## 3 Results and Discussion
### Effect of Liquid Nitrogen Pulsation on DCM Stability  
Figure 8 shows the pressure variation curves at seven monitoring points in the fluid domain under excitation with a frequency of 200 Hz and an amplitude of 1%-\(v_{a}\) (\(v_{a}\) is the average flow velocity of the cooling system). From Figure 8, it can be observed that the pulsation period of pressure at each monitoring point is consistent with the excitation period, and the pulsation amplitude of pressure is more significant at the upstream monitoring points than at the downstream. Firstly, the fluid flow must adhere to the continuity equation, which states that the total mass entering and leaving a unit fluid volume must be equal. On a macro scale, since the pipe is of constant diameter, the velocities in each section are similar. Therefore, the fluctuation in the inlet velocity will cause fluctuations in the velocities in different  
Figure 8: Pressure changes at monitoring points. The x-axis is the time in units of pulsation periods of liquid nitrogen. Figure 7: Calculation of the vibration angle of the second crystal. The green portion represents the crystal’s initial position, the red dashed box represents the position of the crystal after vibration, and the blue line represents the meridian line of the crystal. sections of the pipe, and the trends of these fluctuations are consistent. Next, there is energy loss in the fluid flow process, and the upstream needs to increase the pressure to maintain a smooth flow. (a) When fluid flows through the pipe, energy loss occurs due to the friction between the fluid, the pipe wall, and the internal viscosity. Energy loss is divided into along-pipe resistance loss \(\Delta P_{f}\) and local resistance loss \(\Delta P_{m}\)[21]. The formula for calculating along-pipe resistance is given by Equation (8). (b) When the shape or size of the structural domain changes, turbulent vortices are generated in the fluid domain near the fluid-structure interface, and local relative motion intensifies. This type of motion hinders the flow of the fluid, applying backpressure locally in the opposite direction to the motion, forming back pressure. The force generated by this is called local resistance, and the calculation of local resistance uses Formula (9). Thus, the pipe resistance loss \(\Delta P\) is given by: \(\Delta P=\Delta P_{f}+\Delta P_{m}\). \[\Delta P_{f}=\lambda\frac{l}{d}\cdot\frac{\rho v^{2}}{2} \tag{8}\]  
\[\Delta P_{m}=\zeta\frac{\rho v^{2}}{2} \tag{9}\]  
where \(\Delta P_{f}\) represents the along-pipe resistance loss, \(\Delta P_{m}\) represents the local resistance loss, \(\lambda\) is the dimensionless relative friction factor, \(l\) is the length of the pipe, \(d\) is the equivalent diameter of the pipe, \(\rho\) is the density of liquid nitrogen, \(v\) represents the average velocity of liquid nitrogen across the cross-section of the pipeline, and \(\zeta\) is the dimensionless local resistance coefficient, which is related to the shape and size of the pipe. The turbulent along-pipe resistance coefficient \(\lambda\) is a function of the Reynolds number \(Re\) and the absolute roughness of the pipe wall \(\varepsilon/d\). When Reynolds number \(Re\) is constant, the friction factor \(\lambda\) is numerically positively correlated with \(\varepsilon/d\). Additionally, according to the energy conservation equation and the Bernoulli equation for steady flow, any two points within the fluid should adhere to:  
\[P_{1}+\frac{1}{2}\rho v_{1}^{2}+Z_{1}=P_{2}+\frac{1}{2}\rho v_{2}^{2}+Z_{2}+h _{w(1-2)} \tag{10}\]  
where \(P\), \(\frac{1}{2}\rho v^{2}\) and \(Z\) represent the pressure head, velocity head, and elevation head of the fluid per unit volume, respectively, and \(h_{w(1-2)}\) represents the mechanical energy loss from point 1 to point 2. The total pressure of the fluid is \(P_{total}=P+\frac{1}{2}\rho v^{2}+Z\). When the fluid runs stably, the pressure at the upstream monitoring point is more significant than that at the downstream.