SUBGRID-SCALE STRESS MODELING BASED ON ARTIFICIAL NEURAL NETWORK
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摘要: 亚格子(SGS)应力建模在湍流大涡模拟(LES)中有着极为重要的作用. 传统亚格子应力模型存在相对误差较大、耗散过强等问题. 近年来, 计算机技术的发展使得人工神经网络(ANN)等机器学习方法逐渐成为亚格子应力建模型的新研究范式. 本文着重考虑滤波宽度及雷诺数影响, 在不可压缩槽道湍流中建立了亚格子应力的ANN模型. 该模型以滤波后的直接数值模拟(fDNS)流场物理量及滤波尺度为输入信息, 相应滤波尺度下的亚格子应力为输出量. 通过对不同滤波尺度及不同雷诺数数据的训练, ANN模型能够给出与直接数值模拟(DNS)高度吻合的亚格子应力. 此外, 模型在亚格子耗散等非ANN建模量上也有着优异的预测性能, 与基于DNS获得的对应物理量的相关系数大都在0.9以上, 较梯度模型及Smagorinsky模型有明显提升. 在后验测试中, ANN模型对流向平均速度剖面的预测同样优于梯度模型、Smagorinsky模型及隐式大涡模拟(ILES)等传统LES模型. 在脉动速度均方根预测方面, 除了某些法向位置外ANN模型的性能整体上相对其他3个模型有所提升. 然而, 随着网格尺度的增大ANN模型预测的结果与fDNS结果的偏差逐渐增大. 总之, ANN方法在发展高精度亚格子应力模型上具有很大的潜力.Abstract: Subgrid-scale (SGS) stress modelling can be of particular importance in large-eddy simulation (LES) of turbulent flows. Traditional SGS stress models usually suffer from the drawbacks of large relative errors, excessive dissipations, etc. With the rapid progress in computer technology, machine learning methods such as artificial neural network (ANN) have gradually become a new research paradigm for SGS stress modeling. In the present paper, an ANN is employed to establish the SGS stress model for incompressible turbulent channel flow with particular attention devoted to the effect of filter width and Reynolds number. To this end, the filtered direct numerical simulation (fDNS) flow field and filter width are used as the inputs and the SGS stress at the corresponding filter width as the outputs. After training based on the data at different filter widths and different Reynolds numbers, the SGS stress predicted by ANN model is in acceptable agreement with the direct numerical simulation (DNS) data. Furthermore, excellent performance can also be found in non-modeling quantities of ANN such as SGS dissipation. The correlation coefficients between the ANN-based quantities and those calculated using DNS data are all above 0.9, indicating obvious improvements of the present ANN model over the gradient model and Smagorinsky model. In the a posteriori test, the ANN model can give better predictions on the streamwise mean velocity as compared with a variety of traditional LES models including the gradient model, Smagorinsky model and implicit LES. For the prediction of root-mean-square (RMS) fluctuating velocity, the ANN-based model is generally superior to the other three models except for some specific wall-normal locations. However, the RMS fluctuating velocities predicted by ANN-based model deviate from the fDNS results with the increase of grid size. It is suggested that ANN should have great potential for development of SGS stress models with high accuracy.
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Key words:
- turbulent flow /
- large-eddy simulation /
- artificial neural network /
- subgrid-scale stress /
- filter width
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图 5
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _f}/\varDelta$ = 4时$y$ = 0.9截面上亚格子应力分量${\tau _{11}}$ 及${\tau _{12}}$ 的分布云图. (a,b) DNS; (c,d) ANN模型; (e,f)梯度模型; (g,h) Smagorinsky模型Figure 5. Contours of the SGS stress components
${\tau _{11}}$ and${\tau _{12}}$ at${{R}}{{{e}}_\tau }$ = 180,${\varDelta _f}/\varDelta $ = 4 and$y$ = 0.9. (a,b) DNS; (c,d) ANN model; (e,f) gradient model; (g,h) Smagorinsky model表 1 DNS计算参数
Table 1. Computational parameters of DNS
$ Re $ $ R{e}_{\tau } $ $ {L}_{x} $ $ {L}_{y} $ $ {L}_{z} $ $ \mathrm{\Delta }{x}^{+} $ $\mathrm{\Delta }{y}_{{\rm{min}}}^{+}$ $\mathrm{\Delta }{y}_{{\rm{max}}}^{+}$ $ \mathrm{\Delta }{\mathrm{z}}^{+} $ $ ({N}_{x},{N}_{y},{N}_{z}) $ 2800 180 $ 4{\text{π}} $ 2 $ 2{\text{π}} $ 8.885 0.198 7.112 4.442 (256, 129, 256) 5000 300 $ 2{\text{π}} $ 2 $ {\text{π}} $ 9.752 0.216 7.905 4.876 (192, 193, 192) 7000 395 $ 2{\text{π}} $ 2 $ {\text{π}} $ 9.842 0.128 8.862 4.921 (256, 257, 256) 表 2 相同雷诺数下不同滤波尺度ANN模型训练及测试集
