A study of the spatio-temporal behaviour of bed load transport rate fluctuations

Abstract

This thesis addresses the problem of a statistical description of the transport of sediment as bed load. It highlights the role of fluctuations arising during the transport process, and their impact on macroscopic averages. The results presented here are based on four experimental studies. Two of them were published recently by Bohm et al. [2004] and Roseberry et al. [2012] while the other two were carried out during the thesis. In particular, two high speed cameras were used to automatically reconstruct particle trajectories over a window of approximately 1mlength, continuously over a few minutes. This constitutes, at the time of writing, one of the largest sets of experimental particle trajectory data available. Based on these experiments, two probabilistic models are proposed. The first one offers a macroscopic picture of the fluctuations of particle activity (concentration of moving particles). Based on a model recently proposed by Ancey et al. [2008], it allows for the accurate prediction of the fluctuations observed in the bed load flux. A new formula for the probability density function of the volume-averaged bed load flux is derived and compared to experimental data as well as to existing theory. By slightly modifying the original model, it was also possible to derive the probability density function of the inter-arrival time of particles. The latter shows an unusual bimodal shape due to the effect of collective entrainment. This phenomenon is referred to as the “separation of time scales” [Heyman et al., 2013]. Although providing an accurate picture of the macroscopic fluctuations arising in bed load transport, the first model does not gather information about the spatial behaviour of the bed load flux fluctuations. To remedy this, a new probabilistic model, able to locally describe the transport process, is proposed. This model lies in-between a kinetic description of the transport process and the macroscopic model proposed by Ancey et al. [2008]. In this regard, only the particle positions are treated as random variables, while particle velocities are assumed to be close to the Maxwellian equilibrium distribution. By a careful analysis of first and second moments, in both spatial and temporal dimensions, I prove the occurrence of large correlated structures that strongly perturb the average equilibrium, while not fundamentally modifying it. Moreover, I show that the validity of Taylor’s frozen flow hypothesis for bed load transport is severely called into question by the experimental data, proving the peculiar behaviour of those structures during transport by the mean fluid flow. Finally, a discussion about scales of fluctuations is provided. It follows from experimental results, as well as from theoretical predictions, that local fluctuations may play an important role in both an experimental setup and natural rivers. Indeed, the saturation and the correlation lengths are often so large that fluctuations may interact in a non-trivial way with the boundaries of the system, precluding the use of macroscopic average equations. Finally, this thesis suggests that the inherent fluctuations of bed load transport rates may need to be taken into account in numerical simulations in order to accurately describe the transport process.