Physics-constrained machine learning for generalisable turbulence modelling across flow regimes

In this work we propose a physics-constrained machine learning approach to enhance the generalisability of neural-network-represented turbulence models across different flow regimes. Machine learning-augmented turbulence modelling has emerged to improve the predictive accuracy of the Reynolds-averaged Navier–Stokes method based on high-fidelity data. However, such data-driven models face difficulties in generalising well in different flow scenarios outside training datasets. Here we show that this issue can be alleviated by incorporating physical constraints of the Reynolds stress during the learning process. It is achieved with a physics-constrained learning approach based on the regularised ensemble Kalman method, which allows for model learning from limited velocity data and is flexible in imposing physical constraints. The method is used to learn a nonlinear eddy viscosity model, represented by a tensor-basis neural network, from velocity data in separated flows over periodic hills. Two physical constraints are incorporated during the model learning, i.e. the baseline coefficients to capture the log law of the wall and the algebraic Reynolds stress equation to inform the Reynolds stress anisotropy. Moreover, an adaptive penalty method is developed to enforce physical constraints in the feature space associated with canonical flows, avoiding the incompatibility with the training data from separated flows. Our results show that the proposed method can learn a generalisable turbulence model across different flow scenarios, including similar separated flows, canonical wall-bounded flows and complex flow configurations. Finally, the learned neural-network-based model function is discussed, demonstrating its consistency with the imposed physical constraints.