Rarefied Gas Research Articles

A design optimization method for rarefied and continuum gas flows

A design optimization method applicable to both rarefied and continuum gas flows with good efficiency is proposed in the framework of gas-kinetic theory. The method follows the methodology of topology optimization where the areas of gas and solid are marked by the material density, based on which a fictitious porosity model is used to reflect the effect of the solid on the gas and mimic the diffuse boundary condition on the gas-solid interface. The formula of this fictitious porosity model is modified to make the model work well from the free-molecular flow regime to the continuum flow regime. To find the optimized material density, a gradient-based optimizer is adopted and the design sensitivity is obtained by the continuous adjoint method. To solve the primal kinetic equation and the corresponding adjoint equation, efficient multiscale numerical schemes are constructed. Several airfoil optimization problems are solved to demonstrate the performance of the present design optimization method in wide ranges of flow speed and gas rarefaction. It is found that the thickness of the optimized airfoil first increases and then decreases as the Knudsen number increases, and on the supersonic condition the optimized airfoil has quite different shapes of leading edge for different degrees of gas rarefaction.

Droplet dynamics in a two-dimensional rarefied gas under Kawasaki dynamics

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature β>0 in a large finite box Λβ⊂Z2 whose volume depends on β. Each pair of neighboring particles has a negative binding energy−U<0, while each particle has a positive activation energyΔ>0. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime Δ∈(U,2U) and in the limit as β→∞, how and when the system nucleates, i.e., grows a supercritical droplet somewhere in Λβ. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyze how subcritical droplets form and dissolve on multiple space–time scales when the volume is moderately large, namely, |Λβ|=eΘβ with Δ<Θ<2Δ−U. In the third paper we consider the setting where the volume is very large, namely, |Λβ|=eΘβ with Δ<Θ<Γ−(2Δ−U), where Γ is the energy of the critical droplet in the local model, i.e., when Λβ has a fixed volume not depending on β and particles can be created and annihilated at the boundary, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles move around and interact but are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach: the tube of typical trajectories leading to nucleation is described via a series of events, whose complements have negligible probability, on which the evolution of the gas can be captured by a coarse-grained Markov chain on a space of droplets, which we refer to as droplet dynamics.

Two-level parallel load balancing strategy for accelerating DSMC simulations in near-continuum gases

The Direct Simulation Monte Carlo (DSMC) algorithm is widely employed for simulating rarefied gas flows and is increasingly applied in near-continuum regimes for research and engineering purposes. However, its computational demands, notably load imbalance and extended simulation time, hinder widespread adoption. Addressing these challenges, this paper introduces the Two-Level parallel load balancing strategy. This novel approach combines thread-level and multi-process parallelism to enhance load balancing and reduce simulation time. Key features include a thread-level load-decoupling strategy implemented via OpenMP and a multi-process load balancing mechanism employing distributed memory via MPI. Building upon our previous [Formula: see text] [L. Li, W. Ren and B. Zhang, J. Aeronaut. Astronaut. Aviat. Ser. A 46, 88 (2014)] approach, the load balancing mechanism utilizes Stop At Risk (SAR) criteria for repartitioning with METIS. Additionally, a specialized data transmission mechanism utilizing MPI nonblocking communication minimizes global communication between processes. Validation and evaluation are performed using four hypersonic flow cases around a cylinder and sphere, demonstrating significant improvements. Notably, the proposed strategy achieves [Formula: see text] enhancement over the [Formula: see text] strategy under 512 CPU cores compared to 16 CPU cores, and reduces between-process communication time with [Formula: see text]. These advancements contribute to enhancing the effectiveness of the DSMC algorithm in near-continuum aerodynamic simulations.

