稀薄气体流动模拟的通用合成迭代算法

曾嘉楠; 张颜兵; 李琪; 苏微; 吴雷

doi:10.6052/1000-0992-26-007

稀薄气体流动模拟的通用合成迭代算法

doi: 10.6052/1000-0992-26-007 cstr: 32046.14.1000-0992-26-007

曾嘉楠¹,
张颜兵²,
李琪³,
苏微^{1, 4},
吴雷^2, ,

1.
香港科技大学, 新兴跨学科领域学部, 香港特别行政区 999077
2.
南方科技大学, 力学与航空航天工程系, 深圳 518055
3.
中山大学, 航空航天学院, 深圳 518107
4.
香港科技大学, 数学系, 香港特别行政区 999077

基金项目: 感谢南方科技大学工程多尺度计算中心成员对GSIS方法发展的贡献. 感谢国家自然科学基金原创探索计划项目(基金号: 12450002)资助.

详细信息

作者简介:
吴雷,

通讯作者:
wul@sustech.edu.cn

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- 文章访问数: 52
- HTML全文浏览量: 9
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- 被引次数: 0
出版历程
- 收稿日期: 2026-02-06
- 录用日期: 2026-04-02
- 网络出版日期: 2026-04-08

General synthetic iterative scheme for the simulation of rarefied gas flows

ZENG Jianan¹,
ZHANG Yanbing²,
LI Qi³,
SU Wei^{1, 4},
WU Lei^{2
, ,}

1.
Division of Emerging Interdisciplinary Areas, Academy of Interdisciplinary Studies, The Hong Kong University of Science and Technology, Hong Kong SAR, China
2.
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, 518055 Shenzhen, China
3.
School of Aeronautics and Astronautics, Sun Yat-sen University, Shenzhen, 518107, China
4.
Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong SAR, China

