Intense Condensation of a Molecular Gas

The kinetic problem of intense subsonic condensation of a polyatomic gas on a plane surface is solved by the method of semispatial moments. The contribution of the internal degrees of freedom to the total heat capacity is taken into account. The domains of existence of the Boltzmann equation solution are determined for the given method. Analytic expressions for calculating the gas concentration behind the Knudsen layer depending on the Mach number and temperature are derived for the cases of mon-, di-, and triatomic gases.     

Couette problem for the generalized Krook equation stress-peak effect

The Couette problem is the simplest problem of steady shear flow of rarefied gas in a region bounded by solid surfaces. This problem has been examined in the linear formulation by many authors, using either the linearized Krook equation or the moment methods (see [1]). It has recently been solved by the Monte Carlo method [2].The nonlinear problem of Couette flow with heat transfer for the Krook equation has been solved by reducing the problem to a system of integral equations [3] over a wide range of flat-plate velocities and temperature ratios and by the discrete-velocity method [4] for moderate plate velocities. In this article we solve the same problem for the generalized Krook equation [5] which approximates the Boltzmann equation for a pseudo-Maxwellian gas in accordance with the method suggested by the author [6, 7]. The generalized Krook equation was solved numerically by a modified discrete-velocity method which has been used by the author previously to solve the problem of shock wave structure [8].The primary case examined is that of pseudo-Maxwellian molecules, in which the viscosity is proportional to the temperature. The computations were made for Prandtl numbers of 1 and 2/3 over a wide range of Mach and Knudsen numbers as well as flat-plate temperature ratios. As we would expect, the Prandtl number effect is greatest for small Knudsen numbers. The flow velocity profiles are not very sensitive to variation of the Prandtl number (at least for pseudo-Maxwellian molecules).However, the most interesting result of the study is independent of the Prandtl number. Specifically, it was found that for any sufficiently high flat-plate velocities the friction stress, referred to the corresponding free molecular value, does not change monotonically with variation of the Knudsen number; instead, there is a peak. As far as the author is aware, this nonlinear effect has not been discussed previously in the literature (including articles [3, 4]).     

Half-space problem of the nonlinear Boltzmann equation for weak evaporation and condensation of a binary mixture of vapors

The half-space problem of evaporation and condensation of a binary mixture of vapors is investigated on the basis of the kinetic theory of gases. Assuming the Mach number of the normal component of the flow is small, a solution of the Boltzmann equation that varies slowly in the scale of the molecular mean-free-path (slowly varying solution) is introduced. Then a fluid-dynamic system that describes the behavior of the slowly varying solution is derived by a systematic asymptotic analysis. The analytical expression of the conditions allowing steady evaporation or condensation is derived from that system. We analyze the qualitative difference between the conditions in the evaporation and condensation cases: four conditions are needed in the former case while only one condition is required in the latter case. The present paper extends a earlier contribution of the first author for the BGK-type model equation [S. Takata, Half-space problem of weak evaporation and condensation of a binary mixture of vapors, in: Capitelli M. (Ed.), Rarefied Gas Dynamics, AIP, New York, 2005, pp. 503–508] to the Boltzmann equation. The extension is achieved by considering the linear stability of the far field in the case of evaporation and the H theorem, the monotonic decrease of the flux of Boltzmann's H function, in the case of condensation.     

Nonlinear Knudsen boundary layer on an infinitely thin permeable membrane

The state of a gas in the neighborhood of an infinitely thin permeable membrane whose sides have different temperatures is investigated. The dimensions of orifices in the membrane are much less than the mean molecular free path. The gas temperatures and pressures on both sides of the membrane outside the nonlinear Knudsen layers adjacent to the membrane and variations in these parameters in the Knudsen layers are determined. The investigation is carried out by solving the Boltzmann kinetic equation by means of the direct simulation Monte-Carlo method. The semiempirical Maxwell method is also used for consideration of the molecular flows on both sides of the membrane when analyzing the temperatures and the pressures. The solution to the nonlinear problem is compared with the solution to the linear problem of a jump temperature near a nonisothermal porous body obtained earlier.     

Asymptotic theory of the Boltzmann system,for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.     

基于扰动方程的超音速轴对称射流马赫波辐射研究

超音速不稳定波是导致剪切流失稳和转捩的主要不稳定模态,这种模态以马赫波的形式辐射到远场,从而产生强烈的声场。采用线性稳定性理论和非线性扰动方程(NLDE)分析,计算超音速轴对称射流不稳定波的扰动演化(Ma=2.1),对马赫波辐射进行研究,包括马赫波辐射方向、辐射源位置,以及随斯特劳哈尔数的变化情况。研究结果表明,在超音速轴对称射流中,马赫波沿固定方向辐射向远方,不稳定波相位沿另一方向传播,这两个方向相互正交;马赫波辐射源位置位于不稳定波压力幅值最大处;斯特劳哈尔数St越大,马赫波辐射的能力越强,辐射区域越集中。     

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.      

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.     

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.     

Condensation interaction between aerosol particles

The interaction between aerosol particles driven by the processes of condensation/evaporation on them is considered. It is assumed that the heat-transfer process can proceed in the free-molecular, transition or continuum regime. Asymptotic and numerical simulation methods (the DSMC method for solving a system of nonlinear kinetic Boltzmann equations and the statistical test-molecule method of solving the linear Boltzmann equation) are used. The characteristics of the forces of interaction between two aerosol particles are studied. The interaction forces are determined as functions of the flow regime (of the Knudsen number), the distance between the particles and their shape, the condensed vapor concentration, and the condensation coefficient. Approximations of these functions are constructed.     

