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Atmosphere-Breathing Electric Propulsion systems (ABEP) are currently investigated to utilize the residual atmosphere as propellant for drag-compensating thrusters on spacecraft in (very) low orbits. The key concept for an efficient intake of such a system is to feed a large fraction of the incoming flow to the thruster by a high transmission probability Θ for the inflow while Θ for the backflow should be as low as possible. This is the case for rarefied flows through tube-like structures of arbitrary cross section when assuming diffuse wall reflections inside and after these ducts, and entrance velocities u larger than thermal velocities vth ∝ √kBT/m. The theory of transmission for free molecular flow through cylinders is well known for u = 0, but less research results are available for u > 0. In this paper, the desired theoretical characteristics of intakes for ABEP are pointed out, a short review of transmission probabilities is given, and results of Monte Carlo simulations concerning T are presented. Based on simple algebraic relations, an intake can be optimized in terms of collection efficiency by choosing optimal ducts. It is shown that T depends only on non-dimensional values of the duct geometry combined with v th and u. The simulation results of a complete exemplary ABEP configuration illustrate the influence of modeling quality in terms of inflow conditions and inter-particle collisions. [ABSTRACT FROM AUTHOR]

In order to improve the accuracy of rarefied aerodynamic prediction, a hypersonic rarefied wind tunnel (HRWT) was developed at Japan Aerospace Exploration Agency. While this wind tunnel has been limited to inert gases, such as nitrogen or argon, we recently extended the capability of HRWT to CO2 hypersonic flows for several Mars missions. Compared to our previous N2 cases, the condensation effect may not be negligible for CO2 rarefied aerodynamic measurements. Thus, in this work, we have utilized both experimental and numerical approaches to investigate the condensation and rarefaction effects in CO2 hypersonic nozzle flows. [ABSTRACT FROM AUTHOR]

The REX-Free Flyer is a concept study by the German Aerospace Center (DLR) in an effort to develop an orbital experimental platform to fill the gap between short duration experimental facilities and long-term missions at the International Space Station. The envisioned reusable vehicle shall enable day- and week-long experiments in high-quality weightlessness. The unique sharp-edged geometry promises improved aerodynamic properties. A soft and controlled re-entry shall guarantee a recovery of the experimental setup. The Direct Simulation Monte Carlo (DSMC) method is used to simulate a Mach 20 nitrogen flow around a scaled-down model of REX in the rarefied gas regime. First, a brief overview of the implemented numerical and phenomenological models in the in-house code DSMC code PICLas is given. Simulation parameters ensuring physical results are presented. Consequently, the code is used to investigate the lift, drag, and pitching moment coefficients at five different angles of attack: 0°, 14°, 28°, 40°, and 52°. Simulation results are compared to an approximate method, where acceptable agreement with a deviation of less than 44% can be found considering the underlying assumptions, and to dsmcFoam, where excellent agreement with a deviation of less than 3% between the different DSMC implementations is found. Furthermore, the complex three-dimensional flow environment is investigated and presented in detail for the 52° case. [ABSTRACT FROM AUTHOR]

An attempt has been made to analyze impact of Martian atmosphere parameter uncertainties on entry vehicle aerodynamics for hypersonic rarefied conditions with a DSMC code. The code has been validated by comparing Viking vehicle flight data with present computational results. Then, by simulating flows around the Mars Science Laboratory, the impact of errors of free stream parameter uncertainties on aerodynamics is investigated. The validation results show that the present numerical approach can show good agreement with the Viking flight data. The physical and chemical properties of CO2 has strong impact on aerodynamics of Mars entry vehicles, so it is necessary to make proper corrections to the data obtained with air model in hypersonic rarefied conditions, which is consistent with the conclusions drawn in continuum regime. Uncertainties of free stream density and velocity weakly influence aerodynamics and pitching moment. However, aerodynamics appears to be little influenced by free stream temperature, the maximum error of what is below 0.5%. Center of pressure position is not sensitive to free stream parameters. [ABSTRACT FROM AUTHOR]

