Effective Wall Friction in Wall-Bounded 3D Dense Granular Flows

We report numerical simulations on granular shear flows confined between two flat but frictional sidewalls. Novel regimes differing by their strain localization features are observed. They originate from the competition between dissipation at the sidewalls and dissipation in the bulk of the flow. The effective friction at sidewalls is characterized (effective friction coefficient and orientation of the friction force) for each regime, and its interdependence with slip and force fluctuations is pointed out. We propose a simple scaling law linking the slip velocity to the granular temperature in the main flow direction which leads naturally to another scaling law for the effective friction.

Figure 1

(a) Sketch of the wall-bounded flow configuration. (b) Velocity profiles at the sidewalls along $z$ obtained for $\tilde{V}=10$, $\tilde{M}=0.2$, and different values of the particle-wall friction coefficient: ${\mu }_{\text{pw}}=0.05,0.1,0.2$, and 0.3. Three patterns of localization appear. (c) Fluctuation of the $x$ particle velocities at the wall along $z$ for the same set of parameters. Inset: Ratio of vertical to horizontal fluctuations versus $z$.

Figure 2

Profiles along $z$ of (a) the effective wall friction coefficient ${\mu }_w$ and of (b) ${\mu }_w$ rescaled on the particle-wall friction coefficient ${\mu }_{\text{pw}}$, for $\tilde{V}=10$, $\tilde{M}=0.2$, and different values of ${\mu }_{\text{pw}}=0.05,0.1,0.2,0.3$. ${\mu }_w/{\mu }_{\text{pw}}$ decreases with ${\mu }_{\text{pw}}$. (c) Profiles of the bulk friction coefficient ${\mu }_{xz}={\sigma }_{xz}/p$ along $z$. The labels are the same as those used in Fig. 1.

Figure 3

Polar representation of the probability density function of tangential force orientation at the wall, for $\tilde{V}=1$, $\tilde{M}=0.2$, at several heights $z$ ($=3d$, $9d$, $18d$, and $36d$), for (a) ${\mu }_{\text{pw}}=0.1$ (regime C) and (b) ${\mu }_{\text{pw}}=0.3$ (regime A). Statistics are performed on a $5d$-wide slice centered on the given $z$ value.

Figure 4

Comparison between a velocity scale based on shear rate, $d\partial v_x/\partial z$, and one based on $x$-velocity fluctuations, $\sqrt{T_x}$ for $\tilde{M}=0.2$ and $\tilde{V}=0.1$ (square), $\tilde{M}=0.2$ and $\tilde{V}=1$ (diamond), $\tilde{M}=0.2$ and $\tilde{V}=10$ (downward triangle), and $\tilde{M}=2$ and $\tilde{V}=1$ (circle). The colors (gray scales) correspond to different grain-wall friction coefficients [see Fig. 1]. The fluctuational velocity scale displays a plateau for each simulation for low shear rates. Inset: $\sqrt{T_x}/\dot{\gamma }d$ increases when decreasing inertial number $I$. The labels are the same as those used in Fig. 1.

Figure 5

Rescaled effective wall friction coefficient versus the dimensionless slip parameter $v_x/\sqrt{T_x}$. The inset shows that a power law relationship holds between $[({\mu }_w/{\mu }_{\text{pw}})/(1-{\mu }_w/{\mu }_{\text{pw}})]$ and $v_x/\sqrt{T_x}$. The labels are the same as those used in Fig. 4.