  The present study represents a data-driven turbulent model with Galilean
invariance preservation based on machine learning algorithm. The fully
connected neural network (FCNN) and tensor basis neural network (TBNN) [Ling et
al. (2016)] are established. The models are trained based on five kinds of flow
cases with Reynolds Averaged Navier-Stokes (RANS) and high-fidelity data. The
mappings between two invariant sets, mean strain rate tensor and mean rotation
rate tensor as well as additional consideration of invariants of turbulent
kinetic energy gradients, and the Reynolds stress anisotropy tensor are
trained. The prediction of the Reynolds stress anisotropy tensor is treated as
user's defined RANS turbulent model with a modified turbulent kinetic energy
transport equation. The results show that both FCNN and TBNN models can provide
more accurate predictions of the anisotropy tensor and turbulent state in
square duct flow and periodic flow cases compared to the RANS model. The
machine learning based turbulent model with turbulent kinetic energy gradient
related invariants can improve the prediction precision compared with only mean
strain rate tensor and mean rotation rate tensor based models. The TBNN model
is able to predict a better flow velocity profile compared with FCNN model due
to a prior physical knowledge.