Attempt has been made to analyze applicability of different bridging relations based on the Mars Pathfinder data. The difference of aerodynamics predicted by seventeen gas/surface interaction models is investigated. Free molecular limit Kn effect on aerodynamics is analyzed lastly. The results show that bridging relations, gas/surface interaction models and free molecular limit Kn number influence aerodynamics seriously. Present data predicted by erflog bridging relation shows a good agreement with computational data predicted by Direct Simulation Monte Carlo(DSMC) method than others. Sin-squared bridging relation is more efficient than erf-log one with some accuracy loss which can be avoided most by adjusting free molecular limit Kn number. Erf-log bridging relations is not sensitive to free molecular limit Kn number, but that Kn number value should be above 50 to keep the accuracy in free molecular regime. [ABSTRACT FROM AUTHOR]

With the advent of the new technologies in low pressure environments such as Hyperloop and helicopters designed for Martian applications, understanding the aerodynamic behavior of airfoils in rarefied environments are becoming more crucial. In this paper, verification of rarefied ES-BGK solver and ideas such as prediction of the thermally induced lift and drag reduction in rarefied aerodynamics are investigated. Validation of the rarefied ES-BGK solver with Runge-Kutta discontinous Galerkin method with experiments in transonic regime with a Reynolds number of 73 showed that ES-BGK solver is the most suitable solver in near slip transonic regime. For the quantification of lift generation, A NACA 0012 airfoil is studied with a high temperature surface on the bottom for the lift creation for different Knudsen numbers. It was seen that for lower velocities, continuum solver under predicts the lift generation when the Knudsen number is 0.00129 due to local velocity gradients reaching slip regime although lift coefficient is higher with the Boltzmann ES-BGK solutions. In the second part, the feasibility of using thermal transpiration for drag reduction is studied. Initial study in drag reduction includes an application of a thermal gradient at the upper surface of a NACA 0012 airfoil near trailing edge at a 12-degree angle of attack and 5 Pa pressure. It was seen that drag is reduced by 4 percent and vortex shedding frequency is reduced due to asymmetry introduced in the flow due to temperature gradient causing reverse flow due to thermal transpiration phenomena. [ABSTRACT FROM AUTHOR]

We propose a deterministic method designed for unsteady flows, based on a discretization of the Boltzmann (BGK) equation with local adaptive velocity grids. These grids dynamically adapt in time and space to the variations of the width of the distribution functions. This allows a significant reduction of the memory storage and CPU time, as compared to standard discrete velocity methods, and avoid the delicate problem to construct a priori a sufficient global velocity grid. [ABSTRACT FROM AUTHOR]

The analysis of influence of heterogeneous reactions in rarefied gas flows with dissociation and recombination is carried on for the first time, at least for hydrogen and methane flows. The flow in channels with heterogeneous reaction can be equilibrium and non-equilibrium, depending on a flow rate. Non-equilibrium effects are pronounced as a rule in the space between channel exit and substrate, where the activated gas flow to the surface of diamond deposition is formed. The gas dynamic analysis of gas jet deposition of diamond facilitates the optimization of experiments and their analysis. [ABSTRACT FROM AUTHOR]

Hypersonic chemical non-equilibrium gas flows around blunt nosed bodies are studied in the present paper to investigate the Reynolds analogy relation on curved surfaces. With a momentum and energy transfer model being applied through boundary layers, influences of molecular dissociations and recombinations on skin frictions and heat fluxes are separately modeled. Expressions on the ratio of Cf/Ch (skin friction coefficient to heat flux) are presented along the surface of circular cylinders under the ideal dissociation gas model. The analysis indicates that molecular dissociations increase the linear distribution of Cf/Ch, but the nonlinear Reynolds analogy relation could ultimately be obtained in flows with larger Reynolds numbers and Mach numbers, where the decrease of wall heat flux by molecular recombinations signifies. The present modeling and analyses are also verified by the DSMC calculations on nitrogen gas flows. [ABSTRACT FROM AUTHOR]

The moment methods in rarefied gas dynamics could be divided into generalized hydrodynamic equations (GHE) and extended hydrodynamic equations (EHE), e.g., Burnett equations, Grad equations and R-13 equations, theoretically. Eu firstly developed the GHE based on a non-equilibrium canonical distribution function and demonstrated the thermodynamically consistent of this model. Subsequently, nonlinear coupled constitutive relations (NCCR) was proposed by Myong by omitting the product of heat flux and velocity gradient in GHE to reduce the computational complexity. According to the successful application in 1-D shock wave structure and 2-D flat plate flow, the capability of NCCR has already been demonstrated successfully. The motivation of this study was to investigate the different behavior of NCCR and GHE for monatomic and diatomic gases in one-dimensional shock structure problems. Therefore, argon and nitrogen shock structure was calculated using both GHE and NCCR model up to Ma=50. The 3rd order MUSCL scheme for inviscid term and the 2nd order central difference scheme for viscid scheme were employed to carry out the computations. Finally, the present results including shock wave profile and its qualitative properties by NCCR and GHE are compared with that of DSMC and NS equations. The results showed that the GHE yield 1-D shock wave in much closer agreement with DSMC results than do the NCCR model without considering the computational complexity and efficiency in present cases. [ABSTRACT FROM AUTHOR]

