气体运动论数值算法在微槽道流中的应用研究

简要介绍基于Boltzmann模型方程的气体运动论数值算法基本思想及其对二维微槽道流动问题数值计算的推广，并阐述适用于微尺度流动问题的气体运动论边界条件数值处理方法。通过对压力驱动的二维微槽道流动问题进行数值模拟，将不同Knudsen数下的微槽道流计算结果分别与有关DSMC模拟值和经滑移流理论修正的N—S方程解进行比较分析，表明基于Boltzmann模型方程的气体运动论数值算法对微槽道气体流动问题具有很好的模拟能力。     

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.     

稀薄流到连续流的气体运动论模型方程算法研究

通过引入碰撞松弛参数和当地平衡态分布函数对BGK模型方程进行修正，确定含流态控制参数可描述不同流域气体流动特性的气体分子速度分布函数的简化控制方程。发展和应用离散速度坐标法于气体分子速度空间，利用一套在物理空间和时间上连续而速度空间离散的分布函数来代替原分布函数对速度空间的连续依赖性。基于非定常时间分裂数值计算方法和无波动、无自由参数的NND耗散差分格式，建立直接求解气体分子速度分布函数的气体运动论有限差分数值方法。推广应用改进的Gauss-Hermite无穷积分法和华罗庚－王元提出的以单和逼近重积分的黄金分割数论积分方法等，对离散速度空间进行宏观取矩获取物理空间各点的气体流动参数，由此发展一套从稀薄流到连续流各流域统一的气体运动论数值算法。通过对不同Knudsen数下一维激波管问题、二维圆柱绕流和三维球体绕流的初步数值实验表明文中发展的数值算法是可行的。     

低速二维流动的粒子仿真研究

本文运用信息保存法对低速二维的流动现象进行模拟，考察了低速条件下的有限平板绕流以及微槽道气体流动问题。研究表明：在对低速流动的模拟过程中，运用IP法在能够获得较好的结果的同时，具有比DSMC方法更高的计算效率。     

基于多块对接网格的隐式气体运动论统一算法应用研究

基于玻尔兹曼模型方程的气体运动论统一算法(gas kinetic unified algorithm,GKUA) 给出了一种能模拟从连续流到自由分子流跨流域空气动力学问题的途径. 该算法采用传统计算流体力学技术将分子运动和碰撞解耦处理,若采用显式格式将受格式稳定条件限制,在模拟超声速流动尤其是近连续流和连续流区的流动时计算效率较低. 为了提高计算效率,扩展其工程实用性,采用上下对称高斯-赛德尔(LU-SGS) 方法和有限体积法构造了求解玻尔兹曼模型方程的隐式方法,同时在物理空间采用能处理任意连接关系的多块对接网格技术. 通过模拟近连续过渡区并排圆柱绕流问题,计算结果与直接模拟蒙特卡洛方法模拟值吻合较好,验证了该方法用于跨流域空气动力计算的可靠性与可行性.      

气体化学反应流动的DSMC／EPSM混合算法研究

发展了平衡粒子模拟方法(EPSM)，建立了与高温气体化学反应动力学理论相匹配的：EPSM耦合模型，并通过混合参数进行流区的自动识别，将：EPSM方法与蒙特卡罗直接模拟方法(OSMC)结合，构造了可模拟化学反应流动的DSMC／EPSM混合算法。应用该算法对汲及化学反应的二维高超音速竖板绕流流场进行模拟，将结果与DSMC方法的结果进行比较，验证了新算法对求解化学反应流动的可行性。将混合算法的计算效率与DSMC方法的计算效率进行比较，发现混合算法能够大大提高计算效率。     

绕流直接数值模拟误差分析

对于无边界绕流问题的计算流体力学模拟通常是将物体置于“足够大”的槽道中，而通过不断改变槽道尺寸以及离散网格密度，后验对比方式来检查模拟误差。本文结合多种经典流场理论，提出一种简单的先验误差估计方法来确定槽道尺寸以及相应的网格分布。在此方法中，对于槽道尺寸的确定基于线性叠加原理（即在极小雷诺数下采用Stokes理论解叠加，而在其他雷诺数条件下采用势流理论解叠加），来估计槽道尺寸对绕流结果的影响。而对网格尺寸与分布，则是使用多项式逼近中的基本误差分析工具，应用到速度边界层，远场势流，以及Rankine涡等简单流动，从而确定整个绕流问题中的离散误差。为了验证前面的理论分析结果，本文模拟了相当大雷诺数范围内的二维翼型以及三维圆球绕流，所得数值结果非常好地验证了理论分析。结果表明，对于Stokes流动问题，槽道尺寸需要大约100倍于物体特征尺寸来保证其结果与无边界绕流相差不超过1%；而在雷诺数超过大约100时，槽道尺寸只需10倍（二维绕流）或者5倍（三维绕流）于物体特征尺寸来达到同等精度。在此先验误差估计方法可应用于一般化的绕流问题。     

