarefied systems, not only in the free molecular region and the hydrodynamic region, but also in the transition region, hence covering all the Knudsen number domain within the utilized data. It was also found that by increasing the pressure ratio and decreasing the length to diameter ratio, the transition region shifted to the higher values of the rarefaction parameter. The proposed model is useful in the vacuum gas community for engineering purposes by offering a practical closed form expression for the DSMC data to predict the rarefied gas flow rate.

Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory.

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Modified Patterson temperature jump condition considering viscous heat generation

Recently, new type of the Smoluchowski temperature jump condition has been proposed by considering viscous heat generation in rarefied gas flow simulations, and improved prediction of the surface gas temperature in comparing with DSMC data. Two forms of the Patterson jump condition have been revisited for rarefied gas flow simulations in the literature. In this paper, the modified Patterson temperature jump condition is proposed by considering viscous heat generation. We numerically investigate the effect of the slip velocity, the wall temperature and viscous heat generation on the surface gas temperature for external high-speed rarefied gas flow simulations by using various forms of the Patterson jump condition. Three configurations such as the sharp leading-edge flat plate, the sharp 25–55-deg biconic, and micro-airfoil NACA0012 are selected for simulations with Mach number from 2 to 15.6, and Knudsen number ranging from 0.04 to 0.26. The simulation results obtained show that the modified Patterson jump condition gives good agreement with the DSMC data, and predict temperatures better than those of new type of the Smoluchowski jump condition for all cases considered.

New temperature jump boundary condition in high-speed rarefied gas flow simulations

The effect of the sliding friction has been important in calculating the heat flux of gas flow from the surface since there is some slip over the surface. There has not been any the temperature jump condition including the sliding friction part so far. In this paper, we will propose a new temperature jump condition that includes the sliding friction. Our new temperature jump condition will be evaluated for NACA0012 micro-airfoil in high-speed rarefied gas flow simulations using the CFD method, which solves the Navier-Stokes equations within the OpenFOAM framework with working gas as air. The airfoil case is simulated with various Knudsen numbers from 0.026 to 0.26, and the angles-of-attack (AOAs) from 0-deg to 20-deg. The surface gas temperatures predicting by our new temperature jump condition give good agreements with the DSMC data, especially the NACA0012 micro-airfoil cases with the high Knudsen numbers, Kn = 0.1, and Kn = 0.26 with AOA = 20-deg. for the lower surface.

Effect of the sliding friction on heat transfer in high-speed rarefied gas flow simulations in CFD

Different nature of the heat transfer computation between the CFD and DSMC methods in rarefied gas flow simulations leads to the difference of their heat transfer results. This paper will revisit a formulation of heat transfer in CFD that includes the sliding friction part for a planar surface. The sliding friction was first introduced by Maslen, and used to be omitted in calculating the CFD heat transfer. This formulation will be extended to compute the heat transfer over the curved surfaces such as NACA0012 micro-airfoil and the sharp 25–55-deg. biconic. Two these configurations 1) NACA0012 micro-airfoil (M = 2) with various Kn = 0.026, 0.1 and 0.26 with the angles-of-attack from 0-deg. to 20-deg., and a sharp 25–55-deg. biconic (M = 15.6) are simulated by the CFD method, which solves the Navier-Stokes equations within the OpenFOAM framework. Moreover, the flat plat cases with various wall temperatures (Tw = 77 K and 294 K) are also undertaken to investigate the significance of the sliding friction on the heat transfer. The CFD simulation results of the heat transfer involving the sliding friction give good agreements with the DSMC data, especially the NACA0012 micro-airfoil cases with high Knudsen numbers Kn = 0.1 and 0.26.

