Using theory and experiments, we investigate granular surface avalanching due to material outflow from a narrow silo. The assumed silo geometry is a deep rectangular box, of moderate spanwise width and small gap thickness between smooth front and back walls. A small orifice deep below the free surface lets grains drain out at a constant rate. The resulting granular flows can therefore be assumed quasi-two-dimensional and quasi-steady over most of the surface descent history. To model these flows, we couple a kinematic model of deep granular flow with a dynamic model of shallow surface avalanching. We then compare the calculated flow fields with detailed particle tracking measurements, letting the silo ascend relative to the high-speed camera to increase spatial resolution. The results show that the avalanching surface shape and near-surface flow are controlled by the spanwise gradient in subsidence velocity, and how this gradient is in turn controlled by the height above orifice and the gap thickness. Whereas the deep flow pattern is rate independent, shallow avalanching is paced by the granular rheology.
Figure : Qualitatively different avalanching flow pattern observed in silo of large gap thickness (a) long exposure image; (b) colour map of velocity magnitude.
C.-Y. Hung et al., J. Fluid Mech. (2018), vol. 856, pp. 444-469 (pdf)
We compare laboratory experiments, contact dynamics simulations, and continuum Navier–Stokes simulations with a mu(I) visco-plastic rheology, of the discharge of granular media from a silo with a lateral orifice. We consider a rectangular silo with an orifice of height D which spans the silo width W, and we observe two regimes. For small-enough aperture aspect ratio A=D/W$ the Hagen–Beverloo relation is obtained. For thin-enough silos, we observe a second regime where the outlet velocity varies with W. This new regime is also obtained in the continuum simulations when the friction on side walls is taken into account in a thickness-averaged version of mu(I) + Navier–Stokes (in the spirit of Hele–Shaw flows.
Moreover most of the internal details of the flow field observed experimentally are reproduced when considering this lateral friction. These two regimes are recovered experimentally for a cylindrical silo with a lateral rectangular orifice of height D and arc length W. The dependency of the flow rate on the particle diameter is found to be reasonably described experimentally using two geometrical functions that depend respectively on the number of beads through the two aperture dimensions. This is consistent with 2D discrete simulation results: at the outlet, the volume fraction and the velocities depend on the particle diameter and this behaviour is correctly described by those geometrical functions. A similar dependency is observed in the 2D continuum simulations (pdf).
figure : Dimensionless mass flow rate as a function of D/W for particles of diameter d=190 mum.
Y. Zhou et al., J. of Fluid Mech. (2017) 829, 459-485 (pdf)
L’actualité scientifique « Une formation de rivières fractales lors de l’érosion d’un lit granulaire », relative à l’article Scale-free channeling patterns near the onset of erosion of sheared granular beds publié dans la revue PNAS, a été mise en ligne à la Une du site de l’INSIS à la rubrique « Actualités scientifiques – En direct des laboratoires ».
When fluid flows across a granular substrate, shearing forces detach material from the interface and transport it downstream. Although erosion-deposition constitutes a central geomorphological process that shapes Earth’s landforms, decades of research has failed to yield a complete description of these systems at the microscopic level. A central aspect is the existence of a threshold stress below which erosion stops, whose microscopic underpinning is debated.
We study experimentally the collective dynamics of the moving particles, using a flume apparatus to simulate river flow over a gravel bed, where the system spontaneously evolves toward the erosion onset. We characterize the spatial organization of the erosion flux.
Figure: Typical channeling patterns
We show that near the origin of erosion the flow of particles is spatially heterogeneous, carried by only a few concentrated channels in the bed (see figure) whose distribution is extremely broad, with strongly anisotropic spatial correlations. The distribution of the local flux σ displays scaling near threshold and follows P(σ)≈J/σ, where J is the mean erosion flux. Channels are strongly correlated in the direction of forcing but not in the transverse direction.
Furthermore, we demonstrate that these results support a model in which erosion is ultimately governed by a give and take between channelization, which accelerates erosion, and interactions among particles which tend to interfere with channeling. This model incorporating both the disorder of the static bed and the interactions between mobile particles support that, for laminar flows, erosion is a dynamical phase transition that shares similarity with the plastic depinning transition occurring in dirty superconductors. The methodology we introduce here could be applied to probe these systems as well.
P. Aussillous et al. PNAS (2016) 113: 11788-11793. (pdf)
Discrete particle simulations are used to study two-dimensional discharge flow from a silo using both monodisperse and bidisperse mixtures. The density and the velocity profiles through the aperture are measured. In the monodisperse case, two particles diameters are studied for different outlet diameters. In the bidisperse case, we varied the fine mass fraction of the mixture. In all cases, the density and the velocity profiles are found to follow the same self-similar law. Based on these observations and the previous work of Benyamine et al. (pdf), a physical model is proposed to describe the flow of bidisperse mixtures giving an explicit expression for the flow rate that is in good agreement with the results.
Zhou et al Phys. Rev. E. 92, 062204 (2015) (pdf)
The discharge flow in a cylindrical and a rectangular silo using both monodisperse and bidisperse mixtures of spherical glass beads is studied experimentally. The flow rate is measured using a precision balance for a large range of particles diameters, size ratio and outlet diameters. A simple physical model is proposed to describe the flow of bidisperse mixtures. It gives an expression for the flow rate and predicts that the bulk velocity follows a simple mixture law. This model implies that a mixture diameter cannot be simply dened. Moreover it is shown that bidisperse granular media allows for the transport of coarse particles below their jamming conditions.
Benyamine et al. Phys. Rev. E. 90 (2014) pdf
The mobile layer of a granular bed composed of spherical particles is experimentally investigated in a laminar rectangular channel flow. Both particle and fluid velocity profiles are obtained using particle image velocimetry for different index-matched combinations of particles and fluid and for a wide range of fluid flow rates above incipient motion. A full three-dimensional investigation of the flow field inside the mobile layer is also provided. These experimental observations are compared to the predictions of a two-phase continuum model having a frictional rheology to describe particle–particle interactions. Different rheological constitutive laws having increasing degrees of sophistication are tested and discussed.
Aussillous et al., J. of Fluid Mech.736, 594-615 (2013) pdf