More Information

Corresponding author: wul@sustech.edu.cn

摘要

摘要: 稀薄气体非平衡输运问题广泛存在于航空航天、真空技术、微纳系统及惯性约束聚变等关键领域. 特别是在航天器再入大气层、临近空间超声速飞行等极端过程中, 流动呈现出极强的跨流域特征, 并伴随分子内能激发、化学反应及辐射等复杂的多物理场耦合效应. 这些特征显著提升了动理学求解难度, 使得常规数值方法面临严峻的计算瓶颈, 制约了大规模工程模拟的精度与效率. 针对上述挑战, 本文系统介绍了兼具渐近保持性与快速收敛特性的通用合成迭代算法(general synthetic iterative scheme, GSIS). 该方法的核心在于构造了与动理学方程物理一致的宏观合成方程, 利用具有信息传播效率优势的抛物型宏观系统, 有效引导双曲型动理学方程的演化, 从而突破了计算网格与时间步长受限于分子碰撞尺度的固有瓶颈. 理论分析与数值验证表明, GSIS不仅在连续流极限下能够严格恢复宏观流体力学描述, 更在全流域范围内展现出卓越的迭代收敛效率. 更具优势的是, GSIS框架具有极强的模型兼容性与算法扩展性. 本文通过大量典型算例, 重点展示了该方法在多原子气体、高温辐射、多组分混合及非定常复杂运动等问题中的高精度与高效率表现. 同时, GSIS算法机制可与随机性粒子算法深度融合, 成功实现了对直接模拟蒙特卡洛框架下玻尔兹曼及Enskog方程的显著加速. 此外, 本文还展示GSIS在跨流域流动外形优化, 流动稳定性分析以及湍流-稀薄流耦合问题中的发展, 展现其在临近空间高超声速飞行中气动特性优化, 转捩与湍流等前沿领域的应用前景. GSIS为复杂非平衡稀薄气体流动的多尺度数值模拟提供了重要工具, 并为面向工程应用的高可靠、高效率仿真与优化设计提供了强有力的理论支撑与技术路径.
- 稀薄气体动力学 /
- 玻尔兹曼方程 /
- 多尺度方法 /
- 合成迭代算法 /
- 伴随优化 /
- 流动稳定性
Abstract: Rarefied gas transport is prevalent in critical fields such as aerospace, vacuum technology, micro- and nano-systems, and inertial confinement fusion. Particularly in extreme processes like spacecraft atmospheric reentry and near-space hypersonic flight, the flow exhibits prominent multiscale characteristics, accompanied by complex multiphysics coupling effects including molecular internal energy excitation, chemical reactions, and radiation. These features significantly increase the complexity of kinetic modeling, leading to severe computational bottlenecks for conventional numerical methods and restricting the accuracy and efficiency of large-scale engineering simulations. To address these challenges, this paper systematically introduces the general synthetic iterative scheme (GSIS), a multiscale numerical method characterized by both fast-convergence and asymptotic-preserving properties. The core of this method lies in the construction of macroscopic synthetic equations that are physically consistent with the kinetic equations. By leveraging the superior information propagation efficiency of parabolic macroscopic systems to guide the evolution of hyperbolic kinetic equations, GSIS breaks the inherent bottleneck where computational grids and time steps are constrained by the molecular collision scales, enabling unified and efficient simulation across all flow regimes. Theoretical analysis and numerical validation demonstrate that GSIS not only rigorously recovers the macroscopic fluid dynamics description in the continuum limit, but also exhibits exceptional iterative convergence efficiency across the entire range of Knudsen numbers. Furthermore, the GSIS framework possesses remarkable model compatibility and algorithmic extensibility. Through a variety of typical benchmarks, this paper highlights its high-precision and high-efficiency performance in problems involving polyatomic gases, high-temperature radiation, multi-component mixtures, and unsteady complex flows. Concurrently, the GSIS mechanism can be deeply integrated with stochastic particle algorithms, achieving significant acceleration of the Boltzmann and Enskog equations within the Direct Simulation Monte Carlo framework. Additionally, this paper presents the recent progress of GSIS in multiscale aerodynamic shape optimization, flow stability analysis, and turbulence-rarefaction interactions, showcasing its promising applications in frontier areas such as transition and turbulence in near-space hypersonic flight. Overall, GSIS provides an essential tool for multiscale numerical simulations of rarefied gas flows, and offers strong theoretical support and practical pathways for high-reliability, high-efficiency engineering simulations and optimization.
- Rarefied gas dynamics /
- Boltzmann equation /
- multiscale methods /
- general synthetic iterative scheme /
- adjoint optimization /
- flow stability
注释:

1) 1在远离边界的地方, 方程(19)变为扩散方程. 因此, 该方法在中子输运、辐射等问题上被称为扩散合成加速方法(Diffuse Synthetic Acceleration). 在靠近边界的地方, 即克努森层内, 描述稀薄气体效应的高阶项始终起到非常重要的作用.

2) 2实际上, 按照郭照立等人的一致渐近保持(unified-preserving)概念(Guo et al. 2023), 在GSIS-I框架下, 我们可以构造出渐近保持到超越NSF的宏观方程, 例如Grad系列矩方程(Torrilhon 2016). 然而, 从实际计算的角度来看, 这类方程数量庞大且形式复杂, 难以直接与传统CFD方法兼容.

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图 1 常规迭代求解一维泊肃叶流动(Wang et al. 2018). (左) 收敛到稳态解所需的迭代次数随克努森数的变化. (右) 克努森数为0.001时, 使用不同网格数量计算得到的流体速度剖面. M10表示空间被均分为10个区间

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图 2 通过傅里叶稳定性分析从线性Shakhov模型计算的误差衰减率(Su et al. 2020c, Zhu et al. 2021). 阈值稀薄参数$ \tau_{th} $定义在公式(34)

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图 3 一维泊肃叶流动的合成迭代算法流程图(吴雷和李琪2025)

