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This paper investigates the two-dimensional rolling motion of a single large particle in a shallow water stream down a steep rough bed from both an experimental and a theoretical point of view. The experiment is prototypal of sediment transport on sloping beds. Two theoretical models are presented. The first model uses the mean kinetic energy balance to deduce the average particle velocity and the bounds of the flow-rate range within which a rolling regime occurs. This range is found to be narrow, which means that the fully rolling regime is a marginal mode of transport between repose and saltation. In the second model, the particle state (resting, rolling, saltating) is considered as a random variable, whose evolution constitutes a jump Markov chain. This makes it possible to deduce the mean particle velocity as a function of the flow conditions without explicit mention of its state. The theoretical results are finally compared to the experimental data. The second model provides correct estimates of the particle velocity and the probability of finding the particle in a given state for various flow conditions (bead material, slope, and roughness).

This paper experimentally and numerically investigates the two-dimensional saltating motion of a single large particle in a shallow water stream down a steep rough bed. The experiment is prototypical of sediment transport on sloping beds. Similar to the earlier experimental results on fine particles entrained by a turbulent stream, we found that most features of the particle motion were controlled by a dimensionless shear stress (also called the Shields number) N(Sh) defined as the ratio of the bottom shear stress exerted by the water flow to the buoyant weight of the particle (scaled by its cross-sectional area to obtain a stress). We did not observe a clear transition from rest to motion, but on the contrary there was a fairly wide range of N(Sh) (typically 0.001-0.005 for gentle slopes) for which the particle could be set in motion or come to rest. When the particle was set in motion, it systematically began to roll. The rolling regime was marginal in that it occurred for a narrow range of N(Sh) (typically 0.005-0.01 for gentle slopes). For sufficiently high Shields numbers (N(Sh)>0.3), the particle was in saltation. The mean particle velocity was found to vary linearly with the square root of the bottom shear stress and here, surprisingly enough, was a decreasing function of the channel slope. We also performed numerical simulations based on Lagrangian equations of motion. A qualitative agreement was found between the experimental data and numerical simulations but, from a quantitative point of view, the relative deviation was sometimes substantial (as high as 50%). An explanation for the partial agreement is the significant modification in the water flow near the particle.