Measurements of diffusion coefficient and kinetic diameter of acetone vapor via molecular tagging

The Molecular Tagging (MT) technique is a promising methodology for locally measuring velocity and temperature fields in rarefied gas flows. Recently, Molecular Tagging Velocimetry (MTV) has been successfully applied to gas flows in mini-channels in the continuum regime at high pressure and early slip-flow regime at lower pressure. As the operating pressure decreases, diffusion effects become more pronounced, and in MTV, they hinder the extraction of the correct velocity profile by simply dividing the displacement profile of the tagged molecular line by time of flight. To address this issue, a reconstruction method that considers Taylor dispersion was previously developed to extract the velocity profile, considering the diffusion effects of the tracer molecules within the carrier gas. This reconstruction method successfully extracted the correct velocity profile in the continuum flow regime. However, the method still faces challenges in the slip-flow regime. Since there is currently no consensus in the literature regarding the kinetic diameter value of acetone vapor, the diffusion coefficient estimation is uncertain especially at low pressures. This is why, in this study, we propose an original optical method to measure the diffusion coefficient of acetone vapor. This is achieved by linking the temporal evolution of the spatial photoluminescence distribution of acetone vapor to the diffusion coefficient via the Chapman-Enskog theory. Our research provides measurements of these parameters for a wide range of pressures (0.5–10 kPa) at ambient temperature.

A discrete unified gas kinetic scheme with sparse velocity grid for rarefied gas flows

In this paper, a discrete unified gas kinetic scheme (DUGKS) with a sparse grid method applied in velocity space (DUGKS-SG) is proposed for simulating rarefied gas flows. The DUGKS-SG decomposes the computationally demanding problem into smaller and independent subproblems, thereby reducing the computational costs and exhibiting good parallelism. Several numerical tests, including the two-dimensional Riemann problem and the lid-driven microcavity flow, have been conducted to validate the performance of the DUGKS-SG. Comparisons with the original DUGKS and the Direct Simulation Monte Carlo (DSMC) method demonstrate that DUGKS-SG can provide satisfactory results with improved efficiency. Specifically, a maximum speedup of 9.486 for a 2D case with 7 CPU cores and 13.035 for a 3D case with 8 CPU cores can be achieved. These results suggest that the proposed DUGKS-SG can serve as an efficient numerical method for rarefied gas flow simulations.

Windage loss characterisation for flywheel energy storage system: Model and experimental validation

In this paper, a windage loss characterisation strategy for Flywheel Energy Storage Systems (FESS) is presented. An effective windage loss modelling in FESS is essential for feasible and competitive design. Unlike generic aerodynamic loss models, FESS require particular attention to their unique characteristics i.e., vacuum, small airgaps, high angular speed, and presence of low friction rotor supports.The proposed model is based on several analytical and semi-empirical windage loss solutions for cylindrical and planar surface interactions, harmonised to manage the transition between laminar and turbulent flows. Also, the model is enriched by introducing corrections for rarefied gasses, using kinetic gas theory formulation. Therefore, it is possible the reduce the windage overestimation occurring with Navier–Stokes equation solutions for laminar flow. The model is compared to case studies from the literature featuring different boundary and operating conditions, to check consistency of all the harmonised models. A dedicated experimental test-rig is developed to validate the corrections to the model for rarefied gas at different vacuum levels. Then, a non-invasive characterisation approach for FESS windage losses in self-discharge phase is proposed and validated on the test-rig.

A Novel ES-BGK Model for Non-polytropic Gases with Internal State Density Independent of the Temperature

A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide (CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found.

Experimental and simulative investigations of burr formation in planing of AISI 1045