Nonlinear dynamic system corresponding to rarefied gas flow in a channel is investigated. Under certain conditions corresponding iterative equations may have unstable solutions in some regions of the values of gas-surface interaction parameters [1]. Numerical calculations have demonstrated that a negligible change of one of the parameters of scattering function (less than 1%) near the bifurcation points causes the substantial difference in gas flow in a channel. As well non-random solutions as random solutions can be observed on bifurcation diagrams. Some of obtained solutions have a physical meaning of locking the channel, i.e. its conductivity reduces significantly. The scattering function is supposed to be close to ray-diffuse model, where the ray model, as well as the specular model, determines only one velocity of reflected gas atoms by given incident velocity. The ray-diffuse model has better experimental confirmation in comparison with the specular-diffuse model widely applied in practical DSMC calculations. However, the problem of the empirical confirmation of the obtained numerically effect is still difficult, because the scattering conditions of this type are hardly reproducible experimentally. To verify the bifurcations of simulated type experimentally all considered physical values in the flows are to be set exactly to the same values as detected in our calculations. Adding the extra parameter of scattering function allows us extending the region where considered dynamic system (related to rarefied gas flow) is unstable in the parametric space. Simulated unstable states of the system are close to physical situations observed in experiments. Obtained connection between the parameters of nonlinear dynamic system and momentum exchange coefficients (or accommodation coefficients) makes it possible to express analytic evaluations in terms of aerodynamic characteristics including Knudsen and Mach numbers, temperature factor etc. [ABSTRACT FROM AUTHOR]

The Boltzmann equation is often used to model rarefied gas flow in the transition or kinetic regime for moderate to large Knudsen numbers. However, standard moment methods like Grad's approach lack hyperbolicity of the equations. We point out the failure of Grad's method and overcome the deficiencies with the help of the new hyperbolic moment models called QBME and HME, derived by an operator projection framework. The new model equations are in partially-conservative form meaning that a subset of the equations cannot be written in conservative form due to some changes in these equations. This leads to additional numerical difficulties. The influence of the partially-conservative terms on the solution is analyzed and we present a numerical scheme for the solution of the partiallyconservative PDE systems, namely the PRICE-C scheme by Canestrelli. Furthermore, a shock structure test case is used to compare the accuracy of the different hyperbolic moment models to a discrete velocity reference solution. The results show that the new hyperbolic models achieve higher accuracy than the standard Grad model despite the fact that the model equations cannot be fully written in conservative form. [ABSTRACT FROM AUTHOR]

The dependence of the macro-parameters of the flow on surface roughness of the walls and on geometrical shape of the surface is investigated asymptotically and numerically in a rarefied gas molecular flow at high Knudsen numbers. Surface roughness is approximated in statistical simulation by the model of poly-Gaussian (with probability density as the mixture of Gaussian densities [1]) random process. Substantial difference is detected for considered models of the roughness (Gaussian, poly-Gaussian and simple models applied by other researchers), as well in asymptotical expressions [3], as in numerical results. For instance, the influence of surface roughness on momentum and energy exchange coefficients increases noticeably for poly-Gaussian model compared to Gaussian one (although the main properties of poly-Gaussian random processes and fields are similar to corresponding properties of Gaussian processes and fields). Main advantage of the model is based on relative simple relations between the parameters of the model and the basic statistical characteristics of random field. Considered statistical approach permits to apply not only diffusespecular model of the local scattering function V0 of reflected gas atoms, but also Cercignani-Lampis scattering kernel or phenomenological models of scattering function. Thus, the comparison between poly-Gaussian and Gaussian models shows more significant effect of roughness in aerodynamic values for poly-Gaussian model. [ABSTRACT FROM AUTHOR]

