Kinetic modeling of rarefied molecular gas dynamics
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摘要: 稀薄气体效应，是指气体在特征尺度与其分子平均自由程相当的系统中流动时出现的非平衡效应。相比于单原子气体，分子气体（每个气体分子包含两个及以上原子）流动因同时具有转动、振动等多种自由度的非平衡过程，其稀薄效应更为复杂。分子气体稀薄效应在航空航天、微机电系统和页岩气开采等民生、科技领域广泛存在，而描述该效应的动理学模型与数值模拟方法尚不成熟。本文从单原子气体与玻尔兹曼方程出发，介绍气体动理学建模的相关研究现状，针对分子气体特性详细讨论气体弛豫过程与输运系数的关系。针对典型的稀薄气体流动验证常用的模型方程的精度，并指出直接模拟蒙特卡罗方法在分子气体稀薄流动应用中存在的问题，即在体积黏性确定的情况下无法指定热流弛豫速率及恢复热导率。随后针对此问题，使用吴模型量化研究该弛豫速率在分子气体稀薄流动模拟中导致的宏观量的不确定性，并讨论从分子动力学模拟和瑞利-布里渊散射实验中减小甚至消除不确定性的方法。本文对涉及化学反应的稀薄气体流动建模有指导意义。Abstract: The rarefaction effects arise when the mean free path of gas molecules is comparable to the characteristic flow length. Compared to monatomic gas, the dynamics of rarefied molecular gas, which consists of two or more atoms in one gas molecule, is more complicated, due to the additional non-equilibrium effects in the rotational and vibrational motions. As the major constituent in the atmosphere of Earth and Mars, rarefied molecular gas dynamics has strong applications in aerospace science, micro-electro-mechanical system, shale gas extraction, and so on. However, the corresponding models and simulation methods are not sophisticated. Starting from the monatomic gas and the Boltzmann equation, we review the development of the kinetic modeling of rarefied gas and the essential connection between the transport coefficients and relaxation processes. Then, we focus on the relaxation process and transport coefficients of molecular gas, and introduce typical kinetic models with the systematic assessment of their accuracy. Meanwhile, we discuss the defect of direct simulation Monte Carlo method in the modeling of rarefied molecular gas, i.e., it cannot specify the thermal relaxation rates (hence cannot recover the thermal conductivities) when the bulk viscosity is determined. Finally, we use the Wu model to quantify the uncertainties, and discuss how to reduce or even eliminate the uncertainties based on the data from the molecular dynamics simulation and Rayleigh-Brillouin scattering experiment. This present article is instructive for the modeling of rarefied gas flows involving chemical reactions.
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图 1 典型稀薄气体流动及纳维-斯托克斯方程的预测误差：第一列图显示，当马赫数大于2时，纳维-斯托克斯方程低估正激波的厚度[15]；第二列图显示，当声波振动频率接近气体分子碰撞频率时，纳维-斯托克斯方程低估声波接收器的声强[16-17]；第三列图显示，低压下纳维-斯托克斯方程可能远远低估微纳米通道的质量流量[18]
Figure 1. Typical rarefied gas flows and the incapability of Navier-Stokes equations. Figures in the first column show that, when the Mach number is larger than 2, the Navier-Stokes equations underpredict the thickness of normal shock wave[15]. Figures in the second column show that the Navier-Stokes equations underpredict the sound pressure at the receiver[16-17]. Figures in the last column show that the Navier-Stokes equations might underestimate the mass flow rate significantly[18]
图 2 (左)两体碰撞前后的速度分布：由于动量和能量守恒，碰撞前后的相对速度分布在球体上并且通过球心；(右上)中心力场作用下的经典两体碰撞示意图，其中
$b$ 为瞄准距离，$\boldsymbol{k}$ 为沿两分子间最短距离方向的单位矢量；(右下) 直径为σ的硬球分子的两体碰撞Figure 2. (Left) Velocity redistribution after the binary collision. Due to the conservation of momentum and energy, the pre- and post-collision relative velocities fall in a sphere and pass through the sphere center. (Top right) Classical binary collision between molecules with central force, where