Investigation of the hysteresis phenomena in steady shock reflection using kinetic and continuum methods

The problem of transition of planar shock waves over straight wedges in steady flows from regular to Mach reflection and back was numerically studied by the DSMC method for solving the Boltzmann equation and finite difference method with FCT algorithm for solving the Euler equations. It is shown that the transition from regular to Mach reflection takes place in accordance with detachment criterion while the opposite transition occurs at smaller angles. The hysteresis effect was observed at increasing and decreasing shock wave angle. Received September 1, 1995 / Accepted November 20, 1995     

Gas-kinetic description of shock wave structures by solving Boltzmann model equation

On the basis of the mesoscopic theory of Boltzmann-type velocity distribution function, the modified Boltzmann model equation describing the one-dimensional gas flows from various flow regimes is presented by incorporating the molecular interaction models relating to the viscosity and diffusion cross-sections, density, temperature and the dependent exponent of viscosity into the molecular collision frequency. The gas-kinetic numerical method for directly solving the molecular velocity distribution function is studied by introducing the reduced distribution functions and the discrete velocity ordinate method, in which the unsteady time-splitting method and the NND finite difference scheme are applied. To study the inner flows of non-equilibrium shock wave structures, the one-dimensional unsteady shock-tube problems with various Knudsen numbers and the steady shock wave problems at different Mach numbers are numerically simulated. The computed results are found to give good agreement with the theoretical, DSMC and experimental results. The computing practice has confirmed the good precision and reliability of the gas-kinetic numerical algorithm in solving the highly nonequilibrium shock wave disturbances from various flow regimes.     

Nonlinear Strong Shock Interactions: A Shock-Fitted Approach

This paper addresses nonlinear effects which result from the interaction of shock waves with vortices. A series of experiments are carried out, which involve the interaction of a strong shock wave with a single plane vorticity wave and a randomly distributed wave system. These experiments are first conducted in the linear regime to obtain a mutual verification of theory and computation. They are subsequently extended into the nonlinear regime. A systematic study of the interaction of a plane shock wave and a single vortex is then conducted. Specifically, we investigate the conditions under which nonlinear effects become important, both as a function of shock Mach number, M ₁, and incident vortex strength (characterized by its circulation Γ). The shock Mach number is varied from 2 to 8, while the circulation of the vortex is varied from infinitesimally small values (linear theory) to unity. Budgets of vorticity, dilatation, and pressure are obtained. They indicate that nonlinear effects become more significant as both the shock Mach number and the circulation increase. For Mach numbers equal to 5 and above, the dilatation in the vortex core grows quadratically with circulation. An acoustic wave propagates radially outward from the vortex center. As circulation increases, its upstream-facing front steepens at low Mach numbers, and its downstream-facing front steepens at high Mach numbers. A high Mach number asymptotic expansion of the Rankine--Hugoniot conditions reveals that nonlinear effects dominate both the shock motion and the downstream flow for ΓM ₁ > 1. Received 28 June 1997 and accepted 25 November 1997     

Zero-Dissipation Limit of Solutions with Shocks for Systems of Hyperbolic Conservation Laws

We consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero. Our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers. These interactions generate four different wave patterns: initial layers, shock layers, diffusion waves and coupling waves. We study the propagation and interactions of the four wave patterns by a detailed pointwise analysis. (Accepted February 19, 1998)     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

浅水区海底管道周围海床渗透压力计算研究

我国海上油田开采起步较晚,大部分油田处于浅水区,因此,在设计管道时,应充分考虑由浅水区波浪引起的管道周围海床渗流力。根据浅水波相关假设,考虑自由水面非线性影响,推导出椭圆余弦波的波面方程,在此基础上进一步得到一个关于速度势的表达式,并根据该表达式得出作用于海床表面的波压公式。考虑海床土的压缩性,推导出一阶近似椭圆余弦波作用下浅水区埋置管道周围海床的渗流压力解析解,最后将计算结果与大型水槽试验及以往研究成果作对比。结果表明,在椭圆余弦波的作用下,由一阶椭圆余弦波理论得到的计算结果与试验结果规律基本一致,与相似工况下的现有理论成果数值基本相同,具有一定的可行性和工程价值。     

Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics

Four basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer’s problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier–Stokes–Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.     

Stability of solitary waves in dispersive media described by a fifth-order evolution equation

The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well-known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.     

Invariant sub-manifolds and modes of nonlinear autonomous systems

I.IntroductionIntheengineeringanalysis,themethodsoflinearmodeexpansionareusedtoapproximatethemotionofthenonlinearsystems(suchastheGalerkinmethods).Becauseoflackinggrounds,theresultsarenotreliable.Forthisreason,thestudiesofnonlinearmodesarouseconsiderableinterests.Buttherearemanydifficultiescausedbyitscomplexity.Manyscentistsandengineerspayattentiontothedefinitionandcomputationofnonlinearmodesthathavenotbeensolvedcompletely.Inthe60's,Rosenbergputforwardadefinitionofnonlinearmodesforconservati…     

A numerical investigation of the steady evaporation of a polyatomic gas

The evaporation and condensation of a polyatomic vapor in contact with its condensed phase has received much less attention than the monatomic case. In this paper we investigate the structure of the Knudsen layer formed in the steady evaporation of a vapor whose molecules behave as rigid rotators. The vapor motion is obtained by the numerical solution of the Boltzmann equation by the Direct Simulation Monte Carlo (DSMC) method. The obtained results are also compared with the solutions of a simplified kinetic BGK-like model equation. It is shown that density and temperature drops across the Knudsen layer are reasonably well reproduced by approximate methods proposed in the literature.