Modeling Reynolds stress anisotropy invariants via machine learning

The presentation and modeling of turbulence anisotropy are crucial for studying large-scale turbulence structures and constructing turbulence models. However, accurately capturing anisotropic Reynolds stresses often relies on expensive direct numerical simulations (DNS). Recently, a hot topic in data-driven turbulence modeling is how to acquire accurate Reynolds stresses by the Reynolds-averaged Navier-Stokes (RANS) simulation and a limited amount of data from DNS. Many existing studies use mean flow characteristics as the input features of machine learning models to predict high-fidelity Reynolds stresses, but these approaches still lack robust generalization capabilities. In this paper, a deep neural network (DNN) is employed to build a model, mapping from tensor invariants of RANS mean flow features to the anisotropy invariants of high-fidelity Reynolds stresses. From the aspects of tensor analysis and input-output feature design, we try to enhance the generalization of the model while preserving invariance. A functional framework of Reynolds stress anisotropy invariants is derived theoretically. Complete irreducible invariants are then constructed from a tensor group, serving as alternative input features for DNN. Additionally, we propose a feature selection method based on the Fourier transform of periodic flows. The results demonstrate that the data-driven model achieves a high level of accuracy in predicting turbulence anisotropy of flows over periodic hills and converging-diverging channels. Moreover, the well-trained model exhibits strong generalization capabilities concerning various shapes and higher Reynolds numbers. This approach can also provide valuable insights for feature selection and data generation for data-driven turbulence models. (Figure presented.).