## 5 Influence of Angular Velocity on Hydrodynamic Ram Effect
### Influence of Angular Velocity Magnitude on the Hydrodynamic Ram Effect  
The magnitude of angular velocity in the \(x\) direction directly affects the attenuation of the translation velocity, and determines whether the debris can pass through the satellite tank. Numerical simulations were conducted by setting the angular velocity in the \(x\) direction as \(w_{x}\) = 0.01 rad/\(\mu\)s, \(w_{x}\) = 0.05 rad/\(\mu\)s, \(w_{x}\) = 0.1 rad/\(\mu\)s, \(w_{x}\) = 0.5 rad/\(\mu\)s, \(w_{x}\) = 1 rad/\(\mu\)s, \(w_{x}\) = 2 rad/\(\mu\)s, \(w_{x}\) = 3 rad/\(\mu\)s, and \(w_{x}\) = 4 rad/\(\mu\)s. Figure 16 shows the front and back perforations caused by the impact of debris with different angular velocities in the \(x\) direction. When the debris impacted the satellite tank with the angular velocity of \(w_{x}\) < 2 rad/\(\mu\)s, it could pass through the back wall and the perforation was formed. However, when the debris impacted the satellite tank with the angular velocity of \(w_{x}\) > 3 rad/\(\mu\)s, it could not pass through the back wall. The reason is that with the increase of the angular velocity, the thickness of boundary layer increases, so the cross-section area of the debris enlarges. The kinetic energy of debris brought by the angular velocity cannot overcome the liquid resistance, and the debris velocity decays to zero in the tank. Figure 17 depicts the attenuation curves of the translation velocity when the debris impacted the satellite tank with angular velocities of \(w_{x}\) = 0.01 rad/\(\mu\)s, 0.05 rad/\(\mu\)s, 0.1 rad/\(\mu\)s, 0.5 rad/\(\mu\)s, and 1 rad/\(\mu\)s. When \(w_{x}\) increased from 0.01 rad/\(\mu\)s to 0.05 rad/\(\mu\)s, the residual velocity of debris increased from 237 m/s to 334 m/s, but when \(w_{x}\) increased from 0.05 rad/\(\mu\)s to 1 rad/\(\mu\)s, the residual velocity of debris decreased gradually. When \(w_{x}\) = 0.05 rad/\(\mu\)s, the back perforation was the largest, which is consistent with Figure 16. The physical parameters of the liquid hydrazine were close to that of water. The reason for this phenomenon can be analyzed according to the conclusions of the rotating projectile  
Figure 16: Front and back perforations caused by the impact of debris with angular velocities of \(w_{x}\) = 0.01 rad/\(\mu\)s, 0.05 rad/\(\mu\)s, 0.1 rad/\(\mu\)s, 0.5 rad/\(\mu\)s, 1 rad/\(\mu\)s, 2 rad/\(\mu\)s, 3 rad/\(\mu\)s, and 4 rad/\(\mu\)s.  
entering water. [PERSON]. [PERSON] et al. [25] pointed out the projectile motion remained stable when the angular velocity reached \(w_{l}\). The formula of \(w_{l}\) is as follows:  
\[w_{l}=w_{a}\sqrt{\rho_{l}/\rho_{a}} \tag{6}\]  
where \(\rho_{l}\) is the density of water; \(\rho_{a}\) is the density of air; and \(w_{a}\) is the angular velocity of the projectile moving steadily in the air. When \(0<w_{x}<0.05\) rad/\(\upmu\)s, with the increase in angular velocity, the stability of the debris motion enhanced, and the attenuation effect of liquid resistance on debris velocity was weakened. When \(w_{x}=w_{l}=0.05\) rad/\(\upmu\)s, the debris moved stably in liquid hydrazine and its residual velocity was the largest. As the angular velocity of debris increased, the rotational velocity of the vortex also enlarged. When the rotational kinetic energy brought by the angular velocity of the debris was not enough to maintain the vortex rotation, the kinetic energy brought by the translational velocity of debris will be further consumed, resulting in the attenuation of the translational velocity. When the debris impacts the satellite tank with the angular velocity in the \(y\) and \(z\) direction, the motion laws of debris are similar. In this section, the angular velocity in the \(y\) direction was taken as an example.