Rarefied gas flow in concentric annular tube: Estimation of the Poiseuille number and the exact hydraulic diameter

The fully developed flow of rarefied gases through circular ducts of concentric annular cross sections is solved via kinetic theory. The flow is due to an externally imposed pressure gradient in the longitudinal direction and it is simulated by the BGK kinetic equation, subject to Maxwell diffuse-specular boundary conditions. The approximate principal of the hydraulic diameter is investigated for first time in the field of rarefied gas dynamics. For the specific flow pattern, in addition to the flow rates, results are reported for the Poiseuille number and the exact hydraulic diameter. The corresponding parameters include the whole range of the Knudsen number and various values of the accommodation coefficient and the ratio of the inner over the outer radius. The accuracy of the results is validated in several ways, including the recovery of the analytical solutions at the hydrodynamic and free molecular limits.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

Gas flow through an elliptical tube over the whole range of the gas rarefaction

A rarefied gas flow trough a long tube with an elliptical cross section is studied on the basis of the BGK kinetic model equation in the whole range of the Knudsen number varying from the free molecular regime to the hydrodynamic one. A wide range of the aspect ratio is considered. The mass flow rate is calculated as a function of the pressures on the tube ends.     

基于Boltzmann模型方程的气体运动论统一算法研究

模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程; 研究发展气体运动论离散速度坐标法, 借助非定常时间分裂数值计算方法和NND差分格式, 结合DSMC方法关于分子运动与碰撞去耦技术, 发展直接求解速度分布函数的气体运动论耦合迭代数值格式; 研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法, 由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法. 通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明, 计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好, 证实气体运动论统一算法求解各流域气体流动问题的可行性. 尝试将统一算法进行HPF并行化程序设计, 基于对球体绕流及类``神舟'返回舱外形绕流问题进行HPF初步并行试算, 显示出统一算法具有很好的并行可扩展性, 可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向. 通过将气体运动论统一算法推广应用于微槽道流动计算研究, 已初步发展起可靠模拟二维短微槽道流动数值算法; 通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟, 证实该算法对微槽道气体流动问题具有较强的模拟能力, 可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.      

GAS-KINETIC UNIFIED ALGORITHM FOR BOLTZMANN MODEL EQUATION IN ROTATIONAL NONEQUILIBRIUM AND ITS APPLICATION TO THE WHOLE RANGE FLOW REGIMES

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1：5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10^-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.     

转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1:5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.      

Solution of the problem of a flow of rarefied gas of constant density around a semiinfinite plate by the integral diffusion method

Several papers [1–4] have proposed approximate diffusion models which can be used to examine the transport process in a rarefied gas where the mean free path is large and transport is not determined by the local gradient of the particular quantity.In this paper the integral diffusion model [2] is used to solve the problem of determination of the friction stress and velocity of a flow of an incompressible gas around a plane semi-infinite plate in the whole range of Knudsen numbers. The obtained solution is compared with published solutions and experimental data [9].     

End effects in rarefied gas outflow into vacuum through a long tube

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.     

Surface effects in rarefied gas dynamics: an analysis based on the Cercignani–Lampis boundary condition

The flow of a rarefied gas, in a plane channel, is investigated, with special attention to the gas–surface interaction, modeled by the Cercignani–Lampis kernel that is defined in terms of normal and tangential accommodation coefficients. An analytical version of the discrete-ordinates method is used to solve, in an unified approach, the Poiseuille flow, thermal-creep flow and Couette flow problems with kinetic equations defined in terms of the BGK model. Numerical results for the velocity and heat-flow profiles and flow rates are reported, for a wide range of the accommodation coefficients and the Knudsen number.     

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann–Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily with massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuille- channel flow and pressure-driven gas flows in two-dimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of micro-scale gas flows occuring in the Micro-Electro-Mechanical System (MEMS). The project supported by the National Natural Science Foundation of China (90205009 and 10321002), and the National Parallel Computing Center in Beijing. The English text was polished by Yunming Chen.     

Translational relaxation of a heavy molecule in a light rarefied gas

A study is made of the translational relaxation (in the sense of the loss of the original direction of the velocity) of a molecule of mass M injected into the flow field of a light gas (mass of a molecule m). It is shown that when the gas is sufficiently rarefied and M ? m the spreading of the wave packet of the heavy particle can mask the mean “classical” deviation of the heavy molecule from the original direction as a result of a collision with a light molecule. Therefore, not all the collisions determined by the gas-kinetic cross section are effective for the observed deflection of the heavy molecule. Some consequences of this behavior of heavy molecules are discussed. The obtained restrictions on classical theory must be taken into account when one is considering problems of the gas dynamics of rarefied mixtures.     

