Generalized Bed-Load Function Based on Empirical Data

Abstract

There exist many bed-load functions in the literature to calculate bed-load transport rates, but none of them fit data from low to high shear stress conditions. This research presents a generalized bed-load function based on empirical data. Specifically, the classic power law in high shear stress conditions is extended to low shear stress conditions by applying a complimentary error function (or logistic function) and using Coles’ mathematical idea for the wake law in turbulent boundary layer velocity distribution. The resulting generalized bed-load function agrees well with the classic data sets; it reduces to Huang’s $5/3$ power law in the very low and the high shear stress conditions, and it is numerically close to Paintal’s 16th power law in the transitional regime. It is found that the maximum turbulence-induced lift force and the minimum critical shear stress in the Shields diagram correspond to the inflection point (in terms of logarithmic scale) in the Einstein bed-load diagram, resulting in the most efficient bed-load transport rate. After that, this paper discusses the effects of turbulence-induced lift force, critical shear stress, viscosity, nonlinearity, and uncertainty on bed-load transport. Finally, an example with uncertainty analysis is illustrated for applications.