In the present work we study the cross sections of VV (vibration-vibration), VT (vibration-translation) energy exchanges and state-specific dissociation reactions in N2 and O2 molecules. Applying the inverse Laplace transform to approximations for state-resolved rate coefficients obtained by quasi-classical trajectory calculations we derive analytical expressions for the cross sections. The results are analyzed in a wide range of energies and vibrational levels. It is shown that cross sections of VV transitions increase almost linearly with the energy of the colliding particles. VT exchanges and dissociation reactions manifest threshold behaviour, and their cross sections are non-monotonic. The dissociation threshold is considerably shifted towards the low energy region for high vibrational states. Using the hard sphere model for the dissociation cross section results in significant inaccuracy. The results of our work can be applied in non-equilibrium fluid dynamics while simulating rarefied gas flows using DSMC methods. [ABSTRACT FROM AUTHOR]

In this paper we review emerging applications of rarefied gas dynamics for microscale sensing, actuation, power generation and thermal management. The performance of conventional fluidic devices such as pumps, combustors and heat engines drops with the decrease of characteristic length scale due to greater viscous and heat transfer losses. However, the close coupling between non-equilibrium gas, liquid and solid-state transport and electromagnetic phenomena enables unconventional micro/nanodevices. We specifically consider three distinct examples of devices with non-equilibrium gas-phase transport based on i) very large thermal gradients; ii) increased capillary forces; iii) high electric fields - all of which are generated by scaling down device size by using nano/micromanufacturing techniques. [ABSTRACT FROM AUTHOR]

We investigate heat transfer in a rarefied gas in a lid-driven cavity flow initiated by instantaneously heating and cooling opposite walls for different flow regimes. Volume diffusion model is used as an extension to the standard Navier- Stokes-Fourier set for simulating the flows. Numerical simulations are presented and compared with standard Navier-Stokes- Fourier. For higher Knudsen numbers the volume diffusion model captures non-local equilibrium effects in corners of the cavity that are missed by the Navier-Stokes-Fourier model. It is generally observed that one can use volume diffusion correction to capture disequilibrium effects in high rarefaction regimes. [ABSTRACT FROM AUTHOR]

The accurate numerical prediction of gas flows within micro-nozzles can help evaluate the performance and enable the design of optimal configurations for micro-propulsion systems. Viscous effects within the large boundary layers can have a strong impact on the nozzle performance. Furthermore, the variation in collision length scales from continuum to rarefied preclude the use of continuum-based computational fluid dynamics. In this paper, we describe the application of a massively parallel direct simulation Monte Carlo (DSMC) library to predict the steady-state and transient flow through a micro-nozzle. The nozzle's geometric configuration is described in a highly flexible manner to allow for the modification of the geometry in a systematic fashion. The transient simulation highlights a strong shock structure that forms within the converging portion of the nozzle when the expanded gas interacts with the nozzle walls. This structure has a strong impact on the buildup of the gas in the nozzle and affects the boundary layer thickness beyond the throat in the diverging section of the nozzle. Future work will look to examine the transient thrust and integrate this simulation capability into a web-based rarefied gas dynamics prediction software, which is currently under development. [ABSTRACT FROM AUTHOR]

Complex and non-intuitive flow structures controlled only by thermal gradients can be observed in rarefied gas flows. A force of thermophoretic nature, often referred to as Knudsen or radiometric force, can become dominant in microflow applications. A Microelectromechanical In-plane Knudsen Radiometric Actuator (MIKRA) that exploits these forces has been developed and tested at Purdue. Previous efforts used DSMC to understand the MIKRA flow structure and validate numerical modeling for simple gases. This work investigated more realistic boundary conditions as well as species separation around the MIKRA beams for Xe-He and N2-H2O mixtures. The main goal of this work was to run DSMC simulations of the MIKRA sensor to not only understand how it can be accurately modeled, but to look at future applications for gas mixture sensing as well. [ABSTRACT FROM AUTHOR]

Rarefied gas dynamics are important for a wide variety of applications. An improvement in the ability of general users to predict these gas flows will enable optimization of current, and discovery of future processes. Despite this potential, most rarefied simulation software is designed by and for experts in the community. This has resulted in low adoption of the methods outside of the immediate RGD community. This paper outlines an ongoing effort to create a rarefied gas dynamics simulation tool that can be used by a general audience. The tool leverages a direct simulation Monte Carlo (DSMC) library that is available to the entire community and a web-based simulation process that will enable all users to take advantage of high performance computing capabilities. First, the DSMC library and simulation architecture are described. Then the DSMC library is used to predict a number of representative transient gas flows that are applicable to the rarefied gas dynamics community. The paper closes with a summary and future direction. [ABSTRACT FROM AUTHOR]

