## 3 Results and Discussion
### Effect of Liquid Nitrogen Pulsation on DCM Stability
The distances between the monitoring points in the vertical direction are relatively close, with a maximum gap of only 0.1 m, and the liquid nitrogen flow velocity is 0.995 m/s. Relative to the effects of kinetic and mechanical energy losses, the influence of gravitational potential energy is negligible. Taking the monitoring points (with subscripts \(u\) for upstream and \(d\) for downstream) and the exit position (with subscript \(o\)) as references, the following equations follow:  
\[\frac{P_{u}}{\rho g}+\frac{v^{2}}{2g}+Z_{u}=\frac{P_{o}}{\rho g}+\frac{v^{2}} {2g}+Z_{o}+h_{w(u-o)} \tag{11}\]  
\[\frac{P_{d}}{\rho g}+\frac{v^{2}}{2g}+Z_{d}=\frac{P_{o}}{\rho g}+\frac{v^{2}} {2g}+Z_{o}+h_{w(d-o)} \tag{12}\]  
due to the more significant resistance loss in the upstream pipeline than in the downstream, resulting in \(h_{w(u-o)}>h_{w(d-o)}\), causing \(P_{total(u)}>P_{total(d)}\). However, due to flow velocity fluctuations inside the pipeline, the resistance loss \(\Delta P\) is proportional to the square of the velocity, resulting in mechanical energy loss \(\Delta h_{w}\) changing proportionally with the square of the velocity. When the flow velocity changes, the variation in mechanical energy loss is \(\Delta h_{w(u-o)}>\Delta h_{w(d-o)}\). Combining Equations (11) and (12), \(\Delta P_{total(u)}>\Delta P_{total(d)}\), and consequently, the amplitude of the pressure pulsations is greater upstream than downstream. The pressure pulsation at each point in Figure 8 does not precisely follow the trend of the excitation. Specifically, the positive pressure pulsation amplitude is more significant for each monitoring point than the negative. Taking any monitoring point as an example, when the velocity at this monitoring point changes from \(v_{1}\) to \(v_{2}\), the pipeline's resistance loss changes from \(\Delta P_{1}\) to \(\Delta P_{2}\). This change can be described by Equations (13) and (14):  
\[\Delta P_{2}-\Delta P_{1}=\left(\frac{\lambda\frac{l}{d}\cdot\rho+\zeta\rho}{2 }\right)\cdot\left(v_{2}^{2}-v_{1}^{2}\right) \tag{13}\]  
\[\Delta P_{2}-\Delta P_{1}=\left(\frac{\lambda\frac{l}{d}\cdot\rho+\zeta\rho}{2 }\right)\cdot\left(2v_{1}+\Delta v\right)\cdot\Delta v \tag{14}\]  
where \(\Delta v\) is the magnitude of the velocity change. When the structure and material of the pipe do not change, the friction coefficient \(\lambda\) varies relatively. This study can ignore the change in \(\lambda\) here. From Equation (14), it can be observed that when the magnitude of velocity change \(\Delta v\) is constant, the change in pipeline pressure loss \((\Delta P_{2}-\Delta P_{1})\) is directly proportional to the initial velocity \(v_{1}\) (ignoring the small numerical value of \(\Delta v^{2}\)). Therefore, when the fluid velocity \(v_{1}\) is relatively large, the change in pipeline pressure loss \((\Delta P_{2}-\Delta P_{1})\) is significant, leading to a correspondingly large change in mechanical energy loss \(\Delta h_{\omega}\). This results in a relatively large amplitude of pressure pulsations \(\Delta P_{total}\), consequently causing the amplitude of positive pressure pulsations to be more significant than that of negative ones. In Figure 9, it can be observed that with the increase in the frequency of velocity pulsation, the pulsation amplitude at each point significantly increases. This is mainly due to the rapid transformation of energy in the system. The flow of liquid nitrogen in the cooling system is continuous, and the inflow and outflow are equal. When the inflow velocity decreases, the velocity at the outlet also decreases accordingly, causing the kinetic energy of the liquid nitrogen inside the pipe to transform into potential energy, acting on the internal flow field. As the frequency of velocity pulsation increases, it shortens the duration of energy conversion, reducing the kinetic energy discharged at the system outlet. More kinetic energy is stored internally, leading to a conversion into higher potential energy. This phenomenon is similar to the \"water hammer effect\".