$b$ is the aiming distance, and$\boldsymbol{k}$ is the unit vector along the minimum distance between two colliding molecules. (Bottom right) Binary collision between HS molecules of the diameter σ
图 3 从DSMC模拟中提取热流的弛豫速率：(a)初始状态用于激发平动热流的速度分布函数，横坐标由最概然速率
$v_m$ 归一化； (b)初始状态用于转动热流的转动能分布函数，横坐标由$k_BT_0$ 归一化；(c, d)氮气平动热流和转动热流的弛豫过程，模拟参数为：式(78)中的非弹性碰撞概率$\varLambda=0.25$ ，施密特数$Sc=1/1.33$ ，转动自由度$d_r=2$ ，温度黏性指数$\omega=0.74$ . 图中数据来源于文献[60]Figure 3. Extraction of heat flux relaxation rates from DSMC simulations. The initial distribution of (a) molecular velocity and (b) rotational energy of nitrogen molecules in DSMC, where the abscissas are normalized by
$\sqrt{2k_BT_0/m}$ and$k_BT_0$ , respectively. (c, d) The evolution of heat fluxes and their time derivatives in Nitrogen.${\varLambda}=0.25$ in Eq. (78), the Schmidt number is$Sc=1/1.33$ , the rotational degree of freedom is$d_r=2$ , and the viscosity index is$\omega=0.74$ . The figures are modified from figure 1 in Ref. [60]
图 4 根据理论公式(76)从DSMC模拟中提取的热弛豫速率. 施密特数为
$Sc=1/1.33$ ，转动自由度为$d_r=2$ ; 对N2和HCl，温度黏性指数ω 分别取0.74和1，图中数据来源于文献[60]Figure 4. The extracted rates
$\boldsymbol{A}$ in the relaxation of heat fluxes from the DSMC simulation, as per Eq. (76). The Schmidt number is$Sc=1/1.33$ , and the rotational degree of freedom is$d_r=2$ ; the viscosity index for Nitrogen and hydrogen chloride is ω = 0.74 and 1, respectively. The figures are modified from figure 1 in Ref. [60]
图 5 室温下Eucken因子的解析解(77)与DSMC数值解的对比. DSMC中使用可变软球模型，自扩散系数 D取值为 D'，菱形、正方形、三角形分别表示平动、内能和总Eucken因子，实心点为解析解，空心点为DSMC结果，虚线表示300 K温度下实验测量的总Eucken因子，图中数据来源于文献[54]
Figure 5. The thermal conductivity obtained from the analytical solution (77) and the DSMC at room temperature, as a function of the inelastic collision probability
$\varLambda$ . The variable-soft-sphere model is used and the self-diffusion coefficient D in DSMC takes the value of D' . Filled diamonds, squares and triangles represent the translational, internal and total Eucken factors from Eq. (77), respectively, while the open symbols are the corresponding results from DSMC. Dashed lines show the total Eucken factor obtained from experiments at a temperature of 300 K. The figures are from Ref. [54]
图 6 不同动理学模型预测的马赫数为7的氮气正激波的密度、温度、应力偏量σ22和热流的对比
Figure 6. Comparisons in the density, temperature, stress, and heat flux between different kinetic models for a normal shock in Nitrogen gas with Mach number 7
图 7 平板库特流动中的密度、温度和垂直于流动方向的热流分布. 第一行和第二行的克努森数分别为
${Kn} =0.1、1$ ；基于对称性，另一半区域$-0.5\leqslant x_1\leqslant 0$ 没有显示Figure 7. Profiles of the density, temperature, and heat flux (perpendicular to the plates) in the planar Couette flow of the Knudsen number Kn = 0.1 (first row) and 1 (second row). Due to symmetry, the other half region
$-0.5\leqslant{}x_1\leqslant0$ is not shown
图 8 麦克斯韦妖驱动下的稀薄气体热蠕动, 其中速度和热流均沿
$ x_1 $ 方向. 第二行和第三行图对应的克努森数分别为Kn = 0.1和1，基于对称性，$ 0.5\leqslant{}x_2\leqslant1 $ 的区域没有显示Figure 8. Profiles of the vertical velocity and heat flux in the thermal creep flow driven by the Maxwell demon, where the Knudsen number in the second and third rows is Kn = 0.1 and 1, respectively. Due to symmetry, the other half region