二维自适应非结构网格DSMC并行算法研究

研究了二维自适应非结构网格DSMC并行算法实现的过程.首先提出了一类非结构网格自适应策略,有效降低了网格尺度对计算结果的影响,提高了流场的分辨率;然后基于PC-CLUSTER群机并行体系结构与消息传递库MPI并行环境,利用分区并行思想,设计了非结构网格DSMC并行算法,节约了计算时间.利用For-tran90的动态分配内存技术编制了通用计算程序;最后对过渡流域高超声绕流进行了数值模拟,计算结果初步验证了算法的可行性与有效性.     

Navier-Stokes方程二阶速度滑移边界条件的检验

对微尺度气体流动,Navier-Stokes方程和一阶速度滑移边界条件的结果与实验数据相比,在滑移区相互符合,在过渡领域则显著偏离.为改善Navier-Stokes方程在过渡领域的表现,有些研究者尝试引入二阶速度滑移边界条件,如Cercignani模型,Deissler模型和Beskok-Karniadakis模型.以微槽道气体流动为例,将Navier-Stokes方程在不同的二阶速度滑移模型下的结果与动理论的直接模拟Monte Carlo(DSMC)方法和信息保存(IP)方法以及实验数据进行比较.在所考察的3种具有代表性的二阶速度滑移模型中,Cercignani模型表现最好,其所给出的质量流率在Knudsen数为0.4时仍与DSMC和IP结果相符;然而,细致比较表明,Cercignani模型给出的物面滑移速度及其附近的速度分布在滑流区和过渡领域的分界处(Kn=0.1)已明显偏离DSMC和IP的结果.     

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.     

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.     

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.     

Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Based on the Bhatnagar–Gross–Krook (BGK) Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non‐linear source terms. Based on the unsteady time‐splitting technique and the non‐oscillatory, containing no free parameters, and dissipative (NND) finite‐difference method, the gas kinetic finite‐difference second‐order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm for the gas dynamical problems from various flow regimes is developed. To test the reliability of the present numerical method, the one‐dimensional shock‐tube problems and the flows past two‐dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related flows indicate that both high resolution of the flow fields and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Gas-kinetic unified algorithm for plane external force-driven flows covering all flow regimes by modeling of Boltzmann equation

The nonequilibrium steady gas flows under the external forces are essentially associated with some extremely complicated nonlinear dynamics, due to the acceleration or deceleration effects of the external forces on the gas molecules by the velocity distribution function. In this article, the gas-kinetic unified algorithm (GKUA) for rarefied transition to continuum flows under external forces is developed by solving the unified Boltzmann model equation. The computable modeling of the Boltzmann equation with the external force terms is presented at the first time by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The conservative discrete velocity ordinate (DVO) method is utilized to transform the governing equation into the hyperbolic conservation forms at each of the DVO points. The corresponding numerical schemes are constructed, especially the forward-backward MacCormack predictor-corrector method for the convection term in the molecular velocity space, which is unlike the original type. Some typical numerical examples are conducted to test the present new algorithm. The results obtained by the relevant direct simulation Monte Carlo method, Euler/Navier-Stokes solver, unified gas-kinetic scheme, and moment methods are compared with the numerical analysis solutions of the present GKUA, which are in good agreement, demonstrating the high accuracy of the present algorithm. Besides, some anomalous features in these flows are observed and analyzed in detail. The numerical experience indicates that the present GKUA can provide potential applications for the simulations of the nonequilibrium external-force driven flows, such as the gravity, the electric force, and the Lorentz force fields covering all flow regimes.     

Rarefied gas flow in a triangular duct based on a boundary fitted lattice

The rarefied fully developed flow of a gas through a duct of a triangular cross section is solved in the whole range of the Knudsen number. The flow is modelled by the BGK kinetic equation, subject to Maxwell diffuse boundary conditions. The numerical solution is based on the discrete velocity method, which is applied for first time on a triangular lattice in the physical space. The boundaries of the flow and computational domains are identical deducing accurate results with modest computational effort. Results on the velocity profiles and on the flow rates for ducts of various triangular cross sections are reported and they are valid in the whole range of gas rarefaction. Their accuracy is validated in several ways, including the recovery of the analytical solutions at the free molecular and hydrodynamic limits. The successful implementation of the triangular grid elements is promising for generalizing kinetic type solutions to rarefied flows in domains with complex boundaries using adaptive and unstructured grids.