There is a large amount of experimental and numerical work dealing with dry granular flows (such as sand, glass beads, etc.) that supports the so-called $\ensuremath{\mu} (I)$-rheology. The reliability of the $\ensuremath{\mu} (I)$-rheology in the case of complex transient flows is not fully ascertained, however. From this perspective, the granular column collapse experiment provides an interesting benchmark. In this paper we implement the $\ensuremath{\mu} (I)$-rheology in a Navier–Stokes solver (Gerris) and compare the resulting solutions with both analytical solutions and two-dimensional contact dynamics discrete simulations. In a first series of simulations, we check the numerical model in the case of a steady infinite two-dimensional granular layer avalanching on an inclined plane. A second layer of Newtonian fluid is then added over the granular layer in order to recover a close approximation of a free-surface condition. Comparisons with analytical and semi-analytical solutions provide conclusive validation of the numerical implementation of the $\ensuremath{\mu} (I)$-rheology. In a second part, we simulate the unsteady two-dimensional collapse of granular columns over a wide range of aspect ratios. Systematic comparisons with discrete two-dimensional contact dynamics simulations show good agreement between the two methods for the inner deformations and the time evolution of the shape during most of the flow, while a systematic underestimation of the final run-out is observed. The experimental scalings of spreading of the column as a function of the aspect ratio available from the literature are also recovered. A discussion follows on the performances of other rheologies, and on the sensitivity of the simulations to the parameters of the $\ensuremath{\mu} (I)$-rheology.

Numerical simulations of the collapse and spreading of granular columns onto a horizontal plane using the Contact Dynamics method are presented. The results are in agreement with previous experimental work. The final shape of the deposit appears to depend only on the initial aspect ratio a of the column. The normalized runout distance has a power-law dependence on the aspect ratio a, a dependence incompatible with a simple friction model. The dynamics of the collapse is shown to be mostly controlled by a free fall of the column. Energy dissipation at the base of the column can be described simply by a coefficient of restitution. Hence the energy available for the sideways flow is proportional to the initial potential energy $E_0$. The dissipation process within the sideways flow is approximated well by basal friction, unlike the behaviour of the runout distance. The proportion of mass ejected sideways is shown to play a determining role in the spreading process: as a increases, the same fraction of initial potential energy $E_0$ drives an increasing proportion of the initial mass against friction. This gives a possible explanation for the power-law dependence of the runout distance on a. We propose a new scaling for the runout distance that matches the data well, is compatible with a friction model, and provides a qualitative explanation of the column collapse.

The granular silo is one of the many interesting illustrations of the thixotropic property of granular matter: a rapid flow develops at the outlet, propagating upwards through a dense shear flow while material at the bottom corners of the container remains static. For large enough outlets, the discharge flow is continuous; however, by contrast with the clepsydra for which the flow velocity depends on the height of fluid left in the container, the discharge rate of granular silos is constant. Implementing a plastic rheology in a 2D Navier-Stokes solver (following the µ(I)-rheology or a constant friction), we simulate the continuum counterpart of the granular silo. Doing so, we obtain a constant flow rate during the discharge and recover the Beverloo scaling independently of the initial filling height of the silo. We show that lowering the value of the coefficient of friction leads to a transition toward a different behavior, similar to that of a viscous fluid, and where the filling height becomes active in the discharge process. The pressure field shows that large enough values of the coefficient of friction ( 0.3) allow for a low-pressure cavity to form above the outlet, and can thus explain the Beverloo scaling. In conclusion, the difference between the discharge of a hourglass and a clepsydra seems to reside in the existence or not of a plastic yield stress.

We investigate numerically the transition between static equilibrium and dynamic surface flow of a 2D cohesionless granular system driven by a continuous gravity loading. This transition is characterized by intermittent local dynamic rearrangements and can be described by an order parameter defined as the density of critical contacts, e.g. contacts where the friction is fully mobilized.Analysis of the spatial correlations of critical contacts shows the occurence of "fluidized" clusters which exhibit a power-law divergence in size at the approach of the stability limit. The results are compatible with recent models that describe the granular system during the static/dynamic transition as a multi-phase system.

Using a continuum Navier-Stokes solver with the μ(I) flow law implemented to model the viscous behavior, and the discrete Contact Dynamics algorithm, the discharge of granular silos is simulated in two dimensions from the early stages of the discharge until complete release of the material. In both cases, the Beverloo scaling is recovered. We first do not attempt a quantitative comparison, but focus on the qualitative behavior of velocity and pressure at different locations in the flow. A good agreement for the velocity is obtained in the regions of rapid flows, while areas of slow creep are not entirely captured by the continuum model. The pressure field shows a general good agreement, while bulk deformations are found to be similar in both approaches. The influence of the parameters of the μ(I) flow law is systematically investigated, showing the importance of the dependence on the inertial number I to achieve quantitative agreement between continuum and discrete discharge. However, potential problems involving the systems size, the configuration and "non-local" effects, are suggested. Yet the general ability of the continuum model to reproduce qualitatively the granular behavior is found to be very encouraging.

A modified shallow-water model is presented for the collapse of tall columns of grains. The flow is divided in two parts. Depth-averaged shallow-water equations are applied to a thin horizontally spreading layer which is subjected to Coulombic friction. The falling mass of grains is gradually added to the zone of the initial column during the free-fall time of the column. This ‘rain’ is assumed to have no horizontal momentum. The results obtained here are in agreement with both planar and axisymmetric experiments over a range of aspect ratio $a$. In particular, the runout distance is found to vary as $a^{0.65}$ (planar) and $a^{0.52}$ (axisymmetric). The flow dynamics compares well with discrete simulations which have been successfully compared with experiments.

Discrete numerical simulations are performed to study the evolution of the micro-structure and the response of a granular packing during successive loading-unloading cycles, consisting of quasi-static rotations in the gravity field between opposite inclination angles. We show that internal variables, e.g., stress and fabric of the pile, exhibit hysteresis during these cycles due to the exploration of different metastable configurations. Interestingly, the hysteretic behaviour of the pile strongly depends on the maximal inclination of the cycles, giving evidence of the irreversible modifications of the pile state occurring close to the unjamming transition. More specifically, we show that for cycles with maximal inclination larger than the repose angle, the weak contact network carries the memory of the unjamming transition. These results demonstrate the relevance of a two-phases description -strong and weak contact networks-for a granular system, as soon as it has approached the unjamming transition.

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