## 3 Experimental and Numerical Setup
### Influence of Key Parameters
#### 4.2.3 The Combined Influence of \(\mathrm{d_{i}}\) and \(\mathrm{\mu}\)  
When the frictional coefficient of particles changes simultaneously with the intruder's diameter, we can observe the effect of their combined impact on the scaled drag force. From Figure 11a, there is a systematic increase in the drag force with \(\mathrm{\mu}\) for \(\mathrm{\mu}\) \(<\) 0.9; when \(\mathrm{\mu}\)\(\geq\) 0.9, the drag force seems to be insensitive to further increases in \(\mathrm{\mu}\). Accordingly, the graph in Figure 11b displays the scaled drag force for various friction coefficients and intruder diameters. The results indicate that the value of \(F^{*}\) varies between 34 and 46 when \(\mathrm{\mu}\) is below 0.9 and then decreases gradually with a further increase in \(\mathrm{\mu}\). The value of \(F^{*}\) can be collapsed into two straight lines with \(F^{*}\) = 41.6 and \(F^{*}\) = \(-34\mathrm{\mu}\) + 68, which means that the drag force remains proportional to the square of the intruder diameter; however, it  
Figure 10: Comparison of penetration drag forces values and linear fitting functions for different particle diameters (scattered points represent simulation results, while solid straight lines represent fitted curves). is no longer strictly proportional to the friction coefficient when the friction coefficient is greater than 0.9. Based on the above discussion, we can obtain the piecewise formula for drag force:  
\[F^{*}=\left\{ \begin{aligned} 41.6,&\mu<0.9\\ -34\mu+68,&\mu\geq 0.9\end{aligned} \right. \tag{15}\]  
To elucidate why the drag force ceases to increase beyond a friction coefficient of 0.9, we present vertical profiles of velocity and solid volume fraction in Figure 12, averaged within a cylinder of diameter. These parameters, \(v\) and \(\Phi\), respectively, represent the velocity component along the Z direction and the volume fraction of particles. The area highlighted in Figure 13 was chosen to examine the characteristics of particle flow and the density of force-transmitting particles directly beneath the vertical intruder. The moment the intruder contacts the first particle is designated as \(t=0\) s. The velocity profile of particles, \(v\), demonstrates uniform cessation at a distance of \(2\mathrm{d_{i}}\) from the intruder, as shown in Figure 12a. Consequently, a region of \(2\mathrm{d_{i}}\) was selected for analyzing the solid volume fraction. Observations show that the solid volume fraction significantly diminishes as the friction coefficient decreases to 0.9, beyond which it stabilizes. This suggests that the bulk density of particles in front of the intruder's leading edge remains constant for friction coefficients above 0.9, maintaining a consistent force exerted by the displacement of particles ahead. Figure 11: **(a)** The averaged forces. **(b)** The scaled drag forces of intruder influenced by different friction coefficients and the diameters of intruder (\(d_{\mathrm{c}}:d_{\mathrm{i}}=5\):1, \(d_{\mathrm{p}}=6\) mm). Figure 12: Flow profiles averaged within a cylinder aligned with the intruder axis. **(a)** The vertical grain velocity \(v\) as a function of distance from the intruder surface at \(t=4.0\) s. **(b)** Local packing density under the intruder surface as a function of the friction coefficient. The parameter \(P\) denotes the number of particles contacting with the intruder, as shown in Figure 14a. In vertical penetration, the contact area between the side wall of the intruder and the particle gradually increases, which contributes greatly to the numerical value of \(P\). It is easy to understand that the larger the \(d_{i}\), the greater the side wall surface, and the greater the value of \(P\). When the friction coefficient is set to a minimum value (\(\mu=0\)), the flow properties of the granule bed are more pronounced and the particles are not stacked on the intruder surface, so the number of contact points is relatively small.