Erosion of a granular bed driven by laminar fluid flow

TL;DR

This study investigates how laminar fluid flow causes erosion in a granular bed, revealing a power-law relationship between bed height, fluid flux, and shear stress, and refining the Shields criterion for predicting granular erosion.

Contribution

It introduces a detailed experimental analysis of granular bed erosion under laminar flow, refining the Shields criterion with a new offset and power-law scaling.

Findings

Bed erodes to a rest height proportional to the square root of fluid flux.

Refined Shields criterion with a half-grain diameter offset collapses data.

Granular flux follows a power-law relationship with bed stress.

Abstract

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 $h_r \propto Q^{1/2}$ 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 $1.75 \pm 0.25$. The dynamics of the bed height relaxation are well described by the power law relationship between the granular flux and the bed stress.