Unsteady granular chute flows at high inertial numbers

We study the time-dependent flow behavior of gravity-driven free surface granular flows using the discrete element method and continuum modeling. Discrete element method (DEM) simulations of slightly polydisperse disks flowing over a periodic chute with a bumpy base are performed. A simple numerical solution based on a continuum approach with the inertial-number-based $\mu \text{−}I$ rheology has been proposed to predict the flow dynamics. The results of the continuum model are compared with the DEM simulation results for a wide range of chute inclinations. Solutions for the constitutive model described by the popular Jop-Forterre-Pouliquen (JFP) model as well as the recently proposed modified rheological model using a nonmonotonic variation of $\mu \text{−}I$ are obtained. Our results demonstrate that the popular JFP model reliably predicts the flow at low to moderate inclination angles (i.e., for $I\lesssim 0.5$). However, it fails to predict the flow properties at high inclinations. The modified rheological model, on the other hand, is very well able to predict the time-averaged flow properties for all the inclination angles considered in this study. Accounting for the presence of the slip velocity, layer dilation, and stress anisotropy is found to be crucial for accurate predictions of transient flows at high inertial numbers (i.e., for $I\gtrsim 1$).

Figure 1

Variation of the (a) effective friction coefficient $\mu$ with the inertial number $I$, (b) solids fraction $\varphi$ with $I$, and (c) ratio of normal stress difference to the pressure, $N_1/P$, with $I$ using the model parameters from Patro et al. [32]. Solid lines represent the variation according to the modified rheology. The dashed line represents the fitted line using the JFP model.

Figure 2

Snapshot of simulation of a granular layer flowing under the influence of gravity at an inclination $\theta ={32}^{\circ }$ from the horizontal at any instant. Solid black circles represent the static particles that form the bumpy chute base and solid grey circles represent the flowing disks.

Figure 3

Variation of the (a) velocity $v_x$, (b) shear rate $\dot{\gamma }$, (c) inertial number $I$, (d) solids fraction $\varphi$, (e) shear stress ${\tau }_{yx}$, and (f) viscosity $\eta$ with distance $y$ from the base at different times for $\theta ={24}^{\circ }$. Symbols represent the DEM simulation data and the solid lines represent the JFP model predictions.

Figure 4

Variation of the (a) average velocity of the layer, $v_{\mathrm{avg}}$, (b) average inertial number in the bulk, $I_{\mathrm{bulk}}$, (c) average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, and (d) slip velocity at the base, $v_{\mathrm{slip}}$, with time $t$ ($e_n=0.5$) for different inclinations $\theta$. Symbols represent the DEM simulation data whereas the solid lines in (a)–(c) represent the JFP model predictions.

Figure 5

Variation of the (a) average velocity of the layer, $v_{\mathrm{avg}}$, (b) average inertial number in the bulk, $I_{\mathrm{bulk}}$, (c) average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, and (d) slip velocity at the base, $v_{\mathrm{slip}}$, with time $t$ ($e_n=0.5$) for different inclinations. Symbols represent the DEM simulation data whereas the solid lines in (a)–(c) represent the JFP model predictions.

Figure 6

Variation of the slip velocity $v_{\mathrm{slip}}$ with time $t$ for $e_n=0.5$ corresponding to (a) $\theta ={32}^{\circ }$ and (b) $\theta ={36}^{\circ }$. Black symbols represent the DEM simulation data. The solid red line represents the exponential fit to the DEM data. The brown band represents the modeled variation of the slip velocity used in the theoretical predictions by accounting for the fluctuations in the mean slip velocity.

Figure 7

Variation of the (a) velocity $v_x$, (b) shear rate $\dot{\gamma }$, (c) inertial number $I$, (d) solids fraction $\varphi$, (e) shear stress ${\tau }_{yx}$, and (f) viscosity $\eta$ with distance $y$ from the base at different times for $\theta ={28}^{\circ }$. Symbols represent the DEM simulation data ($e_n=0.5$), solid lines are the predictions of the modified rheology, and dashed lines represent the JFP model predictions.

Figure 8

Variation of the (a) average velocity of the layer, $v_{\mathrm{avg}}$, (b) average inertial number in the bulk, $I_{\mathrm{bulk}}$, and (c) average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, with time $t$ for different inclinations $\theta$. Symbols represent the DEM simulation data ($e_n=0.5$), solid lines are the predictions of the modified rheology, and dashed lines represent the JFP model predictions.

Figure 9

Variation of the (a) velocity $v_x$, (b) inertial number $I$, and (c) solids fraction $\varphi$ with distance $y$ from the base at different times for $\theta ={36}^{\circ }$. Variation of the (d) average velocity of the layer, $v_{\mathrm{avg}}$, (e) average inertial number in the bulk, $I_{\mathrm{bulk}}$, and (f) average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, with time $t$ for even inclinations ${32}^{\circ }\le \theta \le {40}^{\circ }$. Symbols represent the DEM simulation data ($e_n=0.5$), solid lines are the predictions of the modified rheology, and dashed lines represent the JFP model predictions.

Figure 10

Variation of the slip velocity $v_{\mathrm{slip}}$ with (a) the shear rate at the base, ${\dot{\gamma }}_{\mathrm{base}}$, and (b) average inertial number $I_{\mathrm{bulk}}$ on a semilogarithmic scale. The insets show the main figure data on a linear scale over a smaller range. (c) Variation of the ratio of the slip velocity to the average velocity, $v_{\mathrm{slip}}/v_{\mathrm{avg}}$, with the average inertial number $I_{\mathrm{bulk}}$. The data for all three cases correspond to steady-state flow corresponding to $e_n=0.5$.

