Rolling motion of a bead in a rapid water stream

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 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 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).