Scaling laws for velocity profile of granular flow in rotating drums

Abstract

We theoretically and numerically investigate the steady flow of two-dimensional granular materials in a rotating drum using the discrete element method and a continuum model with the $μ(I)$-rheology. The velocity fields obtained from both methods are in quantitative agreement. The granular flow exhibits two distinct regions: a surface flow layer and a static flow regime corresponding to rigid rotation near the drum bottom. The thickness of the surface flow layer increases with the drum diameter and shows a weak dependence on the angular velocity of the drum. Using dimensional analysis of the continuum equations, we analytically identify nondimensional parameters for the velocity profile and the surface flow layer thickness, which lead to scaling laws characterising the flow in rotating drums with low Froude number and large system size. The validity of the scaling laws is confirmed by numerical simulations.

Figures (9)

• Figure 1: Schematic of steady granular flow in a rotating drum. $X$ and $Z$ denote the horizontal and vertical coordinates, respectively, with the origin located at the drum center. The $x$-axis is taken parallel to the free surface, and the $z$-axis is normal to it.
• Figure 2: Velocity fields obtained from the DEM (a) and CFD (b) simulations for $D=150d$ and $\Omega = 1.26 \times 10^{-3} \sqrt{g/d}$. The red solid line represents the free surface. The $x$-axis is taken parallel to the free surface, and the $z$-axis is normal to it.
• Figure 3: (a) Velocity profile $u(z)$ at $x=0$ for $\Omega = 6.28 \times 10^{-3} \sqrt{g/d}$ with $D=150d$, $200d$, and $300d$. (b) Velocity profile $u(z)$ at $x=0$ for $D=150d$ with $\Omega = 1.26 \times 10^{-3} \sqrt{g/d}$ to $1.26 \times 10^{-2} \sqrt{g/d}$. The open and closed symbols represent the results of the DEM and CFD simulations, respectively. The solid lines represent $- \Omega z$ for the rigid rotation.
• Figure 4: Surface flow layer thickness $h$ against the angular velocity $\Omega$ for various values of $D$. The open and closed symbols represent the results of the DEM and CFD simulations, respectively.
• Figure 5: Normalized velocity $u / \Omega D$ against $z/D$ at position $x=0$ for (a) $Fr = 3.00 \times 10^{-3}$, and (b) $Fr = 1.20 \times 10^{-2}$ with various values of $D$. The open and closed symbols represent the results of the DEM and CFD simulations, respectively. The solid line corresponds to the rigid rotation.