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On the basis of the kinetic model, the steady efflux of monatomic gas from a high pressure camera (the Knudsen number $${\text{Kn}} \ll 1$$ ) through a long channel between two parallel plates into a vacuum camera under a constant temperature on the bounding surfaces is studied. Using asymptotic estimates for relatively long channels, the flow domain is divided into three subdomains: (1) a neighborhood of the channel entry, (2) the main part of the flow in the channel that occupies almost all channel length, and (3) a neighborhood of the channel exit. The flow in subdomain (1) is not considered due to its low speed. In the main subdomain (2), the flow is slow and is driven by a low pressure gradient (the diffusion area). In subdomain (3), the flow gets faster, and the gas expands in the channel and in the vacuum camera. In subdomain (2), we have the continuum flow regime, and the well-known results of the linear one-dimensional theory of viscous gas flows in long channels (Poiseuille flow) are used. In the subdomain of fast flow, the full nonlinear kinetic equation (S-model) is used. The condition of asymptotic matching of solutions in two subdomains is replaced by the boundary condition of solution coupling in a certain section the position of which is chosen from the smoothness condition of the full solution of the problem. The kinetic equation is solved by the method of time marching to steady state using the conservative second-order scheme with respect to all variables implemented in Nesvetay software package. The proposed solution method can be considered as a hybrid one because the Navier–Stokes and kinetic equations are solved simultaneously.

A survey of the capabilities of the Nesvetay code as applied to computing the flow of a high-speed monatomic gas around objects of irregular shape for large flight altitudes is given. An implicit numerical method on an arbitrary unstructured grid and a two-level approach to the organization of parallel computations are described. This code is compared with the well-known MONACO and SMILE codes that implement the direct simulation Monte Carlo method.

[Abstract:] UCNS3D is an open-source computational solver for compressible flows on unstructured meshes. State-of-the-art high-order methods and their associated benefits can now be implemented for industrial-scale CFD problems due to the flexibility and highly-automated generation offered by unstructured meshes. We present the governing equations of the physical models employed in UCNS3D, and the numerical framework developed for their solution. The code has been designed so that extended to other systems of equations and numerical models is straightforward. The employed methods are validated towards a series of stringent well-established test problems against experimental or analytical solutions, where the full capabilities of UCNS3D in terms of applications spectrum, robustness, efficiency, and accuracy are demonstrated. Unión Europea; 314139 Unión Europea; 653838 Unión Europea; 823767 Engineering and Physical Sciences Research Council (EPSRC); EP/L000261/1 Engineering and Physical Sciences Research Council (EPSRC); EP/P020259/1 Engineering and Physical Sciences Research Council (EPSRC); EP/G069581/1 Engineering and Physical Sciences Research Council (EPSRC); EP/T518104/1 Engineering and Physical Sciences Research Council (EPSRC); 13794 UKRI Innovate UK; 263261

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years, including both methodological aspects as well as very ambitious applications. In spite of methodological progress, the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem (GRP) remain the centre of attention in the ADER approach. In the original formulation of ADER schemes, the proposed solution procedure for the GRP was based on (i) Taylor series expansion of the solution in time right at the element interface, (ii) subsequent application of the Cauchy–Kowalewskaya procedure to convert time derivatives to functionals of space derivatives, and (iii) solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion. For realistic problems the Cauchy–Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable. In this paper we propose a new class of solvers for the GRP that avoid the Cauchy–Kowalewskaya procedure and result in simpler ADER schemes. This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface. Still relying on a time Taylor series expansion of the solution at the interface, the time derivatives are then easily calculated from the time-reconstruction polynomial. The resulting schemes are called ADER-TR. A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method, thus deducing linear stability regions. Also, via careful numerical experiments, we deduce stability regions for the corresponding non-linear schemes. Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods. This paper is restricted to the one-dimensional scalar case with source term, but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.

