A microstructural approach to bed load transport: mean behaviour and fluctuations of particle transport rates

Abstract

This paper concerns a model of bed load transport, which describes the advection and dispersion of coarse particles carried by a turbulent water stream. The challenge is to develop a microstructural approach that, on the one hand, yields a parsimonious description of particle transport at the microscopic scale and, on the other hand, leads to averaged equations at the macroscopic scale that can be consistently interpreted in light of the continuum equations used in hydraulics. The cornerstone of the theory is the proper determination of the particle flux fluctuations. Apart from turbulence-induced noise, fluctuations in the particle transport rate are generated by particle exchanges with the bed consisting of particle entrainment and deposition. At the particle scale, the evolution of the number of moving particles can be described probabilistically using a coupled set of reaction-diffusion master equations. Theoretically, this is interesting but impractical, as solving the governing equations is fraught with difficulty. Using the Poisson representation, we show that these multivariate master equations can be converted into Fokker-Planck equations without any simplifying approximations. Thus, in the continuum limit, we end up with a Langevin-like stochastic partial differential equation that governs the time and space variations of the probability density function for the number of moving particles. For steady-state flow conditions and a fixed control volume, the probability distributions of the number of moving particles and the particle flux can be calculated analytically. Taking the average of the microscopic governing equations leads to an average mass conservation equation, which takes the form of the classic Exner equation under certain conditions carefully addressed in the paper. Analysis also highlights the specific part played by a process we refer to as collective entrainment, i.e. a nonlinear feedback process in particle entrainment. In the absence of collective entrainment the fluctuations in the number of moving particles are Poissonian, which implies that at the macroscopic scale they act as white noise that mediates bed evolution. In contrast, when collective entrainment occurs, large non-Poissonian fluctuations arise, with the important consequence that the evolution at the macroscopic scale may depart significantly that predicted by the averaged Exner equation. Comparison with experimental data gives satisfactory results for steady-state flows.

Bedrock incision by bedload: insights from direct numerical simulations

“…The bedload flux then reaches a relatively steady value: from this moment we consider that the system is in the permanent regime. Let us note that we still observe relatively large fluctuations of the bedload flux, consistent with what has been observed experimentally (see, for instance, Böhm et al, 2004, andAncey et al, 2006), and which motivated stochastic approaches to bedload transport on top of a sediment layer ( Roseberry et al, 2012;Ancey and Heyman, 2014;Fan et al, 2014). However, we will not investigate these fluctuations in further detail and in the rest of the discussion all results are computed in the permanent regime and only concern the average values of the sediment and energy fluxes.…”

Why and How Do We Study Sediment Transport? Focus on Coastal Zones and Ongoing Methods

“…New types of models are developed, tested and evaluated on simple configurations to improve knowledge of basic mechanisms at a fine scale, e.g., direct numerical simulation of bedform evolution and/or sediment transport [298,299], emerging methods of smoothed particle hydrodynamics for fluid-flow interaction [194]. In-depth theoretical analysis is also ongoing on bedload (e.g., [300,301]), on bedforms [302], on the transition from bedload to suspended load [303,304], on the granular flow rheology in bedload transport [305], amongst others. These works, not yet adapted to the study of natural environments, precede the models evolution and move forward our knowledge of the involved mechanisms.…”

Determinants of modelling choices for 1-D free-surface flow and morphodynamics in hydrology and hydraulics: a review

“…See for example Ancey and Heyman (2014) and Ballio et al (2014) for the various possible formulations of the sediment continuity equation and associated numerical aspects, depending on the strength of the intended coupling with the carrier phase. The authors rather prefer the fluid mechanics type of use of the SV equations for hydroenvironmental applications that necessitate taking maximum advantage of the level of details offered by Eq.…”