Shallow-water continuum modelling of dry granular flows in partailly obstructed chutes

We employ data about a dry granular flow down a 19º smooth-walled chute, partially obstructed at the downstream end, to verify the solution of a shallow-water continuum model. The system of conservation equations is based on depth-averaging the ensemble-averaged mass, momentum and fluctuating kinetic energy equations:

(1)  $\partial_t \left(\phi h \right) + \partial_x \left(\phi h u \right) = - \partial_t z_b$

(2)  $\partial_t \left( \rho h u \right) + \partial_{x} \left( \rho h u^2 \right)  = -\partial_{x} \left( \rho g h^2 / 2 \right) - g \rho h \, \partial_{x} z_b  - \tau_b$

(3) $\partial_{t} z_b = - \left( E(x,t) - D(x,t) \right)$

(4) $P = f(\phi) f(e,k,\phi_c) \rho_g T$

(5) $-Q^\prime + \frac{1}{2}\tau_b u/h - \Gamma = 0$

where $x$ is the distance, $t$ is time, the conservative variables are the elevation of the granular bed, $z_b$, the equivalent depth of flowing granular material $\phi h$ and flow momentum $\rho \phi h$, where $\phi$ is the solid fraction, $h$ the granular depth and $u$ the depth-averaged longitudinal velocity, $\tau_b$ is the wall stress, $E$ and $D$ are the rates of particle pick-up and deposition, respectively, $e$ is the normal coefficient of restitution, $k$ is particle stiffness, $\phi_c$ is the critical solid fraction, $\rho_g$ is the density of the solid particles, $\rho = \rho_g \phi$, $\Gamma$ is the rate of dissipation of fluctuating kinetic energy and $Q^\prime$ is the flux of fluctuating kinetic energy at the bottom wall.  The solid fraction is determined from (4) as a function of the granular pressure $P$ (considered hydrostatic) and the granular temperature $T$.

Preliminary results of simulations with borosilicate spheres ( g/cm³ and coefficient of restitution ), with  and  as tuning parameters, indicate that the celerity of the jamming wavefront is well-reproduced. The jump strength and the head losses are not in full agreement, requiring adjustments in the equation of state (4).

 

Acknowledgements

Portuguese Foundation for Science and Technology (FCT) through the PhD scholarship PD/BD/150693/2020, project PTDC/ECI- EGC/7739/2020 and CERIS funding UIDB/04625/2020.