High-order accurate conservative method for computing the poiseuille rarefied gas flow in a channel of arbitrary cross section

A high-order accurate method is proposed for analyzing the isothermal rarefied gas flow in an infinitely long channel with an arbitrarily shaped cross section (Poiseuille flow). The basic idea behind the method is the use of hybrid unstructured meshes in physical space and the application of a conservative technique for computing the gas velocity. Examples of calculations are provided for channels of various cross sections in a wide range of Knudsen numbers. Schemes of the first-, second-, and third orders of accuracy in space are compared.     

Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes

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.     

Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains

A high-order accurate method for analyzing two-dimensional rarefied gas flows is proposed on the basis of a nonstationary kinetic equation in arbitrarily shaped regions. The basic idea behind the method is the use of hybrid unstructured meshes in physical space. Special attention is given to the performance of the method in a wide range of Knudsen numbers and to accurate approximations of boundary conditions. Examples calculations are provided.     

Application of a momentum method to some rarefied gas flow problems

Zusammenfassung Eine Kombination der Lees-Methode und des Mott-Smith Ansatzes wird auf zwei Probleme der Couette-Strömung zwischen zwei Zylindern angewandt. Es werden Ausdrücke hergeleitet für das Moment im Fall, dass ein Zylinder rotiert, und für den Wärmefluss im Fall, dass beide Zylinder stationär sind, aber verschiedene Temperatur haben; diese Ausdrücke sind für alle Werte der Knudsenzahl gültig.     

Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas

Plane Poiseuille flow of a highly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the Boltzmann equation for a hard sphere molecular gas and the diffuse reflection boundary condition. The behavior of the solution in the regime of large mean free path and small strength of gravity is studied numerically based on the one-dimensional Boltzmann equation derived by means of the asymptotic analysis for a slow variation in the flow direction. It is clarified that the effect of weak gravity on the flow is not negligible when the gas is so rarefied that the mean free path is comparable to the maximum range that the molecules travel along the parabolic path within the channel. When the mean free path is much larger than this range, the effect of gravity that makes the molecules fall plays the dominant role in determining the distribution function, and thus the over-concentration in the distribution function as well as the flow velocity does not increase further even if the mean free path is increased. The upper bound of the flow velocity and the mass flow rate of the gas are obtained as a function of the gravitational acceleration.     

Poiseuille flow of a rarefied gas mixture

The flow of a rarefied binary gas mixture is governed by transport equations which correspond to a coupled set of Fredholm integral equations of the second kind.By means of the Fourier transform technique and by expanding the transformed kernels in terms or orthogonal functions the problem is reduced to the solution of a system of linear algebraic equations.As a significant application, the case of a plane Poiseuille flow of a binary mixture following Sirovich-Hamel's model is considered.The convergence of the method is excellent.

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Riassunto Il flusso di una miscela binaria di gas rarefatti è retto da equazioni del trasporto che corrispondono ad un sistema di equazioni integrali di Fredholm di seconda specie. Mediante la tecnica della trasformata di Fourier e sviluppando i kernel trasformati in termini di funzioni ortogonali, il problema si riduce alla soluzione di un sistema di equazioni algebriche lineari. Quale applicazione significative si considera il flusso piano di Poiseuille di una miscela binaria che segue il modello di Sirovich-Hamel. La convergenza del metodo e'eccellente.

     

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

Simplified Navier-Stokes equations for computing viscous gas flow past long bodies

Simplified Navier-Stokes equations are applied to analyze the flow of supersonic viscous gas at moderately large Reynolds numbers near the lateral surface of long bodies. Numerical integration of the equations is performed by the marching method, stabilizing in each step the solution of the nonstationary system of equations in the longitudinal coordinates. We consider the flow past cylinders and cones with a spherical blunt nose. We investigate the effect of the Reynolds number and the body shape on the flow field, the drag coefficient, and the heat flux. The numerical solutions of simplified and complete Navier-Stokes equations are compared.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 231–239, 1985.     

Computation of rarefied diatomic gas flows through a plane microchannel

A numerical method based on a model kinetic equation was developed for computing diatomic rarefied gas flows in two dimensions. Nitrogen flows through a plane microchannel were computed, and the gas flow rate was constructed as a function of the Knudsen number for various channel lengths.     

