  Although an increased availability of computational resources has enabled
high-fidelity simulations of turbulent flows, the RANS models are still the
dominant tools for industrial applications. However, the predictive
capabilities of RANS models are limited by potential inaccuracy driven by
hypotheses in the Reynolds stress closure. Recently, a Physics-Informed Machine
Learning (PIML) approach has been proposed to learn the functional form of
Reynolds stress discrepancy in RANS simulations based on available data. It has
been demonstrated that the learned discrepancy function can be used to improve
Reynolds stresses in different flows where data are not available. However,
owing to a number of challenges, the improvements have been demonstrated only
in the Reynolds stress prediction but not in the corresponding propagated
quantities of interest. In this work, we introduce the procedures toward a
complete PIML framework for predictive turbulence modeling, including learning
Reynolds stress discrepancy function, predicting Reynolds stresses in different
flows, and propagating to mean flow fields. The process of Reynolds stress
propagation and predictive accuracy of the propagated velocity field are
investigated. To improve the learning-prediction performance, the input
features are enriched based on an integrity basis of invariants. The fully
developed turbulent flow in a square duct is used as the test case. The
discrepancy model is trained on flow fields obtained from several Reynolds
numbers and evaluated on a duct flow at a Reynolds number higher than any of
the training cases. The predicted Reynolds stresses are propagated to velocity
field through RANS equations. Numerical results show excellent predictive
performances in both Reynolds stresses and their propagated velocities,
demonstrating the merits of the PIML approach in predictive turbulence
modeling.