In order to extend the existing approach of computing ionized flows about reentry spacecraft by the Direct Simulation Monte Carlo (DSMC) method, an ionized shock layer in a rarefied gas flow around a reentry vehicle is considered. The effect of various physical processes on the distribution of the electron temperature in the shock layer is analyzed by evaluation of the relevant relaxation and exchange terms in the equation of the electron energy balance. [ABSTRACT FROM AUTHOR]

The three-dimensional hypersonic rarefied gas flow over blunt bodies in the transitional flow regime is studied. The 3D thin viscous shock layer equations are solved by the asymptotic method developed for low Re numbers. The simple analytical solution is obtained for heat transfer and skin friction coefficients as functions of flow parameters and body geometry parameters. The values of these coefficients approach their values in the free molecular flow at unit accommodation coefficient as Reynolds number tends to zero. Comparison with DSMC solutions is carried out. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of the time-dependent rarefied gas flow into vacuum from a circular pipe closed at one end. The problem is studied numerically on the basis of the S-model kinetic equation up to an very large output time. The results demonstrate the dependence of flow pattern and evacuation time on the Knudsen number and pipe's length. [ABSTRACT FROM AUTHOR]

Isothermal, pressure driven rarefied gas flows through long channels with octagonal cross section shapes are analyzed computationally. The capillary is between inlet and outlet reservoirs. The cross section is constant along the axial direction. The boundary condition at the solid-gas interface is assumed to be diffuse reflection. Since the channel is long, the gaseous velocity is small compared to the average molecular speed. Consequently, a linearized description can be used. The flow is described by the linearized Bhatnagar-Gross-Krook kinetic model. The solution of the problem is divided into two stages. First, the local flow field is determined by assuming the local pressure gradient. Secondly, the global flow behavior is deduced by the consideration of the conservation of the mass along the axis of the capillary. The kinetic equation is solved by the discrete velocity method on the cross section. Both spatial and velocity spaces are discretized. A body fitted rectangular grid is used for the spatial space. Near the boundary, first-order, while in the interior part of the flow domain, second-order finite-differences are applied to approximate the spatial derivatives. This combination results into an efficient and straightforward numerical treatment. The velocity space is represented by a Gauss-Legendre quadrature. The kinetic equation is solved in an iterative manner. The local dimensionless flow rate is calculated and tabulated for a wide range of the gaseous rarefaction for octagonal cross sections with various geometrical parameters. It exhibits the Knudsen minimum phenomenon. The flow rates in the octagonal channel are compared to those through capillaries with circular and square cross sections. Typical velocity profiles are also shown. The mass flow rate and the distribution of the pressure are determined and presented for global pressure driven flows. [ABSTRACT FROM AUTHOR]

A stable high-order discontinuous Galerkin scheme which strictly preserves the positivity of the solution is designed to solve the Boltzmann kinetic model equations. The stability is kept by the accuracy of the velocity discretization, the conservation of the collision terms and a limiter. By requiring the time step smaller than the local mean collision time and forcing positive values of velocity distributions on certain points, the limiter can preserve the positivity of the solutions of the cell average velocity distributions. Verification is performed with a normal shock wave at Mach number of 2.05 and a supersonic flow about a 2D cylinder at Mach number of 6.0. [ABSTRACT FROM AUTHOR]

Devices involved in engineering applications, such as vacuum pumps or MEMS, may be made of several moving parts. This raise the issue of the simulation of rarefied gas flow around moving boundaries. We propose a simple process, known as cut cell method, to treat the motion of a solid body in the framework of the deterministic solving of a kinetic equation. Up to our knowledge, this is the first time that this approach has been used for this kind of simulations. The method is illustrated by the 2D and 3D simulations of a Crookes radiometer. [ABSTRACT FROM AUTHOR]

A unified gas-kinetic scheme is developed to simulate the axisymmetric rarefied flow. Numerical results of the cylindrical Couette flow with different Knudsen numbers agree well with existing studies and validate its good performance in all flow regimes. The modified schemes based on simplified source term work much efficiently and the numerical results are satisfactory in near-continuum flow regime. [ABSTRACT FROM AUTHOR]

