  Neural network-based turbulence modeling has gained significant success in
improving turbulence predictions by incorporating high--fidelity data. However,
the interpretability of the learned model is often not fully analyzed, which
has been one of the main criticism of neural network-based turbulence modeling.
Therefore, it is increasingly demanding to provide physical interpretation of
the trained model, which is of significant interest for guiding the development
of interpretable and unified turbulence models. The present work aims to
interpret the predictive improvement of turbulence flows based on the behavior
of the learned model, represented with tensor basis neural networks. The
ensemble Kalman method is used for model learning from sparse observation data
due to its ease of implementation and high training efficiency. Two cases,
i.e., flow over the S809 airfoil and flow in a square duct, are used to
demonstrate the physical interpretation of the ensemble-based turbulence
modeling. For the flow over the S809 airfoil, our results show that the
ensemble Kalman method learns an optimal linear eddy viscosity model, which
improves the prediction of the aerodynamic lift by reducing the eddy viscosity
in the upstream boundary layer and promoting the early onset of flow
separation. For the square duct case, the method provides a nonlinear eddy
viscosity model, which predicts well secondary flows by capturing the imbalance
of the Reynolds normal stresses. The flexibility of the ensemble-based method
is highlighted to capture characteristics of the flow separation and secondary
flow by adjusting the nonlinearity of the turbulence model.