Table 2. Training and test sets of ANN model at the same Reynolds number and different filter widths
$ R{e}_{\tau } $ =
180, 300, 395${\mathrm{\varDelta } }_{\mathrm{f} }/\mathrm{\varDelta }$ Size of dataset test set 2 whole field training set $ \sqrt{6} $ six planes of streamwise test set 3 whole field training set $2\sqrt{3}$ six planes of streamwise test set 4 whole field 表 3 亚格子应力空间平均相关系数
Table 3. The spatial averaged correlation coefficient of SGS stress
${ { {\varDelta } }_{\mathrm{f} } }/{ {\varDelta } }$ DNS & ANN DNS & GRA DNS & SMA $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ 2 0.99 0.98 0.98 0.99 0.65 0.74 0.84 0.63 0.28 0.21 0.30 0.14 3 0.99 0.98 0.98 0.99 0.63 0.72 0.82 0.59 0.27 0.21 0.29 0.12 4 0.98 0.96 0.97 0.97 0.60 0.70 0.80 0.55 0.25 0.22 0.27 0.12 表 4 不同雷诺数和不同滤波尺度下ANN模型训练及测试集
Table 4. Training and test sets of ANN model at different Reynolds numbers and filter widths
Training or test set $ {Re}_{\tau } $ $ {{\varDelta }}_{\mathrm{f}} /\varDelta $$ \mathrm{} $ Size of dataset training set 180 2, $\sqrt{6},\; 3,\; 2\sqrt{3},\; 4$ two planes of streamwise test set 300 2, $\sqrt{6}, \;3,\; 2\sqrt{3}, \;4$ whole field training set 395 2, $\sqrt{6}, \;3,\; 2\sqrt{3}, \;4$ one plane of streamwise 表 5 亚格子应力空间平均相关系数
Table 5. Spatially-averaged correlation coefficient of SGS stress
${\varDelta } _{\mathrm{f}} / {\varDelta }$ DNS & ANN DNS & GRA DNS & SMA $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ $ {\tau }_{11} $ $ {\tau }_{12} $ $ {\tau }_{22} $ $ {\tau }_{33} $ 2 0.91 0.85 0.95 0.95 0.66 0.76 0.83 0.63 0.21 0.21 0.28 0.13 $ \sqrt{6} $ 0.92 0.92 0.92 0.94 0.65 0.75 0.83 0.61 0.20 0.21 0.27 0.13 3 0.87 0.89 0.92 0.90 0.64 0.74 0.82 0.59 0.20 0.21 0.27 0.12 $2\sqrt{3}$ 0.91 0.89 0.95 0.93 0.63 0.73 0.81 0.57 0.19 0.21 0.26 0.12 4 0.92 0.89 0.93 0.92 0.61 0.72 0.80 0.55 0.19 0.21 0.25 0.11 表 6 各个后验算例计算稳定后的
${{R}}{{{e}}_\tau }$ 值Table 6. Values of
${{R}}{{{e}}_\tau }$ for a posteriori test cases at statistically steady state${ {{\varDelta } }_{\mathrm{f} } }/{{\varDelta } }$ $ {Re}_{\tau }=180 $ $ {Re}_{\tau }=300 $ $ {Re}_{\tau }=395 $ ANN SMA GRA ILES ANN SMA GRA ILES ANN SMA GRA ILES 2 179.4 175.5 177.1 177.4 296.6 289.7 293.0 294.4 392.3 388.4 388.0 386.7 3 171.8 169.6 168.1 168.2 283.1 279.3 278.9 281.7 377.2 374.7 375.1 376.0 4 168.2 166.4 160.8 161.8 273.8 271.3 264.0 268.1 365.5 361.3 358.3 360.4 -
[1] Pope S. Turbulent Flows Cambridge: Cambridge University Press, 2000 [2] Yang ZY. Large-eddy simulation: past, present and the future. Chinese Journal of Aeronautics, 2015, 28(1): 11-24 doi: 10.1016/j.cja.2014.12.007 [3] 时北极, 何国威, 王士召. 基于滑移速度壁模型的复杂边界湍流大涡模拟. 力学学报, 2019, 51(3): 754-766 (Shi Beiji, He Guowei, Wang Shizhao. Large-eddy simulation of flows with complex geometries by using the slip-wall model. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 754-766 (in Chinese) doi: 10.6052/0459-1879-19-033 [4] 陈林烽. 基于 Navier-Stokes 方程残差的隐式大涡模拟有限元模型. 力学学报, 2020, 52(5): 1314-1322 (Chen Linfeng. A residual based unresolved-scale finite element modelling for implict large eddy simulation. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1314-1322 (in Chinese) doi: 10.6052/0459-1879-20-055 [5] 吴霆, 时北极, 王士召等. 大涡模拟的壁模型及其应用. 力学学报, 2018, 50(3): 453-466 (Wu Ting, Shi Beiji, Wang Shizhao, et al. Wall-model for large-eddy simulation and its applications. