1.  高超声速多尺度流动模拟

       近年来，新型高超声速飞行器不断涌现，飞行高度突破了传统航空领域的范围，空天一体化趋势明显。跨流域飞行带来的稀薄气体效应不可忽略，对于这类多尺度流场的模拟和气动特性分析，单一尺度的计算方法，如基于NS方程的计算流体力学（CFD）和基于分子的直接模拟蒙特卡洛（DSMC）方法，均不能同时满足精度和效率的要求。课题组与合作者发展了耦合分子运动和碰撞的统一随机粒子（USP）方法[1, 2]，如图1所示，突破了传统粒子方法的时空步长限制，将粒子方法在近连续流域的计算效率提高了几个数量级，并发布了基于USP方法的开源求解器SPARTACUS[3]，可高效模拟三维高超声速跨流域流动, 如图2所示。我们已应用SPARTACUS软件成功模拟了多种高速气体流动，包括多原子气体分子的内能激发等效应[4]。近期，我们将以上多尺度计算思想拓展到了Fokker-Planck模型，发展了多尺度随机粒子（MSP）方法，同样具有时空二阶精度[5]。同时，我们采用USP方法对可压缩衰减各向同性湍流开展了多尺度模拟，阐明了分子热涨落对湍流谱和湍流可预测性的影响[6]。

图1 DSMC和USP方法对比示意图

图2 SPARTACUS算例示意图和结果对比

[1] F. Fei, J. Zhang*, J. Li, and Z. H. Liu*, "A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows," Journal of Computational Physics 400, 108972 (2020).

[2] K. Feng, Z. Cui, P. Tian, and J. Zhang*, "A unified stochastic particle method with spatiotemporal adaptation for simulating multiscale gas flows," Journal of Computational Physics 505, 112915 (2024).

[3] K. Feng, P. Tian, J. Zhang*, F. Fei, and D. Wen, "SPARTACUS: An open-source unified stochastic particle solver for the simulation of multiscale nonequilibrium gas flows," Comput. Phys. Commun. 108607 (2023).

[4] P. Tian, K. Feng, Q. Ma, Z. Li, J. Zhang*, "Unified stochastic particle simulation of polyatomic gas flows using SPARTACUS," Computers & Fluids 265, 105987 (2023).

[5] Z. Cui, K. Feng, Q. Ma, and J. Zhang*, "A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows," Journal of Computational Physics 520, 113458 (2025).

[6] Q. Ma, C. Yang, S. Chen, K. Feng, Z. Cui, and J. Zhang*, "Effect of thermal fluctuations on spectra and predictability in compressible decaying isotropic turbulence," Journal of Fluid Mechanics 987, A29 (2024).

       2. 空天飞行的界面动力学

 空天飞行的非平衡效应不仅存在于气体流动过程，也存在于界面多相作用过程，如壁面滑移和催化复合等。针对高超声速条件下气固相互作用的复杂物理化学过程，课题组采用多尺度研究策略，从微观层次的分子动力学（MD）计算出发，再经过介观层次的分子动理论分析，可从理论上得到宏观层次的滑移边界条件[7, 8, 9]，如图3所示，为气固相互作用模型的构建提供了完整的思路。针对航空飞行的结冰问题，课题组发展了CFD和MD耦合的多尺度计算方法[10, 11]，如图4所示，可精确预测气液固移动接触线的动力学行为。

图3 气固相互作用的多尺度计算与建模

图4 气液固三相界面的多尺度耦合计算

[7] P. Luan, H. Yang, Q. Ma, and J. Zhang*, "A theoretical derivation of slip boundary conditions based on the Cercignani–Lampis–Lord scattering model," Journal of Fluid Mechanics 999, A36 (2024).

[8] J. Zhang*, P. Luan, J. Deng, P. Tian, and T. Liang, "Theoretical derivation of slip boundary conditions for single-species gas and binary gas mixture," Physical Review E 104, 055103 (2021).

[9] J. Deng, J. Zhang*, T. Liang, J. Zhao, Z. Li, and D. Wen, "A modified Cercignani–Lampis model with independent momentum and thermal accommodation coefficients for gas molecules scattering on surfaces," Physics of Fluids 34, 107108 (2022).

[10] J. Zhang*, M. K. Borg, and J. M. Reese, "Multiscale simulation of dynamic wetting," International Journal of Heat and Mass Transfer 115, 886 (2017).

[11] H. Liu, J. Zhang*, P. Capobianchi, M. K. Borg, Y. Zhang, and D. Wen, "A multiscale volume of fluid method with self-consistent boundary conditions derived from molecular dynamics," Physics of Fluids 33, 062004 (2021).

   3. 多尺度计算和机器学习耦合

       在非平衡流动中存在复杂的物理化学过程，导致NS方程中基于经验假设的应力和热流本构关系失效，如何构建非平衡条件下的本构关系和控制方程，是有挑战性的工作。课题组将基于物理的分子模拟和基于数据的机器学习方法相结合，通过数据驱动发现隐藏在分子模拟背后的流体力学控制方程，如图5所示，为微观和宏观耦合提供了新思路[12]。近年来，课题组发展了满足量纲齐次性约束的基因表达式编程算法，结合分子模拟产生的数据，可用于非平衡条件下的本构关系和宏观方程预测，显著提升了宏观方程描述非平衡特征的能力[13,14]。

图5 基于分子模拟的数据驱动发现宏观方程

[12] J. Zhang*, and W. Ma, "Data-driven discovery of governing equations for fluid dynamics based on molecular simulation," Journal of Fluid Mechanics 892, A5 (2020).

[13] H. Xing, J. Zhang*, W. Ma, and D. Wen, "Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations," Physics of Fluids 34, 057109 (2022).

[14] W. Ma, J. Zhang*, et al., "Dimensional homogeneity constrained gene expression programming for discovering governing equations," Journal of Fluid Mechanics 985, A12 (2024).