Shock waves in rapid flows of dense granular materials: theoretical predictions and experimental results

Dry granular avalanche impact force on a rigid wall: Analytic shock solution versus discrete element simulations

The present paper investigates the mean impact force exerted by a granular mass flowing down an incline and impacting a rigid wall of semi-infinite height. First, this granular flow-wall interaction problem is modeled by numerical simulations based on the discrete element method (DEM). These DEM simulations allow computing the depth-averaged quantities-thickness, velocity, and density-of the incoming flow and the resulting mean force on the rigid wall. Second, that problem is described by a simple analytic solution based on a depth-averaged approach for a traveling compressible shock wave, whose volume is assumed to shrink into a singular surface, and which coexists with a dead zone. It is shown that the dead-zone dynamics and the mean force on the wall computed from DEM can be reproduced reasonably well by the analytic solution proposed over a wide range of slope angle of the incline. These results are obtained by feeding the analytic solution with the thickness, the depth-averaged velocity, and the density averaged over a certain distance along the incline rather than flow quantities taken at a singular section before the jump, thus showing that the assumption of a shock wave volume shrinking into a singular surface is questionable. The finite length of the traveling wave upstream of the grains piling against the wall must be considered. The sensitivity of the model prediction to that sampling length remains complicated, however, which highlights the need of further investigation about the properties and the internal structure of the propagating granular wave.

Depth-averaged analytic solutions for free-surface granular flows impacting rigid walls down inclines

In the present paper, flows of granular materials impacting wall-like obstacles down inclines are described by depth-averaged analytic solutions. Particular attention is paid to extending the existing depth-averaged equations initially developed for frictionless and incompressible fluids down a horizontal plane. The effects of the gravitational acceleration along the slope, and of the retarding acceleration caused by friction as well, are systematically taken into account. The analytic solutions are then used to revisit existing data on rigid walls impacted by granular flows. This approach allows establishing a complete phase diagram for granular flow-wall interaction.

Self-organized shocks in the sedimentation of a granular gas

A granular gas in gravity heated from below develops a certain stationary density profile. When the heating is switched off, the granular gas collapses. We investigate the process of sedimentation using computational hydrodynamics, based on the Jenkins-Richman theory, and find that the process is significantly more complex than generally acknowledged. In particular, during its evolution, the system passes several stages which reveal distinct spatial regions of inertial (supersonic) and diffusive (subsonic) dynamics. During the supersonic stages, characterized by Mach>1, the system develops supersonic shocks which are followed by a steep front of the hydrodynamic fields of temperature and density, traveling upward.

Unstable blast shocks in dilute granular flows

Shocks and blasts can be readily obtained in granular flows be they dense or dilute. Here, by examining the propagation of a blast shock in a dilute granular flow, we show that such a front is unstable with respect to transverse variations of the density of grains. This instability has a well-defined wavelength which depends on the density of the medium and has an amplitude which grows as an exponential of the distance traveled. These features can be understood using a simple model for the shock front, including dissipation which is inherent to granular flows. While this instability bears much resemblance to that anticipated in gases, it is distinct and has special features we discuss here.

Blast shock wave mitigation using the hydraulic energy redirection and release technology

A hydraulic energy redirection and release technology has been developed for mitigating the effects of blast shock waves on protected objects. The technology employs a liquid-filled plastic tubing as a blast overpressure transformer to transfer kinetic energy of blast shock waves into hydraulic energy in the plastic tubings. The hydraulic energy is redirected through the plastic tubings to the openings at the lower ends, and then is quickly released with the liquid flowing out through the openings. The samples of the specifically designed body armor in which the liquid-filled plastic tubings were installed vertically as the outer layer of the body armor were tested. The blast test results demonstrated that blast overpressure behind the body armor samples was remarkably reduced by 97% in 0.2 msec after the liquid flowed out of its appropriate volume through the openings. The results also suggested that a volumetric liquid surge might be created when kinetic energy of blast shock wave was transferred into hydraulic energy to cause a rapid physical movement or displacement of the liquid. The volumetric liquid surge has a strong destructive power, and can cause a noncontact, remote injury in humans (such as blast-induced traumatic brain injury and post-traumatic stress disorder) if it is created in cardiovascular system. The hydraulic energy redirection and release technology can successfully mitigate blast shock waves from the outer surface of the body armor. It should be further explored as an innovative approach to effectively protect against blast threats to civilian and military personnel.