The clearance flow in dry-running vacuum pumps is the main loss mechanism. To reduce the clearance mass flow rate in rarefied gas flows, sawtooth structures transversing to the direction of flow can be utilized. However, due to the sawtooth’s structure size, micro-machining is necessary, whereby burr formation is a central challenge. First, the effectiveness of non-idealized sawtooth structures is investigated, demonstrating a high sensibility of the performance regarding the geometry of the tip. Therefore, burr formation in the cutting process must be minimized. For this reason, a 3D finite element (FE) chip formation model capable of predicting the burr formation is developed. An analysis of the burr formation zone showed positive triaxialities; thus, the triaxiality-dependent Johnson–Cook damage model is utilized. To minimize the mesh-induced error, a convergence analysis is conducted, showing no convergence of the maximum burr height. This is caused by the pathological mesh size dependence of local continuum damage models. A comparison of the cutting experiments and simulations revealed a reasonable prediction of cutting forces. In contrast, the passive force is predicted poorly, which is attributed to the underestimation of the ploughing force for non-elastic simulations. The prediction quality regarding the maximum burr height differs for the investigated cutting speeds, which can be explained by a built-up edge and a change in the machine tool compliance. Thereby, an analysis of the burr formation revealed that the burr height is captured by a non-physical remeshing algorithm and that the burr volume might be a more appropriate characteristic.

A second-order particle Fokker-Planck model for rarefied gas flows

The direct simulation Monte Carlo (DSMC) method has become a powerful tool for studying rarefied gas flows. However, for the DSMC method to be effective, the cell size must be smaller than the mean free path, and the time step smaller than the mean collision time. These constraints make it difficult to use the DSMC method in multiscale rarefied gas flows. Over the past decade, the particle Fokker-Planck (FP) method has been studied to address computational cost issues in the near-continuum regime. To capture the main features of the Boltzmann equation, various FP models have been proposed, such as the quadratic entropic FP (Quad-EFP) and the ellipsoidal statistical FP (ESFP). Nevertheless, few studies have clearly demonstrated that the FP method offers a computational advantage over the DSMC method without sacrificing accuracy. This is because conventional particle FP methods have employed first-order accuracy schemes. The present study proposes a unified stochastic particle ESFP (USP-ESFP) model. This model improves the accuracy of shear stress and heat flux predictions. Additionally, a spatial interpolation scheme is introduced to the particle FP method. The numerical test cases include relaxation problem, Couette flows, Poiseuille flows, velocity perturbation, and hypersonic flows around a cylinder. The results show that the USP-ESFP model agrees well with both analytical and DSMC results. Furthermore, the USP-ESFP model is found to be less sensitive to cell size and time step than the DSMC method, resulting in a factor of four speed-up for the considered hypersonic flow around a cylinder.

Efficient parallel solver for rarefied gas flow using GSIS

Recently, the general synthetic iterative scheme (GSIS) has been proposed to find the steady-state solution of the Boltzmann equation in the whole range of gas rarefaction, where its fast-converging and asymptotic-preserving properties lead to the significant reduction of iteration numbers and spatial cells in the near-continuum flow regime. However, the efficiency and accuracy of GSIS have only been demonstrated in two-dimensional problems with small numbers of spatial cells and discrete velocities. Here, a large-scale parallel computing strategy is designed to extend the GSIS to three-dimensional flow problems, including the supersonic flows which are usually difficult to solve by the discrete velocity method. Since the GSIS involves the calculation of the mesoscopic kinetic equation which is defined in six-dimensional phase-space, and the macroscopic high-temperature Navier–Stokes–Fourier equations in three-dimensional physical space, the proper partition of the spatial and velocity spaces, and the allocation of CPU cores to the mesoscopic and macroscopic solvers, are the keys to improving the overall computational efficiency. These factors are systematically tested to achieve optimal performance, up to 100 billion spatial and velocity grids. For hypersonic flows around the Apollo reentry capsule, the X38-like vehicle, and the space station, our parallel solver can obtain the converged solution within one hour.

