Mobility of bidisperse mixtures during bedload transport

The flow of segregated bidisperse assemblies of particles is of major importance for geophysical flows and bedload transport in particular. In the present paper, the mobility of strictly bidisperse segregated particle beds is studied with a coupled fluid discrete element method (DEM). Large particles are initially placed above small ones in a three-dimensional domain inclined at a slope of $10\%$. A gravity-driven water free surface flow induces a downslope shear-driven granular flow of the erodible bed. It is observed that, for the same water flow conditions, the bedload transport rate is higher in the bidisperse configuration than in the monodisperse one. Depending on the Shields number and on the depth of the interface between small and large particles, different transport phenomenologies are observed, ranging from no influence of the small particles to small particles reaching the bed surface due to diffusive remixing. In cases where the small particles hardly mix with the overlying large particles and for the range of studied size ratios $(r<4)$, it is shown that the increased mobility is not a bottom roughness effect, that would be due to the reduction of roughness of the underlying small particles, but a granular flow effect. This effect is analyzed within the framework of the $\mu \left(I\right)$ rheology, modeling the stress to strain rate relation for dense granular flows. It is demonstrated that the buried small particles are more mobile than larger particles and play the role of a “conveyor belt” for the large particles at the surface. Based on rheological arguments, a simple predictive model is proposed for the additional transport in the bidisperse case. It reproduces quantitatively the DEM results for a large range of Shields numbers and for size ratios smaller than 4. The results of the model are used to identify four different transport regimes of bidisperse mixtures, depending on the mechanism responsible for the mobility of the small particles. A phenomenological map is proposed for bidisperse bedload transport and, more generally, for any granular flow on an erodible bed.

Figure 1

A typical numerical setup. Initially $N_l$ layers of large particles ($d_l=6$ mm) are deposited by gravity on $N_s$ layers of small particles ($d_s=3$ mm). The fluid of depth $h_w$ flows by gravity due to the slope angle $\alpha$ and entrains particles.

Figure 2

Illustration of some considered initial configurations. (a) $N_l=2$, (b) $N_l=4$, (c) monodisperse case.

Figure 3

(a) Solid transport rate as a function of the Shields number for all simulations. (b) Increased transport rate in percentage compared with monodisperse configuration.

Figure 4

Transport profiles of each class of particles for different configurations and Shields numbers. (a) $\theta \sim 0.2$, $N_l=4$, (b) $\theta \sim 0.45$, $N_l=2$, (c) $\theta \sim 0.55$, $N_l=1$. The transport profile of the large particles in the monodisperse case for the same Shields number is also plotted for comparison. Note changes in the abscissa scale.

Figure 5

Mean surface diameter as a function of the Shields number and the large particle number of layers. The dashed line shows the transition between a large particle surface state and a mixture surface state.

Figure 6

Comparison of the monodisperse (dotted line) and the bidisperse $N_l=2$ (full line) configuration for $\theta \sim 0.45$. (a) Pressure and shear stress profiles, (b) friction coefficient profiles, (c) inertial number profiles, and (d) velocity profiles. The dotted black line ($\text{..........}$) corresponds to a translation of $\mathrm{\Delta }v=0.383\sqrt{g{d}_l}$ of the monodisperse velocity profile (blue dotted line). The lower horizontal line at $z_1$ ($\text{----}$) separates the quasistatic regime from the flowing dense regime. The upper horizontal line at $z_i$ ($–.–$) shows the transition from small to large particles in the bidisperse configuration.

Figure 7

(a) Dimensionless additional transport measured in the DEM simulations (solid symbols) and predicted by Eq. (23) (open symbols), for different values of the Shields number and $N_l$. Only cases for which the surface is exclusively composed of large particles are presented for readability. (b) Error between the total transport predicted by Eq. (23) and the transport computed with the DEM simulations.

Figure 8

Comparison between idealized (dotted lines) and DEM profiles (full lines) in the monodisperse configuration for $\theta \sim 0.45$. (a) Concentration profiles, (b) granular pressure and shear stress profiles, (c) friction coefficient profiles, and (d) velocity profiles.

Figure 9

(a) Dimensionless additional transport in the bidisperse case obtained with the DEM simulations (solid symbols) and computed with Eq. (29) (open symbols), for different values of Shields number and $N_l$. Only cases for which the surface is composed of large particles only are presented for readability. (b) Error between the total bidisperse transport predicted by Eq. (29) and the measured transport with the DEM simulations.

Figure 10

Mapping of the four different observed phenomenologies in the bidisperse transport process. Each regime is illustrated with a typical simulation picture where the creeping flow has been shaded in gray. Results are plotted in colored squares and split into two classes: blue (predicted transport error less than $10\%$) and brown (larger error).

Figure 11

(a) Additional transport rate predicted by Eq. (29) (orange crosses) and computed from the DEM simulations (blue squares) for different size ratios at $\theta \sim 0.45$ and $N_l=2$. (b) Nondimensional surface diameter in the DEM simulations as defined in Eq. (12) as a function of the size ratio. Cases $r=2.5$ and $r=4$ are illustrated with a picture from the DEM simulation.

Figure 12

Monodisperse case at $\theta \sim 0.45$. (a) Granular pressure from DEM simulation (full line) and computed with analytical expression (A6) (dashed line). (b) Fluid and granular shear stress from DEM simulation (full line) and computed from analytical expression (A9) (dotted line) and expression (A11) (dashed line).