Sediment transport on rippled beds.

We conduct an Euler-Lagrange, direct numerical simulation of a turbulent channel flow at a shear Reynolds number of $Re_{\tau}=180$ over an erodible particle bed.

The particle bed consists of approximately 1.3 million monodisperse particles, resulting in a bed thickness of around 12 to 13 particles.

The particle density and size are chosen to achieve a ratio of 4 for the Shields number to the critical Shields number necessary for incipient motion such that particle transport occurs primarily as bedload.

The simulation is run long enough for ripples to form.

We track the temporal evolution of the particle flux and excess Shields stress for the entire bed as well as for the four regions of a ripple, namely the crest, trough, lee side and stoss side.

We find that the particle flux and excess Shields stress closely match the Wong \& Parker correlation when the particle bed is featureless at early time but diverge from the correlation when ripples form.

This deviation primarily arises from particle transport in the trough and lee side regions.

Conversely, particle transport in the crest and stoss side regions remains largely consistent with the Wong \& Parker correlation.

Additionally, we note that ripples attain a self-similar profile in the shape and near-bed shear stress when they are sufficiently distant from their upstream neighbor.

Any departure from self-similarity occurs when the upstream neighbor gets within close proximity.