An artificial neural network-based quadratic constitutive Reynolds stress model for separated turbulent flows using data-augmented field inversion and machine learning

Reynolds-averaged turbulence models have become one of the most important and popular techniques for practical engineering applications in aeronautics and astronautics. However, the poor performance in the prediction of flow separations restricts its application ranges due to the traditional linearity and equilibrium hypotheses that constitute the equation of Reynolds stress in turbulence modeling. In this study, an artificial neural network-based quadratic constitutive (ANN-QCR) Reynolds stress model is proposed for simulating turbulent flows with separations by using the field inversion and machine learning technique (FIML) with high-fidelity experimental data. In particular, the Reynolds stress is decomposed into linear and non-linear parts, respectively. The former is evaluated by the Spalart-Allmaras model with a correction factor imposed on the production term to account for the non-equilibrium effect, while the latter is a quadratic constitutive term with a self-calibrated factor. These correction factors are predicted by an artificial neural network (ANN) depending on the local flow features. The unified framework of FIML updates the weights of ANN-QCR directly by the gradient-based discrete adjoint method, ensuring consistency between the field inversion and the neural network training. The data-augmented ANN-QCR turbulence model is well validated through several separated turbulent flows induced by adverse pressure gradients, shock wave boundary interfaces, higher angles of attack, and higher Reynolds numbers (Re). With the optimization target at lift coefficients, the established model also improves the predictive performance in other flow quantities, such as drag coefficients and pressure distributions. In addition, this model captures the development of separation bubbles better with the increase in the angle of attack. Benefiting from the compatibility and convergence of forward simulation, the generalization capability of the present ANN-QCR model is successfully validated in various numerical simulations of separated turbulent flow problems across a wide range of attack angles and Reynolds numbers with good accuracy.