Vortices Enhance Diffusion in Dense Granular Flows

This Letter introduces unexpected diffusion properties in dense granular flows and shows that they result from the development of partially jammed clusters of grains, or granular vortices. Transverse diffusion coefficients $D$ and average vortex sizes $\ell$ are systematically measured in simulated plane shear flows at differing inertial numbers $I$ revealing (i) a strong deviation from the expected scaling $D\propto d^2\dot{\gamma }$ involving the grain size $d$ and shear rate $\dot{\gamma }$ and (ii) an increase in average vortex size $\ell$ at low $I$, following $\ell \propto d{I}^{-\frac{1}{2}}$ but limited by the system size. A general scaling $D\propto \ell d\dot{\gamma }$ is introduced that captures all the measurements and highlights the key role of vortex size. This leads to establishing a scaling for the diffusivity in dense granular flow as $D\propto d^2\sqrt{\dot{\gamma }/t_i}$ involving the geometric average of shear time $1/\dot{\gamma }$ and inertial time $t_i$ as the relevant time scale. Analysis of grain trajectories is further evidence that this diffusion process arises from a vortex-driven random walk.

Figure 1

(a) Plane shear geometry with biperiodic boundary conditions (dashed lines). (b) Schematic of Delaunay triangulation and computation of relative velocities for triplet of grains.

Figure 2

Time evolution of mean square displacement $⟨\mathrm{\Delta }{y}^2⟩$ for different values of inertial number $I$ for (a) frictional grains and (b) frictionless grains (system size $H/d=120$). (c) Measured diffusivity $D$ versus inertial number $I$ for systems of differing size with $H/d=20$ (diamonds), 30 (triangles), 60 (circles), and 120 (squares). Hollow and filled symbols represent frictional and frictionless grains, respectively. Inset: variation of the constant $B$ (see text) with different system size.

Figure 3

Granular vortices. (a),(b) Snapshots illustrating granular vortices, i.e., triplets of kinematically jammed grains shown in magenta for $I=0.1$ (a) and 0.001 (b) for $H/d=120$ system with frictional grains. (See Supplemental Material movies [18].) (c) Average vortex size $\ell$ versus inertial number $I$ for frictional grains and frictionless grains in systems of differing size $H$. Symbols represent the same configurations as those in Fig. 2. Inset (i): correlation function $g(r)$ measured in flows of differing inertial numbers ranging from 0.5 to 0.001 for $H/d=120$ system with frictional grains. Inset (ii): same plot as (c) but rescaled with the system size $H$ and a power of inertial number.

Figure 4

Time evolution of mean square displacement $⟨\mathrm{\Delta }{y}^2⟩$ for different values of inertial number $I$ for frictional grains (a) and frictionless grains (b) of system size $H/d=120$. (c) Measured diffusivity $D$ versus inertial number $I$ and vortex size $\ell$ for systems of differing size $H$. Dashed line represents the function $D/(d\ell /t_i)=0.08I$. All the symbols represent the same configurations as those in Fig. 2.