  Most turbulence models used in Reynolds-averaged Navier-Stokes (RANS)
simulations are partial differential equations (PDE) that describe the
transport of turbulent quantities. Such quantities include turbulent kinetic
energy for eddy viscosity models and the Reynolds stress tensor (or its
anisotropy) in differential stress models. However, such models all have
limitations in their robustness and accuracy. Inspired by the successes of
machine learning in other scientific fields, researchers have developed
data-driven turbulence models. Recently, a nonlocal vector-cloud neural network
with embedded invariance was proposed, with its capability demonstrated in
emulating passive tracer transport in laminar flows. Building upon this
success, we use nonlocal neural network mapping to model the transport physics
in the k-epsilon model and couple it to RANS solvers, leading to a PDE-free
eddy-viscosity model. We demonstrate the robustness and stability of the RANS
solver with a neural network-based turbulence model on flows over periodic
hills of parameterized geometries. Our work serves as a proof of concept for
using a vector-cloud neural network as an alternative to traditional turbulence
models in coupled RANS simulations. The success of the coupling paves the way
for neural network-based emulation of Reynolds stress transport models.
