Unsteady granular flows in a rotating tumbler

The characteristics of steady granular flow in quasi-two-dimensional rotating tumblers have been thoroughly investigated and are fairly well understood. However, unsteady time-varying flow has not been studied in detail. The linear response of granular flow in quasi-two-dimensional rotating tumblers is presented for periodic forcing protocols via sinusoidal variation in the rotational speed of the tumbler and for step changes in rotational speed. Variations in the tumbler radius, particle size, and forcing frequency are explored. Similarities to steady flow include the fastest flow occurring at the free surface of the flowing layer and an instantaneous approximately linear velocity profile through the depth. The flowing layer depth varies by 2–5 particle diameters between minimum and maximum rotation rates. However, unsteady forcing also causes the flow to exhibit dynamic properties. For periodic rotational speeds, the phase lag of the flowing layer depth increases linearly with increasing input forcing frequency up to nearly 2.0 rad over 0–20 cycles per tumbler revolution. The amplitude responses of the velocity and shear rate show a resonance behavior unique to the system level parameters. The phase lag of all flow properties appears to be related to the number of particle contacts from the edge of the rotating tumbler. Characterization via step changes in rotational speed shows dynamic properties of overshoot (up to 35%) and rise times on the order of 0.2–0.7 s. The results suggest that the unsteady granular flow analysis may be beneficial for characterizing the “flowability” and “rheology” of granular materials based on particle size, moisture content, or other properties.

Figure 1

Schematic of a half-filled quasi-2D tumbler parameters (front view) with radius $R$, axial thickness $L$, rotation rate $\omega \left(t\right)$, and the coordinate system at the midlength of the flowing layer. The dashed box represents the area of the flowing layer that was investigated at $x=0$. The top view shows the rotating tumbler position relative to the CCD camera, the diffuser plate, and the dual-head laser light source (not to scale).

Figure 2

(Color online) Periodic forcing protocol (top) in terms of rotational speed $\omega$ (solid curve; left axis) and Fr (dashed curve; right axis) for tumbler frequency $n=6$. The streamwise velocity response $u$ (bottom) in the $R=13.9\text{cm}$, $L=0.9\text{cm}$ tumbler with 1 mm particles. Each curve on the velocity plot is for an individual one-particle-diameter-thick layer in the flow with the upper most curve indicating the streamwise velocity at the free surface. The lower curves indicate the velocity response at increasing depths through the flowing layer. Markers are only shown for half of the time steps for clarity, and standard deviations of the ensemble averaging for each layer are shown starting at $t=3.5\text{s}$.

Figure 3

(Color online) Streamwise velocity profile $u\left(y/d\right)$ through the flowing layer for accelerating flow (left) and decelerating flow (right). Conditions are the same as Fig. 2 with $R=13.9\text{cm}$, $L=0.9\text{cm}$, and 1 mm particles. The arrows indicate increasing time.

Figure 4

(Color online) Streamwise velocity profile at each time step (relative to period of oscillation $T=\frac{1}{n{\omega }_{\text{o}}}$) as a function of the flowing layer depth (in terms of number of particle diameters) in the $R=8.6\text{cm}$, $L=1.8\text{cm}$ tumbler with 2 mm particles and $n=13$. The maximum streamwise velocity is $u_{\text{max}}=0.25\text{m}/\text{s}$. The black curve at the zero velocity plane is the calculated flowing layer depth $\delta$.

Figure 5

Phase lag of the sinusoidal response of the flowing layer depth ${\varphi }_{\delta }$ increases linearly as a function of the forcing frequency. (◻) $d=1\text{mm}$ particles and (◇) $d=2\text{mm}$ particles; open markers represent tumbler radius $R=8.6\text{cm}$ and filled markers represent $R=13.9\text{cm}$.

Figure 6

Amplitude of sinusoidal response of the shear rate ${\Lambda }_{\dot{\gamma }}$ as a function of the forcing frequency. (◻) $d=1\text{mm}$ particles and (◇) $d=2\text{mm}$ particles; open markers represent tumbler radius $R=8.6\text{cm}$ and filled markers represent $R=13.9\text{cm}$; the curves for each radius-particle size combination are guides for the eyes.

Figure 7

(Color online) Response of the streamwise velocity scaled by the maximum velocity for the top six single-particle-diameter-thick layers (timing resolution does not provide data at precisely $t/T=1$); the left column is for $R=8.6\text{cm}$ and the right column is for $R=13.9\text{cm}$; the top row is for $d=1\text{mm}$ and the bottom row is for $d=2\text{mm}$. The horizontal dashed lines indicate the unit dimensionless velocity for the tumbler radius and particle size combination (color online). The curve connecting the waveform peaks at different forcing frequencies (red online), indicates the variation in the phase lag of the free surface of the flowing layer. The vertical curve for each forcing frequency (black online) traces the phase lag through the depth.

Figure 8

(Color online) Response of the streamwise velocity for single-particle-diameter-thick layers to a step increase in Fr at time $t=0\text{s}$; $d=1\text{mm}$ particles in the $R=8.6\text{cm}$ tumbler. Markers represent the raw data with curves corresponding to a fit to the general step response function in Eq. (4). Black square markers and the vertical curve indicate the time of the maximum overshoot of the fitting function. Typical standard deviations are indicated between $0.5\le t\le 1.5\text{s}$.

Figure 9

Streamwise velocity (relative to final velocity) at the free surface as a function of time following a step increase in the rotational speed at $t=0$. (◻) $d=1\text{mm}$ particles and (◇) $d=2\text{mm}$ particles; open markers represent tumbler radius $R=8.6\text{cm}$ and filled markers represent $R=13.9\text{cm}$. For clarity of the fitted step response, markers are only shown for half of the time steps.