A probabilistic treatment of fluvial entrainment of cohesionless particles

Abstract

In the traditional treatment of sediment entrainment in fluvial hydrology, equating a turbulent fluid force arising from shear stress with a particle weight (adjusted for buoyancy) gives the condition on critical shear for particle entrainment. This stipulation leads to the entrainment condition that the volume d3 of the largest particles entrained is proportional to the sixth power of the friction velocity. An alternate approach is to consider streams as reservoirs of kinetic energy, analogous to heat baths, from which energy can be removed to lift a particle over a potential energy barrier. Following this analogy, the probability of transferring an energy, E, from the fluid to a particle, is proportional to the Boltzmann factor, exp(‐E/H), where H is the appropriate analog to kT, a unit of thermal energy associated with the heat bath. By making the analogy H = (1/2)ρw(ν*2)d3, where ν* is the friction velocity and ρw is the fluid density, the entrainment condition is recovered in a probabilistic sense, i.e. the probability of entrainment of particles larger than d drops rapidly to zero. The new condition may be interpreted in terms of the likelihood that eddies (similar in size to the particle concerned) transfer their energy to a particle. The advantage of this approach is that it can treat particles of arbitrary size on a bed of arbitrary geometry. Extension of the treatment to such collections of particles allows derivation of a self‐consistent integral equation for the distribution of particle sizes on a bed.