Foreword 1. Introduction 2. Interactions at the grain level 3. The granular solid: statics and elasticity 4. The granular solid: plasticity 5. Granular gases 6. The granular liquid 7. Immersed granular media 8. Erosion and sediment transport 9. Geomorphology References Index.

Granular Media: Between Fluid and Solid

[1] We report an experimental investigation of the motion of bed load particles under steady and spatially uniform turbulent flow above a flat sediment bed of uniform grain size. Using a high-speed video imaging system, we recorded the trajectories of the moving particles and measured their velocity and the length and duration of their flights, as well as the surface density of the moving particles. Our observations show that entrained particles exhibit intermittent motion composed of the succession of periods of “flight” and periods of rest. During one flight, a particle may go through phases of reptation, during which it moves in nearly persistent contact with the rough bed, and phases of saltation, during which it travels sufficiently high above the bed to reach high velocities. The distributions of longitudinal and transverse particle velocities obey a decreasing exponential and a Gaussian law, respectively. Interestingly, these observations are similar to those previously reported for viscous flows. The experimental results presented here support the erosion-deposition model of Charru (2006) and allow the calibration of the involved coefficients. In particular, noting τ*, the Shields number, and τ*c, the threshold Shields number, we find that (1) the surface density of moving particles increases linearly with τ* − τ*c; (2) the average particle velocity increases linearly with τ*1/2 − τ*c1/2, with a finite nonzero value at the threshold; (3) the flight duration scales with a characteristic settling time with no significant dependence on either τ* or the settling Reynolds number; and (4) the flight length increases linearly with τ*1/2 − τ*c1/2. The results presented in this paper should provide a valuable physical framework to describe bed form development in turbulent flows.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2009JF001628

Bed load transport in turbulent flow at the grain scale: Experiments and modeling

An erodible bed sheared by a fluid flow, gas or liquid, is generally unstable, and bed forms grow. This review discusses the following issues, in light of the recent literature: What are the relevant dynamical mechanisms controlling the emergence of bed forms? Do they form by linear instability or nonlinear processes such as pattern coarsening? What determines their timescales and length scales, so different in air and water? What are the similarities and differences between aeolian and subaqueous patterns? What is the influence of the mode of transport: bed load, saltation, or suspension? Can bed forms emerge under any hydrodynamical regime, laminar and turbulent? Guided by these questions, we propose a unified description of bed-form growth and saturation, emphasizing the hydrodynamical regime in the inner layer and the relaxation phenomena associated with particle transport.

Sand Ripples and Dunes

We investigate the entrainment, deposition and motion of coarse spherical particles within a turbulent shallow water stream down a steep slope. This is an idealization of bed-load transport in mountain streams. Earlier investigations have described this kind of sediment transport using empirical correlations or concepts borrowed from continuum mechanics. The intermittent character of particle transport at low-water discharges led us to consider it as a random process. Sediment transport in this regime results from the imbalance between entrainment and deposition of particles rather than from momentum balance between water and particles. We develop a birth–death immigration–emigration Markov process to describe the particle exchanges between the bed and the water stream. A key feature of the model is its long autocorrelation times and wide, frequent fluctuations in the solid discharge, a phenomenon never previously explained despite its ubiquity in both nature and laboratory experiments. We present experimental data obtained using a nearly two-dimensional channel and glass beads as a substitute for sediment. Entrainment, trajectories, and deposition were monitored using a high-speed digital camera. The empirical probability distributions of the solid discharge and deposition frequency were properly described by the theoretical model. Experiments confirmed the existence of wide and frequent fluctuations of the solid discharge, and revealed the existence of long autocorrelation time, but theory overestimates the autocorrelation times by a factor of around three. Particle velocity was weakly dependent on the fluid velocity contrary to the predictions of the theoretical model, which performs well when a single particle is moving. For our experiments, the dependence of the solid discharge on the fluid velocity is entirely controlled by the number of moving particles rather than by their velocity. We also noted significant changes in the behaviour of particle transport when the bed slope or the water discharge was increased. The more vigorous the stream was, the more continuous the solid discharge became. Moreover, although 90% of the energy supplied by gravity to the stream is dissipated by turbulence for slopes lower than 10%, particles dissipate more and more energy when the bed slope is increased, but surprisingly, the dissipation rate is nearly independent of fluid velocity. A movie is available with the online version of the paper.

