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
图 6
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta$ = 2时DNS与ANN模型的亚格子应力平均值及其脉动的均方根剖面Figure 6. Profiles of mean SGS stress and RMS fluctuating SGS stress obtained from DNS and ANN model at
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta$ = 2
图 7
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta $ = 3时各模型与DNS亚格子应力的相关系数剖面(横坐标:$y$ )Figure 7. Profiles of correlation coefficient between modeled SGS stress and DNS data at
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta $ = 3 (x-coordinate:$y$ )
图 8
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta $ = 3时各模型与DNS亚格子应力的相关系数随${y^ + }$ 的变化剖面(横坐标:${y^ + }$ )Figure 8. Profiles of correlation coefficient between modeled SGS stress and DNS data at
${{R}}{{{e}}_\tau }$ = 180,${\varDelta _{\rm{f}}}/\varDelta $ = 3 (x-coordinate:${y^ + }$ )
图 9 亚格子应力空间平均相关系数
Figure 9. Spatially-averaged correlation coefficients between the modeled and DNS SGS stresses
图 10
${{R}}{{{e}}_\tau }$ = 300时3个测试算例的平均亚格子耗散剖面Figure 10. Profiles of mean SGS dissipation for the three test cases at
${{R}}{{{e}}_\tau }$ = 300
图 11
${{R}}{{{e}}_\tau }$ = 300下3个测试算例的平均亚格子反传剖面Figure 11. Profiles of mean SGS backscatter for the three test cases at
${{R}}{{{e}}_\tau }$ = 300
图 12
${{R}}{{{e}}_\tau }$ = 300下3个测试算例的平均亚格子力$ \partial {\tau _{1j}}/\partial {x_j} $ 剖面Figure 12. Profiles of mean SGS force
$ \partial {\tau _{1j}}/\partial {x_j} $ for three test cases at${{R}}{{{e}}_\tau }$ = 300
图 13
${{R}}{{{e}}_\tau }$ = 300下3个测试算例的平均亚格子输运剖面Figure 13. Profiles of mean SGS transport for three test cases at
${{R}}{{{e}}_\tau }$ = 300
图 14 亚格子物理量空间平均相关系数
Figure 14. Spatially-averaged correlation coefficient of the SGS quantities
图 15 DNS与ANN模型的亚格子应力平均值剖面
Figure 15. Profiles of mean SGS stress obtained from DNS and ANN model
图 16 亚格子应力空间平均相关系数
Figure 16. Spatially-averaged correlation coefficient of SGS stress
图 17 各个后验算例计算稳定后
${{R}}{{{e}}_\tau }$ 值的相对误差率Figure 17. The relative errors of
${{R}}{{{e}}_\tau }$ for a posteriori test cases at statistically steady state
图 18 不同模型在3个网格尺度下的流向平均速度剖面
Figure 18. Profiles of mean streamwise velocity obtained using different models at three grid scales
图 19 不同模型在3个网格尺度下脉动速度均方根剖面
Figure 19. Profiles of RMS fluctuating velocity obtained using different models at three grid scales
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) 
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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
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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
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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
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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
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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
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