Improving computational efficiency in DSMC simulations of vacuum gas dynamics with a fixed number of particles per cell

The present study addresses the challenge of enhancing computational efficiency without compromising accuracy in numerical simulations of vacuum gas dynamics using the direct simulation Monte Carlo (DSMC) method. A technique termed ‘fixed particle per cell (FPPC)’ was employed, which enforces a fixed number of simulator particles across all computational cells. The proposed technique eliminates the need for real-time adjustment of particle weights during simulation, reducing calculation time. Using the SPARTA solver, simulations of rarefied gas flow in a micromixer and rarefied supersonic airflow around a cylinder were conducted to validate the proposed technique. Results demonstrate that applying the FPPC technique effectively reduces computational costs while yielding results comparable to conventional DSMC implementations. Additionally, the application of local grid refinement coupled with the FPPC technique was investigated. The results show that integrating local grid refinement with the FPPC technique enables accurate prediction of flow behaviour in regions with significant gradients. These findings highlight the efficacy of the proposed technique in improving the accuracy and efficiency of numerical simulations of complex vacuum gas dynamics at a reduced computational cost.

Sensitivity analysis of the Cercignani - Lampis accommodation coefficients in prototype rarefied gas flow and heat transfer problems via the Monte Carlo method

In rarefied gas dynamics, the Cercignani-Lampis (CL) scattering kernel, containing two accommodation coefficients (ACs), namely the tangential momentum and normal energy ones, is widely employed to characterize gas-surface interaction, particularly in non-isothermal setups, where both momentum and energy may simultaneously be exchanged. Here, a formal and detailed sensitivity analysis of the effect of the CL ACs on the main output quantities of several prototype problems, namely the cylindrical Poiseuille, thermal creep and thermomolecular pressure difference (TPD) flows, as well as the plane Couette flow and heat transfer (Fourier flow), is performed. In each problem, some uncertainties are randomly introduced in the ACs (input parameters) and via a Monte Carlo propagation analysis, the deduced uncertainty of the corresponding main output quantity is computed. The output uncertainties are compared to each other to determine the flow configuration and the gas rarefaction range, where a high sensitivity of the output quantities with respect to the CL ACs is observed. The flow setups and rarefaction regimes with high sensitivities are the most suitable ones for the estimations of the ACs, since larger modeling and experimental errors may be acceptable. In the Poiseuille and Couette flows, the uncertainties of the flow rate and shear stress respectively are several times larger than the input uncertainty in the tangential momentum AC and much smaller than the uncertainty in the normal energy AC in a wide range of gas rarefaction. In the thermal creep flow, the uncertainty of the flow rate depends on the input ones of both ACs, but, in general, it remains smaller than the input uncertainties. A similar behavior with the thermal creep flow is obtained in the TPD flow. On the contrary, in the Fourier flow, the uncertainty of the heat flux may be about the same or even larger than the input ones of both ACs in a wide range of gas rarefaction. It is deduced that in order to characterize the gas-surface interaction via the CL ACs by matching computations with measurements, it is more suitable to combine the Poiseuille (or Couette) and Fourier configurations, rather than, as it is commonly done, the Poiseuille and thermal creep ones. For example, in order to estimate the normal energy AC within an accuracy of 10 %, experimental uncertainties should be less than 4 % in the thermal creep or TPD flows, while may be about 10 % in the Fourier flow.

Effects of geometrical parameters on rarefied gas flows and Knudsen force in the system of triangular-rectangular beams with different temperatures

In the rarefied gas, the gas flows induced by the inhomogeneous temperature fields and the Knudsen force exerting on the immersed structure are potentially applicable to the micro/nano-electromechanical systems (MEMS/NEMS) and aerospace. The system composed of a series of triangle cold beams and rectangle hot beams is proposed, and the effects of the beam geometry, dimensions, and positions for different pressures on rarefied gas flows and Knudsen force are researched based on the direct simulation Monte Carlo (DSMC) method. The results reveal that both the gas flows and Knudsen force are highly sensitive to these factors. Particularly, the applications of triangular cross-section cold beams and large height-to-width ratios are necessary for a larger Knudsen force. Compared to the rectangle cold beam, when the triangle cold beam has an inclination of approximately 20°, the peak value of Knudsen force can increase by 142%. Moreover, Knudsen force monotonically reduces with the distance between the cold and hot beams increasing. However, the effects of the distance between the cold beam and substrate (substrate proximity) on Knudsen force depend on the ambient pressure. In general, an appropriate increase in substrate proximity appears to be a smart way to practically enhance the Knudsen force.