A new form of the second-order temperature jump boundary condition for the low-speed nanoscale and hypersonic rarefied gas flow simulations

The accuracy of numerical simulations of rarefied gas flows, in particular the Navier-Stokes-Fourier (N–S–F) equations, depends on the employed surface boundary conditions. In the literature, the combination of the second-order slip/jump conditions has primarily been used for either the Burnett or the BGK Burnett equations for hypersonic gas flows. In this work, we suggest the second-order temperature jump condition in a new form. The second-order slip/jump conditions are implemented in the framework of OpenFOAM to employ with the N–S–F equations for low-speed nanoscale and hypersonic rarefied gas flows. We investigate both the first and second-order slip/jump boundary conditions for low speed rarefied gas flow in the pressure-driven backward facing step nanochannel as well as hypersonic gas flows over the flat plate and past a circular cylinder in cross-flow. Simulation results show that the combination of the second-order slip/jump (in new form) conditions predicts better surface properties than those of the first-order slip/jump conditions for all cases studied by comparing the Burnett and DSMC data. Especially, the N–S–F simulation results of the second-order slip/jump (in new form) conditions of the cylinder case can capture the Burnett data at Kn = 0.1, while those of the first-order conditions do not.

Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case

This paper is a continuation of our work on the development of multiscale numerical scheme from low-speed isothermal flow to compressible flows at high Mach numbers. In our earlier work [Z. L. Guo et al., Phys. Rev. E 88, 033305 (2013)], a discrete unified gas kinetic scheme (DUGKS) was developed for low-speed flows in which the Mach number is small so that the flow is nearly incompressible. In the current work, we extend the scheme to compressible flows with the inclusion of thermal effect and shock discontinuity based on the gas kinetic Shakhov model. This method is an explicit finite-volume scheme with the coupling of particle transport and collision in the flux evaluation at a cell interface. As a result, the time step of the method is not limited by the particle collision time. With the variation of the ratio between the time step and particle collision time, the scheme is an asymptotic preserving (AP) method, where both the Chapman-Enskog expansion for the Navier-Stokes solution in the continuum regime and the free transport mechanism in the rarefied limit can be precisely recovered with a second-order accuracy in both space and time. The DUGKS is an idealized multiscale method for all Knudsen number flow simulations. A number of numerical tests, including the shock structure problem, the Sod tube problem in a whole range of degree of rarefaction, and the two-dimensional Riemann problem in both continuum and rarefied regimes, are performed to validate the scheme. Comparisons with the results of direct simulation Monte Carlo (DSMC) and other benchmark data demonstrate that the DUGKS is a reliable and efficient method for multiscale flow problems.

On the degree of boundary slip over nonplanar surfaces

The breakdown of the no-slip boundary condition at a fluid–solid interface has been recognized in micro/nanofluidics for many years. However, the relationship between the curvature of the surface and the degree of boundary slip has not been understood sufficiently well. The present study reveals that the degree of slip depends effectively on the surface curvature, which is having an opposing effect over rotating concave and convex surfaces. The results show that as surface curvature increases, the boundary slip becomes negligible over a concave surface, while it becomes increasingly important over a convex surface. In addition, boundary slip formulae are proposed that can accurately predict the boundary slips over convex and concave surfaces. These formulae are found to be in very good agreement with DSMC data for a range of accommodation coefficients and boundary curvatures. The present study then explains the mechanism of the intriguing phenomenon of velocity inversion which has, until the present study, often been mistakenly attributed solely to the effects of boundary curvature.

Langmuir–Maxwell and Langmuir–Smoluchowski boundary conditions for thermal gas flow simulations in hypersonic aerodynamics

The simulation of nonequilibrium thermal gas flow is important for the aerothermodynamic design of re-entry and other high-altitude vehicles. In computational fluid dynamics, the accuracy of the solution to the Navier–Stokes–Fourier (N–S–F) equations depends on the accuracy of the surface boundary conditions. We propose new boundary conditions (called the Langmuir–Maxwell and the Langmuir–Smoluchowski conditions), for use with the N–S–F equations, which combine the Langmuir surface adsorption isotherm with the Maxwell/Smoluchowski slip/jump conditions in order to capture some of the physical processes involved in gas flow over a surface. These new conditions are validated for flat plate flow, circular cylinder in cross-flow, and the flow over a sharp wedge for Mach numbers ranging from 6 to 24, and for argon and nitrogen as the working gases. Our simulation results show that the new boundary conditions give better predictions for the surface pressures, compared with published experimental and DSMC data.