$ 0.5\leqslant{}x_2\leqslant1 $ is not shown
图 9 克努森数为0.6的稀薄气体在二维方腔内的热蠕动
Figure 9. Rarefied gas flow driven by temperature gradient in the solid walls: (a) The streamlines and the contour of translational heat flux in the x1 direction; (b) Velocity profiles at x2 = 0 and 0.5; (b) Velocity profiles at x2 = 0.5; (d) The translational and rotational heat fluxes along the bottom wall
图 11 (第一行)热流弛豫速率对麦克斯韦妖驱动的热蠕动速度和热流分布的影响，以及吴模型中改变热弛豫速率的结果对比. 红色实线对应的
${{\boldsymbol{A}}}$ 数值来自于DSMC，蓝色阴影部分来自吴模型计算结果，对应的取值范围为$ {A_{rt}}\in[-0.3124,0.0] $ 、$ {A_{tr}}\in[-0.1250,0.0] $ ，其他参数为${Kn} = 0.2$ ,$ f_{tr} = 2.365 $ 以及$ f_{rot} = 1.435 $ ；(第二行)总Eucken因子$f_{eu}$ 保持不变，改变平动Eucken因子的结果对比，${Kn} = 0.2$ .图中数据来源于文献[60]Figure 11. (First row) Influence of the thermal relaxation rates in the creep flow driven by the Maxwell demon, and comparison of the results of varying the thermal relaxation rate in the Wu model. Red solid lines are the results with
${{\boldsymbol{A}}}$ extracted from DSMC, blue shade region shows the results from the kinetic model (103), with$ {A_{rt}}\in[-0.3124,0.0] $ and$ {A_{tr}}\in[-0.1250,0.0] $ . Other parameters are${Kn} = 0.2$ ,$ f_{tr} = 2.365 $ and$ f_{rot} = 1.435 $ . (Second row) Influence of the translational Eucken factor, while$f_{eu}$ is fixed. The Knudsen number is${Kn} = 0.2$ . The figures are from Ref. [60]
图 13 基于实验频谱(圆圈)和吴模型预测结果(线条)，提取N2的体积黏性与平动Eucken因子[54]：第一行的实线为计算光谱，通过计算给定
$ f_{tr} $ 和$ Z $ 范围内的最小误差；第二行给出了实验光谱与理论光谱之间的误差. 实验与计算中所用到的参数汇总在表 2中，频率$ f_s $ 由$ v_m/\ell $ 归一化Figure 13. Extraction of the bulk viscosity and translational Eucken factor of Nitrogen from the experimental spectra (circles)[54]. Lines in the first row show the spectra obtained from the Wu model, while those in the second row show the corresponding residuals between the experimental and theoretical spectra. Experimental conditions and extracted parameters are summarized in Table 2. The frequency
$ f_s $ is normalized by$ v_m/\ell $ 表 1 不同动理学模型和DSMC模拟中所使用的Eucken因子与热弛豫速率. N2的Eucken因子为
$f_{eu}=1.993$ Table 1. Eucken factors and heat flux relaxation rates used in different gas kinetic models and DSMC. The total Eucken factor of the Nitrogen at room temperature is
$f_{eu}=1.993$ ES-BGK Rykov Wu DSMC $f_{tr}$ 2.373 2.365 2.365 2.365 $f_{rot}$ 1.424 1.436 1.436 1.436 $A_{tt}$ 0.702 0.705 0.786 0.786 $A_{tr}$ 0 0 –0.201 –0.201 $A_{rt}$ 0 0 –0.059 –0.059 $A_{rr}$ 0.702 0.696 0.842 0.842 
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表 2 自发瑞利-布里渊散射实验中，N2相关参数汇总以及根据吴模型从实验数据中提取的体积黏性与平动/转动Eucken因子(剪切黏性与总热导率均使用国际单位制)
Table 2. Experimental conditions in the spontaneous Rayleigh-Brillouin scattering, and the extracted bulk viscosity, translational/rotational Eucken factors from the Wu model. The SI units are adopted for the shear viscosity and thermal conductivity
实验 $ T $/K $ 10^5\mu_s $ $ 100\kappa $ $ P $/bar $ \omega $ $ \mu_b/\mu_s $ $ f_{tr} $ $ f_{rot} $ (a) 254.7 1.57 2.28 2.563 0.80 0.48 2.16 1.65 (b) 275.2 1.67 2.44 2.784 0.78 0.61 2.25 1.53 (c) 296.7 1.77 2.60 3.000 0.76 0.69 2.31 1.47 (d) 336.6 1.95 2.88 3.400 0.74 0.94 2.43 1.33
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