A probabilistic derivation of the exponential-like distribution of bed load particle velocities

Particles transported as bed load within a specified streambed area possess a distribution of velocities. This distribution figures prominently in describing the rate of sediment transport and the rate of dispersal of particles during transport. We provide a probabilistic derivation of the distributions of streamwise and cross-stream particle velocities under uniform, quasi-steady transport conditions. The formulation centers on defining the ensemble of microstates of particle momenta (the set of possible ways to partition a fixed number of particles into momentum states) subject to the constraint that the sum of the particle momenta in each microstate is fixed, a constraint that is imposed by conditions of equilibrium transport. From this, we obtain the most probable distribution of momentum states, assuming each microstate within the ensemble is equally probable. The analysis suggests that, for small particle numbers, the distribution of velocities is exponential-like but decays faster than an exponential function. For large particle numbers, the distribution is exponential. These distributions are consistent with experimental results from high-speed imaging of sand particles transported as bed load over a planar bed, which reveal that the probability distributions of streamwise and cross-stream particle velocities are exponential-like. These particle velocity distributions also emerge from numerical analyses involving large eddy simulations of turbulent fluid motions and a discrete element method describing particle motions, wherein the large eddy simulations and discrete element method models are fully coupled in momentum.