The unsteady two-dimensional problem of the reflection of a uniform supersonic rarefied-gas flow incident normal on a wall with an orifice and the gas outflow through the orifice (slot or channel) is solved on the basis of the kinetic S-model. The slot influence on the reflection nature, the velocity of the shock wave reflected from the wall, and the formation of a gas jet flowing out into a vacuum is studied as a function of the incident flow velocity, the gas rarefaction degree, and the conditions imposed on the wall. The effect of the wall thickness, that is, the length of the channel, through which the gas flows out, is also investigated. The gas flow rate through the orifice is calculated as the main integral characteristic. The attainment of steady reflection regime in the outflow process is traced. The kinetic equation is solved numerically using the conservative difference method of the second order with respect to all variables.

We study axisymmetric time-dependent problem of reflection of a shock wave running into the wall with a short orifice and associated rarefied gas flow into vacuum through a pipe. The main characteristics under consideration are the mass flow rate through the pipe and its transition to the constant value, the influence of the orifice presence, pipe relative length and Knudsen number on the reflected shock wave velocity. The analysis is based on the numerical solution of the S-model kinetic equation using the second-order accurate finite volume method, implemented into the “Nesvetay-3D” software package.

In near-wall turbulence modeling it is necessary to resolve a thin boundary layer containing high gradients of the solution. An accurate enough resolution of such a layer can take most of the computational time. The situation becomes even worse for unsteady problems. To avoid time-consuming computations,in the present paper a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at theinterface boundary between domains is constructed by transferring the original boundary condition from the wall. For the first time, recently developed unsteady interface boundary conditions of Robin type are used for the unsteady Reynolds Averaged Navier-Stokes equations. The interface boundary conditions contain a memory term which takes into account the nonlocal effect in time to be taken into account for essential unsteady problems. In the case of stationary solutions the new approach automatically reduces to the technique earlier developed for the steady problems. The considered test cases demonstrate that the effect of the memory term can be significant for the accuracy of the near-walldomain decomposition. The criteria for importance of the memory term in the interface boundary condition are formulated and confirmed for turbulent flows.

The influence of the position of the pylon on the characteristics of propeller noise has been studied as applied to environmental noise calculations for future aircraft. Components related to propeller noise itself and to a signal reflected from a pylon have been separated in the overall noise produced by the propeller–pylon system at the blade passing frequency, and the interference of these signals has been investigated. A numerical method has been developed based on matching of the following two computational blocks: a rotating domain in the immediate vicinity of the propeller and the outer static domain comprising the pylon. A noise calculation procedure by the integral Ffowcs Williams and Hawkings method has been implemented with the use of the Green’s function for the convective wave equation.

For the purpose of taking the internal degrees of freedom into account, threetemperature approximating model equations, which are a generalization of the R- and ES–BGKmodels, are proposed for a diatomic gas. The surface pressure, friction, and heat transfer coefficients are compared with the direct simulation Monte Carlo (DSMC) solution in the problem of flow past a cylinder in the super- and hypersonic flow regimes. The dependence of the surface coefficients on the rotational collision number is analyzed.

The paper is devoted to the further development of the numerical methods to solve model kinetic equations and their application to hypersonic rarefied gas flows. Firstly, we verify the accuracy of the approach by comparing our results with the well resolved DMSC calculations for argon and nitrogen. Secondly, computation of an external flow over a model winged re-entry vehicle with orbital free-stream velocity and 100 km altitude is shown. Influence of the spatial mesh type is studied. Finally, scalability of the implicit kinetic solver is demonstrated for up to 256 nodes.

Different parallel distributed-memory versions of Lower-Upper Symmetric Gauss–Seidel (LU-SGS) method for solution of discrete equations in finite-volume framework are compared in terms of parallel structure of algorithms. New version based on multilevel recursive decomposition is proposed.

