Bed load at low Shields stress on arbitrarily sloping beds: Alternative entrainment formulation

According to the Bagnold hypothesis for equilibrium bed load transport, a necessary constraint for the maintenance of equilibrium bed load transport is that the fluid shear stress at the bed must be reduced to the critical, or threshold, value associated with incipient motion of grains. It was shown in a companion paper [Seminara et al., 2002], however, that the Bagnold hypothesis breaks down when applied to equilibrium bed load transport on beds with transverse slopes above a relatively modest value that is well below the angle of repose. An investigation of this failure resulted in a demonstration of its lack of validity even for nearly horizontal beds. The constraint is here replaced with an entrainment formulation, according to which a dynamic equilibrium is maintained by a balance between entrainment of bed grains into the bed load layer and deposition of bed load grains onto the bed. The entrainment function is formulated so that the entrainment rate is an increasing function of the excess of the fluid shear stress at the bed over the threshold value. The formulation is implemented with the aid of a unique set of laboratory data that characterizes equilibrium bed load transport at relatively low shear stresses for streamwise angles of bed inclination varying from nearly 0degrees to 22degrees. The formulation is shown to provide a description of bed load transport on nearly horizontal beds that fits the data as well as that resulting from the Bagnold constraint. The entrainment formulation has the added advantage of not requiring the unrealistically high dynamic coefficient of Coulomb friction resulting from the Bagnold constraint. Finally, the entrainment formulation provides reasonable and consistent results on finite streamwise and transverse bed slopes, even those at which the Bagnold formulation breaks down completely.