  This work presents a review and perspectives on recent developments in the
use of machine learning (ML) to augment Reynolds-averaged Navier--Stokes (RANS)
and Large Eddy Simulation (LES) models of turbulent flows. Different approaches
of applying supervised learning to represent unclosed terms, model
discrepancies and sub-filter scales are discussed in the context of RANS and
LES modeling. Particular emphasis is placed on the impact of the training
procedure on the consistency of ML augmentations with the underlying physical
model. Techniques to promote model-consistent training, and to avoid the
requirement of full fields of direct numerical simulation data are detailed.
This is followed by a discussion of physics-informed and mathematical
considerations on the choice of the feature space, and imposition of
constraints on the ML model. With a view towards developing generalizable
ML-augmented RANS and LES models, outstanding challenges are discussed, and
perspectives are provided. While the promise of ML-augmented turbulence
modeling is clear, and successes have been demonstrated in isolated scenarios,
a general consensus of this paper is that truly generalizable models require
model-consistent training with careful characterization of underlying
assumptions and imposition of physically and mathematically informed priors and
constraints to account for the inevitable shortage of data relevant to
predictions of interest. Thus, machine learning should be viewed as one tool in
the turbulence modeler's toolkit. This modeling endeavor requires
multi-disciplinary advances, and thus the target audience for this paper is the
fluid mechanics community, as well as the computational science and machine
learning communities.