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(3): 453-466 (in Chinese) doi: 10.6052/0459-1879-18-071 [6] Smagorinsky J. General circulation experiments with the primitive equations: I. The basic experiment. Monthly Weather Review, 1963, 91(3): 99-164 doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2 [7] Moin P, Squires K, Cabot W, et al. A dynamic subgrid-scale model for compressible turbulence and scalar transport. Physics of Fluids A:Fluid Dynamics, 1991, 3(11): 2746-2757 doi: 10.1063/1.858164 [8] Germano M, Piomelli U, Moin P, et al. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A:Fluid Dynamics, 1991, 3(7): 1760-1765 doi: 10.1063/1.857955 [9] Lilly DK. A proposed modification of the Germano subgrid-cale closure method. Physics of Fluids A: Fluid Dynamics, 1992, 4(3): 633-635 doi: 10.1063/1.858280 [10] Bardina J, Ferziger J, Reynolds WC. Improved subgrid-scale models for large-eddy simulation//13th Fluid and Plasmadynamics Conference, 1980: 1357 [11] Liu S, Meneveau C, Katz J. On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. Journal of Fluid Mechanics, 1994, 275: 83-119 doi: 10.1017/S0022112094002296 [12] Domaradzki JA, Saiki EM. A subgrid-scale model based on the estimation of unresolved scales of turbulence. Physics of Fluids, 1997, 9(7): 2148-2164 doi: 10.1063/1.869334 [13] Clark RA, Ferziger JH, Reynolds WC. Evaluation of subgrid-scale models using an accurately simulated turbulent flow. Journal of Fluid Mechanics, 1979, 91(1): 1-16 doi: 10.1017/S002211207900001X [14] Zang Y, Street RL, Koseff JR. A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Physics of Fluids A: Fluid Dynamics, 1993, 5(12): 3186-3196 doi: 10.1063/1.858675 [15] Langford JA, Moser RD. Optimal LES formulations for isotropic turbulence. Journal of Fluid Mechanics, 1999, 398: 321-346 doi: 10.1017/S0022112099006369 [16] Langford JA, Moser DR. Optimal large-eddy simulation results for isotropic turbulence. Journal of Fluid Mechanics, 2004, 521: 273 doi: 10.1017/S0022112004001776 [17] Moser RD, Malaya NP, Chang H, et al. Theoretically based optimal large-eddy simulation. Physics of Fluids, 2009, 21(10): 105104 doi: 10.1063/1.3249754 [18] Hughes TJR, Mazzei L, Jansen KE. Large eddy simulation and the variational multiscale method. Computing and Visualization in Science, 2000, 3(1): 47-59 [19] Chai X, Mahesh K. Dynamic-equation model for large-eddy simulation of compressible flows. Journal of Fluid Mechanics, 2012, 699: 385-413 doi: 10.1017/jfm.2012.115 [20] Xie CY, Wang JC, Li H, et al. An approximate second-order closure model for large-eddy simulation of compressible isotropic turbulence. CiCP, 2020, 27(775): 31 [21] Slotnick J, Khodadoust A, Alonso J, et al. CFD vision 2030 study: a path to revolutionary computational aerosciences. Contractor Report, 20140003093, 2014 [22] Krizhevsky A, Sutskever I, Hinton GE. Imagenet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems, 2012, 25: 1097-1105 [23] Sallab AEL, Abdou M, Perot E, et al. Deep reinforcement learning framework for autonomous driving. Electronic Imaging, 2017, 19: 70-76 doi: 10.2352/ISSN.2470-1173.2017.19.AVM-023 [24] Alipanahi B, Delong A, Weirauch MT, et al. Predicting the sequence specificities of DNA-and RNA-binding proteins by deep learning. Nature Biotechnology, 2015, 33(8): 831-838 doi: 10.1038/nbt.3300 [25] Duraisamy K, Iaccarino G, Xiao H. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics, 2019, 51: 357-377 doi: 10.1146/annurev-fluid-010518-040547 [26] 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. 空气动力学学报, 2019, 37(3): 444-454 (Zhang Weiwei, Zhu Linyang, Liu Yilang, et al. Progresses in the application of machine learning in turbulence modeling. Acta Aerodynamica Sinica, 2019, 37(3): 444-454 (in Chinese) doi: 10.7638/kqdlxxb-2019.0036 [27] 谢晨月, 袁泽龙, 王建春等. 