13. Principles of Bedload Transport of Non-cohesive Sediment in Open-Channels

Note that gravity appears in Eq. (13.4) through terms \(u_{*}\) and \(g(\rho^{(g)} - \rho^{(w)} )\); in the latter case it allows for separating clearly the effects of grain weight, directly depending of \(g(\rho^{(g)} - \rho^{(w)} )\), and inertial effects, depending only on \(\rho^{(g)}\) (Yalin 1977).

We define exposed grain as that which, to be able to move, would not have to disturb the position of its neighbours (we ignore, for now, the possibility of collective entrainment).

The ensemble analysis underlying Eq. (13.26) is not possible in practice since it is not feasible to subject the exact same bed to a succession of turbulent flows. The practical way to generate a succession of flow realizations is to expose the bed to a turbulent flow over a sufficiently long but finite time interval, so that all relevant turbulent scales are included. Hence, in practical terms, an empirical probability of entrainment can be obtained for a set of bed locations, coincident with the position of a set of grains that may be substituted over the course of time.

The view of sediment transport as a purely size selective process has been rejected (Parker and Klingeman 1982; Komar 1987). Deterministic bedload discharge formulas that include a dependence of a power of \(\theta - \, \theta_{c}\) require corrections to either the Shields or the critical Shields parameters when applied to the transport of size fractions. Hiding-exposure coefficients have been proposed to reduce/increase \(\theta_{k} - \, \theta_{ck}\) for given size fractions d _k smaller/larger than a reference diameter (Egiazaroff 1965; Ashida and Michiue 1972). As seen in the example of Fig. 13.6, hiding and exposure describe different phenomena. The former is related to local modifications of the flow field which, for smaller size fractions may signify reduced mobility due to sheltering or decreased near bed turbulence (Nelson et al. 1995). The latter expresses the positioning of the grain relatively to its immediate neighbours. Increased exposure and decreased support angle are common of larger size fractions (Sect. 13.3 and, e.g., Komar and Li 1988). Due to this combined effect, in the limit, equal mobility of different size fractions can be nearly attained for gravel mixtures (Parker and Klingeman 1982; Parker et al. 1982), although the range of conditions for this to happen has been shown to be relatively narrow (Wathen et al. 1995; Parker and Toro-Escobar 2002). Hiding/exposure coefficients thus account, in a lumped way, for the differential grain response regarding entrainment, depending on the particular location/diameter of the particle relatively to their neighbours.

Kennedy (1995) believed that Shields (1936) used Kramer’s (1935) category of weak-countable transport conditions to define the incipient motion conditions, thus reducing subjectivity to the notion of “countable”.