  Data-driven turbulence modeling studies have reached such a stage that the
fundamental framework is basically settled, but several essential issues remain
that strongly affect the performance, including accuracy, smoothness, and
generalization capacity. Two problems are studied in the current research: (1)
the processing of the Reynolds stress tensor and (2) the coupling method
between the machine learning turbulence model and flow solver. The first
determines the form of predicting targets and the resulting physical
completeness and interpretability. The second determines the training process
and intrinsic relevance between the mean flow features and Reynolds stress. For
the Reynolds stress processing issue, we perform the theoretical derivation to
extend the relevant tensor arguments of Reynolds stress in addition to the
strain rate and rotation rate. Then, the tensor representation theorem is
employed to give the complete irreducible invariants and integrity basis. In
addition, an adaptive regularization term is employed to enhance the
representation performance. For the coupling issue, an iterative coupling
data-driven turbulence modeling framework with consistent convergence is
proposed. The training data preparation, predicting target selection, and
computation platform are illustrated. The framework is then applied to a
canonical separated flow for verification. The mean flow results obtained by
coupling computation of the trained machine learning model and flow solver have
high consistency with the direct numerical simulation true values, which proves
the validity of the current approach.
