Improving accuracy of turbulence models by neural network

Neural networks of simple structures are used to construct a turbulence model for large-eddy simulation (LES). Data obtained by direct numerical simulation (DNS) of homogeneous isotropic turbulence are used to train neural networks. It is shown that two methods are effective for improvement of accuracy of the model: weighting data for training and addition of the second-order derivatives of velocity to the input variables. As a result, high correlation between the exact subgrid scale stress and the prediction by the neural network is obtained for large filter width; the correlation coefficient is about 0.9 and 0.8 for filter widths 48.8η and 97.4η, respectively, where η is the Kolmogorov scale. The models established by neural networks are close to but not identical with the gradient models. LES with the neural network model is performed for the homogeneous isotropic turbulence and the initial-value problem of the Taylor-Green vortices. The results obtained with the neural network model are in reasonable agreement with those of the filtered DNS. However, symmetry in the latter problem is broken since the neural network model does not possess rigorous symmetry under orthogonal transformations.