基于人工神经网络的湍流大涡模拟方法. 力学学报, 2021, 53(1): 1-16 (Xie Chenyue, Yuan Zelong, Wang Jianchun, et al. Artificial neural network-based subgrid-scale models for large-eddy simulation of turbulence. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1): 1-16 (in Chinese) [28] Ren F, Hu H, Tang H. Active flow control using machine learning: A brief review. Journal of Hydrodynamics, 2020, 32: 247-253 doi: 10.1007/s42241-020-0026-0 [29] 陈海昕, 邓凯文, 李润泽. 机器学习技术在气动优化中的应用. 航空学报, 2019, 40(1): 522480 (Chen Haixin, Deng Kaiwen. Utilization of machine learning technology in aerodynamic optimization. Acta Aeronautica et Astronautica Sinica, 2019, 40(1): 522480 (in Chinese) [30] Brunton SL, Noack BR, Koumoutsakos P. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 2020, 52: 477-508 doi: 10.1146/annurev-fluid-010719-060214 [31] 张伟伟, 寇家庆, 刘溢浪. 智能赋能流体力学展望. 航空学报, 2021, 42(4): 524689 (Zhang Weiwei, Kou Jiaqing, Liu Yilang. Prospect of artificial intelligence empowered fluid mechanics. Acta Aeronautica et Astronautica Sinica, 2021, 42(4): 524689 (in Chinese) [32] Maulik R, San O, Rasheed A, et al. Data-driven deconvolution for large eddy simulations of Kraichnan turbulence. Physics of Fluids, 2018, 30(12): 125109 doi: 10.1063/1.5079582 [33] Maulik R, San O, Rasheed A, et al. Subgrid modelling for two-dimensional turbulence using neural networks. Journal of Fluid Mechanics, 2019, 858: 122-144 doi: 10.1017/jfm.2018.770 [34] Vollant A, Balarac G, Corre C. Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. Journal of Turbulence, 2017, 18(9): 854-878 doi: 10.1080/14685248.2017.1334907 [35] Zhou Z, He G, Wang S, et al. Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network. Computers & Fluids, 2019, 195: 104319 [36] Xie C, Wang J, Weinan E. Modeling subgrid-scale forces by spatial artificial neural networks in large eddy simulation of turbulence. Physical Review Fluids, 2020, 5(5): 054606 doi: 10.1103/PhysRevFluids.5.054606 [37] Yuan Z, Xie C, Wang J. Deconvolutional artificial neural network models for large eddy simulation of turbulence. Physics of Fluids, 2020, 32(11): 115106 doi: 10.1063/5.0027146 [38] Wang Z, Luo K, Li D, et al. Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation. Physics of Fluids, 2018, 30(12): 125101 doi: 10.1063/1.5054835 [39] Xie C, Li K, Ma C, et al. Modeling subgrid-scale force and divergence of heat flux of compressible isotropic turbulence by artificial neural network. Physical Review Fluids, 2019, 4(10): 104605 doi: 10.1103/PhysRevFluids.4.104605 [40] Xie C, Wang J, Li H, et al. Artificial neural network mixed model for large eddy simulation of compressible isotropic turbulence. Physics of Fluids, 2019, 31(8): 085112 doi: 10.1063/1.5110788 [41] Xie C, Wang J, Li K, et al. Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence. Physical Review E, 2019, 99(5): 053113 doi: 10.1103/PhysRevE.99.053113 [42] Sarghini F, De Felice G, Santini S. Neural networks based subgrid scale modeling in large eddy simulations. Computers & Fluids, 2003, 32(1): 97-108 [43] Gamahara M, Hattori Y. Searching for turbulence models by artificial neural network. Physical Review Fluids, 2017, 2(5): 054604 doi: 10.1103/PhysRevFluids.2.054604 [44] Park J, Choi H. Toward neural-network-based large eddy simulation: application to turbulent channel flow. Journal of Fluid Mechanics, 2021, 914: A16 [45] Vreman AW, Kuerten JGM. Comparison of direct numerical simulation databases of turbulent channel flow at Reτ = 180. Physics of Fluids, 2014, 26(1): 015102 doi: 10.1063/1.4861064 [46] Moser RD, Kim J, Mansour NN. Direct numerical simulation of turbulent channel flow up to Reτ = 590. Physics of Fluids, 1999, 11(4): 943-945 doi: 10.1063/1.869966 [47] Sagaut P, Grohens R. Discrete filters for large eddy simulation. International Journal for Numerical Methods in Fluids, 1999, 31(8): 1195-1220 doi: 10.1002/(SICI)1097-0363(19991230)31:8<1195::AID-FLD914>3.0.CO;2-H [48] 周志华. 