## 3 Experimental and Numerical Setup
### Influence of Key Parameters  
#### 4.2.1 Penetrating Velocity  
The critical velocity is 0.443 m/s from \(V_{c}=\sqrt{2 gd_{\text{p}}}\) when the mean diameter of the particle \(d_{\text{p}}\) is 10 mm and the gravity \(g\) is 9.81 m/s\({}^{2}\). The values of \(d_{\text{c }}\) and \(d_{\text{i}}\) are also consistent with the experiments. When \(V<V_{c}\), the penetration can be regarded as the quasi-static regime. For each drag force curve, five individual runs were averaged, as shown in Figure 8.  
The forces acting on the cylinders have three distinguished tendencies at different velocity ranges. From Figure 8a, it can be found that for \(V~{}<V_{c}\) (0.01-0.3 m/s), the system is in the quasi-static regime, where the drag force-depth curves exhibit fewer fluctuations and all drag forces (\(V<V_{c}\)) collapse onto a master line, as predicted by \(F=K_{z}Z\)[15; 25]. Figure 8: Drag force exerted on an intruder impacting at the velocity ranging from (**a**) 0.01 to 2.0 m/s; (**b**) 3.0 to 10.0 m/s. Figure 7: Comparison of the drag forces between numerical simulation and experimental results (penetration velocity \(V\) = 10 mm/s, \(d_{\text{p}}\) = 10 mm). The parameter \(K_{z}\) represents the slope of the fitting line. Additionally, the force fluctuates over a larger area as the cylinder sinks along the penetration direction when the velocity exceeds the critical point of 0.443 m/s (0.5-2.0 m/s). However, when the speed is greater than 3 m/s (Figure 8b), the force fluctuates tremendously and it is independent of depth. The real speed at where the turnaround occurs is shown in Figure 9. It shows the relationship between the average force \(F_{a}\) and the penetrating velocity. For the simulation where \(V<3\) m/s, the drag force is linear with depth, and the average force can be calculated according to Equation (10). When \(V\geq 3\) m/s, the force fluctuates tremendously, so the average force formula is  
\[F_{a}=\frac{\sum_{i=1}^{n}F_{i}}{n} \tag{11}\]  
in which \(n\) represents the number of drag force records in the vertical direction. Figure 9 depicts the average force \(F_{a}\) applied to the cylinder, which can be distinctly classified into three regimes: the quasi-static regime (velocity-independent), the viscous regime (force linearly correlates with velocity), and the inertial regime (force is proportional to the square of the velocity). For scenarios entailing either penetration at a constant speed or an impact with a predetermined initial velocity, it is proposed that the drag force can be uniformly described as follows:  
\[F_{a}=\alpha V^{2}+\beta V+K_{z}Z_{a} \tag{12}\]  
where \(Z_{a}\) represents the average of the penetration distance. When the penetration velocity \(V\) falls below \(V_{sv}\) (with \(V_{sv}\) being the critical velocity transitioning from the quasi-static to the viscous regime), the drag force correlates with depth \(Z\) and is not dependent on \(V\). In the scenario where \(V_{sv}<V<V_{vi}\) (\(V_{vi}\) representing the critical velocity from the viscous to the inertial regime), the force is governed by a viscous term linearly dependent on velocity (\(\beta V\)). For velocities exceeding \(V_{vi}\), the drag force is governed by an inertial term proportional to the square of the velocity (\(\alpha V^{2}\)), where \(\alpha\), \(\beta\), and \(K_{z}\) are constants that adapt based on the velocity regime. The transition velocities \(V_{sv}\) and \(V_{vi}\) can be determined from the intersections of three fitting curves [20]. The critical velocity \(V_{c}=\sqrt{2 gd_{\text{P}}}\approx 0.443\) m/s from the quasi-static regime to the viscous regime is nearly consistent with existing conclusions, which is \(V_{sv}\approx 0.456\) m/s in our situation. The critical velocity \(V_{vi}\) transitioning from the viscous to the inertial regime is approximately 2.88 m/s, which is inconsistent with the theoretical solution derived from classical hydrodynamics models [18]. Figure 9: The relationship between the average force and the penetrating velocity.