Simulation Of Rarefied Gas Flows Research Articles

Wasserstein-penalized Entropy closure: A use case for stochastic particle methods

We introduce a framework for generating samples of a distribution given a finite number of its moments, targeted at particle-based solutions of kinetic equations and rarefied gas flow simulations. Our model, referred to as the Wasserstein-Entropy distribution (WE), couples a physically-motivated Wasserstein penalty term to the traditional maximum-entropy distribution (MED) function, which serves to regularize the latter. The penalty term becomes negligible near the local equilibrium, reducing the proposed model to the MED, known to reproduce the hydrodynamic limit. However, in contrast to the standard MED, the proposed WE closure can cover the entire physically realizable moment space, including the so-called Junk line. We also propose an efficient Monte Carlo algorithm for generating samples of the unknown distribution which is expected to outperform traditional non-linear optimization approaches used to solve the MED problem. Numerical tests demonstrate that given moments up to the heat flux—that is, information equivalent to that contained in the Chapman-Enskog distribution—the proposed methodology provides a reliable closure in the collision-dominated and early transition regimes. Applications to greater rarefaction demand information from higher-order moments, which can be incorporated within the proposed closure.

Thermal rarefied gas flow simulations with moving boundaries based on discrete unified gas kinetic scheme and immersed boundary method

Rarefied gas flows in complex geometries with heat transfer and moving boundaries have received remarkable attention due to the prevailing of micro/nano science and engineering applications. This study presents an immersed boundary (IB)-based discrete unified gas kinetic scheme (DUGKS) for the simulation of complex moving boundary thermal rarefied flows. The fluid and temperature fields are solved according to the DUGKS with Shakhov collision model (SDUGKS), which is capable of modeling non-unit Prandtl number problems in all Knudsen regimes. The IB approach, featured by its superior performance to incorporate complicated boundary constraints on Cartesian grids, is applied here to handling the interaction between the micro gas and curved boundaries. Specifically, for the IB adopted in this study, we use the least square method for the reconstruction of the distribution functions near the boundary. The accuracy of the IB-SDUGKS is explored comprehensively in six benchmark problems, including the micro lid-driven cavity flow, thermal equilibrium gas in a circular domain, thermal transpiration in an enclosure, moving shuttle in a rarefied gas, movement of a piston with pressure differences and particle free motion in a lid-driven cavity. The present results agree well with the corresponding analytical and direct simulation Monte Carlo (DSMC) solutions, illustrating the versatility of the present IB-SDUGKS complex thermal rarefied flows.

Further acceleration of multiscale simulation of rarefied gas flow via a generalized boundary treatment

The recently-developed general synthetic iterative scheme (GSIS) is efficient in simulating multiscale rarefied gas flows due to the coupling of mesoscopic kinetic equation and macroscopic synthetic equation: for linearized Poiseuille flow where the boundary flux is fixed at each iterative step, the steady-state solutions are found within dozens of iterations in solving the gas kinetic equations, while for general nonlinear flows the iteration number is increased by about one order of magnitude, caused by the incompatible treatment of the boundary flux for the macroscopic synthetic equation. In this paper, we propose a generalized boundary treatment (GBT) to further accelerate the convergence of GSIS. The main idea is, the truncated velocity distribution function at the boundary, similar to that used in the Grad 13-moment equation, is reconstructed by the macroscopic conserved quantities from the synthetic equation, as well as the high-order correction of non-equilibrium stress and heat flux from the kinetic equation; therefore, in each inner iteration solving the synthetic equation, the explicit constitutive relations facilitate real-time updates of the macroscopic boundary flux, driving faster information exchange in the flow field, and consequently achieving faster convergence. Moreover, the high-order correction derived from the kinetic equation can compensate the approximation by the truncation and ensure the boundary accuracy. The one-dimensional Fourier flow, two-dimensional hypersonic flow around a cylinder, three-dimensional pressure-driven pipe flow and the flow around the hypersonic technology vehicle are simulated. The accuracy of GSIS-GBT is validated by the direct simulation Monte Carlo method, the previous versions of GSIS, and the unified gas-kinetic wave-particle method. For the efficiency, in the near-continuum flow regime and slip regime, GSIS-GBT can be faster than the conventional iteration scheme in the discrete velocity method and the previous versions of GSIS by two- and one-order of magnitude, respectively.

