Spatial correlations in bed load transport: Evidence, importance, and modeling

Abstract

This article examines the spatial dynamics of bed load particles in water. We focus particularly on the fluctuations of particle activity, which is defined as the number of moving particles per unit bed length. Based on a stochastic model recently proposed by Ancey and Heyman (2014), we derive the second moment of particle activity analytically, that is, the spatial correlation functions of particle activity. From these expressions, we show that large moving particle clusters can develop spatially. Also, we provide evidence that fluctuations of particle activity are scale dependent. Two characteristic lengths emerge from the model: a saturation length sat describing the length needed for a perturbation in particle activity to relax to the homogeneous solution and a correlation length c describing the typical size of moving particle clusters. A dimensionless Péclet number can also be defined according to the transport model. Three different experimental data sets are used to test the theoretical results. We show that the stochastic model describes spatial patterns of particle activity well at all scales. In particular, we show that c and sat may be relatively large compared to typical scales encountered in bed load experiments (grain diameter, water depth, bed form wavelength, flume length, etc.) suggesting that the spatial fluctuations of particle activity have a nonnegligible impact on the average transport process.

Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation

“…The particle diffusivity D u in (5)-(6) arises from the second moment of the probability density function of particle displacements [36] and has been recently quantified near incipient motion conditions by Heyman et al [43]. The parametric dependence on the flow variables remains unexplored to our knowledge and, because of this, we treat D u as a degree of freedom in our study.…”

“…In the first set of simulations (Section 4.2) we consider a prescribed steady uniform flow and solve (5) for the continuous Poisson density b(x, t). The numerical solutions are compared with the theoretical solutions built by Ancey et al [3,4] and Heyman et al [43] and with additional numerical simulations using simpler numerical schemes. Then, in Section 4.3, we focus attention on the deterministic part of the model equations.…”

“…Next, Section 4 is devoted to the description of the hybrid finite-differences/finitevolume method and the numerical simulations. Accuracy and performance are evaluated by comparing numerical simulations with available theoretical solutions [3,4,43]. We also study the evolution of infinitesimal disturbances on a uniform base flow down an inclined plane, which leads to pattern formation and anti-dunes.…”

Two-Dimensional Numerical Simulation of Bed-Load Transport of a Finite-Depth Sediment Layer: Applications to Channel Flushing

“…As stated by Furbish et al (2012), the bulk sediment transport rate is the sum of the instantaneous sediment transport rate plus the contribution of particle diffusivity. However, because the parametric dependence of particle diffusivity with flow variables is challenging (Bohorquez and Ancey 2015;Heyman et al 2014Heyman et al , 2016, its contribution to the sediment rate is not considered in this work.…”

Scales of collective entrainment and intermittent transport in collision-driven bed load

“…Starting with Ancey et al (2008b), a series of models for bed-load transport have been proposed that posit that particles are often entrained collectively rather than individually at low mean transport rates. These models propose modifications of Einstein's entrainment function that take this correlated behavior into account through the introduction of a collective entrainment rate, µ, that leads to a characteristic correlation length scale, l c (Ancey et al (2008b); Heyman et al (2013);Ma 30 et al (2014);Heyman et al (2014)). As the mean transport rate is lowered, modeled values for µ and l c must increase in order to reproduce the observed growth in variance of bed-load activity (i.e., the number density of moving grains).…”