Effects of particle size of mono-disperse granular flows impacting a rigid barrier

Abstract

Understanding the interaction between complex geophysical flows and barriers remains a critical challenge for protecting infrastructure in mountainous regions. The scientific challenge lies in understanding how grain stresses in complex geophysical flows become manifested in the dynamic response of a rigid barrier. A series of physical flume tests were conducted to investigate the influence of varying the particle diameter of mono-dispersed flows on the impact kinematics of a model rigid barrier. Particle sizes of 3, 10, 23 and 38 mm were investigated. Physical tests results were then used to calibrate a discrete element model for carrying out numerical back-analyses. Results reveal that aside from considering bulk characteristics of the flow, such as the average velocity and bulk density, the impact load strongly depends on the particle size. The particle size influences the degree of grain inertial stresses which become manifested as sharp impulses in the dynamic response of a rigid barrier. Impact models that only consider a single impulse using the equation of elastic collision warrant caution as a cluster of coarse grains induce numerous impulses that can exceed current design recommendations by several orders of magnitude. Although these impulses are transient, they may induce local strucutral damage. Furthermore, the equation of elastic collision should be adopted when the normalized particle size with the flow depth, δ/h, is larger than 0.9 for Froude numbers less than 3.5.

Appendices

Appendix 1: Comparison of numerical 2D and 3D analysis

1.1 Input parameters and modelling procedures

The 3D numerical model is geometrically similar to the physical model tests (Fig. 8). The storage container, channel walls and barrier are modelled as planar and rigid elements. A linear contact model was adopted, consisting of a linear spring dashpot. Impact kinematics and dynamics were measured along the height of the barrier. The input parameters and modelling procedures are reminiscent to those used in the 2D simulations. Each simulation included the generation of a target mass of 20 kg. The total number of elements N is calculated as follows:

$$N = \frac{{M_{\text{T}} }}{{M_{\text{p}} }}$$

(12)

where $M_{\text{T}}$ is the total weight of sample which is equal to 20 kg, $M_{\text{p}}$ is the mass of a single particle in 3D case, and can be calculated from Eq. 4:

$$M_{\text{p}} = \rho V_{\text{p}} = \rho \left( {\frac{4}{3}\pi r^{3} } \right)$$

(13)

where $\rho$ and $r$ are the density and radius of each element, respectively.

Fig. 8

3D Numerical model setup: a top and side view; b 3D view

After the generation of particles, a gravitational acceleration of 9.81 m/s2 was applied to the computational domain and elements were allowed to free fall in the storage container until the ratio of average unbalanced force to the average contact forces converged was less than 1% difference (Cui et al. 2016). The storage container gate was then deleted to simulate dam break initiation mechanism to release the discrete elements downslope into the rigid barrier.

1.2 Interpretation and discussion

Results show that for 3D simulation, the impact load exhibit a disparity of about 20% less compared to 2D simulation. The difference is attributed predominantly to side wall effects, which is the most apparent from observing the geometry of the flow fronts in both the 2D and 3D simulations. Furthermore, the 2D simulations exhibit several distinct impulses after the initial peak impulse. By contrast, the 3D simulations did not have such observation. This is because of the enhanced degree of freedom for the 3D simulations. The additional degree of freedom enhances the attenuation of flow energy in the direction perpendicular to the flow.

To further elucidate the impact mechanism pertaining to the 2D simulations, the computed flow kinematics between the first and second impacts (computational time from 0.719 s to 0.794 s) are shown in Fig. 9a–f. The arrow of each particle indicates a velocity vector. From the computed results, it is evident that two particles in particular (Particles No. 1 and No. 2) impact the rigid barrier almost simultaneously (Fig. 9a, b). After the first impact, Particles No. 1 and No. 2 rebound and collide with the incoming particles, specifically Particle No. 3 (Fig. 9c, d). After this collision, Particle No. 1 bounces back towards the barrier and induces a second impact (Fig. 9e, f).

Fig. 9

Impact mechanism between first and second impact on rigid barrier (30°) when $\delta /h$ = 0.43: a t = 0.719 s; b t = 0.734 s; c t = 0.749 s; d t = 0.764 s; e t = 0.779 s; f t = 0.794 s

The observed dynamics lead to two distinct impact forces as shown in Fig. 10. By contrast, this phenomenon is not as evident in the 3D simulations because a larger number of particles and greater momentum transfer in the lateral direction lead to lower impact loads for the 3D case compared to the 2D case.

Fig. 10

Comparison of the impact load history on rigid barrier (30°) when $\delta /h$ = 0.43

Appendix 2: Calibration of frictional parameter

Before the experimental flume test, the direct shear test is carried out for glass spheres with 3 mm diameter, following the standard ASTM standard (ASTM 2011). The samples were filled in a standard shear box measuring 50.8 $\times$ 50.8 $\times$ 21.5 mm (width $\times$ length $\times$ height), after which a constant compaction effort was applied to the samples, which brought them to a dense state and similar densities. The samples were tested at 45, 90 and 180 kPa normal stresses in a strain-controlled mode at a shear rate of 0.02 mm/min. Figure 11 shows the frictional angle for 3-mm-diameter glass sphere is 20 degree, which corresponds to the inter-element frictional coefficient of 0.36.

Fig. 11

Results of direct shear test on 3-mm-diameter glass sphere assembly

The interface-element frictional coefficient of the glass sphere was measured in Choi et al. (2016) by means of tilting tests referring to Pudasaini and Hutter (2007), Mancarella and Hungr (2010), Jiang and Towhata (2013). A cylindrical container of 200 mm diameter filled with granular material to a height of 100 mm was placed on the channel. The channel was inclined until the container began to slide. At this moment, the angle of the channel was recorded as the interface friction angle. The recorded angle 22 degree is then converted to interface-element frictional coefficient as 0.4 in DEM calculations.