Simulation of rarefied gas flows using physics-informed neural network combined with discrete velocity method

The linearized Bhatnagar–Gross–Krook equation is widely used to describe low-speed rarefied gas flows and can be solved numerically using deterministic methods such as the discrete velocity method (DVM). With the help of physics-informed neural networks (PINNs), which have experienced rapid development in recent years, we develop a new surrogate model that combines the PINNs and DVM methods (PINN–DVM) to simulate rarefied gas flows. In the proposed PINN–DVM model, the linearized Bhatnagar–Gross–Krook equation is directly encoded into the residual of an artificial neural network. A new loss function for the boundary condition based on the impermeable diffusion model is proposed to accurately capture the velocity slip and temperature jump at the boundary. PINN–DVM overcomes the limitations of conventional numerical methods, such as complex mesh generation and derivative calculations. Furthermore, the proposed model is enhanced by a self-adaptive strategy based on the neural tangent kernel, which enables the loss weight for each component of the loss function to be adjusted adaptively. Four representative numerical cases of Couette flow, Fourier heat transfer flow, triangular duct flow, and cavity flow demonstrate the superiority of the proposed PINN–DVM model in solving rarefied gas flows compared with original PINNs.

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.

Scalable Simulation of Pressure Gradient-Driven Transport of Rarefied Gases in Complex Permeable Media Using Lattice Boltzmann Method

Accurate representations of slip and transitional flow regimes present a challenge in the simulation of rarefied gas flow in confined systems with complex geometries. In these regimes, continuum-based formulations may not capture the physics correctly. This work considers a regularized multi-relaxation time lattice Boltzmann (LB) method with mixed Maxwellian diffusive and halfway bounce-back wall boundary treatments to capture flow at high Kn. The simulation results are validated against atomistic simulation results from the literature. We examine the convergence behavior of LB for confined systems as a function of inlet and outlet treatments, complexity of the geometry, and magnitude of pressure gradient and show that convergence is sensitive to all three. The inlet and outlet boundary treatments considered in this work include periodic, pressure, and a generalized periodic boundary condition. Compared to periodic and pressure treatments, simulations of complex domains using a generalized boundary treatment conserve mass but require more iterations to converge. Convergence behavior in complex domains improves at higher magnitudes of pressure gradient across the computational domain, and lowering the porosity deteriorates the convergence behavior for complex domains.

SPARTACUS: An open-source unified stochastic particle solver for the simulation of multiscale nonequilibrium gas flows

Recently, a unified stochastic particle (USP) method [14] has been developed for the simulation of multiscale nonequilibrium gas flows. Compared with the conventional stochastic particle methods, such as the direct simulation Monte Carlo (DSMC) method, USP can be applied with much larger time steps and cell sizes as it couples the effects of particle movements and collisions. To extend the applicability of USP to complex nonequilibrium gas flows, we develop a USP solver referred to as SPARTACUS within the framework of SPARTA, which is a widely used software based on the DSMC method for the simulation of rarefied gas flows. Inheriting from SPARTA, SPARTACUS is parallelized with the message passing interface (MPI), and it is open-source under the GNU General Public License (GPL) and released in an online repository with essential documentation, tutorial examples, and benchmark cases. To evaluate the performance of SPARTACUS, the test cases cover from one to three dimensional flows with a wide range of the Knudsen number and Mach number, including Couette flow, lid-driven cavity flow, hypersonic flow past a cylinder, and supersonic flow around a blunt body. The results obtained by SPARTACUS in the rarefied and continuum regimes are in good agreement with DSMC and computational fluid dynamics (CFD) results, respectively. Moreover, SPARTACUS has significant computational efficiency advantage over traditional DSMC solvers, especially for three-dimensional nonequilibrium gas flows. In addition, SPARTACUS shows comparable parallel efficiency with SPARTA, so it is promising to be applied with high performance computing for the simulation of complex multiscale gas flows encountered in engineering problems. Program summaryProgram Title: SPARTACUSCPC Library link to program files:https://doi.org/10.17632/pyb343w7zv.1Developer's repository link:https://github.com/KKFeng/spartacusLicensing provisions: GNU General Public License version 2Programming language: C++External routines/libraries: SPARTA (http://sparta.sandia.gov/)Nature of problem: Multiscale simulation of nonequilibrium gas flows using unified stochastic particle method.Solution method: Unified Stochastic Particle (USP) method.