机器学习. 北京: 清华大学出版社, 2016Zhou Zhihua. Machine Learning. Beijing: Tsinghua University Press, 2016 (in Chinese) [49] LeCun Y, Bengio Y, Hinton G. Deep learning. Nature, 2015, 521(7553): 436-444 doi: 10.1038/nature14539 [50] Rumelhart DE, Hinton GE, Williams RJ. Learning representations by back-propagating errors. Nature, 1986, 323(6088): 533-536 doi: 10.1038/323533a0 [51] Goodfellow I, Bengio Y, Courville A, et al. Deep Learning. Cambridge: MIT Press, 2016 [52] Loshchilov I, Hutter F. Decoupled weight decay regularization. arXiv preprint arXiv: 1711.05101, 2017. [53] Georgiadis NJ, Rizzetta DP, Fureby C. Large-eddy simulation: current capabilities, recommended practices, and future research. AIAA Journal, 2010, 48(8): 1772-1784 [54] Scotti A, Meneveau C. A fractal model for large eddy simulation of turbulent flow. Physica D: Nonlinear Phenomena, 1999, 127(3-4): 198-232 [55] Park N, Lee S, Lee J, et al. A dynamic subgrid-scale eddy viscosity model with a global model coefficient. Physics of Fluids, 2006, 18(12): 125109 doi: 10.1063/1.2401626 [56] Chumakov SG. Scaling properties of subgrid-scale energy dissipation. Physics of Fluids, 2007, 19(5): 058104 doi: 10.1063/1.2735001 [57] Härtel C, Kleiser L, Unger F, et al. Subgrid-scale energy transfer in the near-wall region of turbulent flows. Physics of Fluids, 1994, 6(9): 3130-3143 doi: 10.1063/1.868137 [58] Piomelli U, Yu Y, Adrian RJ. Subgrid-scale energy transfer and near-wall turbulence structure. Physics of Fluids, 1996, 8(1): 215-224 doi: 10.1063/1.868829 [59] Balarac G, Le Sommer J, Meunier X, et al. A dynamic regularized gradient model of the subgrid-scale scalar flux for large eddy simulations. Physics of Fluids, 2013, 25(7): 075107 doi: 10.1063/1.4813812 [60] Leonard A. Energy cascade in large-eddy simulations of turbulent fluid flows. Advances in Geophysics, 1975, 18: 237-248 [61] Völker S, Moser RD, Venugopal P. Optimal large eddy simulation of turbulent channel flow based on direct numerical simulation statistical data. Physics of Fluids, 2002, 14(10): 3675-3691 doi: 10.1063/1.1503803 [62] Van Driest ER. On turbulent flow near a wall. Journal of the Aeronautical Sciences, 1956, 23(11): 1007-1011 doi: 10.2514/8.3713 [63] Boris JP, Grinstein FF, Oran ES, et al. New insights into large eddy simulation. Fluid Dynamics Research, 1992, 10(4-6): 199 doi: 10.1016/0169-5983(92)90023-P [64] Visbal M, Morgan P, Rizzetta D. An implicit LES approach based on high-order compact differencing and filtering schemes//16th AIAA Computational Fluid Dynamics Conference, 2003: 4098 [65] Park N, Yoo JY, Choi H. Toward improved consistency of a priori tests with a posteriori tests in large eddy simulation. Physics of Fluids, 2005, 17(1): 015103 doi: 10.1063/1.1823511 [66] Komen EMJ, Camilo LH, Shams A, et al. A quantification method for numerical dissipation in quasi-DNS and under-resolved DNS, and effects of numerical dissipation in quasi-DNS and under-resolved DNS of turbulent channel flows. Journal of Computational Physics, 2017, 345: 565-595 doi: 10.1016/j.jcp.2017.05.030 [67] Haering SW, Lee M, Moser RD. Resolution-induced anisotropy in large-eddy simulations. Physical Review Fluids, 2019, 4(11): 114605 doi: 10.1103/PhysRevFluids.4.114605 -