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图 4 一维库埃特流动的常规迭代(a)和合成迭代(b)对比(Su et al. 2019). 稀薄参数为$ \delta_{rp}=100 $. 图例中的数字代表迭代步数

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图 5 两偏心圆柱间的库埃特流的速度云图和流线(Su et al. 2020b). GSIS和常规迭代的结果分别绘制在每个小图的左右区域. (a, b) $ \delta_{rp}=1000 $. 虚线是速度无滑移边界下的NSF方程的速度等值线. (c, d) $ \delta_{rp}=10 $. 流体速度已用外圆柱速度$ u_w $归一化

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图 6 GSIS单次迭代流程: 首先求解一次动理学方程(38)获得速度分布函数$ f^{k+1/2} $; 随后求解$ M $次合成方程(42)得到宏观量$ {\boldsymbol{W}}^{k+1} $; 最后按式(43)对速度分布函数进行更新得到$ f^{k+1} $. 经数十次迭代循环后, 即可收敛至稳态解

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图 7 GSIS并行计算(Zhang et al. 2024): 计算区域在物理空间上划分为$ N_x $个子域. 在每个物理子域内, 进一步分配$ N_v $个处理器, 以实现动理学方程在离散速度空间上并行求解. 宏观合成方程则在每个物理子域上独立求解

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图 8 类X-38飞行器表面的当地克努森数$ {\rm{Kn}}_{{\rm{local}}} $. 来流状态为$ {\rm{Kn}}=0.00443 $(特征长度取$ L=0.1 $米), $ {\rm{Ma}}=8 $, 攻角$ 20^\circ $

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图 9 阿波罗返回舱周围的气体-辐射耦合流动模拟. (a)通过傅里叶稳定性分析计算常规迭代与GSIS中误差衰减率随$ {\rm{Kn}}_{{\rm{gas}}}|\theta| $的变化, 其中情形A-C对应不同的$ \left({\rm{Kn}}_{{\rm{gas}}}/ {\rm{Kn}}_{{\rm{photon}}}\right)\times\tilde{\sigma}_R $组合, 具体参数可参考文献(Zeng et al. 2026). (b) 不同Kn下的GSIS与常规迭代残差收敛历程的对比. (c-h) $ {\rm{Kn}}=0.5 $时的温度分布及沿流向的热流密度分布. $ {\rm{Kn}}_{{\rm{photon}}} = 10 $, $ {\tilde{\sigma}_R} = 0.015 $, $ {\rm{Ma}}=15 $

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图 10 质量比为10的两组分喷管流动(Zeng et al. 2024). (a, b)基于各组分当地声速定义的轻、重组分马赫数等值面分布. (c, d)中心线上的数密度与水平速度分布, 其中方形与三角形符号对应轻组分与重组分, 速度以轻组分最概然速率归一化

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图 11 在GSIS计算框架中引入任意拉格朗日-欧拉方法与重叠网格技术, 模拟方腔流动中的颗粒运动(Zeng et al. 2025). (a) 初始时刻, 在方腔中心放置一个直径为$ 0.14L $、密度是气体密度10倍的圆形颗粒. 颗粒表面与流场网格采用重叠网格技术进行耦合. (b) 结合刚体三自由度方程求解得到的颗粒运动轨迹

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图 12 登陆器非定常模拟(Zeng et al. 2025). (a, b) 重叠网格装配结果, 白色和红色单元为插值单元. (c-f) 不同时间步下登陆器的姿态, 流场速度及地面压力分布. (g) 不同时刻(以相位$ \phi=180^\circ (\omega t+\text{π})/\text{π} $表征), 地面所受压力在$ y=0, x\in[-100, 200] $上的分布

