An analytical discrete-ordinates solution of the S-model kinetic equations for flow in a cylindrical tube

An integro-differential form of the linearized S-model kinetic equations for describing flow in a cylindrical tube is projected in such a way as to yield a pair of coupled transport equations that defines the desired velocity and heat-flow profiles. This system is then solved symbolically to yield a pair of coupled integral equations for the physical quantities required. At this point some transformations are carried out to yield a restatement of the original problem in terms of a “pseudo-problem” defined by plane-geometry variables. An analytical version of the discrete-ordinates method is then used to solve the pseudo-problem, and so, after both MATLAB and FORTRAN versions of the developed algorithm are implemented, results thought to be highly accurate are obtained for the case of diffuse reflection from the walls of a cylindrical tube. In addition to the velocity and heat-flow profiles, for the cases of Poiseuille flow and thermal-creep flow, the velocity slips, the heat-flow profiles evaluated at the wall, the particle-flow rates and the heat-flow rates for these two problems are reported for selected values of the tube radius.     

Generalized boundary conditions for the S-model kinetic equations basic to flow in a plane channel

An analytical version of the discrete-ordinates method is used to solve the classical problems of Poiseuille flow and thermal-creep flow in a plane channel. The kinetic theory for the rarefied-gas flow is based on the S model (a generalization of the BGK model), and in addition to the use of the diffuse-specular reflection model (based on a single accommodation coefficient) for describing particle scattering from the channel walls, the Cercignani-Lampis model defined in terms of normal and tangential accommodation coefficients is implemented. The established solution is tested numerically, and results for the velocity and heat-flow profiles, the particle-flow rate and the heat-flow rate thought to be correct to many significant figures are reported for various values of the channel width.     

Beyond direct simulation Monte Carlo (DSMC) modelling of collision environments

ABSTRACT

Direct simulation Monte Carlo (DSMC) models have been successfully adopted and adapted to describe gas flows in a wide range of environments since the method was first introduced by Bird in the 1960s. We propose a new approach to modelling collisions between gas-phase particles in this work – operating in a similar way to the DSMC model, but with one key difference. Particles move in a mean field, generated by all previously propagated particles, which removes the requirement that all particles be propagated simultaneously. This yields a significant reduction in computation effort and lends itself to applications for which DSMC becomes intractable, such as when a species of interest is only a minor component of a large gas mixture.     

The linearized Boltzmann equation: a concise and accurate solution of the temperature-jump problem

Polynomial expansion procedures, along with an analytical discrete-ordinates method, are used to solve the temperature-jump problem based on a rigorous version of the linearized Boltzmann equation for rigid-sphere interactions. In particular, the temperature and density perturbations and the temperature-jump coefficient are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that can be considered a new standard of reference.     

DSMC方法的压力边界条件实现

提出了一种实现DSMC压力边界条件的新方法,新方法不仅可以避免传统"通量法"造成的计算发散问题,又较之"入口平均法"有更快的收敛速度.使用这种方法,对不同压差驱动下等壁温微通道内的气体流动进行了模拟,并与传统的基于连续介质假设的滑移理论所得结果进行了比较.     

液晶投影仪液晶板自动汇聚系统的对焦算法分析

在建立液晶投影仪的液晶板自动聚焦合色系统基础上,对三种不同的自动对焦算法进行了理论模拟和实验分析,并选择了适合于液晶板自动对焦的算法.实验表明该算法满足汇聚系统的应用要求.     

圆筒内氢等离子体非定常化学反应羽流场DSMC分析

利用直接模拟Monte Carlo方法研究圆筒侧壁注入氢等离子体羽流场在8×10^-6s内的非定常流动特性.根据Bird的化学反应模型考虑离解-复合反应模型和电荷转移反应模型.在流场中注入H₂、H、金属原子X、H₂⁺和H⁺五种组分,研究离解-复合反应对流场中粒子分布和密度的影响,结果表明离解-复合反应使H₂数密度降低,H数密度增加,说明在流场中H₂的离解反应速率大于H的复合反应速率.加入电荷转移反应后H₂⁺数密度降低,H⁺数密度增加,对其他组分数密度没有显著影响.     

Searching for a near neighbor particle in DSMC cells using pseudo-subcells

LeBeau et al. (2003) [4] introduced the ‘virtual-subcell’ (VSC) method of finding a collision partner for a given DSMC particle in a cell; all potential collision partners in the cell are examined to find the nearest neighbor, which becomes the collision partner. Here I propose a modification of the VSC method, the ‘pseudo-subcell’ (PSC) method, whereby the search for a collision partner stops whenever a ‘near-enough’ particle is found, i.e. whenever another particle is found within the ‘pseudo-subcell’ of radius δ centered on the first particle. The radius of the pseudo-subcell is given by δ = Fd_n, where d_n is the expected distance to the nearest neighbor and F is a constant which can be adjusted to give a desired trade-off between CPU time and accuracy as measured by a small mean collision separation (MCS). For 3D orthogonal cells, of various aspect ratios, d_n/L ≈ 0.746/N^0.383 where N is the number of particles in the cell and L is the cube root of the cell volume. There is a good chance that a particle will be found in the pseudo-subcell and there is a good chance that such a particle is in fact the nearest neighbor. If no particle is found within the pseudo-subcell the closest particle becomes the collision partner.     