The two-dimensional time-dependent problem of rarefied gas flow in a plane channel, formed by parallel plates of finite length and closed at one end, is solved on the basis of the kinetic S-model. The flow develops as a result of rupture of a diaphragm which separates the gas at rest in the channel and the gas at rest in a reservoir of infinite volume. The effect of gas deceleration at the channel walls under the conditions of diffuse molecular reflection from the channel walls and end face is studied. Decay of a shock wave and disappearance of a homogeneous flow zone behind the shock wave is traced for three variants of conditions at the channel inlet: (1) gas enters the channel from a reservoir of infinite length and width (as the basic variant), the simultaneous motion in the reservoir and channel being studied; (2) the high-pressure reservoir represents a usual channel section; and (3) the motion of the gas in the reservoir is not considered at all, instead of this, the boundary conditions of the evaporation-condensation type under the conditions of gas at rest in the reservoir are imposed in the inlet cross-section.

The paper is devoted to the development of the numerical approaches to solve model kinetic equations as applied to computing high-speed rarefied gas flows over three-dimensional geometries. The use of a very non-uniform unstructured velocity mesh is proposed, and the influence of the velocity mesh resolution is studied in detail. The test problems include a simple flow over a planar circular cylinder and a three-dimensional flow over a model winged re-entry vehicle.

The range of validity of various linear kinetic modeling approaches simulating rarefied pressure driven gas flow through circular tubes is computationally investigated by comparing the flowrates obtained by the linear approaches with the corresponding nonlinear ones. The applicability margins of the linear theories in terms of the parameters determining the flow (gas rarefaction, pressure ratio, tube aspect ratio) are specified, provided that the introduced deviation norm is smaller than a specific value. The work is motivated by the fact that computational effort is significantly reduced when linear, instead of nonlinear, kinetic modeling is implemented. It is found that the range of validity of the linear solutions is much wider than the expected one, as defined by their formal mathematical constrains and it remains valid in a range of parameters, where the DSMC method and nonlinear kinetic modeling become computationally inefficient, resulting in great computational savings.

The paper is a short review of the recent results of the author in the development of numerical methods and parallel computer code Nesvetay to solve kinetic equations with approximate (model) collision integrals. The efficiency and robustness of the methods are demonstrated by computing \(M_\infty =25\) rarefied gas flow over a model winged geometry of a spacecraft.

A two-level OpenMP + MPI parallel implementation is used to numerically solve a model kinetic equation for problems with complex three-dimensional geometry. The scalability and robustness of the method are demonstrated by computing the classical gas flow through a circular pipe of finite length and the flow past a reentry vehicle model. It is shown that the two-level model significantly speeds up the computations and improves the scalability of the method.

The kinetic S-model is used to study the unsteady rarefied gas flow through a plane channel between two parallel infinite plates. Initially, the gas is at rest and is separated by the plane x = 0 with different pressure values on opposite sides. The gas deceleration effect of the channel walls is studied depending on the degree of gas rarefaction and the initial pressure drop, assuming that the molecules are diffusely reflected from the boundary. The decay of the shock wave and the disappearance of the uniform flow region behind the shock wave are monitored. Special attention is given to the gas mass flux through the cross section at x = 0, which is computed as a function of time. The asymptotic behavior of the solution at unboundedly increasing time is analyzed. The kinetic equation is solved numerically by applying a conservative finite-difference method of second-order accuracy in space.

The numerical study of one-dimensional gas expansion under pulsed evaporation into vacuum is carried out on the basis of the direct simulation Monte Carlo method, the exact Boltzmann kinetic equation, and the S-model kinetic equation. The results are presented for various levels of evaporation intensity, defined by the amount of evaporated material. Special attention has been paid to the calculation of the average axial energy of particles, the velocity vector of which deviates from the axis by no more than a small prescribed angle α. This characteristic of the flow is important for analysis of time-of-flight distributions in pulsed laser ablation. It is found that for intense evaporation, the average axial energy has a maximum as a function of time. The presented results allow us to establish the relative accuracy of the considered kinetic approaches for various flow regimes.

