Bed-Load Transport Equation for Sheet Flow

When open-channel flows become sufficiently powerful, the mode of bed-load transport changes from saltation to sheet flow. Where there is no suspended sediment, sheet flow consists of a layer of colliding grains whose basal concentration approaches that of the stationary bed. These collisions give rise to a dispersive stress that acts normal to the bed and supports the bed load. An equation for predicting the rate of bed-load transport in sheet flow is developed from an analysis of 55 flume and closed conduit experiments. The equation is $i_b=\omega$ where $i_b=\mathrm{immersed}$ bed-load transport rate; and $\omega =\mathrm{flow}$ power. That $i_b=\omega$ implies that $e_b=\tan \alpha =u_b/u\text{,}$ where $e_b=\mathrm{Bagnold}’s$ bed-load transport efficiency; $u_b=\mathrm{mean}$ grain velocity in the sheet-flow layer; and $\tan \alpha =\mathrm{dynamic}$ internal friction coefficient. Given that $\tan \alpha \approx 0.6$ for natural sand, $u_b\approx 0.6u\text{,}$ and $e_b\approx 0.6\text{.}$ This finding is confirmed by an independent analysis of the experimental data. The value of 0.60 for $e_b$ is much larger than the value of 0.12 calculated by Bagnold, indicating that sheet flow is a much more efficient mode of bed-load transport than previously thought.