CONSTRUCRION AND ANALYSIS OF A NEW COMPUTABLE MODEL FOR BOLTZMANN EQUATION
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摘要: 临近空间飞行器因各部件尺寸差异较大, 在高空高马赫数条件下飞行会出现多流区共存的多尺度复杂非平衡流动现象, 流场中的气体分子速度分布函数与当地位置、流场分子速度、气体密度、流动速度、温度、热流矢量、应力张量等相关. 通过分析玻尔兹曼方程的一阶查普曼−恩斯科近似解, 构造了一种同时考虑热流矢量和应力张量影响、满足玻尔兹曼方程高阶碰撞矩的跨流域统一可计算模型方程, 并在数学上分析了其守恒性、H定理等基本属性, 证明了新模型方程与玻尔兹曼方程的相容性, 给出了新模型与现有模型如沙克霍夫(Shakhov)模型等的递进关系, 基于碰撞动力学确定了各流域统一气体分子碰撞松弛参数表达式. 在气体动理论统一算法中采用新建模型及现有模型模拟了一维激波结构、二维近空间飞行环境平板和多体圆柱干扰流动, 并与直接模拟蒙特卡洛方法对比分析, 结果表明在流场中粘性效应显著的区域新建模型能更好地捕捉激波位置, 尤其是在激波内部新模型模拟的宏观参数分布与直接模拟蒙特卡洛方法结果符合更好, 验证了新模型的有效性和可靠性, 同时说明在非平衡显著的流动区域碰撞松弛模型受多参数共同作用的影响.
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关键词:
- 玻尔兹曼方程 /
- 气体动理论统一算法 /
- 查普曼−恩斯科近似解 /
- 考虑热流矢量与应力张量影响的碰撞模型 /
- 气体分子碰撞松弛参数
Abstract: Due to large differences of geometric scale between the components of near space vehicles, the multi-scale complex non-equilibrium flow phenomenon will appear in many flow field regions when vehicles flying at high Mach number and high altitudes. In those regions the gas molecular velocity distribution functions are related to the local molecular velocities and macroscopic parameters, such as velocities, temperatures, heat flux vectors and stress tensors. By analyzing the first-order Chapman−Enskog approximate solution of Boltzmann equation, a new computable collision relaxation model is constructed, which considers the influence of heat flux vector and stress tensor, and satisfies the high-order collision moments of Boltzmann equation. The basic properties such as conservation law and H theorem are analyzed mathematically. The compatibility between the new model equation and Boltzmann equation is proved. The relationships between the new model and old models such as Shakhov and Belyi models are given. The expression of the collision relaxation parameter is determined by using molecular collision dynamics. As examples, one dimensional shock profiles and two dimensional flows around a flat plate and two side-by-side cylinders in near space environments are simulated by gas kinetic unified algorithm with different models. By comparing with results of DSMC method, it shows that in one dimensional problems the results of Shakhov model with heat flux is better than the new model because of smaller 1D shear stress, but in two dimension the new model can capture the position of shock wave better than the other two models due to higher dimensional shear stresses leading to more distinct viscosity effect. Especially for macro parameters in shock waves, the results of the new model accord with those of DSMC better. Then the validity and reliability of the new model is verified. These results illuminate that the collision relaxation model is influenced by multi-parameters together in the flow field when the non-equilibrium effects are quite distinct. -
图 1 不同模型模拟的激波轮廓与直接模拟蒙特卡洛方法对比
Figure 1. Comparison of shock profiles simulated by GKUA with different models and DSMC
图 2 不同模型模拟的无量纲热流和切应力对比
Figure 2. Comparison of non-dimensional heat fluxes and shear stresses simulated by GKUA with different models
图 3 本文模拟的温度云图与直接模拟蒙特卡洛方法结果对比
Figure 3. Comparison of temperature contours simulated by GKUA with new model and DSMC
图 4
$x $ = 0.1 m截面上压力和温度沿y方向不同碰撞模型的统一算法计算值与直接模拟蒙特卡洛方法模拟结果的对比Figure 4. Comparison of pressure and temperature simulated by GKUA with different collision models and DSMC along the direction of y on the section of
$x $ = 0.1 m
图 5
$x $ = 0.5 m截面上压力和温度沿y方向不同碰撞模型统一算法计算结果与直接模拟蒙特卡洛方法模拟值比较Figure 5. Comparison of pressure and temperature simulated by GKUA with different collision models and DSMC along the direction of y on the section of
$x $ = 0.5 m
图 6
$x $ = 0.8 m截面上压力和温度沿y方向不同碰撞模型统一算法模拟结果与直接模拟蒙特卡洛方法的对比Figure 6. Comparison of pressure and temperature simulated by GKUA with different collision models and DSMC along the direction of y on the section of
$x $ = 0.8 m
图 7 驻点线上不同碰撞模型统一算法计算的温度分布与直接模拟蒙特卡洛方法模拟值比较
Figure 7. Comparison of temperature simulated by GKUA with different collision models and DSMC along the stagnation line
图 8 3种模型方程计算得到的温度分布云图与直接模拟蒙特卡洛方法结果的对比
Figure 8. Comparison of temperature contours simulated by GKUA with three collision models and DSMC
图 9
$y $ = 0 m截面上压力、温度和速度沿$ x $ 方向不同碰撞模型模拟结果与直接模拟蒙特卡洛方法的对比Figure 9. Comparison of pressure, temperature and velocity simulated by GKUA with different collision models and DSMC along the direction of x on the section of
$y $ = 0 m表 1 不同模型/方法所得轴向力系数及其偏差与法向力系数
Table 1. The axial and normal force coefficients by different models and DSMC method and errors between axial force coefficients
Model/Method Ca Error/% Cn DSMC 1.287 0 — 3.82 × 10−4 New model 1.3458 4.57 5.76 × 10−3 Shakhov model 1.3760 6.92 8.25 × 10−3 Belyi model 1.3261 3.04 3.90 × 10−3 
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