Multiphysics Simulations of Rarefied Gas Flows Using DSMC with Quasi-Nanbu Scheme

Multiphysics Simulations of Rarefied Gas Flows Using DSMC with Quasi-Nanbu Scheme

Efficient simulation of rarefied gas flow past a particle: A boundary element method for the linearized G13 equations

We develop a novel boundary integral formulation for the steady linearized form of Grad's 13-moment (G13) equations applied to a uniform flow of rarefied gas past solid objects at low Mach numbers. Changing variables leads to a system of boundary integral equations that combines integral equations from Stokes flow and potential theory. The strong coupling between the stress deviator and heat flux featured by the G13 equations demands adding a boundary integral equation for the pressure. We specialize the integral equations for an axisymmetric flow with no swirl and derive the axisymmetric fundamental solutions for the pressure equation, seemingly absent in the Stokes-flow literature. Using the boundary element method to achieve a numerical solution, we apply this formulation to streaming flow of rarefied gas past prolate or oblate spheroids with their axis of symmetry parallel to the free stream, considering various aspect ratios and Knudsen numbers—the ratio of the molecules' mean free path to the macroscopic length scale. After validating the method, we obtain the surface profiles of the deviations from the unperturbed state of the traction, heat flux, pressure, temperature, and slip velocity, as well as the drag on the spheroid, observing convergence with the number of elements. Rarefaction phenomena, such as temperature jump and polarization, Knudsen effects in the drag, and velocity slippage, are predicted. This method opens a new path for investigating other gas non-equilibrium phenomena that can be modeled by the same set of equations, such as thermophoresis, and has application in nano- and microfluidics.

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.

Gas-surface interaction in rarefied gas flows through long capillaries via the linearized Boltzmann equation with various boundary conditions

Reliable gas–surface interaction modeling is of major importance in consistent simulation of rarefied gas flows with applications in vacuum technology and micro-electromechanical systems. The effectiveness of the most prominent kinetic boundary conditions, namely the ones by Maxwell, Cercignani-Lampis and Epstein, is elaborated by numerically solving the classical plane Poiseuille and thermal creep flows via the linearized Boltzmann equation, subject to these boundary conditions. The flow rates are provided in a wide range of the gas rarefaction for many values of the accommodation coefficients of each scattering kernel, identifying in parallel, the sensitivity of the flow rates in small variations of the free parameters. A comparison with available experimental data is performed to deduce that the Cercignani-Lampis and Epstein scattering kernels, compared to the Maxwell one, are more suitable to characterize coupled momentum and energy accommodation modes. With these boundary conditions, the accommodation coefficients may be extracted by fitting computations with measurements, in the Poiseuille flow in a wide range of gas rarefaction, but in the thermal creep flow, experimental data under rarefied conditions, as high as possible, are recommended. The present work may be useful in determining the measurement conditions and uncertainties, allowing accurate extraction of the accommodation coefficients.

Molecular beam scattering of a diatomic molecule from a solid surface in case of strong rotational non-equilibrium

In this work, a numerical study on molecular beam scattering of a nitrogen molecule from a graphite surface has been performed. The study was carried out using the molecular dynamics method. The focus of the study is mainly placed on investigating the scattering dynamics in the case of strong rotational non-equilibrium, defined here as a state in which rotational temperature of a molecule strongly deviates from the room temperature. To that end, the incident beam velocity and initial rotational energy of nitrogen molecules have been varied greatly in order to capture a broad range of possible initial states. The obtained results provide valuable insight into the nature of energy transfer occurring during collisions and help to quantify the intensity of rotational–translational coupling between inner kinetic modes in gas–surface collisions. Consequently, the collected data can potentially be used for more accurate characterization of the respective phenomena and improve the quality of boundary models used in rarefied gas flow simulations.

