Bed Load Transport: From Individual Particle Forces To Group Transport Behavior

The stochasity of bed load has been a key research topic for more thanseveral decades, which is important and challenging. The stochastic motion ofbed load particles eventually results from the fluctuating forces exerted onindividual particles, and thus deduce that in small and immediate temporal andspatial scales, the velocities, step lengths, step times and waiting times are allrandom variables, and in large temporal and spatial scales these randomvariables result in diffusion of a group of particles.This paper focuses on the contact load (i.e., sliding and rolling are the mainregime for particle motion). A framework based on statistical mechanics isdeveloped. From the analysis of actual forces exerted on one single particle, theforces are break into two parts: deterministic part and stochastic part, and thus astochastic differential equation, which is called Langevin Equation (LE) isobtained. From the Langevin Equation describing the forces exerted on onesingle particle, the Fokker-Planck Equation (FPE), which describes theevolution of the Probability Density Function (PDF) of a group of particles isderived, as a result, the characteristics of micro-scale (individual particle) andmacro-scale (a group of particles) are connected, and could be described at therespective scales. Based on the above achievements, waiting time is introducedto the LE and the Episodic Langevin Equation (ELE) is obtained, which enableus to simulate a large number of particles in both large temporal and spatialscales, and both advection and diffusion characteristics are studied.The steady solutions of the FPEs show exponential-like PDFs of particlevelocities (i.e. PDFs for both positive and negative particles decay exponentially)in both streamwise and cross-stream directions, and fit well with theexperimental data. The steady solutions of the FPEs theoretically explain thatthe exponential-like PDFs of particle velocities result from the forces exerted onindividual particles: the friction force together with the stochastic force deducesthe exponential-like PDF of the velocities, and the deterministic downstreamforce makes the PDF to be asymmetric. From the velocities time series simulated by the LE, step characteristics areextracted and the relationship of step lengths and step times fit well with theexperimental data, both show5/3power law. As a result, step lengths and steptimes are correlated and thus the temporal and spatial relations of particlemotions are coupled.The forces exerted on moving particles could be described well by the LEbut those on still particles could not be. Multi-valued friction force is a mainchallenge for describing forces exerted on still particles. Waiting times areintroduced to the LE and the ELE is obtained, from which the episodic motionof particles can be simulated. From the ELE with different kind of waiting times(heavy/thin tailed), normal and anomalous advection and diffusioncharacteristics for both uniform particles and a mixture of particles are studied.The simulated results illustrated that (1) for uniform particles, the PDF ofstep lengths is heavy tailed, but super-diffusion could not necessarily show,which is limited by thin tailed velocities. The diffusion characteristic isdetermined by the tail of waiting times, sub-, super-and normal diffusion couldall occur and only very heavy tailed waiting times could deduce sub-diffusion.Advection characteristics will not be affected if step times are ignored butdiffusion characteristics will be.(2) For a mixture of particles, identicaldistributed of random variables, which is a basic assumption for random walkmodels is no more satisfied, so the heavy tailed waiting times for a mixture ofparticles deduce completely different characteristics of both advection anddiffusion from uniform particles, e.g. ballistic always shows in large temporaland spatial scales. Transport of a mixture of particles in large temporal andspatial scales is deterministic downstream sorting, rather than anomalousdiffusion.Bed elevation dynamics are also studied from experiments conducted in alarge scale flume. The experiments contains two kinds of bed surface materiale.g. gravel mixed with sand and sand. The increment of elevation series shownexponential-like distribution, but the spectrum characteristics were not the sameas the time series simulated by the LE we used. The slopes of spectrum of bedelevation series for gravel mixed with sand and sand are-2and-3respectively.The absolute value for gravel mixed with sand is smaller than sand, which reveals slower decay of energy, it is concluded that gravel mixed sand canproduce structure, and thus make the surface relative steady to resist flow.