For modeling turbulent flow, the near-wall domain decomposition (NDD) approach initially proposed by the second author and recently developed in a number of papers proved to be very efficient. It leads to a non-overlapping domain decomposition with a Robin-to-Dirichlet map between an inner (near-wall) and outer regions. The regions are linked with each other via interface boundary conditions of Robin type which equivalently replace both the boundary conditions at the wall and simplified governing equations in the inner region. As has been shown, this approach can reduce the computational time by one order of magnitude while retaining sufficiently high accuracy. In the current paper, for the first time the technique is extended to compressible gas flows. In addition, it is modified to include an exact domain decomposition applied to the original Reynolds-averaged Navier-Stokes equations (RANS) without any simplifications near the wall. The efficiency and accuracy of the algorithm are demonstrated on a number of test cases with the use of the Spalart-Allmaras turbulence model for compressible flows implemented in the in-house code “FlowModellium”. Apart from the approximate NDD (ANDD) based on the thin boundary layer model, for the first time an exact NDD (ENDD) is implemented. The interface boundary conditions in both ANDD and ENDD approaches are consistent. Thereby, the ENDD can effectively complete the ANDD approach when it is needed.

Resolution of a near-wall boundary layer is one of hot topics in turbulence modeling. It often requires most of computational time. A non-overlapping domain decomposition method has turned out to be very efficient to tackle this problem for engineering applications. The method is much more universal than conventional approaches based on wall functions. In our approach, the interface boundary conditions are of Robin type and obtained via transfer of the boundary conditions from the wall to the interface boundary. In the current chapter, the technique is extended to the time-dependent Navier-Stokes equations. As a result, the interface boundary condition remains to be of Robin type but it becomes nonlocal in time since contains a memory term. Unsteady effects are analyzed in the chapter.

Numerical solution of the Boltzmann equation for stationary high-speed flows around complex three-dimensional bodies is an extremely difficult computational problem. This is because of high dimension of the equation and lack of efficient implicit methods for the calculation of the collision integral on arbitrary non-uniform velocity grids. Therefore, the use of the so-called model (approximate) kinetic equations appears to be more appropriate and attractive. This article uses the numerical methodology recently developed by the second author which includes an implicit method for solving the approximating kinetic equation of E.M. Shakhov (S-model) on arbitrary unstructured grids in both velocity and physical spaces. Since most of model equations have a well-known drawback associated with the velocityindependent collision frequency it is important to determine the deviations of solutions of these equations from the solution of the complete Boltzmann equation or DSMC for high-speed gas flows. Our recent comparison of the DSMC and S-model solutions for monatomic gases with a soft interaction potential shows good agreement of surface coefficients of the pressure, heat transfer and friction, which are most important for industrial applications. In this paper, we compare the solution of model equations and the Boltzmann equation for the problem of supersonic gas flow around a cylinder when molecules interact according to the law of hard spheres. Since this law of molecular interaction is the most rigid, the difference in solutions can show the maximum error that can be obtained by using model equations instead of the exact Boltzmann equation in such problems. Our high-fidelity computations show that the use of model kinetic equations with adaptation in phase space is very promising for industrial applications.

The time-dependent problem of the development of a rarefied flow in a plane channel formed by two parallel infinite plates is considered on the basis of a kinetic model. At the initial moment of time the gas at rest occupies half the channel and borders on a vacuum. The effect of gas deceleration at the channel walls is studied as a function of the gas rarefaction degree under the conditions of the diffuse reflection of molecules from the boundary. Emphasis is placed on the calculations of the time dependence of the integral gas mass flux through the boundary separating the gas from the vacuum at the initial moment. The deceleration of the conditional expansion wave front and the conditional front of gas expansion into the vacuum is traced. The kinetic equation is numerically solved using a first-order difference method.

The paper is devoted to the comparative study of Shakhov model kinetic equation and Direct Simulation Monte Carlo (DSMC) solutions as applied to high-speed flow of a monatomic gas over generic three-dimensional space vehicle under angle of attack. The corresponding calculations are carried out using Nesvetay and SMILE software packages, developed by the authors. Satisfactory agreement is shown for surface distribution of pressure, friction and heat transfer coefficients despite very different modeling approaches and computational meshes employed. It is thus shown that the use of the model kinetic equations with approximate collision integrals provides accurate results for high-speed external monatomic gas flows over complex three-dimensional geometries at acceptable computational cost.

