## 4 Results
### Tube Length
#### 4.2.2 The Influence of Tube Length in Cavity-Double Tube Flow  
This subsection quantifies the influence of \(l_{2}/d\) on the inertial forces within the cavity by analyzing the relative variation in the gas quantity \(\overline{m}\). Figure 15 illustrates the effect of the increase in \(l_{2}/d\) on the variation history of \(\overline{m}\). Using the oscillation-related parameters at \(l_{2}/d=0\) as a comparative baseline, it is observed that as \(l_{2}/d\) increases, the parameters associated with the variation history of \(\overline{m}\) change slightly, exhibiting an overall decreasing trend. By examining the distribution of \(Ma\) within the cavity at moments \(t_{s1}\) and \(t_{s2}\) in Figure 16, we further investigate the mechanism by which \(l_{2}/d\) influences the inertial forces within the cavity.  
To analyze the impact of changes in \(l_{2}/d\), we examine the variations in the distribution range of \(Ma\) within the cavity at moments \(t_{s1}\) and \(t_{s2}\), as illustrated in Figure 16. Due to the differing positions of the tube within the flow path, an increase in \(l_{2}/d\) does not allow the region within the tube to be equivalent to the CIDZ range; rather, it increases the volume of the flow path, thereby diminishing the influence of inertial forces within the flow path.  
Figure 17 compares the variation in the axial force-related parameter \(\overline{m}\) within the cavity as \(l_{2}/d\) increases from 0 to 1.5. As \(l_{2}/d\) increases, the oscillation amplitudes and periods of \(\overline{F_{A}}\) and \(\overline{F_{B}}\) exhibit slight decreases, while the phase difference remains relatively unchanged, resulting in minimal impact on \(\Delta\overline{F}\). A detailed comparison of the parameters is presented in Table 10. Due to the differing positions of the tube, the variation of \(l_{2}/d\) primarily influences the sudden change boundary response process by altering the flow path volume, without directly affecting the CIDZ. Consequently, the impact on the evolution of axial loads within the cavity is less significant than that of \(l_{1}/d\).  
Figure 15: (**a**) Evolution of \(\overline{m}\); (**b**) Comparison of oscillation amplitude of \(\overline{m}\); (**c**) Comparison of period of \(T\).  
Figure 16: Comparison of flow field \(Ma\) variation in different \(l_{2}/d\) cavity-double tube flow path. (**a**) discharge process; (**b**) charge process.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(l_{2}/d\) & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.5 \\ \hline \(\left(\overline{F_{A}}\right)_{\text{min}}\) & 0.53778 & 0.54016 & 0.54193 & 0.54351 & 0.54903 & 0.55422 \\ \hline \(\left(\overline{F_{A}}\right)_{\text{max}}\) & 0.75796 & 0.75639 & 0.75278 & 0.75211 & 0.74862 & 0.75141 \\ \hline \(\left(\overline{F_{B}}\right)_{\text{min}}\) & 0.53236 & 0.53588 & 0.53500 & 0.53843 & 0.54354 & 0.55173 \\ \hline \(\left(\overline{F_{B}}\right)_{\text{max}}\) & 0.76373 & 0.75958 & 0.75918 & 0.75351 & 0.75026 & 0.75167 \\ \hline \(\left(\overline{A\overline{F}}\right)_{\text{min}}\) & \(-0.04752\) & \(-0.06072\) & \(-0.06080\) & \(-0.06080\) & \(-0.06081\) & \(-0.06080\) \\ \hline \(\left(\overline{A\overline{F}}\right)_{\text{max}}\) & 0.00756 & 0.00742 & 0.01745 & 0.01420 & 0.00624 & 0.01903 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison of parameters for different end face relative axial load variation processes.