Density dependence for the dielectric permittivity of simple gases and liquids

The present work is devoted to the analysis of polarization effects in atomic systems of the argon type. Special attention is given to the manifestation of the polarizabilities of two particles in the density dependence of the dielectric permittivity of a system and its effective molecular polarizability. It is shown that the effective polarizability of a molecule in the liquid state of a system is mainly caused by deviation of the molecular distribution in the first coordination layer from the spherical one. Due to this, the atomic polarizability inherent to rarefied gas increases practically twice near the triple point. The interconnection between the atomic polarizability and the proper volume of an atom is discussed in detail. It is established that the optimal size of an atom follows from the kinematic shear viscosity measurements.

On the increased interfacial resistance of hydrogen in carbon nanotube arrays and its effect on gas mixture separation

We outline a surface scattering kernel for rarefied gas flows through ideally ordered nanomaterials, such as high aspect ratio carbon nanotubes. The derived model allows for a comparison of the tangential momentum accommodation coefficient, and, hence, the total effective friction, for different species of gases as a function of the particle diameter. This surface kernel is incorporated with a Fokker–Planck model as an approximation to transport of a rarefied gas through ideally ordered carbon nanotubes. The results of this analysis predict that H2 experiences higher friction in such systems in comparison with larger molecules such as CH4. The results are proposed as a potential explanation of the reduced gas transport of hydrogen gas in nanoporous systems.

Saturated phase change threshold of plume flow gas species under low temperature and rarefied environment

Gas species in high altitude plume flow are all exposed to low temperature, thus making possible phase change processes on those other than H2O vapor. In order to obtain saturated threshold of gas species, which is normally abundant in plume flow, this study establishes Gibbs phase calculation method for CO2, CO, H2, and N2. The non-continuum effect caused by rarefied gas environment is quantitatively calculated by ratio between molecular free path scale and characteristic length of investigated object. Validation on Gibbs method is guaranteed by comparison between calculated results and data from American National Standards Institute (ANSI), while non-continuum effect method is verified by revealing essential reason for gas-solid phase change processes resembling droplet condensation phenomenon observed in high altitude environment. According to the results, non-continuum effect only influences gas-solid boundary. Phase change for rarefied gas species can only occur under low temperature. Saturated threshold deflects toward solid phase region below specific molecules number density, thus decreasing gas to solid phase change trend. Both increasing on characteristic length and existence of other gas components contamination decreases non-continuum effect. Variation on non-continuum effect shows biggest influence on CO2 saturated phase change threshold, while shows smallest influence on that of N2.

Numerical modeling of the heat and mass transfer of rarefied gas flows in a double-sided oscillatory lid-driven cavity

In micro/nano research and engineering applications, internal flows of rarefied gas in confined geometries with moving parts are of great interest. The rarefied gaseous flows in an oscillating lid-driven cavity are investigated numerically in this work, taking into account both the parallel and antiparallel motion of the driving walls. The discrete unified gas kinetic scheme (DUGKS) is applied to the evolution of the flow and temperature fields. The Shakhov collision model is incorporated into DUGKS to account for flows of non-unit Prandtl numbers. The validity of the Shakhov-DUGKS is confirmed in both single-sided oscillating and steady lid-driven cavity flows. The effects of gas compressibility, rarefaction, and lid oscillation frequency on the mass and heat transport are then thoroughly examined. The corresponding parameter spaces of the Mach, Knudsen and Stokes numbers are 0.16 ≤ Ma ≤ 1.2, 0.075 ≤ Kn ≤ 10 and 0.5 ≤ St ≤ 5, respectively. The shear force and Nusselt number Nu on the cavity walls, as well as the velocity, temperature, and heat flux lines in the cavity, are measured and analyzed. The results suggest that the anti-Fourier heat transfer phenomenon might also occur in the rarefied double-sided oscillating cavity flow. Nu is prone to non-simple harmonic oscillation in its evolution. Furthermore, in the parallel motion as opposed to the anti-parallel motion, the left wall experiences a greater intensity of heat transmission. Compressibility, expansion cooling, and viscous heating compete to provide the complicated flow and heat transfer characteristics found in cavities.

