Onset of grain motion in eroding subaqueous bimodal granular beds

We report experimental measurements of the onset of grain motion in fully submerged erodible beds containing grains of two sizes driven by a turbulent shear flow. Although we find that a traditional Shields-number framework successfully accounts for grain-size effects in beds composed of grains of only a single size, our results show that bimodal beds are distinct in a way that cannot be captured by a Shields-number rescaling. In particular, we find that large grains in a bimodal bed are mobilized into bedload transport by stresses that would normally be subcritical for them but that the behavior of small grains is less significantly affected. By analyzing higher-order statistics, we reveal the key role played by the granular contact and force networks in the bed near onset and clarify the impact of grain-size polydispersity on the transition from a static to an eroding bed. Our results have implications for modeling sediment transport in natural systems where polydispersity is unavoidable.

Figure 1

Velocity profiles as measured from PTV for five representative flow rates. The solid lines are fits to the data using the method described by Rodríguez-López et al. [25], from which we extract the friction velocity and wall shear stress.

Figure 2

(a) Mean grain velocity $\langle u_g\rangle$ averaged over all identified grains in unimodal beds plotted as a function of the bed shear stress ${\tau }_w$ for small grains (circles) and large grains (triangles). Error bars show the standard error computed over six trials. (b) The same data as in (a) but scaled by the flow velocity evaluated one grain diameter above the bed $u\left(D_i\right)$ and plotted as a function of ${\mathrm{\Theta }}_i-{\mathrm{\Theta }}_{i,c}$, where ${\mathrm{\Theta }}_i$ is the grain-specific Shields number and ${\mathrm{\Theta }}_{i,c}$ is the estimated critical Shields number.

Figure 3

Comparison of the mean grain velocity $\langle u_g\rangle$ in unimodal beds (blue circles and triangles) and bimodal beds (red squares and diamonds) for (a) small grains and (b) large grains. The unimodal data are the same as in Fig. 2. For the bimodal data, $\langle u_g\rangle$ is computed by averaging over all identified grains of a particular size, and the error bars show the standard error computed over 11 trials. Grain velocities are scaled by the mean flow velocity evaluated at one grain diameter above the bed as in Fig. 2, and data are plotted as a function of the small- and large-grain Shields numbers ${\mathrm{\Theta }}_s$ and ${\mathrm{\Theta }}_l$, respectively. Data for small grains in unimodal and bimodal beds (a) collapse when plotted in this way, but the same is not true for the large grains.

Figure 4

Representative velocity probability density functions (PDFs) for the case of small grains in bimodal beds (solid lines); results for other experimental cases are qualitatively similar. Data are shown for ${\mathrm{\Theta }}_s=0.018$, 0.024, 0.030, 0.037, and 0.045, from bottom to top. The dashed lines are fits of Eq. (2).

Figure 5

Independent parameters in our model velocity PDF (that is, $B, u_g^{*}, \sigma$, and $\zeta$) as determined from fits to the data for the four experimental cases considered as a function of the difference between the grain-specific Shields number ${\mathrm{\Theta }}_i$ and the corresponding estimated critical value ${\mathrm{\Theta }}_{i,c}$. Data are shown for small grains in unimodal beds (circles), small grains in bimodal beds (squares), large grains in unimodal beds (triangles), and large grains in bimodal beds (diamonds). The error bars represent standard errors, computed in the same way as in Figs. 2 and 3. The parameters $B$ and $\zeta$ are dimensionless by construction. The typical speed of a moving grain $u_g^{*}$ is scaled by the flow velocity one grain diameter above the bed $u\left(D_i\right)$, as above, and the typical velocity fluctuation $\sigma$ is scaled by the friction velocity $u_{*}$.