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. Limiting solutions corresponding to the one-dimensional free-molecular and hydrodynamic flow regimes are also considered. The results demonstrate the dependence of flow pattern and evacuation time on the Knudsen number.

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. Two cases are studied: wider pipe followed by narrower pipe (converging configuration) and the narrower pipe followed by the wider (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 two 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.

The problem of rarefied gas flow into vacuum through a short circular pipe is studied numerically by solving the Boltzmann kinetic equation. Comparison of the results obtained with the exact and S-model collision integrals is presented across a large range of Knudsen numbers. Computed values of mass flow rate are also compared against the DSMC results and experimental data from existing literature.

The paper is devoted to the study of a rarefied gas flow through a finite length conical pipe into vacuum. The problem is solved in the complete geometrical setup, which included not only the pipe, but also high- and low-pressure reservoirs. The analysis is based on the direct numerical solution of the Boltzmann kinetic equation with the S-model collision integral in three space dimensions. The method of the solution is based on the recent implicit total variation diminishing (TVD) method on unstructured spatial meshes. It is conservative with respect to the collision integral and work across all flow regimes. The results are provided for various ratios of the outlet and inlet diameters, pipe's lengths and Knudsen numbers. The computed flow rates are compared with the case of the circular pipe of constant radius as well as an approximate method for very long pipes. The influence of the pipe geometry on the flow field is also examined. The presented results can be used as a benchmark for calculations by other methods and codes.

This special issue is dedicated to recent developments concerning modern numerical methods for time dependent partial differential equations that can be used to simulate complex nonlinear flow and transport processes and tries to cover a wide spectrum of different applications and numerical approaches. Most of the models under consideration here are nonlinear hyperbolic conservation laws, which are a very powerful mathematical tool to describe a large set of problems in science and engineering. Hyperbolic PDE are based on first principles, such as mass, momentum and total energy conservation as well as the second law of thermodynamics. These principles are universally valid and have been successfully used for the simulation of a very wide class of problems, ranging from astrophysics over geophysics to complex flows in engineering and computational biology. Most of the seemingly different applications listed above have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher-order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise from the nonlinearities that allow the formation of non-smooth solutions such as shock waves, material interfaces or other discontinuities, but nonlinear conservation laws can simultaneously also contain smooth features like sound waves and vortex flows. Even nowadays the key challenge for the construction of successful numerical methods for nonlinear hyperbolic PDE is therefore to capture at the same time shock waves and discontinuities robustly without producing spurious oscillations and exhibit little numerical dissipation and dispersion errors in smooth regions of the solution. For this reason, the development of robust and accurate numerical methods for nonlinear flow and transport problems remains a great challenge even after several decades of successful research, which started back in the 1940s and 1950s with the pioneering works of von Neumann [1] and Godunov [2] . The present special issue is published on the occasion of the 70th birthday of Prof. Eleuterio F. Toro and in honor of his outstanding contributions to the fields of applied mathematics, scientific computing and computational physics with his well-known numerical methods for the solution of hyperbolic conservation laws and their application to computational fluid dynamics, computational physics and computational biology.