A lightweight, conservation-moment-based implicit gas kinetic Lax–Wendroff scheme for all Knudsen number isothermal gas flows

This work presents an efficient implicit gas kinetic Lax–Wendroff scheme for steady isothermal gas flows in all Knudsen number (Kn) regimes. In the scheme, the discrete velocity Bhatnagar–Gross–Krook model equations (DVE) and the associated conservation moment equations (CME) are coupled and solved by matrix-free implicit schemes. Thanks to obtaining the fluxes of the CME by multiplying the fluxes of the DVE with a projection matrix and utilizing the equilibrium distribution functions at the new time predicted by the CME, both the complicated macro fluxes reconstruction of the CME and the calculation of the Jacobian matrix of the equilibrium distribution functions are not needed in the scheme, which makes the scheme lightweight. Moreover, to enhance the accuracy of the predicted equilibrium distribution functions at the new time to improve the convergence, a symmetric Gauss–Seidel scheme with inner iterations is used to solve the CME system. Due to the coupling between the DVE and the CME, the highly efficient implicit scheme for the CME drives the DVE system to converge quickly for the continuum and near-continuum flows. Furthermore, to verify the accuracy and high efficiency of the proposed implicit scheme, comparison studies of several two-dimensional isothermal rarefied gas flow cases simulated by the present implicit scheme and the explicit gas kinetic Lax–Wendroff scheme are also provided. The numerical results show that the present implicit scheme can be as accurate as its explicit counterpart with one to two orders times speed-up in all Kn number flow regimes.

Development of a simple calculation method for the conductance of rarefied gas flow in cylindrical pipe applicable to divertor exhaust investigation of nuclear fusion reactor

ABSTRACT A gas pressure distribution calculation method for viscous laminar and intermediate flows is developed. By introducing the pseudo-conductance for viscous laminar flow, linearized equations describing pressure distribution are obtained. In intermediate flow, the pressure distribution is iteratively obtained. The validity of the method is confirmed by demonstrating a pressure in a long-tube flow path with various flow rates. The result smoothly connects theoretical curves of molecular flow and viscous laminar flow between two regions. Then, the calculation method is applied to an actual device and is compared to the measurement. The comparison between the calculation and measurement shows the qualitative agreement and reveals the requirement for a more detailed implementation of the actual conditions.

Grain polydispersity and non-sphericity effects on gas flow through granular beds using measurements and modelling

ABSTRACT The quality of cometary surface activity simulations and erosion models of icy moons depends on a good knowledge of the surface layer permeability to gas flow. Therefore, we study various models of the Knudsen diffusion coefficient and the viscous permeability, which are used to describe the flow of rarefied gases through porous materials. Usually, these models are expressed for monodisperse packed beds. In this work, we describe a generalization to polydisperse packed beds and compare them with experimental results. In addition, we analyse non-spherical packings to test how well the recently developed models are applicable. For this purpose, the gas flow parameters of these samples are measured in a dedicated measurement set-up. Special attention had to be paid to biases in measuring the porosity and the pressure drop in the sample, which are discussed in detail. Our measurements confirm that the Knudsen diffusion coefficient is inversely proportional to the specific surface area of the grains and that the viscous permeability is inversely proportional to the specific surface area squared. Further, we were able to identify a relation between the gas flow parameters, represented by a parameter β, which seems to be an indicator of the mean orientation of the grains. The findings give further evidence of the importance of the grain size distribution and the grain shape for rarefied gas flow. In particular, the results show that the standard polydisperse model is not sufficient when a considerable part of the pore space consists of traps or other rarely percolated parts.