Equation for the force experienced by a wall overflowed by a granular avalanche: experimental verification

The present paper deals with the force experienced by a wall overflowed by a granular avalanche. First, we briefly report laboratory tests on dry granular avalanches overflowing a wall down a rough channel. In the first step, the thickness and velocity of the control flows without a wall are measured. In the second step, a wall is mounted to obstruct the flow and the normal force experienced by the wall is measured. Then a set of equations based on depth-averaged momentum conservation, making it possible to derive the time-varying force on the wall, is described. The model was proposed and calibrated in two-dimensional (2D) discrete numerical simulations in an earlier work [Chanut, Faug, and Naaim, Phys. Rev. E 82, 041302 (2010)]. This model takes into account the fact that a quasistatic stagnant zone is established upstream of the wall and coexists with an inertial flowing zone above. For a large range of slopes, the model's prediction is successfully compared to experimental data with a reasonable estimation of the incoming flow density and in spite of some rough assumptions made to describe the dynamics of the dead zone. Finally, the results are analyzed with regard to previous 2D discrete numerical simulations and we discuss future work to be undertaken on the dynamics of the dead zone established upstream of the wall.

Time-varying force from dense granular avalanches on a wall

We studied avalanches of cohesionless granular materials down a rough inclined plane and overflowing a wall normal to the incoming flow and to the bottom. This paper focuses on the transient time-varying mean force exerted by the granular stream on the obstacle at various slope inclinations. A nearly triangular dead zone is formed upstream of the obstacle. It largely contributes to the overall force signal at low slope inclinations. It also drives the residual force corresponding to the avalanche tail until its standstill whatever the slope inclination. An analytical hydrodynamic model based on depth-averaged momentum conservation was successfully developed for steady-flow conditions to predict the steady-state force computed from discrete numerical simulations [T. Faug, R. Beguin, and B. Chanut, Phys. Rev. E 80, 021305 (2009)]. The basic equations of the model are briefly reviewed and adapted to transient time-varying flows. The modified hydrodynamic model quite accurately represents the force peak produced by the granular avalanche flows computed from discrete numerical simulations reported in previous studies. A fitting procedure is needed to represent the decrease of the force after the force peak, thus quantifying the different contributions to the mean force on the wall. We show that the weight of each contribution is largely dependent on the slope inclination.

Laboratory observations of self-excited dust acoustic shocks

Repeated, self-excited dust acoustic shock waves (DASWs) have been observed in a dc glow discharge dusty plasma using high-speed video imaging. Two major observations are reported: (1) The self-steepening of a nonlinear dust acoustic wave (DAW) into a saw-tooth wave with sharp gradient in dust density, very similar to those found in numerical solutions of the fully nonlinear fluid equations for a nondispersive DAW [B. Eliasson and P. K. Shukla, Phys. Rev. E 69, 067401 (2004)], and (2) the collision and confluence of two DASWs.

Mean steady granular force on a wall overflowed by free-surface gravity-driven dense flows

We studied free-surface gravity-driven recirculating flows of cohesionless granular materials down a rough inclined plane and overflowing a wall normal to the incoming flow and to the bottom. We performed two-dimensional spherical particle discrete element simulations using a linear damped spring law between particles with a Coulomb failure criterion. High-frequency force fluctuations were observed. This paper focuses on the mean steady force exerted by the flow on the obstacle versus the macroscopic inertial number of the incoming flow, where the inertial number measures the ratio between a macroscopic deformation time scale and an inertial time scale. A triangular stagnant zone is formed upstream of the obstacle and sharply increases the mean force at low incoming inertial numbers. A simple hydrodynamic model based on depth-averaged momentum conservation is proposed. This analytical model predicts the numerical data fairly well and allows us to quantify the different contributions to the mean force on the wall. Beyond this model, our study provides an example of the ability of simple hydrodynamic approaches to describe the macroscopic behavior of an assembly of discrete particles not only in terms of kinematics but also in terms of forces.