Adopting the numerical results from the Navier-Stokes equation and the UGKS method as the training data of the flow field samples, the Driven Nonlinear Constitutive Relations (DNCR) method constructs a high-dimensional complex nonlinear regression model for the heat flux/stress tensor and flow field characteristic parameters. The macro conservation equations are also solved by coupling data-driven nonlinear constitutive relations, and the numerical solution to the rarefied non-equilibrium flow to be predicted is obtained. However, not all features (such as velocity, pressure, and density) and outputs (such as heat flow and stress tensor) of the existing DNCR method are rotation invariants, and therefore the training model cannot be directly applied to rotated or translated coordinate systems. Aiming at this defect, this paper constructs a set of new features and outputs with the rotation invariant assumption, and optimizes all the training set features by combining the prediction results of the typical examples and the weight feedback. Meanwhile, parameter selection and optimization for the extreme random tree is conducted to improve the prediction range and generalization performance of the regression model, finally proposing an improved DNCR based on rotation invariants. The improvement in the new method is evaluated through two-dimensional hypersonic flow around the cylinder and lid driven flow under different flow conditions with different configurations compared with the original one. The results show that the use of rotation invariants can significantly improve the ability of the training model to adapt to coordinate system rotation and shape changes, enabling the DNCR with better generalization ability.
[1] TSIEN H S. Superaerodynamics, mechanics of rarefied gases[J]. Journal of the Aeronautical Sciences, 1946, 13(12):653-664.
[2] BIRD G A. Molecular gas dynamics[M]. Oxford:Clarendon Press, 1976.
[3] XU K, HUANG J C. A unified gas-kinetic scheme for continuum and rarefied flows[J]. AIP Conference Proceedings, 2011, 1333(1):525-530.
[4] 张伟伟, 寇家庆, 刘溢浪. 智能赋能流体力学展望[J]. 航空学报, 2021, 42(4):524689. ZHANG W W, KOU J Q, LIU Y L. Prospect of artificial intelligence empowered fluid mechanics[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(4):524689(in Chinese).
[5] 任峰, 高传强, 唐辉. 机器学习在流动控制领域的应用及发展趋势[J]. 航空学报, 2021, 42(4):524686. REN F, GAO C Q, TANG H. Machine learning for flow control:applications and development trends[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(4):524686(in Chinese).
[6] 张伟伟, 朱林阳, 刘溢浪, 等. 机器学习在湍流模型构建中的应用进展[J]. 空气动力学学报, 2019, 37(3):444-454. ZHANG W W, ZHU L Y, LIU Y L, et al. Progresses in the application of machine learning in turbulence modeling[J]. Acta Aerodynamica Sinica, 2019, 37(3):444-454(in Chinese).
[7] TRACEY B D, DURAISAMY K, ALON-SO J J. A machine learning strategy to assist turbulence model development[C]//53rd AIAA Aerospace Sciences Meeting. Reston:AIAA, 2015.
[8] WANG J X, WU J L, XIAO H. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data[J]. Physical Review Fluids, 2017, 2(3):034603.
[9] WU J L, XIAO H, PATERSON E. Physics-informed machine learning approach for augmenting turbulence models:A comprehensive framework[J]. Physical Review Fluids, 2018, 3(7):074602.
[10] ZHU L Y, ZHANG W W, KOU J Q, et al. Machine learning methods for turbulence modeling in subsonic flows around airfoils[J]. Physics of Fluids, 2019, 31(1):015105.
[11] LING J L, KURZAWSKI A, TEMPLETON J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance[J]. Journal of Fluid Mechanics, 2016, 807:155-166.
[12] 李廷伟, 张莽, 赵文文, 等. 面向稀薄流非线性本构预测的机器学习方法[J]. 航空学报, 2021, 42(4):524386. LI T W, ZHANG M, ZHAO W W, et al. Machine learning method for correction of rarefied nonlinear constitutive relations[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(4):524386(in Chinese).
[13] ZHANG J, MA W J. Data-driven discovery of governing equatios for fluid dynamics based on molecular simulation[J]. Journal of Fluid Mechanics, 2020, 892:A5.
[14] 李航. 统计学习方法[M]. 北京:清华大学出版社, 2012. LI H. Statistical learning method[M]. Beijing:Tsinghua University Press, 2012(in Chinese).
[15] BREIMAN L. Random forests[J].Machine Learning, 2001, 45(1):5-32.
[16] 许允之,王舒萍. 基于随机森林算法的徐州雾霾回归预测模型[Z]. 中国北京:20196. XU Y X, WANG S P. Regression prediction model of xuzhou haze based on stochastic forest algorithm[Z]. China Beiing:20196(in Chinese).
[17] LI X, WANG Z J, WANG L Y, et al. A multi-dimensional context-aware recommendation approach based on improved random forest algorithm[J]. IEEE Access, 2018, 6:45071-45085.
[18] EVANS J, WATERSON B, HAMILTON A. Forecasting road traffic conditions using a context-based random forest algorithm[J]. Transportation Planning and Technology, 2019, 42(6):554-572.
[19] 常琦, 杨维希, 赵恒, 等. 基于多传感器的裂纹扩展监测研究[J]. 航空学报, 2020, 41(2):223336. CHANG Q, YANG W X, ZHAO H, et al. A multi-sensor based crack propagation monitoring research[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(2):223336(in Chinese).
[20] 冯蕴雯, 潘维煌, 刘佳奇, 等. 基于机器学习的飞机动力装置运行可靠性[J]. 航空学报, 2021, 42(4):524732. FENG Y W, PAN W H, LIU J Q, et al. Operational reliability of aircraft power plant based on machine learning[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(4):524732(in Chinese).
[21] GEURTS P, ERNST D, WEHENKEL L. Extremely randomized trees[J]. Machine Learning, 2006, 63(1):3-42.
[22] 刘沙, 王勇, 袁瑞峰, 等. 统一气体动理学方法研究进展[J]. 气体物理, 2019, 4(4):1-13. LIU S, WANG Y, YUAN R F, et al. Advance in unified methods based on gas-kinetic theory[J]. Physics of Gases, 2019, 4(4):1-13(in Chinese).
[23] BHATNAGAR P L, GROSS E P, KROOK M. A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems[J]. Physical Review, 1954, 94(3):511-525.
[24] XIAO H, WU J L, LAIZET S, et al. Flows over periodic hills of parameterized geometries:A dataset for data-driven turbulence modeling from direct simulations[J]. Computers & Fluids, 2020, 200:104431.
[25] JOHN B, GU X J, EMERSON D R. Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow:A parallel DSMC study[J]. Computers & Fluids, 2011, 45(1):197-201.
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