On the behavior of a slightly rarefied gas mixture over plane boundaries

Summary The behavior of a slightly rarefied gas mixture bounded by plane boundaries is investigated on the basis of the linearized Boltzmann equation ofB-G-K type for gas mixtures under the diffusive boundary condition. A useful result of the present analysis is that the macroscopic equations and the appropriate boundary conditions in terms of slip and jump are obtained together with the Knudsen-layer corrections near the boundaries. This system of equations makes possible the treatment at fluid dynamic level for various problems of gas mixtures with plane geometry which require kinetic theory consideration. As an application of this system, some basic flow problems of a slightly rarefied gas mixture, namely, Couette flow, thermal slip flow and diffusion slip flow between two plates are taken up. The total velocity distributions of these concrete problems are explicitly obtained for the first time, and their dependence on the properties and concentration of the component gases in the mixture are clarified in some detail.

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Zusammenfassung Das Verhalten einer verdünnten Gasmischung, bei kleiner aber nicht vernachlässigbarer Knudsen Zahl, zwischen zwei parallelen Platten wird analytisch untersucht. Die linearisierten Boltzmann Gleichungen desB-G-K Typs für Gasmischungen mit diffusiven Randbedingungen werden angewendet. Aus der vorliegenden Untersuchung resultieren brauchbare makroskopische Gleichungen mit den zugehörigen Randbedingungen, — mit Gleitgeschwindigkeit und Temperatursprung formuliert, — sowie die Korrekturen für die wandnahe Knudsen-Schicht. Verschiedene Strömungen von Gasmischungen, bei denen gas-kinetische Effekte eine Rolle spielen, entlang ebener Begrenzungen können mit den Gleichungssystemen auf strömungsmechanischem Niveau behandelt werden. Das System wird auf die Couette Strömung und die thermal-slip und diffusion-slip Strömungen angewendet. Zum ersten Mal werden die Geschwindigkeitsprofile dieser elementaren Strömungen explizit berechnet. Der Einfluß der Gaseigenschaften und Konzentrationen auf diese Profile werden weitgehend erklärt.

     

Unsteady rarefied gas flow in a microchannel driven by a pressure difference

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.     

Magnetohydrodynamic flow of a rarefied gas near an accelerated porous plate

An investigation is made of the flow of an electrically conducting rarefied gas due to the time-varying motion of an infinite porous plate, the gas being permeated by a transverse magnetic field. The suction is taken to be a constant and the magnetic lines of force are taken to be fixed relative to the fluid. The effects of magnetic field, rarefaction parameter, suction parameter are shown by means of some tables. The expressions of the skin friction for the two particular cases have also been obtained.     

Numerical study of the Couette flow of a diatomic rarefied gas

The Couette flow is numerically studied using a model kinetic equation for a diatomic rarefied gas (nitrogen). The boundary condition set on the wall takes into account that the molecular rotational energy passes into translational energy when the molecule interacts with the wall. For comparison purposes, the Couette flow is computed using the classical diffuse model of the gas-wall interaction. A comparison of the results obtained with both types of boundary conditions shows that the computed parameters of the Couette flow coincide only for sufficiently low Knudsen numbers. This suggests that transitions between rotational and translational energy in the gas-wall interaction have to be taken into account in the boundary condition.     

Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively.     

Superconvergence for a mixed finite elmenent method for elastic wave propagation in a plane domain

Summary A higher order mixed finite element method is introduced to approximate the solution of wave propagation in a plane elastic medium. A quasi-projection analysis is given to obtain error estimates in Sobolev spaces of nonpositive index. Estimates are given for difference quotients for a spatially periodic problem and superconvergence results of the same type as those of Bramble and Schatz for Galerkin methods are derived.     

Efficient algorithms for computing the maximum distance between two finite planar sets

An 0(n log n) algorithm is presented for computing the maximum euclidean distance between two finite planar sets of n points. When the n points form the vertices of simple polygons this complexity can be reduced to 0(n). The algorithm is empirically compared to the brute-force method as well as an alternate 0(n²) algorithm. Both the 0(n log n) and 0(n²) algorithms run in 0(n) expected time for many underlying distributions of the points. An ?-approximate algorithm can be obtained that runs in $\text{0(n +}\text{1}\text{?}\text{)}$ worst-case time.     

A finite cutting plane method for facial disjunctive programs

This paper shows that any linear disjunctive program with a finite number of constraints can be transformed into an equivalent facial program. Based upon linear programming technique, a new, finite cutting plane method is presented for the facial programs.

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Zusammenfassung Die Arbeit zeigt, daß jedes lineare disjunktive Optimierungsproblem mit endlich vielen Restriktionen in ein äquivalentes Fazetten-Problem transformiert werden kann. Auf der Grundlage von linearer Optimierungstechnik wird für das Fazetten-Problem ein neues, endliches Schnittebenenverfahren vorgestellt.

     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.