Computational methods for pore-scale simulation of rarefied gas flow

Direct simulation at the pore-scale is crucial to unravel rarefaction effect on gas transport in tight porous media. To satisfy the dual demands on modeling accuracy and computation effort, an appropriate method must be chosen. This work, therefore, evaluates four numerical methods for pore-scale rarefied gas flows in a two-dimensional (2D) model porous media over a wide range of rarefaction. These methods are the incompressible Navier-Stokes equation with the first-order velocity-slip boundary condition, and three gas-kinetic solvers i.e. a finite-difference (FD) iterative solver for the linearized Bhatnagar, Gross and Crook (BGK) model kinetic equation, a finite-volume (FV) solver for the non-linear Shakhov model, and an open-source direct simulation Monte Carlos (DSMC) solver. The benchmark cases cover the characteristic Knudsen number ranging from 0.0231 to 4.62 while the characteristic Reynolds numbers are kept to be less than 1.0. All the solvers are developed in OpenFOAM, except the FD solver, allowing us to investigate the effect of local grid refinements using the automatic Cartesian grid generator in OpenFOAM. The flow fields and the apparent permeabilities predicted by all the solvers have been compared in detail. Besides, the computational time of these solvers is measured and analyzed to demonstrate the relative cost of the three kinetic solvers. It is found that the FD solver is the most efficient one and gives accurate results over the whole range of Knudsen number. Finally, this study also evaluates the feasibility of the recently developed discrete unified gas-kinetic scheme (DUGKS), the algorithm in the aforementioned non-linear Shakhov model equation solver, for pore-scale rarefied gas flow. It is found the results predicted by this solver agree well with the other two kinetic solvers.

Monte Carlo simulation of rarefied gas flow between parallel flat plates using simple scattering models

Monte Carlo simulation of rarefied gas flow between parallel flat plates using simple scattering models

General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows

The general synthetic iterative scheme (GSIS) is extended to find the steady-state solution of the nonlinear gas kinetic equation, resolving the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredient of GSIS is the tight coupling of gas kinetic and macroscopic synthetic equations, where the constitutive relations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated from the velocity distribution function; however, their constructions are simpler than our previous work (Su et al., 2020 [28]) for linearized gas kinetic equations. On the other hand, solutions of macroscopic synthetic equations are used to accelerate the evolution of gas kinetic equation at the next iteration step. A rigorous linear Fourier stability analysis of the present schemes in periodic system shows that the error decay rate of GSIS can be smaller than 0.5, which means that the deviation to steady-state solution can be reduced by 3 orders of magnitude in 10 iterations. Other important advantages of the GSIS are: (i) it does not rely on the specific form of Boltzmann collision operator, and (ii) it can be solved by sophisticated techniques in computational fluid dynamics, making it amenable to large scale engineering applications. In this paper, the efficiency and accuracy of GSIS are demonstrated by a number of canonical test cases in rarefied gas dynamics, covering different flow regimes.

On the conservative property of particle-based Fokker–Planck method for rarefied gas flows

The Fokker–Planck-type approximation of the full Boltzmann equation has aroused intense research interest due to its potential for the stochastic particle simulation of rarefied gas flows. The ellipsoidal statistical Fokker–Planck (ES-FP) model treats the evolution of molecular velocity as a continuous stochastic process, and it satisfies the basic requirements for a proper gas-kinetic model including the H-theorem and an adjustable Prandtl number. The ES-FP model can be numerically implemented with computational particles in a Monte Carlo manner. Two different particle ES-FP schemes are presented. The first scheme utilizes the exact stochastic integral solution of the Langevin equations corresponding to the ES-FP equation and couples free-molecular moves and intermolecular collisions. The second scheme is designed to enforce the conservation of momentum and energy during the numerical simulation based on the decoupled algorithm and the analysis of the specific conditions for the conservative property. Numerical tests are conducted to demonstrate the performances of different schemes. In the simulation of a homogeneous gas system, the ES-FP scheme without enforcement of conservation leads to unphysical variation in the momentum and loss in energy, whereas the conservative ES-FP scheme strictly maintains the momentum and energy of the system. For the Mach 6 flows over the leading edge of a flat plate and over a round-nosed blunt body, the non-conservative ES-FP scheme underestimates the shock angle and the shock standoff distance, makes inaccurate predictions of aerodynamic force and heating, and produces low-temperature anomalies in front of the shock waves. In comparison with the results given by the direct simulation Monte Carlo method, the results of the conservative ES-FP simulations show satisfactory accuracy for the flow fields as well as the distributions of pressure, friction, and heat flux on the wall surfaces.

