Characterization of base roughness for granular chute flows

Base roughness plays an important role in the dynamics of granular flows but is still poorly understood due to the difficulty of its quantification. For a bumpy base made of spheres, at least two factors should be considered in order to characterize its geometric roughness, namely, the size ratio of flow to base particles and the packing arrangement of base particles. In this paper, we propose an alternative definition of base roughness, $R_a$, as a function of both the size ratio and the distribution of base particles. This definition is generalized for random and regular packings of multilayered spheres. The range of possible values of $R_a$ is presented, and optimal arrangements for maximizing base roughness are studied. Our definition is applied to granular chute flows in both two- and three-dimensional configurations, and is shown to successfully predict whether slip occurs at the base. A transition is observed from slip to nonslip conditions as $R_a$ increases. Critical values of $R_a$ are identified for the construction of a nonslip base at various angles of inclination.

Figure 1

Case setup. (a) Chute flow in experiments; (b) 3D and (c) 2D periodic elements in DEM.

Figure 2

Base generation. Random packing in 3D by (a–c) base thickness and (d–f) packing density; regular packing by spacing in (g–i) 3D and (j–l) 2D. Note that the regular packing in (g) is streamwise and in (h) is spanwise. Flow is directed down the incline.

Figure 3

Basal effect. (a–c) Typical velocity profiles for Φ=1.0, 0.5, 2.0, respectively. (d–f) Flow snapshots near base (with $H_b/d=1.8$) for $\mathrm{\Phi }=1.0$, 0.5, 2.0, respectively.

Figure 4

Base velocity as a function of (a) base thickness and (b) packing density. Symbols represent size ratio: $\mathrm{\Phi }=2.0\left(▵\right),1.25(\circ ),1.0(\square ), 0.67(\diamond ),0.5(+),0.4(\times )$. Cases annotated by arrow circles in (a) have the same base thickness but different packing density, and in (b) have similar base surface $\left({\eta }_s\right)$ but different underneath layers.

Figure 5

Discretization of a surface made of spheres. (a) Original layer, (b) projection and discretization, (c) a discretized triangle.

Figure 6

Definition of local roughness. Larger balls represent flowing particles and smaller balls the fixed base particles. Dashed balls are hidden in 3D. Dashed lines link centroids of balls. The arrow points to the flow direction. (a) An arbitrary void with area $A_i$ in 3D, (b) an equilateral void with area $A_m$ in 3D (most stable with respect to flow), (c) an arbitrary void with length $L_i$ in 2D, (d) a void with length $L_m$ in 2D (most stable with respect to flow).

Figure 7

Critical void area in (a) 3D and (b) 2D situations.

Figure 8

A base of multiple layers (2D for clarity). Replace $L$ by $A$ in 3D.

Figure 9

Values of $R_a$ for different size ratio. Dashed lines with filled and empty squares are lower bounds in 3D and 2D, respectively. Solid lines with filled and empty triangles are upper bounds in 3D and 2D, respectively. Dotted line marks ${\mathrm{\Phi }}_0=0.155$.

Figure 10

Typical optimal packing $(\mathrm{\Phi }=1)$. Circles with solid and dashed borders are base and flow particles, respectively. Shadows distinguish base particles in different layers. (a) Top view of a one-layer optimal base in 3D. (b) Top view of a two-layer optimal base in 3D, where base thickness is around $1.82d$. (c) Side view of a one-layer optimal base in 2D. (d) Side view of a two-layer optimal base in 2D, where base thickness is around $1.87d$.

Figure 11

Transition of slip and nonslip condition in (a) 3D $(\theta ={30}^{\circ })$ and (b) 2D $(\theta ={25}^{\circ })$. Symbols represent size ratio $\mathrm{\Phi }=2.0\left(▵\right),1.25(\circ ),1.0(\square ), 0.67(\diamond ),0.5(+),0.4(\times )$, respectively. Error bars result from a series of different random seeds in simulations.

Figure 12

Slip and nonslip condition at different angles of inclination. Inset: base velocity as a function of base thickness for $\mathrm{\Phi }=1.0$ at different inclinations.

Figure 13

Phase diagram for different values of slope angle and base roughness. Symbols represent regimes of sliding (circle), transition (triangle), and nonslip (square), respectively. Color map indicates the value of $v_b/v_s$.

Figure 14

Simplified definition vs rigorous definition. Horizontal bars represent the range of local roughness, $R_{ai}$, obtained when layer averaging is performed.

Figure 15

Effect of packing orientation $(\mathrm{\Phi }=1.0,\varepsilon =0;R_a=0.58)$. Streamwise order [also Fig. 2] is made of triangles at $\pi /6$ and $\pi /2$, while spanwise order [also Fig. 2] is made of triangles at 0 and $\pi /3$, with respect to the flow direction.