## 4 Results
### Tube Length
#### 4.2.1 The Influence of Tube Length in Cavity-Single Tube Flow  
Study quantifies the effect of \(l_{1}/d\) on the inertial force within the cavity using the relative gas change \(\overline{m}\). Figure 12 illustrates the variation of \(\overline{m}\) under different \(l_{1}/d\) conditions. Using the oscillation-related parameters at \(l_{1}/d=0\) as a comparative baseline, during the 2 increase of \(l_{1}/d\) from 0 to 1.5, \(\overline{m}_{Tough}\) decreases linearly, while \(\overline{m}_{Crest}\) and \(T\) both show a notable decrease followed by an increase. When \(l_{1}/d\leq 0.25\), the changes in \(\overline{m}_{Crest}\) and \(T\) are relatively minor; however, when \(0.25<l_{1}/d\), the amplitude of changes in \(\overline{m}_{Crest}\) and \(T\) increases. By combining the distribution of \(Ma\) in the cavity at moments \(t_{s1}\) and \(t_{s2}\) in Figure 13, the differences in CIDZ and EIDZ under varying \(l_{1}/d\) conditions are further analyzed to elucidate the mechanism by which \(l_{1}/d\) affects the inertial force within the cavity. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Cavity-Single Tube Flow**} & \multicolumn{6}{c}{**Cavity-Double Tube Flow**} \\ \hline \(l_{1}/d\) & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.5 & & & 1.5 & & & \\ \(l_{2}/d\) & & & & 0 & & 0.25 & 0.5 & 0.75 & 1 & 1.5 \\ \(t_{char}/\)s & 1.80 & 1.88 & 1.95 & 2.02 & 2.09 & 2.24 & 2.24 & 2.31 & 2.38 & 2.45 & 2.52 & 2.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparative analysis of the effects of length of tube on operating conditions. Figure 12: (**a**) Evolution of \(\overline{m}\); (**b**) Comparison of oscillation amplitude of \(\overline{m}\); (**c**) Comparison of Period of \(T\). When \(l_{1}/d\leq 0.25\) occurs, the mutation of \(Ma\) does not occur at the position of the flow path contraction; instead, the tube structure acts as a hole within the flow path. During the venting process, the distance from EIDZ to CIDZ increases with \(l_{1}/d\), while the highest \(Ma\) region is located within EIDZ. Consequently, the influence of EIDZ on CIDZ diminishes as \(l_{1}/d\) increases. At this point, \(A/V_{Vessel}\)\({}^{\frac{3}{2}}\) remains constant, and the axial range of CIDZ  
Figure 13: Comparison of flow field \(Ma\) variation in different \(l_{1}/d\) cavity-single tube flow path (a) discharge process; (b) charge process. decreases with increasing \(l_{1}/d\), resulting in a slight reduction in the inertial forces within the cavity and a shorter venting duration. This also impacts the axial load response within the cavity, leading to an increased oscillation period for \(\overline{F_{B}}\), a greater phase difference between \(\overline{F_{A}}\) and \(\overline{F_{B}}\), and an increase in \(\Delta\overline{F}\). During the inflation process, the range of EIDZ expands with \(l_{1}/d\), causing \(\overline{m}_{Tough}\) to decrease. When \(0.25<l_{1}/d\) occurs, the mutation of \(Ma\) takes place at the flow path contraction, and the tube structure serves as a conduit within the flow path. During the venting process, the positions of CIDZ and EIDZ remain unchanged. At this stage, due to the continuity of flow, the region within the tube becomes equivalent to CIDZ, thereby enlarging its area. Meanwhile, the highest \(Ma\) region is situated within the tube and exerts a significant influence on CIDZ. As \(l_{1}/d\) increases, the highest \(Ma\) initially rises and then declines, leading to a corresponding increase followed by a decrease in its impact on CIDZ. Consequently, during the venting process, the inertial forces within the cavity are affected by the coupling of \(l_{1}/d\) and the highest \(Ma\), generally increasing with \(l_{1}/d\), although the rate of increase varies. The enlargement of the CIDZ area shortens the oscillation period of \(\overline{F_{B}}\), reduces the phase difference between \(\overline{F_{A}}\) and \(\overline{F_{B}}\), and results in a decrease in \(\Delta\overline{F}\).