DSMC simulation of rarefied gas flow over a 2D backward-facing step in the transitional flow regime: Effect of Mach number and wall temperature

Rarefied gas flow over a backward-facing step (BFS) is often encountered in separating flows prevalent in aerodynamic flows, engine flows, condensers, space vehicles, heat transfer systems, and microflows. Direct Simulation Monte Carlo (DSMC) is a powerful tool to investigate such flows. The purpose of this research is to assess the impact of Mach number and wall temperature on the flow and surface properties in the transitional flow regime. The Mach numbers considered are 5, 10, 25, 30, and the ratio of the temperature of the wall to that of freestream considered are 1, 2, 4, 8. The Reynolds number for the cases studied is 8.6, 17.2, 43, and 51.7, respectively. Typically the flow properties near the wall are found to increase with both Mach number and wall temperature owing to compressibility and viscous dissipation effects. The variation in flow properties is more sensitive to Mach number than the wall temperature. The surface properties are found to decrease with Mach number and increase with wall temperature. Moreover, in the wake of the step, the vortex’s recirculation length is reasonably independent of both free stream Mach number and wall temperature, whereas it decreases with Knudsen number.

Variance-reduction kinetic simulation of low-speed rarefied gas flow through long microchannels of annular cross sections

In micro/nano-devices, the low-speed transport of mass, momentum, and energy through long-ducts is frequently encountered, thereby necessitating scientific investigations. Here, long-ducts of various annular cross sections conducting low-speed gas flows under the influence of a small pressure gradient are considered, in order to understand how the mass flow rate is affected by rarefaction, variations in the radius ratio, and eccentricity of annular geometries. The Boltzmann model equation is treated by a low-variance formulation and simulated by a stochastic kinetic particle-based approach, which addresses the deviation of the molecular distribution function from equilibrium to reduce computational cost significantly. An efficient parallel solver has also been developed and utilized in this research, which is validated against the reported results in the literature. The efficient kinetic particle treatment provides a powerful simulation tool to reveal multi-scale flow physics, which is essential to develop and optimize micro/nano-fluidic devices.

Comparison of existing and extended boundary conditions for the simulation of rarefied gas flows using the Lattice Boltzmann method

Accurate imposition of boundary conditions (BCs) is of critical importance in fluid flow computation. This is especially true for the Lattice Boltzmann method (LBM), where BC imposition is done through operations on populations rather than directly on macroscopic variables. While the regular Cartesian structure of the lattices is an advantage for flow simulation through complex geometries such as porous media, imposition of correct BCs remains a topic of investigation for rarefied flows, where slip BCs need to be imposed. In this work, current kinetic BCs from the literature are reviewed for rarefied flows and an extended version of a technique that combines bounce-back and diffusive reflection (DBB BC) is proposed to solve such flows that exhibit effective viscosity gradients. The extended DBB BC is completely local and addresses ambiguities as regards to the definition of boundary populations in complex geometries. Numerical tests of a rarefied flow through a slit were performed, confirming the intrinsic second-order convergence of the proposed extended DBB BC. It settles a long-standing debate regarding the convergence of BCs in rarefied flows. Good agreement was also found with existing numerical schemes and experimental data.

A velocity space hybridization-based Boltzmann equation solver

In the present work, a new method for simulation of rarefied gas flows is proposed, a velocity-space hybrid of both a DSMC representation of particles and a discrete velocity quasi-particle representation of the distribution function. The hybridization scheme is discussed in detail, and is numerically verified for two test-cases: the BKW relaxation problem and a stationary Maxwellian distribution. It is demonstrated that such a velocity-space hybridization can provide computational benefits when compared to a pure discrete velocity method or pure DSMC approach, while retaining some of the more attractive properties of discrete velocity methods. Further possible improvements to the velocity-space hybrid approach are discussed.