Analysis of three-dimensional rarefied gas flowsin microdevices (micropipes, micropumps etc) and over re-entry vehicles requires development of methods of computational modelling. One of such methods is the direct numerical solution of the Boltzmann kinetic equation for the velocity distribution function with either exact or approximate (model) collision integral. At present, for flows of monatomic rarefied gas the Shakhov model kinetic equation, also called S-model, has gained wide-spread use. The equation can be regarded as a model equation of the incomplete thirdorder approximation. Despite its relative simplicity, the S-model is still a complicated integrodifferential equation of high dimension. The numerical solution of such an equation requires high-accuracy parallel methods.The present work is a review of recent results concerning the development and application of three-dimensional computer package Nesvetay-3D intended for modelling of rarefied gas flows. The package solves Boltzmann kinetic equation with the BGK (Krook) and Shakhov model collision integrals using the discrete velocity approach. Calculations are carried out in non-dimensional variables. A finite integration domain and a mesh are introduced in the molecular velocity space. Next, the kinetic equation is re-written as a system of kinetic equations for each of the discrete velocities. The system is solved using an implicit finite-volume method of Godunov type. The steady-state solution is computed by a time marching method. High order of spatial accuracy is achieved by using a piece-wise linear representation of the distribution function in each spatial cell. In general, the coefficients of such an approximation are found using the least-square method. Arbitrary unstructured meshes in the physical space can be used in calculations, which allow considering flows over objects of general geometrical shape. Conservative property of the method with respect to the model collision integral is guaranteed by means of a special procedure to calculate macroscopic variables. Another important part of the numerical method is the fast solution of the linear system for time increments of the distribution function. The solution is based on the LU-SGS approach so that the number of operations is linearly proportional to the number of cells in the spatial mesh. Large problems can be solved on hundreds of CPU cores using the Message Passing Interface (MPI).Performance and robustness of the numerical method and computer code are illustrated on a number of problems, including rarefied gas flows through microchannels into vacuum and external flows over re-entry vehicles on the high altitude of flight. Rarefied gas flows through simple and composite channels of circular cross sectional area are considered. Comparisons with the results of other authors and experimental data are shown.Good spatial mesh convergence of the method is demonstrated. For the flow in a composite channel the formation of a Mach disk is shown. One of the examples of external flow calculation is the analysis of rarefied gas flow over model winged re-entry space vehicle (RSV), which is a three-dimensional object of a very complex shape. The RSV is proposed by Central Aerohydrodynamic Institute (TsAGI). Its geometry includes a blunt fuselage, sweptwings, keel and flap. The flow pattern over the RSV and surface pressure distribution are shown.At present the work is under way to incorporate an implicit numerical scheme for time-dependent rarefied gas flows, an adaptation algorithm for the efficient modelling of hypersonic flows over complex objects and extension to the diatomic gas modelling on the bases of the Rykov kinetic model.

The problem of steady outflow of rarefied gas from a reservoir into vacuum through a long cylindrical tube of circular cross-section at a constant temperature is considered on the basis of the kinetic S-model. The kinetic equation is solved numerically using a conservative second-order method. The basic calculated quantity is the gas flow rate through the channel; the flowfields are also studied. The solutions obtained are compared with the known results.

The paper is devoted to the study of a rarefied gas flow through a circular pipe caused by small pressure differences between the reservoirs attached to the ends of the pipe. The analysis is based on of the direct numerical solution of the Boltzmann kinetic equation with the linearized model collision integral of Shakhov. The solution of the problem is computed for a wide range of the rarefaction parameter values and is compared with the kinetic and continuum solutions for an infinitely long circular pipe as well as with the flow into vacuum.

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.

A parallel multiblock implementation of a second-order accurate implicit numerical method based on solving a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The performance of the method is illustrated by computing test examples of gas flows in a circular pipe in a wide range of Knudsen numbers. The convergence rate and scalability of the method are analyzed depending on the number of blocks in the spatial grid.

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.

The paper presents a numerical analysis of the nonlinear rarefied gas flow through a long planar channel of finite length. The solution is constructed for the arbitrary pressure and temperature drops, including flow into vacuum. The obtained results for the flow rates are compared with the linearized solutions in the large range of degree of gas rarefaction. The boundaries of the validity of the linearized approximation are established.

Flow of a diatomic rarefied gas in a capillary tube of infinite length and an arbitrary cross-section under a given small pressure gradient (Poiseuille flow) or a small temperature gradient (thermal creep) is studied on the basis of a kinetic model that takes account for the rotational degrees of freedom of molecules (R-model). Numerical investigation is carried out for flows between parallel flat plates and in a circular capillary tube at the gas parameters corresponding to nitrogen. The main calculated quantity is the gas flow rate through a tube cross-section. The results are compared with the corresponding data obtained on the basis of the S-model.