A Cartesian grid-based unified gas kinetic scheme is developed. In this approach, any oriented boundary in a Cartesian grid is represented by many directional boundary points. The numerical flux is evaluated on each boundary point. Then, a boundary flux interpolation method (BFIM) is constructed to distribute the boundary effect to the flow evolution on regular Cartesian grid points. The BFIM provides a general strategy to implement any kind of boundary condition on Cartesian grid. The newly developed technique is implemented in the unified gas kinetic scheme, where the scheme is reformulated into a finite difference format. Several typical test cases are simulated with different geometries. For example, the thermophoresis phenomenon for a plate with infinitesimal thickness immersed in a rarefied flow environment is calculated under different orientations on the same Cartesian grid. These computational results validate the BFIM in the unified scheme for the capturing of different thermal boundary conditions. The BFIM can be extended to the moving boundary problems as well. [ABSTRACT FROM AUTHOR]

The paper describes an Adaptive Mesh in Phase Space (AMPS) technique for solving kinetic equations with deterministic mesh-based methods. The AMPS technique allows automatic generation of adaptive Cartesian mesh in both physical and velocity spaces using a Tree-of-Trees data structure. We illustrate advantages of AMPS for simulations of rarefied gas dynamics and electron kinetics on low temperature plasmas. In particular, we consider formation of the velocity distribution functions in hypersonic flows, particle kinetics near oscillating boundaries, and electron kinetics in a radio-frequency sheath. AMPS provide substantial savings in computational cost and increased efficiency of the mesh-based kinetic solvers. [ABSTRACT FROM AUTHOR]

Numerical simulations of rarefied gas flows are generally made with DSMC methods. Up to a recent period, deterministic numerical methods based on a discretization of the Boltzmann equation were restricted to simple problems (1D, linearized flows, or simple geometries, for instance). In the last decade, several deterministic solvers have been developed in different teams to tackle more complex problems like 2D and 3D flows. Some of them are based on the full Boltzmann equation. Solving this equation numerically is still very challenging, and 3D solvers are still restricted to monoatomic gases, even if recent works have proved it was possible to simulate simple flows for polyatomic gases. Other solvers are based on simpler BGK like models: they allow for much more intensive simulations on 3D flows for realistic geometries, but treating complex gases requires extended BGK models that are still under development. In this paper, we discuss the main features of these existing solvers, and we focus on their strengths and inefficiencies. We will also review some recent results that show how these solvers can be improved: - higher accuracy (higher order finite volume methods, discontinuous Galerkin approaches) - lower memory and CPU costs with special velocity discretization (adaptive grids, spectral methods) - multi-scale simulations by using hybrid and asymptotic preserving schemes - efficient implementation on high performance computers (parallel computing, hybrid parallelization) Finally, we propose some perspectives to make these solvers more efficient and more popular. [ABSTRACT FROM AUTHOR]

In the current work, the impact of shear work due to the slip velocity on non-equilibrium heat transfer in a pressure driven micro/nanochannel is evaluated under constant wall heat flux boundary condition. The DSMC method is employed as numerical tool. Implementation of the wall heat flux in the DSMC method is performed using the "modified Iterative" technique which in general eliminates any difficulty of implementation in adiabatic conditions and improves the solution convergence. We investigate the effects of rarefaction, property variations and compressibility. The numerical results show that shear stress on the walls significantly affects all aspects of the flow behavior and heat transfer through micro/nano channels such as heat flux rates. We also analyze the counter gradient heat flow phenomenon appearing at the cooling conditions. It is observed that viscous dissipation affects the heat flux applied to the walls and may overcome the wall heat flux, .i.e., in the case of low cooling wall heat flux condition, shear work may completely heat the flow field. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of a rarefied gas flow through a composite circular pipe into vacuum. The pipe is made of two cylindrical sections of different diameters: narrower section is followed by the wider one (diverging configuration). The analysis is based on the direct numerical solution of the Boltzmann kinetic equation with the S-model collision integral. The results are presented for several length to radius ratios and a large range of Knudsen numbers. The main computed characteristic is the mass flow rate through the pipe. The dependence of the flow field on pipe's geometry and Knudsen number is established. Formation of special features of the flow, such as recirculation zones and a Mach disk, is studied. [ABSTRACT FROM AUTHOR]