Numerical simulation of bedform evolution on granular sediment beds via Eulerian–Lagrangian approach in low-flow-rate horizontal duct transport

Using an Eulerian–Lagrangian approach, we numerically investigate sediment pattern formation during low-flow-rate transport of granular particles in a horizontal duct. Particles introduced at the inlet sediment to form a rising bed near the entrance, which transitions into a horizontal section further downstream. Incipient ripples emerge progressively based on the inclined bed near the inlet, then successively collapse, ultimately giving way to a larger bedform that develops downstream of the ripple sequences. This larger bedform propagates independently downstream over the underlying horizontal bed section. Increasing the flow rate, duct’s height or width, or introducing downward duct inclination promotes bed flattening along the entire duct length. This reduces the distinction between ripples and bedforms, favoring more uniform bed patterns. Our observations reveal three different types of ripple interaction during the coarsening process from ripple array to bedform: splitting, fusion, and the formation of a small sandbag downstream of the parent ripple. We quantify pattern characteristics, including period, wavelength, migration velocity, and particle transport rate as functions of the Shields number. Notably, particle transport rates — whether derived from individual patterns or from creeping of the ripple array — consistently follow the power-law relationship established by Meyer-Peter and Müller. Furthermore, we analyze the morphology of the bed profile in the inclined section near the entrance of the conduct based on the skin friction of the bed surface, as well as the distributions of boundary shear stress along the interface of an individual bedform. For the bedforms with the downstream bed higher than the upstream, the surface shear stress profiles along the bedforms show a distinct downward-pointing cusp near the trough. This feature is analogous to the stress pattern observed upstream of a bump.