Erosion of a granular bed driven by laminar fluid flow

Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed driven by a laminar fluid to give us new insights into the relationship between hydrodynamic stress and surface granular flow. A closed cell of rectangular cross-section is partially filled with glass beads and a constant fluid flux Q flows through the cell. The refractive indices of the fluid and the glass beads are matched and the cell is illuminated with a laser sheet, allowing us to image individual beads. The bed erodes to a rest height h(r) which depends on Q. The Shields threshold criterion assumes that the non-dimensional ratio theta of the viscous stress on the bed to the hydrostatic pressure difference across a grain is sufficient to predict the granular flux. Furthermore, the Shields criterion states that the granular flux is non-zero only for theta > theta(c). We find that the Shields criterion describes the observed relationship h(r) proportional to Q(1/2) when the bed height is offset by approximately half a grain diameter. Introducing this offset in the estimation of theta yields a collapse of the measured Einstein number q* to a power-law function of theta - theta(c) with exponent 1.75 +/- 0.25. The dynamics of the bed height relaxation are described well by the power-law relationship between the granular flux and the bed stress.