Simulation of gas flows in micro/nano systems using the Burnett equations

The gaseous flow characteristics in micro/nano systems are studied using the Burnett equations. The Burnett equations are the second-order approximation of the Chapman–Enskog solution to the Boltzmann equation. This set of equation is appropriate to describe the rarefied gas flows. Velocity slip and temperature jump boundary conditions are applied on solid surface. Several techniques, including the usage of relaxation method on slip values and Burnett terms, are introduced to increase the stability of the Burnett equations. Hence, convergent results at Knudsen number up to 0.5 are achieved for the first time. The results of the Burnett equations are first verified using the corresponding experimental and the DSMC data. The Navier-Stokes and the Burnett equations give almost the same result when the flow is in slip regime (Kn<0.0l). But when the flow is in transition regime (0.1<Kn<0.5), the results of Navier-Stokes equations deviate from DSMC data while those of Burnett equations still agree very well. The Burnett equations are then used to study the gaseous flows in micro/nano systems. The pressure driven plane Poiseuille and the backward-facing step flows, the flows through micro filters are then studied. Different inlet to outlet pressure ratios, size effects, shape effects and boundary conditions are analyzed. The competitive relations of rarefied and compressible effects in micro gas flows are discussed.

Velocity Inversion In Cylindrical Couette Gas Flows

We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. In comparison to a planar wall, we find that the mean free path for a convex surface is higher at the wall and exhibits a sharper gradient within the Knudsen layer. In contrast, a concave wall exhibits a lower mean free path near the surface and the gradients in the Knudsen layer are shallower. The Navier-Stokes constitutive relations and velocity-slip boundary conditions are modified based on a power-law scaling to describe the mean free path, in accordance with the kinetic theory of gases, i.e. transport properties can be described in terms of the mean free path.Velocity profiles for isothermal cylindrical Couette flow are obtained using the power-law model. We demonstrate that our model is more accurate than the classical slip solution, especially in the transition regime, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomena of velocity-inversion. Our results are compared with conventional hydrodynamic models and direct simulation Monte Carlo data. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations.

Computational study of a rarefied gas flow through a long circular pipe into vacuum

The paper presents an analysis of the non-linear rarefied gas flow through a circular pipe into vacuum. The main attention is given to the case of the large length to radius ratio. The problem is studied on the basis of the numerical solution of the S-model kinetic equation. Results are compared with the available DSMC data for short tubes as well as with the asymptotic solution corresponding to the infinitely long pipe.

Variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states and its bridge function study

This paper is a research on the variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states by using theoretical analysis and numerical simulation methods. The newly developed near space hypersonic cruise vehicles have sharp noses and wingtips, which desires exact and relatively simple methods to estimate the stagnation point heat flux. With the decrease of the curvature radius of the leading edge, the flow becomes rarefied gradually, and viscous interaction effects and rarefied gas effects come forth successively, which results in that the classical Fay-Riddell equation under continuum hypothesis will become invalid and the variation of stagnation point heat flux is characterized by a new trend. The heat flux approaches the free molecular flow limit instead of an infinite value when the curvature radius of the leading edge tends to 0. The physical mechanism behind this phenomenon remains in need of theoretical study. Firstly, due to the fact that the whole flow regime can be described by Boltzmann equation, the continuum and rarefied flow are analyzed under a uniform framework. A relationship is established between the molecular collision insufficiency in rarefied flow and the failure of Fourier’s heat conduction law along with the increasing significance of the nonlinear heat flux. Then based on an inspiration drew from Burnett approximation, control factors are grasped and a specific heat flux expression containing the nonlinear term is designed in the stagnation region of hypersonic leading edge. Together with flow pattern analysis, the ratio of nonlinear to linear heat flux Wr is theoretically obtained as a parameter which reflects the influence of nonlinear factors, i.e. a criterion to classify the hypersonic rarefied flows. Ultimately, based on the characteristic parameter Wr, a bridge function with physical background is constructed, which predicts comparative reasonable results in coincidence well with DSMC and experimental data in the whole flow regime.