## 3 Numerical Example  
### Granular Flume Flow Strike Mast-Like Obstacle  
We simulated the laboratory experiments reported by [PERSON] et al. (2017), in which grains were initially stored in a compartment and then released within several seconds to simulate a steady granular stream in the chute (Figure 1). The released granular stream would strike rectangular obstacles, which varied in size, i.e., D = 10 mm, 20 mm, 40 mm, and 80 mm, in separate simulations. In the laboratory experiments, the height of the compartment opening was 7 cm, so we fixed the inflow boundary condition as \(h_{0}=0.07\) m in the simulation. Additionally, the natural boundary conditions for the flow velocity were also specified at the inflow and outflow boundaries. The earth pressure coefficients (2b,c) were calculated using the estimated angle of internal friction \(\varphi=35^{\circ}\) and the coefficient of basal friction \(\delta=25^{\circ}\), providing the earth pressure coefficient in the \(x\)- and \(y\)-directions, as follows:  
\[K_{x_{act/pus}}=2\sec^{2}\varphi\left(1\mp\left(1-\cos^{2}\varphi/\cos^{2} \delta\right)^{1/2}\right)-1=0.298/5.662, \tag{10a}\] \[K_{y_{act/pus}}=\frac{1}{2}\left\{K_{x}+1\mp\left[(K_{x}-1)^{2}+4\tan^{2} \delta\right]^{1/2}\right\}=\left\{ \begin{array}{cl}0.067&\partial u/ \partial x\geq 0,\partial v/\partial y\geq 0\\ 1.234&\partial u/\partial x\geq 0,\partial v/\partial y<0\\ 0.954&\partial u/\partial x<0,\partial v/\partial y\geq 0\\ 5.708&\partial u/\partial x<0,\partial v/\partial y<0\end{array} \right., \tag{10b}\]  
Figure 1: Schematic of the laboratory installation; grains initially stored in a compartment were released from a 70 mm high opening to simulate a granular stream striking a rectangular obstacle during the flow process. The length of the chute used in the experiment reported by [PERSON] et al. [29] was 7.5 m. We cut out a 2 m long section of the chute near where the obstacle was located and developed the configurational evolution of the granular stream striking a square obstacle (D = 40 mm) as viewed from the top (Figure 2). Obvious jumps in the avalanche depth (i.e., shock wave phenomena) upstream from the obstacle at the front of the flow were evident. Downstream of the obstacle, the granular flow parted into two streams, and a \"shallow vacuum area\" between the two streams behind the obstacle was formed. In Figure 3 (\(t\) = 3.5 s) and Figure 4 (\(t\) = 4.5 s), the top-view configurations of the fan-shaped stream generated by the granular flow impacting the obstacle highly depend on the width of the obstacle. The simulated opening between the two granular streams grew as the side length of the obstacle increased. The depth of the granular flow 0.5 m upstream of the obstacle was tracked for \(t\) = 10 s from when the flow started, and was compared with the continuous regression curve obtained from the experimental data [30] (Figure 5). The flow depth in the simulation within \(t\) = 3 s to 6.5 s was slightly smaller than that in the experiment; however, in general, the simulation results coincided well with the data. The simulated granular flow reached a relatively stable state from \(t\) = 4.5 s to 10 s, whereas the depth of the laboratory granular flow stabilized at around 11.5 mm from \(t\) = 4 s to 6.5 s, and then declined gradually to 1.8 mm at \(t\) = 10 s. The mean square root between the calculated and experimental data was accessed, as shown in Figure 5. The mean square root of the height before \(t\) = 6 s was about 1.7991 mm and, during whole process, it was 15.0258 mm. The difference between the simulation and the experiment mainly arose because the experimental granules stocked in the container diminished over time, and the inflow rate in the experiment decreased after \(t\) = 6.5 s. In contrast, the inflow rate remained constant in our simulation. We also found that the simulated configurations of the flow remained the same after roughly \(t\) = 4.5 s after reaching a \"stable state\".