Figure 11

Variation of (a) the average velocity of the layer, $v_{\mathrm{avg}}$, (b) the average inertial number in the bulk, $I_{\mathrm{bulk}}$, and (c) the average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, with time $t$ at different inclinations $\theta$. Symbols represent the DEM data for $e_n=0.1$ whereas the solid lines represent the theoretical predictions obtained from the modified rheology. Inset shows the results for low to moderate inclinations. The results for higher inclination angles are reported in the main figure. Slip velocity at the base, $v_{\mathrm{slip}}$, as a function of time for inclination (d) $\theta ={34}^{\circ }$, (e) $\theta ={38}^{\circ }$, and (f) $\theta ={42}^{\circ }$. The red line shows the exponential fit to the simulation data (shown using black circles). The brown band shows the variation of $5\%$ from the mean velocity and is able to capture the slip velocity data at all times.

Figure 12

Variation of the average velocity of the layer, $v_{\mathrm{avg}}$, with time $t$ for $e_n=0.5$ at (a) $\theta ={20}^{\circ }, {24}^{\circ }$, and ${28}^{\circ }$ and (b) $\theta ={32}^{\circ }$ and ${36}^{\circ }$. The lines are the theoretical predictions from the modified rheological model. Solid lines are predictions considering variable flowing layer thickness, while dashed lines are predictions considering a constant flowing layer thickness.

Figure 13

Variation of the (a) velocity $v_x$, (b) inertial number $I$, (c) solids fraction, and (d) pressure $P$ with distance $y$ from the base at different times for $e_n=0.5$ at $\theta ={36}^{\circ }$. The solid lines are the theoretical predictions from the rheological model considering variable flowing layer thickness. The dashed lines are the theoretical predictions from the rheological model considering constant flowing layer thickness.

Figure 14

Variation of the average velocity of the layer, $v_{\mathrm{avg}}$, with time $t$ for $e_n=0.5$ at $\theta ={20}^{\circ }, {24}^{\circ }, {28}^{\circ }, {30}^{\circ }$, and ${32}^{\circ }$. Solid lines are predictions considering normal stress difference, while dashed lines are predictions without considering the effect of normal stress difference.

Figure 15

Snapshots of the system configuration at different times for $e_n=0.5$ at $\theta ={42}^{\circ }$.

Figure 16

Steady-state oscillating behavior of (a) bulk flowing layer thickness $h_{\mathrm{bulk}}$ and (b) center of mass $y_{\mathrm{com}}$ at different inclinations for restitution coefficient $e_n=0.5$.

Figure 17

(a) Variation of the Reynolds number Re with the average inertial number in the bulk, $I_{\mathrm{bulk}}$, for restitution coefficient $e_n=0.5$. The inset in (a) shows the variation of the Reynolds number (Re) with the average inertial number ($I_{\mathrm{bulk}}$) in the semilogarithmic scale. Black circles correspond to $h_{\max }$ whereas the red triangles correspond to $h_{\mathrm{bulk}}$ for the layer thickness in Reynolds number calculation. (b) Variation of the slip velocity $v_{\mathrm{slip}}/\sqrt{g{h}_{\min }}$ and $\mathrm{\Delta }{h}_{\mathrm{bulk}}/d$ (on the right ordinate) with the average inertial number $I_{\mathrm{bulk}}$ at steady state.

Figure 18

Fast Fourier transform analysis of the center-of-mass data for all inclination angles.

Figure 19

Fast Fourier transform analysis of the center of mass (black line), bulk flowing layer thickness (red line), and kinetic energy (blue line) for inclination angle (a) $\theta ={30}^{\circ }$, (b) $\theta ={36}^{\circ }$, and (c) $\theta ={42}^{\circ }$.

Figure 20

(a) Variation of the dominant frequency $f$ corresponding to different inclination angles $\theta$. (b) Variation of the amplitude $A$ corresponding to the dominant frequency $f$ [shown in (a)] for different inclination angles. See text for details.

Figure 21

(a) Variation of effective friction coefficient $\mu$ with inertial number $I$ for restitution coefficient $e_n=0.5$. Black circles represent the DEM data up to inclination $\theta ={28}^{\circ }$. The solid red line represents the fitted line of the form $\mu \left(I\right)=aI+b$. Variation of the (b) average velocity of the layer, $v_{\mathrm{avg}}$, (c) average inertial number in the bulk, $I_{\mathrm{bulk}}$, and (d) average solids fraction in the bulk, ${\varphi }_{\mathrm{bulk}}$, with time $t$ for $e_n=0.5$ at different inclinations. Symbols represent the DEM simulation data and the solid lines represent the analytical predictions by Parez et al. [51].

Figure 22

(a) Comparison of the numerical solution (shown using symbols), analytical one-term solution (dashed line), and analytical three-term solution (solid lines) at different times for $\theta ={24}^{\circ }$. (b) Comparison of the numerical solution (shown using symbols) and analytical one-term solution (solid lines) for different angles.

Figure 23

Variation of the (a) normal stress ${\sigma }_{yy}$, (b) normal stress ${\sigma }_{xx}$, and (c) pressure $P$ with distance $y$ from the base at different times for $\theta ={28}^{\circ }$. Symbols represent the DEM simulation data ($e_n=0.5$) and solid lines are the predictions of the modified rheology.

Figure 24

Variation of the (a) normal stress ${\sigma }_{yy}$, (b) normal stress ${\sigma }_{xx}$, and (c) pressure $P$ with distance $y$ from the base at different times for $\theta ={36}^{\circ }$. Symbols represent the DEM simulation data ($e_n=0.5$) and solid lines are the predictions of the modified rheology.