Rarefied gas flow into a vacuum from a plane long channel closed at one end

The time-dependent problem of rarefied gas flow into a vacuum from a plane long channel closed at one end is solved on the basis of the kinetic S-model. The effect of diffuse molecular reflection from the channel walls on the flow velocity and the process of channel cavity vacuumization is studied as a function of the channel length and the extent of gas rarefaction under the condition that the wall temperature is maintained to be constant. The kinetic equation is solved numerically using a conservative finite-difference method of the second order of accuracy in spatial coordinates. The possibility of simplification of the problem for long times by means of reduction to the diffusion process is considered.     

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

Rarefied hypersonic flow in a plane channel with a surface glow discharge

The gasdynamic structure of a hypersonic molecular nitrogen flow in a plane channel whose opposite surfaces are segmented electrodes for generating a continuous surface glow discharge is investigated using a two-dimensional computational model. The electrodynamic structure of the surface glow discharge in the hypersonic rarefied gas flow (distributions of the charged particle concentrations, current density, and electric potential) is studied. A two-dimensional conjugate electrical-gasdynamic model consisting of the continuity, Navier-Stokes, and energy conservation equations and the chargedparticle continuity equations in the ambipolar approximation is proposed. The real thermophysical and transport properties of molecular nitrogen are taken into account. It is shown that using a surface glow discharge in a hypersonic rarefied gas flow makes it possible effectively to modify the shock-wave flow structure and hence to consider this type of discharge as additional tool for controlling rarefied gas flows.     

Kinetic analysis of longitudinal Couette flow between coaxial cylinders

On the basis of a model kinetic equation, the rarefied gas flow between coaxial circular cylinders, of which the outer one is at rest while the inner one travels along its symmetry axis at a constant velocity, is studied. The problem is solved numerically in both the linear and nonlinear formulations by an implicit conservative method of second-order accuracy. The effect of the rarefaction, the cylinder radius ratio, and the inner cylinder velocity on the flow parameters is investigated. The limits of applicability of the linearized kinetic equation are established.     

Rotational relaxation of nitrogen in a viscous shock layer at low reynolds numbers

The viscous shock-layer model is used to examine relaxation of rotational degrees of freedom of molecular nitrogen in flow of a rarefied gas near the stagnation flow line around a sphere. It is shown that in the strongly smeared shock-wave region the rotational degrees of freedom can exhibit substantial nonequilibrium, leading to the increase of temperature and an increase of shock-layer thickness as compared with the equilibrium values. The influence of rotational relaxation on the shock-wave structure is discussed, and boundaries are found for the flow regions when rotational relaxation plays on important role,A comparison is made between the results of numerical calculations and experimentally obtained density profiles available in the literature near the stagnation line in flow of a rarefied gas over a sphere [1, 2]. Quite good agreement is obtained between the results of the calculation and experimental data over a wide range of Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1977.     

Asymptotic solution of the two-dimensional equations for a thin viscous rarefied shock layer on a cold surface

Hypersonic rarefied flow past blunt bodies is studied in the continuum-free-molecular transition regime. On the basis of an asymptotic analysis three rarefied gas flow patterns are established depending on the relation between the relevant parameters of the problem. In the first regime corresponding to a cold surface asymptotic solutions of the equations of a thin viscous shock layer are derived at low Reynolds numbers in the axisymmetric and plane cases. Simple analytical expressions for the pressure and the heat transfer and friction coefficients are obtained as functions of the freestream parameters and the body geometry. With decrease in the Reynolds number the coefficients approach the values corresponding to free-molecular flow. In this regime a similarity parameter for the hypersonic rarefied flow past bodies is determined. The asymptotic solutions are compared with numerical solutions and the results of direct statistical simulation by the Monte Carlo method.     

Investigation of slow monatomic gas flows past a circular cylinder

Slow low-Knudsen-number monatomic-gas flow past a circular cylinder is numerically investigated on the basis of a model kinetic equation. The gas flow is described by a new kinetic equation, from which the continuum equations for slow nonisothermal gas flows containing temperature stresses follow rigorously. It is shown that a closed convective-flow region arises near a nonuniformly heated cylinder in a slow gas flow if the flow impinges on the hot side of its surface. Using a new model of the Boltzmann equation makes it possible to study gas flows both in continuum and rarefied flow regimes.