Lagrangian and Eulerian Description of Bed Load Transport

Abstract

Sediment particles transported as bed load undergo alternating periods of motion and rest, particularly at weak flow intensity. Bed load transport can be investigated by either following the motion of individual particles (Lagrangian approach) or observing the phenomenon at prescribed locations (Eulerian approach). In this paper, the Lagrangian and Eulerian descriptions are merged into a unifying framework that includes definitions for quantities used to describe the kinematics of particle motion, as well as the relationships among these quantities. The alternation of motion and rest is represented by two complementary descriptions: (i) proportion of motion, indicating either the relative time spent in motion by an individual particle or the relative number of moving particles; (ii) persistence of motion, indicating the extent to which the process consists of relatively few long periods of motion or of many short ones. The framework only involves first moments of the key quantities. The conceptual developments are tested against results from an experiment with weak bed load transport, demonstrating the soundness of the approach. From an operational point of view, a Lagrangian observation is difficult to perform, since the particle motion is usually investigated for finite spatial domains (e.g., a measurement window within a laboratory or natural reach). Strategies to overcome such limitations are described, suggesting the possibility of obtaining unbiased mean values for Lagrangian descriptors. The proposed framework can be used in any study aimed at parameterizing the kinematic properties of bed load particles as functions of the hydrodynamic conditions.

Sand particle velocities over a subaqueous dune slope using high‐frequency image capturing

“…Local bedload transport rate is the sum of the product of individual particle volume in motion and the streamwise velocity of the particle (Shim and Duan, 2017;Ballio et al, 2018). However, those predictors are limiting our understanding of local sediment transport over non-plain dune beds, for which gradients in sediment transport exist.…”

“…However, those predictors are limiting our understanding of local sediment transport over non-plain dune beds, for which gradients in sediment transport exist. Local bedload transport rate is the sum of the product of individual particle volume in motion and the streamwise velocity of the particle (Shim and Duan, 2017;Ballio et al, 2018). No disruption of flow is the major advantage of using imagery to quantify sediment transport.…”

Diffusive Regimes of the Motion of Bed Load Particles in Open Channel Flows at Low Transport Stages

“…For the tests carried out in the straight flume the mean resting time of particles that moved at least once ranged between τ r = 4 and 5 s, resulting in an average τ r u* / d = 65, which is close to the largest dimensionless time explored. The reported mean resting time is likely affected by data censorship (Ballio et al, ). It can be argued that the investigated area in the straight flume was too short to enable direct observation of global ranges.…”

Event‐scale pebble mobility observed by RFID tracking in a pre‐Alpine stream: a field laboratory

“…Papanicolaou et al, 2002;Ancey et al, 2006;Radice et al, 2009;Roseberry et al, 2012;Heyman et al, 2014). Ballio et al (2018) recently introduced a proportion of motion that, in a Lagrangian sense, corresponds to the fraction of time for which a bedload particle is moving while, in a Eulerian sense, corresponding to the relative number of moving particles within a fixed area of observation. All these quantities (concentration, activity, proportion) may be retrieved from each other by means of appropriate scaling factors.…”