Data-driven turbulence modeling has been considered an effective method for improving the prediction accuracy of Reynolds-averaged Navier–Stokes equations. Related studies aimed to solve the discrepancy of traditional turbulence modeling by acquiring specific patterns from high-fidelity data through machine learning methods, such as artificial neural networks. The present study focuses on the unsmoothness and prediction error problems from the aspect of feature selection and processing. The selection criteria for the input features are summarized, and an effective input set is constructed. The effect of the computation grid on the smoothness is studied. A modified feature decomposition method for the spatial orientation feature of the Reynolds stress is proposed. The improved machine learning framework is then applied to the periodic hill database with notably varying geometries. The results of the modified method show significant enhancement in the prediction accuracy and smoothness, including the shape and size of separation areas and the friction and pressure distributions on the wall, which confirms the validity of the approach.

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Field inversion and machine learning are implemented in this study to describe three-dimensional (3-D) separation flow around an axisymmetric hill and augment the Spart-Allmaras (SA) model. The discrete adjoint method is used to solve the field inversion problem, and an artificial neural network is used as the machine learning model. A validation process for field inversion is proposed to adjust the hyperparameters and obtain a physically acceptable solution. The field inversion result shows that the non-equilibrium turbulence effects in the boundary layer upstream of the mean separation line and in the separating shear layer dominate the flow structure in the 3-D separating flow, which agrees with prior physical knowledge. However, the effect of turbulence anisotropy on the mean flow appears to be limited. Two approaches are proposed and implemented in the machine learning stage to overcome the problem of sample imbalance while reducing the computational cost during training. The results are all satisfactory, which proves the effectiveness of the proposed approaches.

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Aerodynamic rules and knowledge are often obtained through theoretical research and experiments, which have contributed greatly to aircraft design. For example, Korn's equation predicts the airfoil drag divergence Mach number using the airfoil maximum thickness and lift coefficient. It is very helpful in the aircraft initial design. But it neither reveals the key factors of fluid features on the drag divergence, nor contributes to the detailed design. This paper generates a supercritical airfoil database that covers the typical free stream Mach number, angle of attack, lift coefficient, and geometry of modern transonic commercial aircraft. Correlation screening and multivariate regression are carried out to discover knowledge about the airfoil drag divergence Mach number and pressure distribution features. A new linear correlation is discovered and validated by existed airfoil databases. Compared with Korn's equation, the discovered correlation reduces the maximum prediction error by approximately 40%. It indicates that the drag divergence Mach number can be increased by obtaining a shock wave that is further upstream in the detailed design. Furthermore, it enables the cruise performance and drag divergence Mach number to be predicted with only one simulation of the cruise point, which will greatly save the computational cost of optimizations.