/pdf/entrainment-and-motion-of-coarse-particles-in-a-shallow-16lk1bibre.pdf

Entrainment and motion of coarse particles in a shallow water stream down a steep slope

We present a mesoscale erosion/deposition model, which differs from previous landscape evolution models equations by taking explicitly into account a mass balance equation for the streamflow. The geological and hydrological complexity is lumped into two basic fluxes (erosion and deposition) and two averaged parameters (unit width discharge q and stream slope s). The model couples the dynamics of streamflow and topography through a sediment transport length function x(q), which is the average travel distance of a particle in the flow before being trapped on topography. This property reflects a time lag between erosion and deposition, which allows the streamflow not to be instantaneously at capacity. The so-called x-q model may reduce either to transport-limited or to detachment-limited erosion modes depending on x. But it also may not. We show in particular how it does or does not for steady state topographies, long-term evolution, and high-frequency base level perturbations. Apart from the unit width discharge and the settling velocity, the x(q) function depends on a dimensionless number encompassing the way sediment is transported within the streamflow. Using models of concentration profile through the water column, we show the dependency of this dimensionless coefficient on the Rouse number. We discuss how consistent the x-q model framework is with bed load scaling expressions and Einstein's conception of sediment motion.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2008JF001146

Fluvial erosion/transport equation of landscape evolution models revisited

Experiments are reported on the dynamics of a bed of particles sheared by a viscous Couette flow in an annular channel, with emphasis on the distributions of particle velocities, durations and lengths of the small saltation flights, and surface density of the moving particles. The velocity distributions are shown to decay approximately exponentially, with mean value, U p , equal to 0.1 γd, where y is the shear rate and d is the particle diameter. The duration of the flights does not depend on the shear rate, and is equal to 15 times the settling time d/V S , where V S is the Stokes settling velocity. Starting from an initially loosely packed bed, the surface density of the moving particles, N p , was observed to decrease slowly over several days, unlike their velocity which remains constant with time. This decay is related to the increase of the threshold shear rate for particle motion, and corresponds to rearrangement of the particles near the bed surface (armouring). When the stationary state is reached, N p depends linearly on the shear rate, so that the particle flow rate, Q p = N p U p , is a quadratic function of the shear rate. Two theoretical models are proposed to account for these observations. In the first one, the erosion and deposition rates are modelled using the two hydrodynamic time scales: the inverse shear rate γ -1 for the erosion rate, and the settling time d/V S for the deposition rate. This model accounts for the linear dependence of Np on the shear rate. The second model was developed to capture the slow decrease of Np, by considering the trapping of moving particles into troughs of the bed. This trapping model does recover the main features observed experimentally, although the characteristic time for the decrease of Np still remains too short. Our observations are, finally, compared to existing numerical and experimental studies on turbulent flows.

Erosion and deposition of particles on a bed sheared by a viscous flow

A new approach to studying ripple formation in steady flow is proposed, based on a model for the erosion and deposition of the particles. Higher shear stresses at the crests lead to a loss of particles from there, which can suppress the instability induced by fluid inertia. This model accounts for recent observations that ripples do not grow when the fluid viscosity is increased, a behaviour not present in the classical approach based on an algebraic dependence of the particle flux on the bottom shear stress.

Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow

A new model for the erosion and deposition of particles, developed in Part 1, is now applied to oscillating flows to give an analytic expression for the growth rate. We find that ripples can grow in oscillating flows when they do not in steady flows of the same magnitude, this being due to a suppression in rapidly oscillating flows of the erosion by higher shear rates on the crests.

/pdf/ripple-formation-on-a-particle-bed-sheared-by-a-viscous-32359xaepm.pdf

F. Charru

Papers

Erosion and deposition of particles on a bed sheared by a viscous flow

Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow

Ripple formation on a particle bed sheared by a viscous liquid. Part 2. Oscillating flow