  The local granular rheology is investigated numerically in turbulent bedload
transport. Considering spherical particles, steady uniform configurations are
simulated using a coupled fluid-discrete-element model. The stress tensor is
computed as a function of the depth for a series of simulations varying the
Shields number, the specific density and the particle diameter. The results are
analyzed in the framework of the $\mu(I)$ rheology and exhibit a collapse of
both the shear to normal stress ratio and the solid volume fraction over a wide
range of inertial numbers. Contrary to expectations, the effect of the
interstitial fluid on the granular rheology is shown to be negligible,
supporting recent work suggesting the absence of a clear transition between the
free-fall and turbulent regime. In addition, data collapse is observed up to
unexpectedly high inertial numbers $I\sim2$, challenging the existing
conceptions and parametrization of the $\mu(I)$ rheology. Focusing upon bedload
transport modelling, the results are pragmatically analyzed in the $\mu(I)$
framework in order to propose a granular rheology for bedload transport. The
proposed rheology is tested using a 1D volume-averaged two-phase continuous
model, and is shown to accurately reproduce the dense granular flow profiles
and the sediment transport rate over a wide range of Shields numbers. The
present contribution represents a step in the upscaling process from
particle-scale simulations toward large-scale applications involving complex
flow geometry.
