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基于人工神经网络的亚格子应力建模

吴磊 肖左利

吴磊, 肖左利. 基于人工神经网络的亚格子应力建模. 力学学报, 2021, 53(10): 2667-2681 doi: 10.6052/0459-1879-21-356
引用本文: 吴磊, 肖左利. 基于人工神经网络的亚格子应力建模. 力学学报, 2021, 53(10): 2667-2681 doi: 10.6052/0459-1879-21-356
Wu Lei, Xiao Zuoli. Subgrid-scale stress modeling based on artificial neural network. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2667-2681 doi: 10.6052/0459-1879-21-356
Citation: Wu Lei, Xiao Zuoli. Subgrid-scale stress modeling based on artificial neural network. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2667-2681 doi: 10.6052/0459-1879-21-356

基于人工神经网络的亚格子应力建模

doi: 10.6052/0459-1879-21-356
基金项目: 国家自然科学基金资助项目(91852112)
详细信息
    作者简介:

    肖左利, 研究员, 主要研究方向: 湍流理论与数值模拟. E-mail: z.xiao@pku.edu.cn

  • 中图分类号: O357

SUBGRID-SCALE STRESS MODELING BASED ON ARTIFICIAL NEURAL NETWORK

  • 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.

     

  • 图  1  ANN模型框架

    Figure  1.  The framework of ANN model

    图  2  槽道流计算区域示意图

    Figure  2.  Schematic for computational domain of turbulent channel flow

    图  3  DNS计算结果

    Figure  3.  DNS results

    图  4  前馈人工神经网络示意图

    Figure  4.  Schematic of the feedforward ANN

    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}}$${\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}) $
    2800180$ 4{\text{π}} $2$ 2{\text{π}} $8.8850.1987.1124.442(256, 129, 256)
    5000300$ 2{\text{π}} $2$ {\text{π}} $9.7520.2167.9054.876(192, 193, 192)
    7000395$ 2{\text{π}} $2$ {\text{π}} $9.8420.1288.8624.921(256, 257, 256)
    下载: 导出CSV

    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
    下载: 导出CSV

    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 & ANNDNS & GRADNS & 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} $
    20.990.980.980.990.650.740.840.630.280.210.300.14
    30.990.980.980.990.630.720.820.590.270.210.290.12
    40.980.960.970.970.600.700.800.550.250.220.270.12
    下载: 导出CSV

    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 set1802, $\sqrt{6},\; 3,\; 2\sqrt{3},\; 4$two planes of streamwise
    test set3002, $\sqrt{6}, \;3,\; 2\sqrt{3}, \;4$whole field
    training set3952, $\sqrt{6}, \;3,\; 2\sqrt{3}, \;4$one plane of streamwise
    下载: 导出CSV

    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 & ANNDNS & GRADNS & 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
    下载: 导出CSV

    表  6  各个后验算例计算稳定后的${{R}}{{{e}}_\tau }$

    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 $
    ANNSMAGRAILESANNSMAGRAILESANNSMAGRAILES
    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
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-07-26
  • 录用日期:  2021-09-17
  • 网络出版日期:  2021-09-18
  • 刊出日期:  2021-10-26

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