  Despite their well-known limitations, RANS models remain the most commonly
employed tool for modeling turbulent flows in engineering practice. RANS models
are predicated on the solution of the RANS equations, but these equations
involve an unclosed term, the Reynolds stress tensor, which must be modeled.
The Reynolds stress tensor is often modeled as an algebraic function of mean
flow field variables and turbulence variables. This introduces a discrepancy
between the Reynolds stress tensor predicted by the model and the exact
Reynolds stress tensor. This discrepancy can result in inaccurate mean flow
field predictions. In this paper, we introduce a data-informed approach for
arriving at Reynolds stress models with improved predictive performance. Our
approach relies on learning the components of the Reynolds stress discrepancy
tensor associated with a given Reynolds stress model in the mean strain-rate
tensor eigenframe. These components are typically smooth and hence simple to
learn using state-of-the-art machine learning strategies and regression
techniques. Our approach automatically yields Reynolds stress models that are
symmetric, and it yields Reynolds stress models that are both Galilean and
frame invariant provided the inputs are themselves Galilean and frame
invariant. To arrive at computable models of the discrepancy tensor, we employ
feed-forward neural networks and an input space spanning the integrity basis of
the mean strain-rate tensor, the mean rotation-rate tensor, the mean pressure
gradient, and the turbulent kinetic energy gradient, and we introduce a
framework for dimensional reduction of the input space to further reduce
computational cost. Numerical results illustrate the effectiveness of the
proposed approach for data-informed Reynolds stress closure for a suite of
turbulent flow problems of increasing complexity.