Peristaltic transport in a slip flow

Continuum-mechanic derivation of the entrainment of rarefied gases induced by a surface wave along walls (or peristaltic transport) in a confined parallel-plane microchannel is conducted by the perturbation method. Both no-slip and slip flow cases are investigated with the former ones matched with the previous approach by Fung and Yih. Critical reflux values due to first order slip-flow effects become trivial for the free pumping case, and decrease due to second order slip-flow effects after we compared them with no-slip cases. Received 27 August 1999 and Received in final form 10 January 2000     

Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics

We show that, in the theory of extended thermodynamics, rarefied monatomic gases can be identified as a singular limit of rarefied polyatomic gases. Under naturally conditioned initial data we prove that the system of 14 field equations for polyatomic gases in the limit has the same solutions as those of the system of 13 field equations for monatomic gases where there exists no dynamic pressure. We study two illustrative examples in the process of the limit, that is, the linear waves and the shock waves in order to grasp the asymptotic behavior of the physical quantities, in particular, of the dynamic pressure.     

Investigation of hypersonic flows through a cavity with sweepback angle in near space using the DSMC method

Near space has been paid more and more attentionin recent years due to its military application value.However,flow characteristics of some fundamental configurations(e.g.,the cavity) in near space have rarely been investigated due to rarefied gas effects,which make the numerical simulation methods based on continuous flow hypothesis lose validity.In this work,the direct simulation Monte Carlo(DSMC),one of the most successful particle simulation methods in treating rarefied gas dynamics,is employed to explore flow characteristics of a hypersonic cavity with sweepback angle in near space by considering a variety of cases,such as the cavity at a wide range of altitudes 20-60 km,the cavity at freestream Mach numbers of 6-20,and the cavity with a sweepback angle of 30°-90°.By analyzing the simulation results,flow characteristics are obtained and meanwhile some interesting phenomena are also found.The primary recirculation region,which occupies the most area of the cavity,causes pressure and temperature stratification due to rotational motion of fluid inside it,whereas the pressure and temperature in the secondary recirculation region,which is a small vortex and locates at the lower left corner of the cavity,change slightly due to low-speed movement of fluid inside it.With the increase of altitude,both the primary and secondary recirculation regions contract greatly and it causes them to separate.A notable finding is that rotation direction of the secondary recirculation region would be reversed at a higher altitude.The overall effect of increasing the Mach number is that the velocity,pressure,and temperature within the cavity increase uniformly.The maximum pressure nearby the trailing edge of the cavity decreases rapidly as the sweepback angle increases,whereas the influence of sweepback angle on velocity distribution and maximum temperature within the cavity is slight.     

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.     

The role of the dynamic pressure in stationary heat conduction of a rarefied polyatomic gas

The effect of the dynamic pressure (non-equilibrium pressure) on stationary heat conduction in a rarefied polyatomic gas at rest is elucidated by the theory of extended thermodynamics. It is shown that this effect is observable in a non-polytropic gas. Numerical studies are presented for a para-hydrogen gas as a typical example.     

具有局部稀薄气体效应的高超声速尖锥气动加热特征研究

以微钝尖锥为飞行器前缘模型,采用基于分子运动论的DSMC方法模拟不同前缘曲率半径的尖锥在高超声速来流下的气动热环境,计算驻点热流率,并与Fay-Riddell公式和其他修正理论作对比,研究具有局部稀薄气体效应的高超声速尖锥气动加热特征及其变化规律.发现修正的Cheng参数适合作为工程上判断驻点区域稀薄气体效应影响大小的判据.     

Convergence characteristics of a domain decomposition scheme for approximation of quantum forces

We investigate the convergence characteristics of a domain decomposition scheme to approximately compute quantum forces in the context of semiclassical Bohmian mechanics. The study is empirical in nature. Errors in the approximate quantum forces are compiled while relevant parameters in the numerical scheme are systematically changed. The compiled errors are analyzed to extract underlying trends. Our analysis shows that the number of Bohmian particles used in discretization has relatively weak influence on the error, while the length scale of interaction among subdomains controls the error in most cases. More precisely, the overall numerical error decreases as the length scale of interaction among subdomains decreases, if the error due to the truncation of the tail of the probability density distribution is adequately controlled. Our results suggest that it may be necessary to develop an efficient method to effectively control the error due to the truncation of the tail. Further studies, especially rigorous mathematical ones, should follow to understand and improve the behavior of the scheme.     

Chapman-Enskog and Burnett solutions for a simple, rigid-sphere gas: Numerical solutions using a subtraction technique

Chapman-Enskog and Burnett solutions are obtained for a simple, rigid-sphere gas through application of a subtraction technique to the linearized Boltzmann equation. The numerical results obtained converge and compare well with results reported previously using other techniques. This method is straightforward and may prove quite useful in addressing several other similar problems in the kinetic theory of gases.     

A continuum model of gas flows with localized density variations

We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows.     

Kinetic theory analysis of a binary mixture reacting on a surface

A mixture of two rarefied gases is considered between two parallel planes. On one side of the domain evaporation/condensation conditions are imposed, while on the other side accommodation at the temperature of the wall and chemical equilibrium conditions are considered. The small Knudsen number asymptotics of this problem is performed at the formal level, and fluid-dynamic equations are derived and then solved numerically. We discuss the possible appearance of a ghost effect in this situation.     

层间稀薄气体传热对多层绝热材料性能的影响分析

通过建立的热量传递模型,分析了不同的气体稀薄程度(Knudsen数)时,气体传热对多层绝热材料有效热导率和各层温度分布的影响。分析表明:由多层绝热材料真空度变化引起的稀薄气体传热量波动较大,在10—60层/cm层密度范围,真空度低于100Pa时,Kn数属于自由分子状态区域和中间压强区域,此时材料的有效热导率随残留气体热适应系数的增大而减小,并随着真空度的降低而增大;当残留气体为空气时,为保证多层材料的绝热性能,尽量维持真空度不低于10-2Pa。同时分析表明,为有效降低低真空下稀薄气体传热对多层绝热性能的影响,可以采用综合热适应系数较低的气体置换夹层中的空气,以减少低真空多层绝热材料的有效热导率,改善绝热性能。