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图 13 攻角20°、$ {\rm{Kn}}=0.01 $、$ {\rm{Ma}}=2 $条件下, 在满足升力与体积不降低约束下, 优化NASA类X-38再入飞行器外形, 实现减阻21.8% (Zhang et al. 2025). (第一行)优化前(左)后(右)网格. (第二、三行)从左至右、从上至下, 各子图依次展示初始外形以及第一至第五次优化迭代后的外形, 及对应的压力场. (第四行)优化前后(分别用灰色和黄色表面表示)外形对比

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图 14 (a, b)在线性泊肃叶流问题中, 每隔$ N $步常规迭代作用一次GSIS后得到的速度剖面(Luo and Wu 2024). 参考解由GSIS在稠密网格下获得(Wu et al. 2017); 而其余结果则是在空间区域采用21个非均匀分布的网格点离散得到的. (c)DIG的误差衰减率

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图 15 DIG加速DSMC的流程图. 该周期重复进行直到收敛(Luo and Wu 2024). 与GSIS流程不同之处在于, 在一个DIG迭代周期里, DSMC需要执行$ N_d $次

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图 16 DIG模拟方柱绕流(Hu et al. 2025a). (a) 方柱附近的计算网格, 上半部分为DSMC, 下半部分为DIG. (b) 驻点线上的DIG与DSMC的网格尺寸. (c, d) 驻点线上的温度收敛历史. Ma=5, Kn=0.01

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图 17 分子质量比为100、来流马赫数为10的二元混合气体的圆柱绕流(Luo et al. 2026). 第一行和第二行的结果分别对应全局克努森数为0.1和0.01. 线条和背景云图分别代表DIG和DSMC结果. (第三行)驻点线上的宏观量

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图 18 多孔介质中稠密气体流动(Hu et al. 2025b). (第一行)Enskog数(En, 分子直径与平均自由程之比)为1时的密度等值线图(虚线白线和有色背景分别表示DIG和DSMC结果)以及速度流线. (第二、三行)在$ x=0 $处的密度和水平动量密度. 左列和右列对应的克努森数分别为0.05和0.5

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图 19 稀薄来流和逆向湍流射流的相互作用(Tian and Wu 2025). (a, b) 通过GSIS(上半部分)和GSIS-SST(下半部分)模拟得到的温度云图(以来流温度归一化)和流线. (c, d) 表面压力和热流

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图 20 稀薄来流和侧向湍流射流的相互作用(Tian et al. 2026). (a) GSIS-SST模拟的70公里高度、15°攻角时的温度等高线和侧向射流(位于$ x_1=0.1 {\rm{m}} $, $ x_2=0 {\rm{m}} $)的流线. (b) 模型尖前缘模型下表面的俯仰力矩系数

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表 1 不同克努森数下喷管流动模拟的计算开销. 常规迭代与GSIS模拟三维流动, DSMC模拟二维轴对称流动; 所有模拟均在256个计算核上运行

Kn	DSMC		常规迭代		GSIS		加速比核时
Kn	核时	迭代步数	核时	迭代步数	核时	迭代步数	加速比核时
0.1	5	303	75.8	61	29.9	4.9	2.5
0.01	1030	3939	1007.6	50	26.1	78.8	38.6
0.001	-	$>15 \; 000 $	$>3837 $	50	25.8	$> $300	$> $148.7

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表 2 常规迭代与GSIS方法在平面声波问题中迭代次数的比较(Wu 2026). $ m $为外迭代步数, $ \Sigma k $表示总迭代次数, $ p $由NSF方程预测的特征值给出.

波数$ K $	行波					熵波
	常规迭代		GSIS		改进GSIS	常规迭代		GSIS		改进GSIS
	$ m $	$ \Sigma k $	$ m $	$ \Sigma k $	$ \Sigma k $	$ m $	$ \Sigma k $	$ m $	$ \Sigma k $	$ \Sigma k $
0.01	4	736534	4	89	7	10	1459784	9	341	20
0.1	6	11505	6	98	11	8	11840	11	315	20
1	21	767	21	250	28	18	554	18	231	29

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