The paper is devoted to the development of an efficient deterministic framework for modelling of three-dimensional rarefied gas flows on the basis of the numerical solution of the Boltzmann kinetic equation with the model collision integrals. The framework consists of a high-order accurate implicit advection scheme on arbitrary unstructured meshes, the conservative procedure for the calculation of the model collision integral and efficient implementation on parallel machines. The main application area of the suggested methods is micro-scale flows. Performance of the proposed approach is demonstrated on a rarefied gas flow through the finite-length circular pipe. The results show good accuracy of the proposed algorithm across all flow regimes and its high efficiency and excellent parallel scalability for up to 512 cores.

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.

The linearized kinetic S-model is used to study the nonisothermal steady rarefied gas flow driven by differences in pressure and temperature in a plane channel between long finite parallel plates joining two tanks of infinite volume. An efficient composite (asymptotic) method is developed: a one-dimensional asymptotic solution corresponding to an infinitely long channel is constructed in the middle part of the computational domain, while a solution of the two-dimensional kinetic equation matched with the middle-part asymptotic solution is constructed near the ends of the channel. The latter solution is found numerically by a high-order accurate conservative method. The basic quantity to be computed is the gas flow rate through the channel. Characteristic flow features are also investigated. The resulting solutions are compared with previously known results.

An efficient numerical algorithm for calculating rarefied gas flows in planar microchannels on the basis of the Boltzmann kinetic equation with the linearized S-model collision integral is presented. The algorithm consists of a high-order spatial discretisation on unstructured meshes, conservative procedure to calculate macroscopic quantities and efficient one-step implicit time evolution method. It therefore works across all flow regimes from the free-molecular to nearly continuum ones and provides rapid convergence to steady state solutions. The parallel implementation is provided via MPI programming paradigm. The performance of the method is illustrated by computing the flow for length to the width ratios for up to 10^3 in the wide range of Knudsen numbers. Numerical estimates of efficiency of implicit time marching and parallel performance as well as convergence studies in both physical and velocity spaces are provided. The end effects and other essentially two-dimensional features of the flow are analysed and a detailed comparison with the case of an infinitely long channel is carried out. The presented numerical data can be used as reference for future studies of nonlinear flows as well as for comparison with other numerical approaches and flow models.

The paper presents a linear high-order method for advection–diffusion conservation laws on three-dimensional mixed-element unstructured meshes. The key ingredient of the method is a reconstruction procedure in local computational coordinates. Numerical results illustrate the convergence rates for the linear equation and a non-linear hyperbolic system with diffusion terms for various types of meshes.

An isothermal steady rarefied gas flow in a long channel (tube) of elliptical or rectangular cross-section under the action of a given pressure gradient (Poiseuille flow) is studied on the basis of the Bhatnagar-Gross-Krook model. The solution is obtained using a conservative higher-order method. The velocity field in a channel cross-section is investigated as a function of the rarefaction degree and the cross-section geometry parameters. The main calculated function is the gas flow rate through the tube. The solutions obtained are compared with the available results.

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered.

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.

The method based on the numerical solution of a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The basic idea behind the method is the use of a second-order accurate TVD scheme on hybrid unstructured meshes in physical space and a fast implicit time discretization method without iterations at the upper level. The performance of the method is illustrated by computing test examples of three-dimensional rarefied gas flows in variously shaped channels in a wide range of Knudsen numbers.

The linearized kinetic BGK model is used to study the steady Poiseuille flow of a rarefied gas in a long channel of rectangular cross section. The solution is constructed using the finite-volume method based on a TVD scheme. The basic computed characteristic is the mass flow rate through the channel. The effect of the relative width of the cross section is examined, and the difference of the solution from the one-dimensional flow between infinite parallel plates is analyzed. The numerical solution is compared to available results and to the analytical solution of the Navier-Stokes equations with no-slip and slip boundary conditions. The limits of applicability of the hydrodynamic solution are established depending on the degree of rarefaction of the flow and on the ratio of the side lengths of the channel cross section.